Dissertationen zum Thema „Nursing – Mathematics – Problems, exercises, etc“

Despite the fact that problem posing has been recommended as a useful mathematical activity in the curriculum documents of several countries, research about the types of problem-posing situations in which students could be involved, and about the effects of these problem-posing activities on students' mathematical performance is limited. The application of problem posing in school mathematics has been hindered by the absence of a framework which links problem posing, problem solving and mathematics curricula

Over a period of nine months in 1999, a longitudinal teaching intervention was undertaken with Grade 3 learners in the Fort Beaufort district, Eastern Cape. Working in the interpretive paradigm, the intervention focussed on: - the development of the common fraction concept, - the relevance of the hierarchy of Murray and Olivier’s Four Levels of Development in common fractions and - whether learners’ informal knowledge could be utilised in developing this concept. Using the Problem-centred approach to teaching mathematics, problems set in reallife contexts were used as vehicles for learning. Learners were required to discuss, reflect and make sense of the mathematics they were doing. Participant observation, completed worksheets and unstructured interviews with learners, formed the primary method of data collection. Learners’ work was analysed and classified according to the method used and manner in which the solution was notated. Results showed that learners were able to achieve a significant degree of success in developing a stable common fraction concept. Learners were afforded opportunities to construct their own ideas and to develop a deeper understanding of the concept. Many methods used were based on their informal knowledge of sharing. Learners made sense of realistic problems using drawings, and invented their own procedures. Apart from Level One, Phase Three, all Murray and Olivier’s Levels of Development could be identified during the research. This study will provide educators with valuable information on how learners solve mathematical problems involving fractions and how informal knowledge can be used as a foundation on which to build.

This thesis is concerned principally with tangential geometry and the applications of these concepts to tangentially symplectic foliations. The subject of tangential geometry is still at an elementary stage. The author here systematises current concepts and results and extends them, leading to the definition of vertical connections and vertical G-structures. Tangentially symplectic foliations are then characterised in terms of vertical symplectic forms. Some significant particular cases are discussed.

>Magister Scientiae - MSc
The aim of the thesis is to research the difficulties that first year mainstream mathematics students at UWC experience when solving Related Rates problems in calculus. In chapter 2, an in-depth study was made of the nature of Related Rates problems by studying a number of examples. The findings of this study are summarized in section 2.12. The study adopted the same model of the solution of all types of Related Rates that was used by Martin (2000) for the solution of geometric Related Rates problems. In chapter 3 of this thesis, many examples were used to illustrate how the seven step solution procedure of the Standard Solution model is applied. In the literature review in chapter 4, the underlying concepts which underpin Related Rates problems are identified and specific examples of research on each of these concepts are given. For example, the review of the literature on word problems is done comprehensively and covers extensively the range of issues involved in this topic. Drawing on the work in chapter 2 on the nature of Related Rates problems, it is explained in chapter 5 why this study is underpinned by Constructivism as a theoretical basis. Chapter 6 of the thesis is devoted to answering the 3 research questions posed in chapter 1. The thesis contains many worked examples of Related Rates problems which can be used by the lecturers assigned to the MAT105 course.

The purpose of this study is to investigate various solution strategies employed by Grade 7 learners and their teachers when solving a given set of mathematical tasks. This study is oriented in an interpretive paradigm and is characterised by qualitative methods. The research, set in nine schools in the Eastern Cape, was carried out with nine learners and their mathematics teachers and was designed around two phases. The research tools consisted of a set of 12 tasks that were modelled after the Third International Mathematics and Science Study (TIMSS), and a process of clinical interviews that interrogated the solution strategies that were used in solving the 12 tasks. Aspects of grounded theory were used in the analysis of the data. The study reveals that in most tasks, learners relied heavily on procedural understanding at the expense of conceptual understanding. It also emphasises that the solution strategies adopted by learners, particularly whole number operations, were consistent with those strategies used by their teachers. Both learners and teachers favoured using the traditional, standard algorithm strategies and appeared to have learned these algorithms in isolation from concepts, failing to relate them to understanding. Another important finding was that there was evidence to suggest that some learners and teachers did employ their own constructed solution strategies. They were able to make sense of the problems and to 'mathematize' effectively and reason mathematically. An interesting outcome of the study shows that participants were more proficient in solving word problems than mathematical computations. This is in contrast to existing research on word problems, where it is shown that teachers find them difficult to teach and learners find them difficult to understand. The findings of this study also highlight issues for mathematics teachers to consider when dealing with computations and word problems involving number sense and other problem solving type problems.

Alguns instrumentos avaliativos, aplicados aos alunos brasileiros do ensino fundamental (EF), têm mostrado que eles apresentam um fraco desempenho na resolução de situações -problemas de matemática. Segundo os indicadores, os alunos têm concluído a 48 série do Ensino Fundamental (EF) com sérias deficiências em relação aos conteúdos específicos de Matemática. Quando procuramos justificativas para as lacunas existentes na apreensão dos conteúdos da Matemática, existe uma tendência a atribuir grande parte do fracasso dos alunos às metodologias baseadas em técnicas e métodos tradicionais. Visando a mudar o quadro, reformas curriculares apontam para uma abordagem baseada na exposição do aluno a situações-problemas, estimulando o discente a propor soluções com base nas relações existentes entre o conhecimento matemático, o conteúdo correntemente apresentado e situações concretas do dia-a-dia. Contudo, resultados de avaliações de desempenho de alunos no que diz respeito à Matemática têm revelado que o aluno pode demonstrar dificuldades para compreender as situações-problemas a ele apresentados, o que indica que as dificuldades podem não se limitar simplesmente à falta de conhecimento matemático. O presente Trabalho teve como objetivo investigar se a competência leitora dos alunos interfere na compreensão dos enunciados dos problemas de Matemática. Para o desenvolvimento da Pesquisa, utilizamos, além dos princípios metodológicos de resolução de situações-problemas, considerações teóricas sobre a leitura, acrescidas às contribuições da Lingüística Textual. Para alcançarmos os objetivos do Trabalho, realizamos uma investigação em uma turma de quarta série do EF numa escola municipal de Jaboatão dos Guararapes- PE. Quanto à metodologia, foram realizadas duas atividades para avaliar a compreensão leitora dos alunos e outra para verificar as dificuldades desses alunos quanto à resolução dos problemas matemáticos. Também foi empregado o Teste de Cloze, cujo resultado foi utilizado para agrupar os alunos. Os grupos foram então caracterizados com relação entre a compreensão leitora e à resolução dos problemas matemáticos para avaliarmos a relação entre a compreensão leitora e a resolução dos problemas. A partir da análise das habilidades de cada grupo, pôde-se observar que o baixo desempenho apresentado na resolução dos problemas não está ligado apenas a limitações de cunho eminentemente matemático, mas há uma relação com a baixa competência em leitura observada
Some estimatory instruments, applied to Brasilian basic teaching (BS) students, have shown up displayed that they present a weak performance in mathematics situations problems solution. According to indicators, the students have concluded their BS fourth series, with serious deficiencies regarding to mathematics specifical contents. As we search justifications for these existing gaps in these contents apprehension, understanding, there is a propensity to ascribe most of these failures to methodologies based on traditional know-hows techiniques and methods. Aiming at changing this frame situation, some school curricula reformulations improvements point out to an approaching founded, based on the student exposure to situations-problems, stimulating, encouraging the student, the learning people proposing solutions based on existing rapports between mathematical knowledge, the currently presented content and day by day concrete situations. However, his/her performance evaluations results on Mathematics have revealed displayed that the student con show out manifest some difficulties to understand the situations-problems presented to him/her, which denotes that his/her difficulties can go beyond the mathematical knowledge absence dack. This work aimed at researching whether the students reading skill competency interferes in the mathematics problems enuntiates comprehension. For this research performance, we have used employed beyond the situations-problems solution methodological principles, theoretical considerations regarding to reading, enlarged added- by the textual linguistics contributions. In order to achieve attain this work aims, we have fulfilled an investigation in a BS fourth Series shift, belonging to Jaboatão dos Guararapes/PE `S teaching municipal network. Regarding to the methodology, we have accomplished two activities in order to evaluate the students reading comprehension and another one in order to verify these difficulties as for the mathematical problems solution. We have also employed Clozes Test of which result was used in order to put together the students of the groups were then characterized as for the reading comprehension and mathematical problems solution. From each group habilities, skills analysis, we were able observing that the low performance, presented in the problems solution is connected only with straightly mathematical nature limitations, but there is a strict relation with low skill, competency in observed reading

Electronic portfolio for mathematical problem solving in the elementary school is an authentic assessment tool for teachers and students to utilize in evaluating mathematical skills. It is a computer-based interactive software program to allow teachers to easily access student work in the problem solving area for assessment purposes, and to store multimedia work samples over time.

Orientador: Marcia Regina Ferreira de Brito
Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Educação
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Resumo: A presente pesquisa explorou as relações entre o desempenho no ENADE 2005, o desempenho no EGRAF e o raciocínio dedutivo de 141 estudantes ingressantes e concluintes da Ciência da Computação quando envolvidos em atividades que exploraram o pensar matemático. Os instrumentos foram o exame do ENADE, o exame do EGRAF e o teste psicológico GfRLD. Além disso, doze desses sujeitos foram submetidos a uma testagem individual de alguns componentes da habilidade matemática e a representação mental, além das variáveis estudadas para os 141 estudantes. Para a habilidade e a representação foram utilizados cinco problemas da série XIX-B de Krutetskii e uma entrevista semi-estruturada. Durante a aplicação do teste de Krutetskii, os doze estudantes foram solicitados a pensar em voz alta. Foram feitas anotações complementando informações não captadas via gravação de voz e imagem. Os principais resultados mostraram que nove dos doze estudantes foram considerados mais habilidosos e apresentaram uma tendência para representar os problemas de maneira proposicional. Essa representação sempre que usada por estudantes que dominavam os elementos do problema e suas relações levavamnos ao sucesso em suas soluções. Os desempenhos desses estudantes no ENADE não indicaram estar associados ao fato de serem ingressantes ou concluintes, mas relacionados ao desempenho em nível acadêmico. O raciocínio dedutivo diferiu significativamente entre estudantes concluintes e ingressantes e, dentre os bem e mal sucedidos em atividades acadêmicas. De uma maneira geral, os 141 estudantes tenderam a apresentar associação entre os desempenhos do ENADE e os do EGRAF, que avaliam as inteligências fluido-cristalizadas e cristalizadas, respectivamente
Abstract: he present research explored the relations between the performance in ENADE 2005, the performance in EGRAF and the deductive reasoning of 141 Freshman and Senior students of the Computer Science course when involved in tasks that explored the mathematic thinking. The instruments were the ENADE exam, the EGRAF exam and the psychological test GfRLD. In addition, twelve of those students were submitted to an individual test of some mathematical ability components and the mental representation, over there the variables studied for the 141 students. For the ability and the representation five of Krutetskii's serie XIX- B series problems and a semi structurated interview were used. During the application of the Krutetskii's test, the twelve students were invited to think out loud. Notes were made to complement information not captured by the voice and image videotape. The main results showed that nine out of twelve students were considered more skilful and presented a tendency to represent the problems in a propositional way. This representation when used by students that dominated the elements of the problem and its relations took them to success in their solutions. Those students' performances in ENADE indicated no association with the fact that they were Freshmen or Senior students, but they were related with the level academic performance. The deductive reasoning differed significantly between Freshmen and Senior students and, between well and bad succeeded in academic tasks. In general, the 141 students tended to present association between ENADE performance and EGRAF performance, that evaluate the fluid-crystallized and crystallized intelligences, respectively
Doutorado
Psicologia, Desenvolvimento Humano e Educação
Doutor em Educação

Thesis (MPhil) -- Stellenbosch University, 2001.
Bibliography
ENGLISH ABSTRACT: Few news subjects or events can be comprehensively covered in the media without numbers being used. Indeed, most reports are essentially 'number stories', or could be improved through the judicious use of numbers. Despite this there are frequent complaints about poor levels of numeracy among journalists. Although numbers are fundamental to virtually everything they write, the most superficial review of South African newspapers indicates that most encounters between journalists and numbers of any sort are uncomfortable, to say the least. Reporters shy away from using numbers, and frequently resort to vague comments such as "many", "more", "worse" or "better". When reports do include numbers, they often don't make sense, largely because journalists are unable to do simple calculations and have little understanding of concepts such as the size of the world's population, a hectare, or a square kilometer. They frequently use numbers to lend weight to their facts without having the numerical skills to question whether the figures are correct. Numeracy is not the ability to solve complicated mathematical problems or remember and use a mass of complicated axioms and formulas; it's a practical life skill. For journalists it is the ability to understand the numbers they encounter in everyday life - percentages, exchange rates, very large and small amounts - and the ability to ask intelligent questions about these numbers before presenting them meaningfully in their reports. This thesis is not a compendium of all the mathematical formulas a journalist could ever need. It is a catalogue of the errors that are frequently made, particularly in newspapers, and suggestions to improve number usage. It will hopefully also serve to make journalists aware of the potential of numbers to improve reporting and increase accuracy. This thesis emphasises the importance of basic numeracy for all journalists, primarily by discussing the basic numerical skills without which they cannot do their job properly, but also by noting the concerns of experienced journalists, mathematicians, statisticians and educators about innumeracy in the media. Although the contents of this thesis also apply to magazine, radio and television journalists, it is primarily aimed at their counterparts at South Africa's daily and weekly newspapers. I hope the information contained herein is of use to journalists and journalism students; that it will open their eyes to the possibility of improving number usage and thereby reporting, serve as encouragement to brush up their numerical skills, and help to shed light on the numbers which surround them and which they use so readily.
AFRIKAANSE OPSOMMING: Min nuusonderwerpe of -gebeure kan in beriggewing tot hul reg kom sonder dat enige getalle gebruik word. Trouens, die meeste berigte is in wese 'syferstories', of kan verbeter word deur meer sinvolle gebruik van syfers. Tog is daar vele klagtes oor joemaliste se gebrekkige syfervaardigheid. Ten spyte van die ingeworteldheid van getalle in haas alles wat hulle skryf, toon selfs die mees oppervlakkige ondersoek na syfergebruik in Suid-Afrikaanse koerante joemaliste se ongemaklike omgang met die meeste syfers. Hulle is skugter om syfers te gebruik, en verlaat hulle dikwels op vae kommentaar soos "baie", "meer", "erger" of "beter". Indien hulle syfers gebruik, maak die syfers dikwels nie sin nie: meermale omdat joemaliste nie basiese berekeninge rondom persentasies en statistiek kan doen nie, en min begrip het vir algemene groothede soos die wereldbevolking, 'n hektaar of 'n vierkante kilometer. Hulle sal dikwels enige syfer gebruik omdat hulle meen dit verleen gewig aan hul feite en omdat hulle nie die syfervaardigheid het om dit te bevraagteken nie. Syfervaardigheid is nie die vermoe om suiwer wiskunde te doen of 'n magdom stellings en formules te onthou en gebruik nie; dis 'n praktiese lewensvaardigheid, die vermoe om die syferprobleme wat die daaglikse roetine oplewer - persentasies, wisselkoerse, baie groot en klein getalle- te verstaan en te hanteer. Hierdie tesis is nie 'n versameling van alle berekeninge wat joemaliste ooit sal nodig kry nie; maar veel eerder 'n beskrywing van die potensiaal van syfers om verslaggewing te verbeter en joemaliste te help om ag te slaan op die getalle rondom hulle en die wat hulle in hul berigte gebruik. Die doel van die tesis is om die belangrikheid van 'n basiese syfervaardigheid vir alle joemaliste te beklemtoon, veral die basiese syfervaardighede waarsonder joemaliste nie die verslaggewingtaak behoorlik kan aanpak nie, te bespreek, en ook om ervare joemaliste, wiskundiges, statistici en opvoeders se kommer oor joemaliste se gebrek aan syfervaardigheid op te teken. Hoewel alles wat in die tesis vervat is, ewe veel van toepassing is op tydskrif-, radio- en televisiejoemaliste, val die klem hoofsaaklik op hul ewekniee by Suid-Afrikaanse dag- en weekblaaie. Ek hoop die inligting hierin vervat sal van nut wees vir praktiserende joemaliste en joemalistiekstudente om hulle bewus te maak van die moontlikhede wat bestaan om syfergebruik, en uiteindelik verslaggewing, te verbeter en as aanmoediging dien om hul syfervaardigheid op te skerp.

This study is based on a qualitative investigation framed within an interpretive paradigm, and aims to investigate the extent to which question design affects the solution strategies adopted by children when solving linear number pattern generalisation tasks presented in pictorial and numeric contexts. The research tool comprised a series of 22 pencil and paper exercises based on linear generalisation tasks set in both numeric and 2-dimensional pictorial contexts. The responses to these linear generalisation questions were classified by means of stage descriptors as well as stage modifiers. The method or strategy adopted was carefully analysed and classified into one of seven categories. A meta-analysis focused on the formula derived for the nth term in conjunction with its justification. The process of justification proved to be a critical factor in being able to accurately interpret the origin of the sub-structure evident in many of these responses. From a theoretical perspective, the central role of justification/proof within the context of this study is seen as communication of mathematical understanding, and the process of justification/proof proved to be highly successful in providing a window of understanding into each pupil’s cognitive reasoning. The results of this study strongly support the notion that question design can play a critical role in influencing pupils’ choice of strategy and level of attainment when solving pattern generalisation tasks. Furthermore, this study identified a diverse range of visually motivated strategies and mechanisms of visualisation. An awareness and appreciation for such a diversity of visualisation strategies, as well as an understanding of the importance of appropriate question design, has direct pedagogical application within the context of the mathematics classroom.

Orientador: Anamaria Gomide
Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: Tornar o ensino dos conteúdos de matemática significativo para os alunos do Ensino Médio é um dos desafios que os atuais professores brasileiros tem enfrentado durante suas aulas. Diante disso, considerando que a matemática é uma ciência de observação e abstração e que as funções reais fazem parte do currículo de matemática do 1º ano do Ensino Médio, propomos como metodologia de ensino a abordagem das principais funções reais ¿ afim, quadrática, exponencial, logarítmica ¿ e suas respectivas representações gráficas a partir de situações-problema do cotidiano. Posteriormente a identificação de qual expressão se adequa à situação, discutimos as principais características dessa função e como elas se relacionam com o cotidiano descrito. Além disso, apresentamos alguns exercícios inéditos e de processos de seleção universitária, que tanto exemplificam qual tipo de questão pode ser usada como avaliação, como demonstram o alinhamento deste método com o Exame Nacional do Ensino Médio (Enem), principal avaliação do governo federal deste segmento da educação básica. Por fim, discutimos uma importante função real presente nas ciências biológicas, a equação logística. Após relembrar historicamente como ocorreu sua conceituação aplicamos a metodologia proposta apresentando a expressão a partir de situações-problema. Feito isto, analisamos as possíveis variações gráficas e o que cada um significa do ponto de vista prático
Abstract: Make the teaching of significant Mathematical content for High School students is one of the challenges that the current Brazilian teachers have faced during their classes. Therefore, considering that Mathematics is a science of observation and abstraction and the real functions are part of the Mathematics curriculum of the 1st year of High School, we propose as a teaching methodology addressing the major real functions - linear, quadratic, exponential, logarithmic - and their graphical representations from everyday problem situations. Later determining which expression suits the situation, we discuss the main features of this function and how they relate to the described daily. In addition, we present some exercises, unpublished and university selection processes that both exemplify the kind of question can be used as evaluation, as demonstrated alignment of this method with the National Secondary Education Examination (Enem), the main assessment of the federal government this segment of basic education. Finally, we discuss an important real function present in the Biological Sciences, the logistic equation. After recall occurred historically as its conceptualization apply the proposed methodology showing the expression from problem situations. This done, we analyze the possible graphical variations and what each means from a practical point of view
Mestrado
Matemática em Rede Nacional
Mestre em Matemática em Rede Nacional

CAPES
Este trabalho traz um relato a respeito da Hipótese de Riemann, com o objetivo de tornar os conceitos referentes a esse problema acessíveis ao professor da educação básica, que pretenda abordá-los em sala de aula quando tratar de conteúdos a ele relacionados. A pesquisa foi inteira bibliográfica, apoiada em sua grande parte em textos de História da Matemática, tornando este trabalho divulgador dos problemas que ocupam parte das pesquisas matemáticas deste século, em especial da Hipótese de Riemann.
This study presents a report about the Riemann Hypothesis, leaving the underlying concepts behind this problem more accessible to a high school teacher. The literature review was based mainly on History of Mathematics texts. This research aims to study significant topics of mathematical research throughout this century, particularly to popularize the Riemann Hypothesis.

Como indicam os Parâmetros Curriculares Nacionais, um dos objetivos do Ensino Fundamental no Brasil é o de que os alunos sejam capazes de questionar a realidade formulando problemas e tratando de resolvê-los (PCN, 1998, p. 27). Na mesma perspectiva, um dos propósitos do terceiro ciclo, que corresponde ao sexto ano do Ensino Fundamental, em Matemática, é o de que os alunos sejam capazes de resolver situações-problema envolvendo números naturais, inteiros, racionais e a partir delas ampliar e construir novos significados para as operações aritméticas (op. cit., p. 64). Assim, a Matemática pode dar sua contribuição à formação do cidadão ao proporcionar a construção de estratégias, a comprovação e a justificativa de resultados (op. cit., p. 27) no desenvolvimento da capacidade para resolver problemas, sejam eles dessa ou de qualquer outra área do conhecimento. O ensino de Matemática não tem só a função evidente de propiciar o desenvolvimento de competências referentes ao manuseio das mais diversas habilidades matemáticas, mas deve ter também a preocupação de promover o desenvolvimento de capacidades como comunicação, argumentação e validação de processos (PCN, 1998, p. 56). Essas, por sua vez, necessitam das habilidades de interpretação e expressão escrita e/ou falada. Aprender a resolver problemas matemáticos na escola é deparar-se com um mundo de conceitos que envolvem leitura e compreensão, tanto da língua materna como da linguagem matemática. A resolução de problemas exige compreensão leitora. Para essa compreensão, o aluno precisa de um referencial linguístico e, para expressar os dados em sentenças matemáticas, de um referencial de linguagem matemática, ambos adequados a cada situação-problema a que for exposto. Oferecer ao aprendiz oportunidades de compreensão do enunciado de problemas, por certo o auxiliarão não só a resolvê-los como também a ampliar e aperfeiçoar o estabelecimento de inferências e de conexões lógicas. Há vários estudos sobre as dificuldades em leitura e sobre as dificuldades na resolução de problemas, separadamente, mas poucos aproximam essas duas áreas do conhecimento. O objetivo desta pesquisa é o de verificar como os mecanismos coesivos, presentes em enunciados de problemas de aritmética, podem se constituir fatores intervenientes na compreensão leitora desses enunciados. Pensa-se ser possível, a partir daí, vislumbrar aproximações entre os estudos sobre língua materna e linguagem matemática, no que tange à compreensão de enunciados de problemas aritméticos. Parte-se do pressuposto de que a não compreensão do enunciado de problemas aritméticos compromete a conversão dos dados apresentados em linguagem matemática e, por conseguinte, a resolução desses problemas.
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As the Parâmetros Curriculares Nacionais indicate, one of the purposes of Elementary Schools in Brazil is that students should be able to question reality by formulating problems and trying to solve them (PCN, 1998, p. 27). In that same perspective, one of the purposes in Mathematics for the third cycle, which corresponds to the 6th grade in Elementary School, is that students should be able to solve problem-situations involving, natural numbers, whole numbers, and rational numbers and from those situations be able to enhance and build new meanings for arithmetic operations (op. cit., p. 64). Thus, Mathematics can give its contribution to citizens, by providing the construction of strategies, the evidence and justification of results (op. cit., p. 27) towards the development of the capacity of solving problems, whether they belong to this or any other area of knowledge. Teaching Mathematics does not only have the obvious function of providing the development of competences related to handling with the most varied mathematical abilities, but it must also be concerned with the promotion of the development of abilities such as communication, argumentation, and process validation (PCN, 1998, p. 56). These abilities, on their turn, require abilities of written and/or spoken expression and interpretation. Learning to solve mathematical problems at school means facing a world of concepts that involves reading and comprehension both of one‟s native language and of mathematical language. Solving problems requires reading comprehension. For that comprehension, students need to have some linguistic references and to express data in mathematical sentences they need to have some mathematical references, which should be appropriate according to each problem-situation they are exposed to. Offering learners opportunities to understand the problem utterances should certainly help them not only solve the problems but also to widen and improve their ability to establish inferences and logical connections. Many studies have been carried out about reading and about difficulties in solving problems, although very few have put these two areas of knowledge together. The purpose of this study is to verify how cohesive mechanisms, which are present in the utterances of arithmetic problems, can become intervenient factors in the reading comprehension of those utterances. The author believes it is possible from that point of view to catch a glimpse of ways of making studies of native language get closer to studies of mathematical language in what concerns the comprehension of arithmetical problem utterances. The study starts from the assumption that if the arithmetic utterance is not understood, that compromises the conversion of the data presented in mathematical language and, hence, compromises solving those problems.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
O presente trabalho tem como objetivo auxiliar professores de Educação Básica a abordar alguns conceitos básicos de Cálculo Diferencial e Integral no terceiro ano do ensino médio, utilizando os softwares wxMaxima e Geogebra. Nossa proposta visa resgatar o ensino do Cálculo no âmbito escolar, mas de uma maneira diferente da tradicional, utilizando a tecnologia em nosso favor, como uma ferramenta facilitadora no processo ensino-aprendizagem de conceitos de limites, derivadas e integrais. A ideia é que toda a parte algébrica e gráfica, que exige conhecimentos matemáticos específicos, seja feita pelos softwares e que os estudantes aprendam a interpretar as soluções que as ferramentas nos fornecem e a conhecerem um pouco mais sobre a matemática e suas aplicações.
The present work aims to help Basic Education teachers to approach some basic concepts of Differential and Integral Calculus in the third year of high school using the software wxMaxima and Geogebra. Our purpose is to recover the teaching of Calculus in the school context, but in a different way from the traditional one, using the technology in our favor, as a facilitating tool in the teaching-learning process of boundary, derivative and integral concepts. The idea is that all the algebraic and graphic part, which requires specific mathematical knowledge, is done by software and that students learn to interpret the solutions that the tools provide us and to know a little more about mathematics and its applications.

A research report submitted to the Faculty of Science, University of Witwatersrand, in partial fulfilment for the degree of Masters of Mathematics Education by coursework and research report. Johannesburg, March 2017.
The current mathematics curriculum in South Africa require that learners are provided with opportunities to develop abilities to be methodical, to generalise, to make conjectures and try to justify and prove their conjectures. These objectives call for the use of teaching strategies and tasks that support learners’ participation in the development of mathematical thinking and reasoning. This means that teachers have to be cautious when selecting tasks and deciding on teaching strategies for their classes. Tasks differ in their cognitive and difficulty levels and opportunities they afford for learner to learn mathematics competently. The levels of tasks selected by the teachers; the kinds of questions asked by the teachers during the implementation of the selected tasks and how the questions asked by the teachers and the teachers’ actions at implementations affected the levels of the tasks were the focus of this research report. The study was carried out in one high poverty high school in South Africa. Two teachers were observed teaching and each teacher taught their allocated grades. One teacher was observed teaching Grade 9s while the other taught Grade 11s. Both teacher taught number patterns at the time their lessons were observed. The research was qualitative. Methods of data collection and instruments included lesson observations; collection of tasks used in the observed classes, audio-taping and field notes. Pictures of the teachers’ work and copies of learners’ workbooks also provided some data. The analysis of data shows that the teachers not only selected and used lower-level cognitive demand and ‘easy’ tasks, that did not support mathematical thinking, but also did not lift up the levels and/or maintain the ‘difficulty levels’ of the task at implementation. Teachers were unable to initiate class discussions. Their teaching focused on ‘drill and practice’ learning and teaching practices.
LG2017

M.Sc., Faculty of Science, University of the Witwatersrand, 2011
This study serves to explore the notion of context familiarity and how it affects the way learners perform in closed and open-ended problems in Mathematical Literacy (ML). The learners’ performances in this study are based on how well they were able to do the following: select the relevant data from the given tables; select the appropriate mathematics and execute them with precision; relate the mathematical solution back to the context in order to understand the problem better. The key findings indicate that more familiar contexts provide better opportunities for learners to: select the relevant data from given tables; select and execute the relevant mathematical tools; and relate the mathematical solution back to the context.

M.Cur.
Critical thinking is the ideal of a democratic citizenry of a country. Intellectual liberation is the characteristic of the human mind, but it is learned as the individual takes a conscious step into developing his/her critical thinking skills. The virtues of a critical thinker are open-mindedness, intellectual integrity, intellectual empathy, intellectual courage, accommodation for ambiguity, ability to metacognise and acceptance of one's faults in thinking and a willingness to adapt one's beliefs and values. The nursing profession has a dire need for critically thinking practitioners, who will be able to make calculated judgements that brings about life-saving decisions. Nursing education need to adopt an educational approach that integrates development and facilitation of critical thinking in students. Scholars support the ideal of development and facilitation of critical thinking in the classroom as well as in practice. The purpose of this study is to describe strategies that can be used to overcome obstacles in the facilitation of critical thinking in nursing education. To realize this purpose the researcher undertook a qualitative, exploratory, descriptive and contextual research approach (Mouton 1998). The research design is conducted in 2 phases. The recommendations arising out of the study are based on the findings that there is an urgent need to make a concerted effort to facilitate critical thinking in students and further research on how to integrate critical thinking in the curriculum as well as appropriate evaluation of the skills.

M.Ed.
This study investigates junior primary teachers' conception of problem-centred Mathematics teaching with the view to describe and also explain their conception. The rationale for the investigation is derived from the move being made in South African Mathematics teaching from traditional teaching to constructivist and problem-centred teaching. Teachers have had to change from being authoritative and focused on the product to become facilitators of the learning process. This move implies that teachers have to put aside most of what they have been doing up till now to be able to adopt constructivist ways of thinking. Problem-centred teaching is in strong contrast to these traditional teaching methods. Such a shift in paradigm could prove to be traumatic for teachers and pupild. The report of the study commences with a theory framework in which constructivism is clearly explicated. The constructivist view of knowledge, with the relationship between public knowledge and personal knowledge and the forming of personal knowledge is discussed. The focus then shifts to learning through cognitive restructuring which is facilitated by assimilation and accommodation. In the constructivist view, learning is also facilitated by social interaction and reflection. Both the processes and the relationship between social interaction and reflection are discussed. Most importantly, learning is facilitated through constructivist teaching, but successful teaching depends on teachers'conception thereof. Conception is described as one of the most important components of teachers' personal teaching theory. Teachers use their personal teaching theory to reflect on teaching and learning. The literature review is concluded with a discussion on the nature of constructivist teaching and the role of the teacher in such a teaching model. The theory framework is complemented by a chapter on the design of the research, substantiating the choice of format and methods of data collection and analyses. The data is reported in the final chapter in which examples of raw data from transcriptions and sketches are presented. Finally, the consolidated data is interpreted. - The most significant finding of this study is that junior primary teachers in this group have a negative conception of problem-centred Mathematics teaching. It appears that the most important reason for their negative conception is that they were not adequately equipped for the contructivist approach towards Mathematics teaching. This study then proposes that for teachers to be able to teach from a constructivist paradigm they need relevant constructivist training, more support from the experts, but also more support from each other. They need to change their teaching conception to a constructivist conception of teaching.

A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand, Johannesburg. December 2014.
The National Curriculum and Assessment Policy Statement document (CAPS) (Department of Basic Education, 2012: 4) outlines some of the salient skills related to the modelling of word problems that Mathematics learners are expected to acquire during the course of their learning. One such critical skill is the learners' ability to represent and describe situations in algebraic language, formulae and expressions. In line with this broad objective, the present study investigated learners' linguistic knowledge and skills in translating from word to linear algebraic representations. Particularly focussing on errors learners made during the translation process, a class of 40 Grade 11 learners at one school in Gauteng West District were selected and given a written test on word problems. To gain access to learner misunderstandings leading to the observed errors and their sources, five learners were selected for interviews that were audio recorded. A Mixed-method Sequential Explanatory Design was used in the study. The major finding of this present study was that learners committed more semantic errors than syntactic and schematic errors suggesting that they (learners) had challenges in understanding the language used in the word problems. The major sources of semantic errors were found to be learners' lack of vocabulary knowledge, inexperience in interacting with expository text structures and their lack of syntax awareness. In addition to the above finding, learners exhibited limited metacognitive skills essential in word problem solving as well as their inability to reflect on the appropriateness of their written algebraic representations for all categories of word problems. The study concludes by providing recommendations on how errors committed in word problem solving can be attended to

A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand. Johannesburg, 22 August 2017.
Attainment in problem solving in mathematics is one of the cognitive levels that is used to guide the assessments in South Africa. Unfortunately, it is one of the levels in which most students do not perform well. This research investigates whether the approach taken in an intervention focused on discussing problem solving shows potential for improving individual learning outcomes? A pre-test and post-test on problem solving questions was administered to two Grade 8 classes from a private school in a Northern suburb of Johannesburg. The intervention 1 class wrote a pre-test and thereafter the class was taught how to deal with mathematical problems and later the class wrote a post-test. The intervention 2 class only wrote the pre-test and the post-test, with the usual kind of teaching. Not only did the intervention 1 class average improve by 10% from 38% on the pre-test to 48% on the post-test but also the class improved in terms of using models to solve problems. On the other hand, the intervention 2 class improved by 14% from 24% to 38%. However, while the intervention 2 class had an improvement in terms of using models to solve problems, the improvement was not substantial. The results also show that mathematical problem solving involving ratios remain a challenge for the grade 8 students in my school and that more work needs to be done to ensure success in mathematical problems involving ratios. Keywords: Problem-solving; intervention 2 class; intervention 1 class, cognitive levels
LG2018

Mathematics plays a pertinent role in physics. Students' understanding of this role has significant implications in their understanding of physics. Studies have shown that some students prefer the use of mathematics in learning physics. Other studies show mathematics as a barrier in students' learning of physics. In this study the role of mathematics in students' understanding of electricity problems was examined. The study undertakes a qualitative approach, and is based on an intepretivist research paradigm. A survey administered to students was used to establish students' expectations on the use of mathematics in physics. Focus group interviews were conducted with the students to further corroborate their views on the use of mathematics in physics. Copies of students' test scripts were made for analysis on students' actual work, applying mathematics as they were solving electricity problems. Analysis of the survey and interview data showed students' views being categorised into what they think it takes to learn physics, and what they think about the use of mathematics in physics. An emergent response was that students think that, problem solving in physics means finding the right equation to use. Students indicated that they sometimes get mathematical answers whose meaning they do not understand, while others maintained that they think that mathematics and physics are inseparable. Application of a tailor-made conceptual framework (MATHRICITY) on students work as they were solving electricity problems, showed activation of all the original four mathematical resources (intuitive knowledge, reasoning primitives, symbolic forms and interpretive devices). Two new mathematical resources were identified as retrieval cues and sense of instructional correctness. In general, students were found to be more inclined to activate formal mathematical rules, even when the use of basic or everyday day mathematics that require activation of intuitive knowledge elements and reasoning primitives, would be more efficient. Students' awareness of the domains of knowledge, which was a measure of their understanding, was done through the Extended Semantic Model. Students' awareness of the four domains (concrete, model, abstract, and symbolic) was evident as they were solving the electricity questions. The symbolic domain, which indicated students' awareness of the use of symbols to represent a problem, was the most prevalent.
Science and Technology Education
D. Phil. (Mathematics, Science and Technology Education (Physics Education)))

Community based service-learning is one of the fastest growing reforms in higher education, especially in the field of health care. The increased interest in this phenomenon is based on the demands by government and society that higher education institutions should be more responsive to the needs of the community. Literature, however, reflects that service learning lacks a sound theoretical base to guide teaching and learning due to limited research in this area. This study was, therefore, aimed at exploring the phenomenon knowledge construction in basic nursing programmes in selected South African nursing schools with the intention to generate a middle range theory that may be used to guide the process of knowledge construction in community-based service-learning programmes. This study adopted a qualitative approach and a grounded theory research design by Strauss and Corbin. Two university-based schools of nursing were purposively selected to participate in the study. There were a total number of 16 participants. The collection of data was intensified by the use of multiple sources of data (participant observation, documents analysis and in-depth structured interviews). The data analysis process entailed three phases; open, axial and selective coding. The results of the study revealed that the phenomenon “knowledge construction” is conceptualised as having specific core characteristics, which include the use of authentic health-related problems, academic coaching through scaffolding, academic discourse-dialogue and communities of learners. The findings showed that there are a number of antecedent conditions and contextual circumstances contributing to how knowledge is constructed in a community based service learning programme. The process of knowledge construction emerged as cyclical in nature, with students, facilitators and community members having specific roles to play in the process. A number of intervening variables were identified that had an influence on the expected outcomes on knowledge construction in community based service learning programmes. These findings led to the generation of a conceptual model. Knowledge construction according to this model takes place in an environment which is characterised by interactive learning, collaborative learning, actively learning and inquiry-based learning through continuous reflective learning processes. The main concepts in this conceptual model include concrete learning experiences, continuous reflection, problem posing, problem analysis, knowledge deconstruction and knowledge generation, knowledge verification, knowledge generation, testing of generated knowledge and evaluation of generated knowledge. The sub-concepts include learning through senses, an initial situation, health-related triggers, social interaction, reflection-in action, reflection-on action, hypotheses generation, conceptualisation of learning experiences, information validation and community interventions. Recommendations were categorised into education and training of academic staff, application of the model and further research with regard to quality assurance in CBSL programmes as well as the use of other research designs for similar studies.
Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2011.

The low grades that students score in some statistical units in Kenyan universities is of great concern and has evoked research interest in the teaching of some of the units and the students’ learning of the statistical content. The aim of the study was to investigate the difficulties undergraduate students experience in the learning of bivariate normal distribution in a Kenyan university. The research also aimed to answer the following research questions on the difficulties undergraduate students encounter in the learning of bivariate normal distribution. The first research question was based on the reasons why students find learning of bivariate normal distribution difficult and the second research question was to find the reasons why students experience such difficulties in learning bivariate normal distribution. The target population for this study included lecturers teaching statistics in the university, and second- and third- year students enrolled or who have previously completed the probability and statistics III unit, where the bivariate normal distribution content is covered. In selecting students for the study, the simple random sampling technique was employed while convenient sampling was used to select lecturers who participated in the study. A mixed methods design was adopted for this study where both quantitative and qualitative data was collected. A total of 175 students and six lecturers participated in this research study. All students who participated in the study did a bivariate normal distribution test (Appendix 1) designed by the researcher and then filled in a questionnaire (Appendix 2). The lecturers who participated in the study filled in an open-ended questionnaire (Appendix 3). The results showed that undergraduate students have difficulties in learning bivariate normal distribution. This is because most of them could neither state the bivariate normal distribution nor solve any of the application questions on the content. The students find it difficult to learn and comprehend the bivariate normal distribution equation with its many parameters and constants of the two random independent variables. The results also showed that students could not state the normal distribution equation nor could they solve questions on the normal distribution, which forms the foundational knowledge required for effective learning of the bivariate normal distribution content. ii Based on the results, the study recommended that emphasis should be placed on the basic and foundational knowledge of the normal distribution content and its applications before teaching bivariate normal distribution in probability and statistics III. In addition, it is recommended that all students should be involved in the learning of basic content to enable them to understand all parameters and constants in the equations and their applications. The study also recommends that lecturers revise the foundational knowledge and content related to the bivariate normal distribution before introducing and teaching the bivariate normal distribution content. This study also recommends that the university should consider a change of curriculum by teaching the bivariate normal distribution, as an introductory course to the unit under the multivariate distributions in statistics, in third year of the students’ studies. ; ;
Mathematics Education
M. Sc. (Mathematics Education)