Research Report submitted to the Wits School of Education, Faculty of Science, University of the Witwatersrand, Johannesburg In partial fulfilment of the requirements For the degree of Master of Science (Mathematics Education) Johannesburg, 2017 / South Africa’s performance in mathematics education is ranked amongst the world’s worst. This performance is not only alarming at an international level, but also nationally. Annual National Assessments (ANA) conducted by the Department of Education have showed that the level of mathematics across the foundation and intermediate phase is poor with a pronounced dip in performance at a Grade 4 level (Department of Basic Education, 2014). Multiplication and division are common challenging areas that contribute to this poor performance. This is concerning as mathematics is globally recognised as a key competence for providing access to higher education and developing a country’s society and economy. My study, aimed at exploring multiplicative reasoning with Grade 4 learners through structured problem solving, is focused on the learning of multiplication and division within the context of an intervention concentrated on developing learners’ ability to model multiplicative situations. Shifts in the use of models were investigated following a smallscale intervention in which different modelling approaches (particularly ratio modelling) were introduced and developed. A control group was used to determine the usefulness of the intervention. Questions which I sought to answer were: (a) what kinds of multiplicative reasoning (models) are Grade 4 learners using prior to intervention, (b) what changes, if any, are seen in overall performance, across the intervention and control group, in the post-test, and, (c) what kinds of differences in model use were associated with the shifts in performance? The main dataset comprised of 61 pre- and post-test scripts across three Grade 4 classes in a former Model C school in a Johannesburg district. A sample of 15 interviews were also conducted across the classes. Document analysis and transcription notes were used to analyse data with a Realistic Mathematics Education (RME) framework informing my analysis. Findings from my study reveal that prior to intervention, Grade 4 learners presented limited multiplicative models which were predominantly confined to traditional algorithms. After the small-scale intervention, learners used a broader range of models with an emerging take up of ratio models. The success rate associated with the models presented by learners also improved. Limited and/or no changes in model use and their respective success rates were seen in the control group suggesting that the intervention program was useful. These findings suggest that, as a future recommendation, it would be worthwhile to investigate the outcomes of running a similar intervention in less privileged settings. / MT 2018

At last, this dissertation considers another marketing implication of this number-location association, namely the compatibility effect. In experiment 7, we find that people perceive a discount as more attractive when the two prices are actually posited in compatible locations (original price-right side; discounted price-left side) than in incompatible locations. Similarly, experiment 8 demonstrates that people are more likely to patronage a supermarket when the supermarket's slogan about low price is shown on the left side of a display than on the right side, and this effect is mediated by the subjective fluency feeling people felt at the time they process the advertisement. / Given a display, people usually think that large numbers should be located on the top or on the right hand side of the display, whereas small numbers should be posited at the bottom or on the left (Wood and Fischer 2008). / Given this number-location association, this dissertation secondly intends to apply it to the field of marketing, and to use three experiments to explore how and why location of product image can influence people's price judgment. The results of experiment 4 show that consumers think that the market price of a product is higher if the product's image is shown on the right side of a display than on the left side; experiment 5 and experiment 6 further indicate that the location of product image can only influence consumers' price judgment, but cannot influence quality judgment. / Key Words: Number-location association, Simulation, Perceptual Symbol Systems (PSS), Price perception. / This dissertation firstly aims to provide new evidence for this number-location association. Experiment 1 demonstrates that people incorrectly remember that large numbers appear to the right of the locations they actually were shown while small numbers appear to the left ofthe locations than they actually were presented; experiment 2 and experiment 3 show that people estimate there are more pieces in a pile of object when the pile of object is presented on the right side of a display than on the left side. / Cai, Fengyan. / Adviser: King Man Hui. / Source: Dissertation Abstracts International, Volume: 73-03, Section: A, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 139-144). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest dissertations and theses, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Probabilities, an essential dimension of risk communication, can be presented in various formats including frequencies (e.g., 1 in 10), percentages (e.g., 10%), or verbal phrases (e.g., unlikely); the literature is mixed concerning which format best supports comprehension. Additionally, it is not well understood how people who vary in their level of numeracy understand those probabilities. The goal of the present three-phase within-participant study was to understand how the factors of format and numeracy influence comprehension and mental representations of probabilities for younger and older adults. Overall, the results of this research clearly indicated that comprehension and mental representation of health risk probabilities are influenced by format, age, and numeracy. To best support comprehension and comparison of health risk probabilities for younger adults and healthy older adults with varying numeracy, percent format should be used.

Thesis (Ph. D.)--Ohio State University, 2008. / Title from first page of PDF file. Includes bibliographical references (p. 93-99).

Thesis (Ed.S.)--Mississippi State University. Department of Curriculum and Instruction. / Title from title screen. Includes bibliographical references.

Thesis (M.Phil.) - University of Queensland, 2005. / Includes bibliography.

Thesis (MEd (Education))--Cape Peninsula University of Technology, 2016. / The poor performance of learners in mathematics has long been a matter of concern in South Africa. One certain fact from the Annual National Assessment (ANA) results is that the problem starts in the Foundation Phase (FP) with number concepts. The aim of this study was to explore how five Foundation Phase teachers located in challenging socio-economic school contexts in the Western Cape used mathematical resources to promote teaching for understanding of the important number concept area in CAPS. These resources included humans, materials, culture and time. The research was located within the interpretive qualitative research paradigm and used a case study approach. The participants in the study included five FP teachers teaching Grades 1 to 3 at two schools in the Western Cape. Data was collected through lesson plan analysis, lesson observations and semi-structured interviews. The data collected was then analysed through the lens of Vygotsky’s socio-cultural theory. Socio-cultural theory maintains that knowledge is best acquired if it is mediated by language, more knowledgeable others and physical tools. Vygotsky believed that knowledge is first acquired interpersonally, then intrapersonally, as learners first learn from others, then internalise or individualise knowledge while going through the four stages of the Zone of Proximal Development (ZPD). The findings of this study revealed that teaching for understanding was often compromised by teaching to enable learners to pass assessments. Teachers understood the importance of using resources to teach number concepts in the Foundation Phase, but inclined to rote teaching with work drills in preparation for assessments such as the Annual National Assessment (ANA) and the systemic assessment. Resources were often used when learners struggled to understand concepts and as calculation tools. This study supports the view from the literature that the way in which resources are used affects the teaching and learning of number concepts. It recommends that teachers should read and follow the CAPS mathematics document, as it clearly states what resources to use and how. This study further recommends that more research on the use of resources to teach mathematics in other content areas should be done.

Thesis (DEd)--Stellenbosch University, 1995. / ENGLISH ABSTRACT: The rationale for this study essentially is the perceived and reported misconceptions in algebra that exist within pupils in the Junior Secondary phase. These misconceptions are the direct result of the incomplete mastery of algebra. The purpose of this research is to attempt to contribute towards pupils' complete mastery of algebra and the ensuing elimination of certain pupil misconceptions, by trying to increase primary school pupils' level of awareness of the general number properties. Primary school pupils who learn arithmetic in a problem based environment intuitively use the general number properties to execute arithmetical calculations, while pupils in the traditional teaching approach are taught standard algorithms which they must apply, often without comprehension. In the latter, the existence of the number properties is concealed. When these pupils officially encounter the general number properties for the first time in standard 4 or 5, they are probably not capable of integrating these new concepts into their existing knowledge structure. This may lead to incomplete mastery and ensuing misconceptions, which can become worse when pupils must apply exactly the same number properties in algebra to execute algebraic manipulations. The fact that primary school pupils in the problem based approach intuitively apply the number properties, earlier led to the hypothesis that these pupils possess a higher level of awareness of the number properties. However, research has indicated that these pupils possess only a moderately higher level of awareness. This has led to this study during which a specific attempt is made to take pupils' intuitive knowledge of the number properties, which they spontaneously apply, as a point of departure, and to develop it to such an extent that they become explicitly aware of the existence and nature of these fundamental concepts in mathematics. The technique how to link up with pupils' intuitive knowledge and to increase their level of awareness of the number properties inside a problem based environment, was developed during the first two years of this study. At the end of this period, a suitable teaching strategy was formulated. This strategy is essentially based on the following parameters: Group discussion, the use of calculators, and creating a cognitive disequilibrium. During the third year of this study this strategy was implemented at a number of schools. During this phase research concentrated on the distributive property only. Judging by the results, it appears as if a considerable increase in level of awareness has taken place within these pupils. During this study a hierarchical model of levels of awareness was also formulated. This model was used as a guide in the attempt to increase pupils' level of awareness of the number properties. / AFRIKAANSE OPSOMMING: Die rasionaal vir hierdie studie is hoofsaaklik gelee in die waargenome en gerapporteerde wanbegrippe in algebra wat by leerlinge in die Junior Sekondere fase bestaan. Hierdie wanbegrippe is die direkte gevolg van die onvolledige beheersing van algebra. Die doel van hierdie navorsing is om 'n hydrae te probeer lewer tot leerlinge se volledige beheersing van algebra, en die gepaardgaande uitskakeling van bepaalde leerlingwanbegrippe, deurdat gepoog word om laerskoolleerlinge se vlak van bewustheid van die algemene bewerkingseienskappe te verhoog. Laerskoolleerlinge wat probleemgebaseerde rekenkunde-onderrig ontvang, benut intu'itief die algemene bewerkingseienskappe ten einde rekenkundige berekeninge uit te voer, terwyl aan laerskoolleerlinge in die tradisionele onderrigbenadering standaardalgoritmes onderrig word wat hulle dikwels sonder begrip moet toepas. In laasgenoemde geval word die bestaan van die bewerkingseienskappe erg versluier. Wanneer hierdie leerlinge in standerd 4 of 5 vir die eerste keer amptelik met die algemene bewerkingseienskappe in aanraking kom, is hulle waarskynlik nie in staat om hierdie nuwe begrippe met hul bestaande kennisstruktuur te integreer nie. Dit kan tot onvolledige beheersing en gevolglike wanbegrippe lei, wat kan vererger wanneer leerlinge presies dieselfde eienskappe in algebra moet benut ten einde algebra!ese manipulasies uit te voer. Die feit dat laerskoolleerlinge in die probleemgebaseerde benadering intu'itief die bewerkingseienskappe toepas, het tevore reeds tot die hipotese dat hierdie leerlinge oor 'n hoer vlak van bewustheid van die bewerkingseienskappe beskik, aanleiding gegee. Navorsing het egter aangetoon dat daar slegs 'n matige hoer vlak van bewustheid by hulle bestaan. Dit het tot hierdie studie aanleiding gegee waartydens spesifiek gepoog word om leerlinge se intuitiewe kennis van die bewerkingseienskappe wat hulle spontaan aanwend ten einde rekenkundige bewerkings uit te voer, as vertrekpunt te neem, en te ontwikkel sodat hulle eksplisiet bewus sal raak van die bestaan en wese van hierdie grondliggende waarhede in wiskunde. Die tegniek hoe om by laerskoolleerlinge se intui'tiewe kennis aan te sluit en hul vlak van bewustheid van die bewerkingseienskappe binne 'n probleemgebaseerde omgewing te verhoog, is gedurende die eerste twee jaar van hierdie studie ontwikkel. Aan die einde van hierdie periode is 'n toepaslike onderrigstrategie geformuleer. Hierdie strategie steun sterk op die volgende parameters: Groepsbespreking , die benutting van sakrekenaars, en die skep van 'n kognitiewe disekwilibrium. Gedurende die derde jaar van hierdie studie is hierdie onderrigstrategie in verskeie skole toegepas. Daar is gedurende hierdie fase slegs op die distributiewe eienskap gekonsentreer. Volgens die resultate wil dit voorkom asof daar 'n aansienlike verhoging in die betrokke leerlinge se vlak van bewustheid van die distributiewe eienskap plaasgevind het. Gedurende hierdie studie is ook 'n hierargiese model van vlakke van bewustheid geformuleer aan die hand waarvan die poging aangewend is om leerlinge se vlak van bewustheid van bewerkingseienskappe te verhoog.

This study considered use of practical work as one of the strategies that may be used to teach and learn fraction concepts in primary school Mathematics. Although an educator and learners were participants in the study, the focus was mainly on the learners. The class educator’s perception of practical work was investigated and the results confirmed the assumption that most educators use minimal or no practical work when teaching learners fractions. The researcher carried out an experiment with learners to find out whether they saw any value in doing practical work. Data collection instruments used were an observation schedule which was collated by the researcher in teaching four lessons, written responses of learners to a series of activities they did as class work and their responses to interview questions. Data collected from learners confirmed that practical work did have value in the teaching of fraction concepts, especially addition of fractions. Other than confirming the value of practical work, much other valuable data emerged from the findings. The data have important implications for the teaching and learning of fractions, especially addition of fractions, teacher training in practical work and also further research. These are intended to improve teaching of fractions, particularly addition of fractions. / M. Ed. University of KwaZulu-Natal, Durban 2013.

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2018-07-27T13:29:56Z No. of bitstreams: 1 Aline Tafarelo Tracanella.pdf: 2762008 bytes, checksum: a0ecbfb9e128d24bccdf1a07c2c5e734 (MD5) / Made available in DSpace on 2018-07-27T13:29:57Z (GMT). No. of bitstreams: 1 Aline Tafarelo Tracanella.pdf: 2762008 bytes, checksum: a0ecbfb9e128d24bccdf1a07c2c5e734 (MD5) Previous issue date: 2018-05-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / As soon as children begin their school life, they already carry with them an idea about the numbers and operation of the Decimal Number System (DNS). However, this knowledge need to be systematized, extended and deepened appropriately in order to assist in the construction of other mathematical concepts. Given this problem, the present research aims to investigate the mobilized knowledge of the positional value in the DNS and the understanding of the characteristics of number zero in the same system by students of the fourth year of Elementary School. Therefore, it is done a brief historical context to rescue how the development of this kind of knowledge by ancient people has developed over time. As theoretical contributions, it is used the researches of Piaget & Szeminska, and of Kamii on the constructions of the number concept by the students. Regarding to the acquisition of the properties of the DNS, it is discussed the researches of Fayol, Lerner & Sadovsky as well as Zunino, who also studies the issue of the number zero in this system. To achieve the research objective, it is adopted the qualitative methodology, since the focus of it is on the mobilized knowledge by the students in the search for a solution to proposed activities. It was also developed an instrument with six exercises involving the positional value and the number zero, based on the proposed sequence in the Brandt version. One week after an application of the instrument, it was conducted a semistructured interview, which was of very important to understand the answers provided by the students. In the analysis and discussion of the obtained data, it is understand that the students mobilized knowledge about the numerical sequence and the criteria of comparison pointed out by Lerner & Sadovsky. In addition to these mobilized knowledge, the participants also used the contextualization of activities to justify their responses, using a comparison with everyday situations, such as, for example, age observation among children. Regarding the number zero, it was analyzed the meanings attributed to this number by the students during interviews. During the research phases, all students stated that zero “worth nothing”, but they have provided justifications that meet the historical facts pointed out in the brief contextualization carried out in the third chapter of the research. It is also noted that the participants are building their knowledge about DNS, presenting an unstable knowledge that changes according to the question asked regarding the proposed situation. The results found in this research indicate that the work with DNS needs to be continuous throughout the initial years of Elementary School, as the students continue to build their knowledge about DNS and expand their understanding of the number zero in the years after the literacy cycle / Assim que as crianças iniciam sua vida escolar, já carregam consigo alguma ideia sobre os números e sobre o funcionamento do Sistema de Numeração Decimal (SND). Todavia esses conhecimentos precisam ser sistematizados, ampliados e aprofundados adequadamente, para auxiliar na construção de outros conceitos matemáticos. Diante dessa problemática, a presente pesquisa tem por objetivo investigar que conhecimentos são mobilizados por alunos do quarto ano do Ensino Fundamental acerca do valor posicional no SND e sobre a compreensão do número zero nesse mesmo sistema. Para isso, buscamos em uma breve contextualização histórica resgatar como se deu o desenvolvimento desses saberes por povos antigos no decorrer do tempo. Como aportes teóricos, nos baseamos nas pesquisas de Piaget e Szeminska e de Kamii sobre a construção do conceito de número pelos alunos. Com relação à aquisição das propriedades do SND, discorremos sobre as pesquisas de Fayol e de Lerner e Sadovsky, bem como de Zunino, que aborda também a questão do número zero nesse sistema. Para atender ao objetivo da pesquisa, adotamos a metodologia de cunho qualitativo, pois o foco da investigação está nos conhecimentos mobilizados pelos educandos na busca por uma solução para as atividades propostas. Elaboramos um instrumento com seis exercícios envolvendo o valor posicional e o número zero, baseado na sequência proposta na tese de Brandt. Uma semana após a aplicação do instrumento, realizamos uma entrevista semiestruturada, que foi de suma importância para compreender com maior clareza as respostas fornecidas pelos alunos. Na análise e discussão dos dados obtidos, compreendemos que os estudantes mobilizaram conhecimentos acerca da sequência numérica e dos critérios de comparação apontados por Lerner e Sadovsky. Além desses conhecimentos mobilizados, os participantes também recorreram à contextualização das atividades para justificar suas respostas, usando a comparação com situações cotidianas, como, por exemplo, a observação da idade entre crianças. Com relação ao número zero, analisamos os significados atribuídos a esse número pelos alunos durante as entrevistas. Durante as fases da pesquisa, todos os educandos afirmaram que o zero “não vale nada”, mas trouxeram justificativas que vão ao encontro dos fatos histórico apontados na breve contextualização realizada no primeiro capítulo da investigação. Notamos também que os participantes estão construindo seus conhecimentos acerca do SND, apresentando um conhecimento não estável, ou seja, que se altera de acordo com a pergunta feita referente à situação proposta. Os resultados encontrados nessa pesquisa apontam que o trabalho com o SND precisa ser contínuo, durante todos os anos iniciais do Ensino Fundamental, pois os alunos continuam construindo seus conhecimentos acerca do SND e ampliando sua compreensão sobre o número zero nos anos posteriores ao ciclo de alfabetização