The purpose of this research was to examine children's unmanipulated self-reports of their internalized representation of numbers and the relationship of the spatio-organizational patterns that are represented by the children's drawings to children's ability to solve basic addition problems. Also of interest were possible changes that occurred in children's spatio-organizational patterns as a result of age, mathematics achievement or gender. It was hypothesized that children whose drawings demonstrated more structured spatio-organizational patterns would achieve a higher number of correct answers on a timed test of basic addition problems. It was also hypothesized that the structure of the spatio-organizational patterns that children drew would be influenced by age, gender and mathematics achievement. The results of this exploratory study of children’s unmanipulated internalized constructs of number provided some interesting results. The children were asked to image specific numbers of dots for numerals from 4 through 13 and then to draw a representation of their images. The representations were categorized according to the structure of spatio-organizational patterns. The analyses revealed that the patterns had more structure for older children. Multiple regression analyses also indicated that the correctness of the cardinality of the number of dots imaged was the most frequently occurring variable that had a significant effect on the Imagery Scores. Less than five of more than 450 students expressed any difficulty with the imagery task and then only as it related to one of the ten numerals they were asked to image. The students were asked to image at the foundational level of imagery--reproductive imagery (Piaget & Inhelder, 1971). Because the research task developed for the students did not involve anticipatory images, those requiring transformations or movement, these imaging tasks were not influenced by the children's IQ or mathematics achievement. According to Piaget and Inhelder, children's ability to use anticipatory images indicates that children are developing an operational understanding and use of imagery. The children in this study were not asked to do anticipatory imaging. This may account for the negative relationship of the Imagery Scores to the fifth-grade students’ math percentile scores and the positive relationship between Imagery scores and mathematics percentile scores for the primary level students. The imagery tasks requested of the students were not of sufficient difficulty to relate to any mathematical operations or logio-mathematical thinking for older children. The ability of children to produce reproductive images which have varying degrees of spatio-organizational patterns was demonstrated by this study. Future studies need to address the higher level of anticipatory images. If students were asked to image a specific number of dots and then to image adding another quantity of dots to the original image, would the spatio-organizational patterns used by children in this transformation process change or transform the image? Are there specific spatio-organizational patterns that more easily allow children to develop anticipatory images that use mathematical operations? Are there children who have developed static reproductive images, and as a result, have created internalized constructs that inhibit their future understanding and development of higher level mathematical concepts? / Ed. D.

ENGLISH ABSTRACT: Number properties, amongst others the commutative, associative and distributive properties and general rearrangement principles, form the building blocks of manipulative algebra. Research and observation have shown that sec~mdary school pupils do not sufficiently master manipulative algebra, i.e. they do not possess sufficient mastery towards the nature, meaning, functionality and logic of algebraic manipulations. They are hence not aware that algebraic manipulations are based on the number properties, on the one hand because they were not given sufficient opportunity to experience algebra as generalised arithmetic when they were introduced to algebra, and on the other hand because the number properties, about which young children possess intuitive knowledge, were never explicated for them. This study investigates the level of awareness of several number properties present in senior primary (especially standard 3) pupils, and utilises a few activities to attempt to lead pupils towards a higher level of awareness. In addition this study attempts to determine whether pupils who follow the experimental primary mathematics curriculum (project pupils) possess a higher level of awareness than pupils who follow the traditional curriculum (nonproject pupils). As part of the latter effort, two investigation methods are utilised with regards to specifically the distributive property, i.e. clinical interviews and questionnaires. This also serves as part of a wider effort to design a measuring instrument with which possible differences between the learning outcomes of project and non-project pupils can be measured .. From the results of this study, it seems to appear that the large majority of pupils are explicitly aware of the commutative properties of addition and multiplication and the general rearrangement principles, to a lesser extent with regards to a minus sign before brackets, and that there does not exist a significant difference about the level of awareness towards these properties between project and non-project pupils. With regards to the distributive property, there appears to exist a considerable amount of difference in the level of awareness between project and non-project pupils, the first mentioned being the higher. However, the opinion is expressed that the level of awareness among project pupils is not high enough, and that project pupils must be given sufficient opportunity in (at least standards 4 and 5) to explicate this property for themselves. Finally, a model of the levels of awareness, based on results of this study, is proposed. / AFRIKAANSE OPSOMMING: Getaleienskappe, waaronder die kommutatiewe, assosiatiewe en distributiewe eienskappe en algemene herrangskikkingsbeginsels, vorm die boustene van manipulatiewe algebra. Navorsing en waarneming het aan die lig gebring dat hoerskoolleerlinge manipulatiewe algebra nie na behore beheers nie, dit wil se hulle beskik nie oor voldoende beheersing ten opsigte van die aard, betekenis, funksionalteit en logika van algebraise manipulasies nie. Hulle is dus nie daarvan bewus dat algebraiese manipulasies op die getaleienskappe berus nie, enersyds omdat hulle nie tydens die kennismaking met manipulatiewe algebra genoegsaam in die geleentheid gestel is om algebra as veralgemeende rekenkunde te ervaar nie, en andersyds omdat die getaleienskappe, waaroor jong kinders intuitiewe kennis besit, nooit vir hulle geeksplisiteer is nie. Hierdie studie stel ondersoek in na senior primere (hoofsaaklik standerd 3) leerlinge se vlak van bewustheid van enkele getaleienskappe, en benut enkele aktiwiteite om leerlinge na 'n hoer vlak van bewustheid daarvan te probeer lei. Hierbenewens word probeer om vas te stel of daar by leerlinge wat die eksperimentele primere wiskunde-kurrikulum volg (projekleerlinge) 'n hoer vlak van bewustheid aanwesig is as by leerlinge wat die tradisionele kurrikulum volg (nie-projekleerlinge). As· deel van laasgenoemde poging, word twee ondersoekmetod~s gevolg ten opsigte van spesifiek die distributiewe eienskap, naamlik kliniese onderhoude en vraelyste. Dit dien ook as deel van 'n breer poging om 'n meetinstrument te ontwerp waarmee moontlike verskille tussen die leeruitkomste van projek- en nie-projekleerlinge gemeet kan word. Dit wil uit die bevindinge van hierdie studie voorkom asof die oorgrote meerderheid leertinge eksplisiet bewus is van die kommutatiewe eienskappe ten opsigte van optelling en vermenigvuldiging en die algemene herrangskikkingsbeginsels, in 'n mindere mate ten opsigte van die minusteken voor hakies, en dat daar nie 'n noemenswaardige verskil in die vlak van bewustheid oor hierdie eienskappe by projek- en nie-projekleerlinge bestaan nie. Sover dit die distributiewe eienskap betref, lyk dit asof daar 'n redelike verskil in die vlak van bewustheid by projek- en nie-projekleerlinge is met eersgenoemde die hoogste. Tog word die mening uitgespreek dat die vlak van bewustheid by projekleerlinge nie hoog genoeg is nie, en dat hulle in minstens standerd 4 en 5 in die geleentheid gestel moet word om hierdie getaleienskap vir hulself te eksplisiteer.

National and international studies show that the standard of mathematics teaching and learning in South Africa is very low compared to other countries. These statistics are worrying because mathematics is one of the ‘gatekeeper’ subjects that determine learners’ access to higher learning and a better future. My study, aimed at exploring two Foundation Phase teachers’ selection and use of examples and representations when teaching number, forms part of a longitudinal study currently underway within the Wits Maths Connect Primary (WMC-P) Project. One of the broad aims of the WMC-P Project is to improve primary teachers’ mathematics content knowledge and also to see this translated into improved pedagogy on the ground. This qualitative study was carried out within the WMC-P Project’s 20-Day in-service training course and one of the ten government schools participating in the broader study. My study aimed to build on research that has been carried out on teachers’ use of examples and representations with a focus on the South African terrain. The dataset comprised of two Foundation Phase teacher’s pre-tests, course-work tasks, field notes, and transcripts of observed lessons. Data was analysed using an analytical framework based on current literature related to examples and representations within mathematics teaching. Findings from my study show possible associations between a higher content knowledge score and the extent of a teacher’s example space and more coherent connections between different representational forms. More studies around this topic are needed because research shows that teachers’ examples and representations in mathematics teaching are important for good teaching and conceptual understanding.

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

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

Recent research has elucidated the sustained benefits of early mathematics instruction. With growing concern about the performance of South Africa’s senior learners in mathematics, it is imperative to look at long-term solutions within the education process. One such solution may be to focus on improved mathematics instruction as early as preprimary school. However, children at this young age are not typically suited to formal teaching. Alternative methods of mathematics instruction must be considered for maximum and effective impact. The study was conducted to test the notion that not all early methods of mathematics instruction are equal. During the empirical research approximately 200 preprimary school children in three different socio-economic environments (urban higher SES, township and rural) were tested after experiencing a teacher-guided play-based mathematics teaching intervention, or after experiencing a worksheet-based or free-flow play-based curriculum. The test performance of the participants was primarily compared to find relations between teaching methods and early mathematics performance. The study found that a teacher-guided play-based curriculum is superior to other curriculums in the instruction of mathematics in all educational settings, regardless of socioeconomic background. / Psychology of Education / M. Ed. (Psychology of Education)

Recent research has elucidated the sustained benefits of early mathematics instruction. With growing concern about the performance of South Africa’s senior learners in mathematics, it is imperative to look at long-term solutions within the education process. One such solution may be to focus on improved mathematics instruction as early as preprimary school. However, children at this young age are not typically suited to formal teaching. Alternative methods of mathematics instruction must be considered for maximum and effective impact. The study was conducted to test the notion that not all early methods of mathematics instruction are equal. During the empirical research approximately 200 preprimary school children in three different socio-economic environments (urban higher SES, township and rural) were tested after experiencing a teacher-guided play-based mathematics teaching intervention, or after experiencing a worksheet-based or free-flow play-based curriculum. The test performance of the participants was primarily compared to find relations between teaching methods and early mathematics performance. The study found that a teacher-guided play-based curriculum is superior to other curriculums in the instruction of mathematics in all educational settings, regardless of socioeconomic background. / Psychology of Education / M. Ed. (Psychology of Education)