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.

Thesis Supervisor: Dr. Marv Westrom The purpose of this study was to investigate the effectiveness of the software program Visual Fractions in teaching basic fraction concepts and the effect that student control over the construction of fraction diagrams had on their learning. The Visual Fractions program provides a diagram and two fractions in numeric form. The diagram consists of a figure divided into partitions with some of the partitions shaded. One fraction represents the shaded parts of the whole and the other represents the unshaded parts. Students can control the total number of partitions and whether each is shaded. Manipulating the diagram changes the value of the fractions. A Non-interactive (crippled) version of the software was designed to eliminate the user-control aspect of the program. Users of this program could click to generate a new fraction, but had no control over the choice of fraction. The computer randomly generated a new fraction and displayed the corresponding diagram each time. A third treatment, Fraction Flash Cards, was designed to simulate the Noninteractive version of the program, without the computer. The students received Flash Cards containing images of the computer-generated fraction diagrams. The study consisted of a pilot project during which data collection techniques were tested and revised and the main study. Sixty-four subjects were taken from four intact classes of grade four students. The students were randomly assigned to one of the three Treatment Groups or the Control Group. Three different sets of data were collected: a pretest and postest on fractions, structured interviews, and field notes taken by the researcher during the treatment process. In Treatment Group One, students used the Interactive Version of Visual Fractions. Here, students could create fractions at their command. There is evidence to suggest that this type of interactive control is a critical factor in learning (Merrill, 1987). In Treatment Group Two, students used the Noninteractive version of the software. Students could control the rate of observing fractions and fraction diagrams, but not the value of the fraction. Students in Treatment Group Three used the Flash Cards. Motivation appears to strongly affect one's ability to learn and children appear to be highly motivated to use computers. The purpose of this treatment was to control for any achievement gain that may have been due to the novelty of using computers. The four Groups were compared using analysis of variance with repeated measures. Significance at the 0.01 level was found for the tests and the interaction. A study of the interaction showed that there was no significant difference between the gains of the Visual Fractions Noninteractive Group, the Flash Card Group, or the Control Group. However the gain achieved by the Visual Fractions Interactive Group was significant. From this study, it is clear that the Visual Fractions Interactive program which provides students the opportunity to construct fraction diagrams with immediate feedback, is an effective method of teaching fractions. / Education, Faculty of / Graduate

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The Namibian government through the Ministry of Education seeks for teachers to meet the demands and rise to the challenges of the post-independence basic education system. Therefore the University of Namibia Education Faculty must ensure through the training of student teachers for both lower, upper primary and secondary school level that these demands are met. Research shows that one of the major challenges in teacher training programs, specifically in the mathematics education programmes, lies with the student teachers’ difficulties in dealing with various topics in mathematics, particularly fractions. The teaching of fractions is a challenging area at primary level and lays the foundation for understanding different topics in upper grades such as algebra, ratio and proportion, statistics and probability, to mention just a few. This study focused on an exploration of teaching fractions by five student teachers during their teaching practice.The main purpose was to gain an understanding of the experiences of the level 5-7 student teachers with regard to their practice of fraction teaching; their experience of the mathematics education course with respect to the teaching of fractions; and their prior experience related to fractions and fraction teaching at Grade 6 level. The findings from this research showed that student teachers viewed mathematics, specifically fractions, as just about doing calculations, doing drill and practice as in this way learners’ understanding of fractions is enhanced. The importance of using rules and procedures was emphasized as paving the way to mastering fractions. The data analysis also revealed that student teachers employed different teaching approaches when they considered learners’ prior knowledge. However, they misinterpreted the learner-centred approach in terms of group work and class discussion in their teaching of fractions. The data also revealed that some student teachers had an understanding of fractions as they managed to redirect their learners from applying some fraction misconceptions. The findings further showed that due to lack of understanding most of the participants experienced fear and anxiety in their fraction teaching; attempted an approach then reverted to old practices used in schools; taught fractions as symbolic representations where no meaning was involved; used incorrect terminologies such as ‘1 over 4’ or ‘1 out of 4’ instead of a ‘quarter’ or ‘one-fourth’. It is further revealed that the some student teachers viewed a fraction as a pair of two different whole numbers and can be broken apart. The findings of this study yielded one main implication for the teaching of mathematics: that the teacher training programs, specifically for mathematics teachers, must take into consideration the weaknesses and strengths of the student teachers shown in this study to properly train and prepare them to become effective mathematics teachers.

Acompanha: Tecnomatemática - ensino de matemática e informática / Esta pesquisa teve como objetivo analisar a utilização de um site como ferramenta tecnológica de auxílio para o ensino de frações no 6º Ano do ensino fundamental, utilizando as TIC (Tecnologias da Informação e Comunicação). O site intitulado Tecnomatemática foi desenvolvido com base nas dificuldades encontradas por professores no uso das TIC. A realização da pesquisa ocorreu em sete momentos, desenvolvidos na sala dos professores e no laboratório de informática, usando a metodologia qualitativa e interpretativa, contando com a participação de nove professores de Matemática de uma escola estadual do município de JacarezinhoPR. A coleta de dados ocorreu a partir de avaliações, atividades em momentos semanais e questionários aplicados aos professores em momento de hora-atividade coletiva. Além disso, a pedido de um dos professores participantes, aplicamos a pesquisa com seus alunos do 6º ano. Após a obtenção dos dados, a análise foi realizada apontando como positiva a utilização do site pelos professores e alunos, mostrando que o uso do software JClic disponibilizado pelo site Tecnomatemática pode contribuir para o ensino e a aprendizagem Matemática, no que se refere ao conteúdo de frações. / The objective of this research was to analyze the use of a website as a technological aid tool for the teaching of fractions in the 6th year of elementary school, using ICT (Information and Communication Technologies). The website titled Tecnomathematics was developed based on the difficulties encountered by teachers in the use of ICT. The research was carried out in seven moments, developed in the teachers room and in the computer lab, using the qualitative and interpretative methodology, with the participation of nine mathematics teachers from a state school in the municipality of Jacarezinho-PR. The data collection was based on evaluations, activities at weekly moments and questionnaires applied to teachers in time of collective activity. In addition, at the request of one of the participating teachers, we applied the survey to their 6th grade students. After obtaining the data, the analysis was carried out aiming positively the use of the site by the teachers and students, showing that the use of the JClic software provided by the Tecnomathematics website can contribute to the teaching and learning Mathematics, regarding the content of Fractions.

Thesis (M. Ed. (Curriculum Studies)) -- University of Limpopo, 2021 / The purpose of the research was to explore challenges encountered by teachers when teaching fractions to learners in grade 2 at schools in the Koloti Circuit. A qualitative approach including other research techniques, such as observation, document analysis and interviews, were mutually employed to collect data during the study. A pluralistic approach was employed when gathering data to enhance triangulation and further intensify the merits of the probed facts. Purposive sampling was employed to choose three grade 2 teachers from various schools to serve as participants in this study. The following challenges were uncovered as a result of the research: learners are from child-headed families; there is a high rate of learner absenteeism; parents do not attend consultative meetings; there is a high rate of teacher time-offs; there is a lack of teacher pedagogical content knowledge in mathematics; teachers have to teach in multi-grade classroom; teachers lacking background knowledge on implementation of inclusive classrooms resulting in inadequate support to learners with learning barriers; classrooms are over-crowded; uneven partitioning of circular representations was discovered, and there is lack of content-related workshops. The following were some of the recommendations that were established to counteract the challenges uncovered during the research study, namely: establishment of after-care centres with qualified tutors; provision of state paid security services in schools; teachers should create learner support materials to aid in the teaching of fractions to the learners; teachers should notify parents when their children are absenting themselves from school on regular basis without valid reasons; teachers should notify parents about the benefits of attending consultative meetings; memorial services for teachers should be conducted after learner contact time, competent and/or qualified teachers in arithmetic should be assigned to educate the subject; small schools should be merged to curb multi-grate teaching at schools; more classrooms should be provided to avoid over-crowding in the classrooms, and more content workshops should be organised in order to assist teachers who experience challenges with the teaching of fractions. In conclusion, the outcomes of this study could aid grade 2 teachers in instilling knowledge of fractions into their learners, crafted on the suggested recommendations that are drawn subsequent the challenges facing teachers were singled out, in order to improve learner performance in fractions. This could result in better performance by learners in mathematics at various schools.

Thesis (MEd)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Many learners, even at high school level, have difficulty with fractions and computations involving fractions. A report from the Department of Basic Education (DBE, 2012c: 15) has highlighted that the lack in basic fraction sense was one of the areas of concern that contributed to the low achievement in matriculation mathematics examinations in 2012. Fractions play an important role in our ever-advancing technological society. Many occupations today rely heavily on the ability to compute accurately, proficiently, and insightfully with fractions. High school learners’ understanding or the lack thereof is carried over to their tertiary studies and workplaces. It is for that reason that in this dissertation, the learning and understanding of fractions and their role in the high school curriculum are studied through a critical literature review. Fractions are compound constructs and can therefore be interpreted in many different ways, depending on the area of study within mathematics. The concept of fractions consists of five sub-constructs, namely, part-whole, ratio, operator, quotient, and measure (Behr, Lesh, Post, & Silver, 1983; Kieren, 1980). This thesis starts with discussion of the background of the study and its importance. Thereafter the elements that assist in the understanding of the fraction concept is discussed. Then, the five different sub-constructs are elaborated on, and how these different sub-constructs are used in the high school curriculum is demonstrated. The conclusion offers some implications for classroom teaching and mathematics teachers’ professional development. / AFRIKAANSE OPSOMMING: Talle leerders, tot op hoërskool vlak, ervaar probleme met breuke en berekeninge met breuke nie. ‘n Verslag van die Departement van Basiese Onderwys (DBE, 2012c: 15) het beklemtoon dat die gebrek aan basiese breuk vaardighede een van die oorsake was wat daartoe gely het dat die prestasie in die 2012 matriek wiskunde eksamen so laag was. Breuke speel ‘n belangrike rol in ons voortdurende tegnologiese voor uitgaande samelewing. Talle beroepe vandag is grootliks afhanklik van die akkurate, bekwame en insiggewende berekeninge van breuke. Hoërskool leerders se begrip, of die gebrek daaraan word oorgedra na hul tersiêre studies en werksplekke. Dit is vir dié rede dat hierdie tesis die leer en begrip van breuke en hul rol in die hoërskool kurrikulum bestudeer deur middel van ‘n kritiese literatuur studie. Breuke is ‘n saamgestelde konsep en kan vir hierdie rede op verskillende wyses geïnterpreteer word, afhangende van die area van studie in wiskunde. Die konsep van ‘n breuk bestaan uit vyf sub-konstrukte, naamlik deel-van-‘n-geheel, ‘n verhouding, operateur, kwosiënt en meting (Behr, Lesh, Post, & Silver, 1983; Kieren, 1980). Hierdie tesis begin met ‘n bespreking oor die agtergrond van hierdie studie en die belangrikheid daarvan. Daarna word die faktore wat bydra tot die verstaan van die breuk konsep. Dit word gevolg deur ‘n uitbreiding op die vyf verskillende sub-konstrukte en waar hierdie verskillende sub-konstrukte in die hoërskool kurrikulum voorkom. Die bevinding bied ‘n paar implikasies vir onderrig. Hierdie studie fokus nie op die ontwerp van enige take of ander leermateriaal vir ‘n intervensie program nie, maar konsentreer op die belangrike kwessies rondom breuke. My hoop is dat die bevindinge van hierdie studie implikasies inhou vir wiskunde onderwysers se professionele ontwikkeling deur hul te motiveer om nuwe leerondersteuningsmateriaal te ontwikkel en die aanbieding van breuke in klaskamers aan te pas sodat die begrip van breuke by leerders ten volle ontwikkel kan word.

The study examined the use of multiple representations such as the real world, written symbols, spoken symbols, diagrams and manipulatives by learners to promote the learning of percentages and fractions through social interaction. This investigation was carried out through a teaching and learning programme which was developed and implemented by me, the researcher. The effect of the implemented programme was the main focus of the research. The qualitative study was oriented in the interpretive paradigm – a paradigm that seeks to understand the meaning attached to human actions. Twenty learners participated in the implementation of the programme and 9 learners were selected for focus group interviews. The purpose of the interviews was to explore learners' understanding and feelings about the use of multiple representations in the learning of percentages and fractions through social interactions. The other tools employed in this study were pre-and-post diagnostic tests, observations, learners' work and a journal. The pre-test was used to determine learners' prior knowledge for the program design and implementation, while the post-test and learners' work were used to analyze the effect of the programme. Observations were used to investigate how multiple representations promoted or did not promote the learning of percentages and fractions. The teacher's journal was to record and reflect on any relevant information gathered on each lesson observed. The data shows that the effective use of multiple representations helped learners learn the concept of percentages and fractions better. Learners were able to look at representations in useful ways; multiple representations made some aspects of the concept clear; and multiple representations enabled learners to correct errors. Through the interaction between the teacher and learners, the following was found: all the learners changed words to change focus; learners made links between multiple representations; the learners deepened their concepts of percentages and fractions; learners could convert between fractions using multiple representations; learners could work out percentages of a quantity; and learners could express one quantity as a percentage of another. Furthermore, through the interaction between learners and learners all learners could identify more equivalent fractions of an initial fraction which was given to them; and they could increase and decrease a quantity by a given percentage. On the basis of this research, it can be concluded that the programme promoted the learning of percentages and fractions through three effective methodologies. The first methodology consisted of the effective use of multiple representations; the second methodology concerned the interaction between the teacher and learner during the learning process and the last methodology related to the interaction between the learners - interactions that were not strongly mediated by the teacher. I would recommend that teachers use these three effective approaches when teaching percentages and fractions to promote the learning of the concepts.

The main purpose of the research was to investigate the influence of scientific calculators on Grade 8 South African learner's understanding of fractions in learning mathematics. Quasi-experimental quantitative research methods were used. A sampling frame was selected using non probability sampling technique. A total of 15 learners in each group were randomly selected for an experimental and control group for the study. Both groups were taught fraction concepts by different teachers for the same duration and at the same time. The experimental group used a calculator as a learning aid while the control group used the traditional paper pencil method. Two tasks (post-test and assignment) were administered to both groups and a questionnaire to the experimental group. The results indicated that the scientific calculator has a positive influence in learner's conceptual understanding of fractions in mathematics as reflected in their performance. / Mathematics Education / M. Ed. (Mathematics Education)

Fractions are very hard to learn. As the joke goes, “Three out of two people have trouble with fractions”. Yet the invention of a notation for fractions is very ancient, dating back to Babylonians and Egyptians. Moreover, it is thought that ratio representation is innate. And obviously, fractions are part of our everyday life. We read them in recipes, we need them to estimate distances on maps or rebates in shops. In addition, fractions play a key role in science and mathematics, in probabilities, proportions and algebraic reasoning. Then why is it so hard for pupils to understand and use them? What is so special about fractions? As in other areas of numerical cognition, a fast-developing field in cognitive science, we tackled this paradox through a multi-pronged approach, investigating both adults and children.<p>Based on some recent research questions and intense debates in the literature, a first behavioural study examined the mental representations of the magnitude of fractions in educated adults. Behavioural observations from adults can indeed provide a first clue to explain the paradox raised by fractions. Contrary perhaps to most educated adults’ intuition, finding the value of a given fraction is not an easy operation. Fractions are complex symbols, and there is an on-going debate in the literature about how their magnitude (i.e. value) is processed. In a first study, we asked adult volunteers to decide as quickly as possible whether two fractions represent the same magnitude or not. Equivalent fractions (e.g. 1/4 and 2/8) were identified as representing the same number only about half of the time. In another experiment, adults were also asked to decide which of two fractions was larger. This paradigm offered different results, suggesting that participants relied on both the global magnitude of the fraction and the magnitude of the components. Our results showed that fraction processing depends on experimental conditions. Adults appear to use the global magnitude only in restricted circumstances, mostly with easy and familiar fractions. <p>In another study, we investigated the development of the mental representations of the magnitude of fractions. Previous studies in adults showed that fraction processing can be either based on the magnitude of the numerators and denominators or based on the global magnitude of fractions and the magnitude of their components. The type of processing depends on experimental conditions. In this experiment, 5th, 6th, 7th-graders, and adults were tested with two paradigms. First, they performed a same/different task. Second, they carried out a numerical comparison task in which they had to decide which of two fractions was larger. Results showed that 5th-graders do not rely on the representations of the global magnitude of fractions in the Numerical Comparison task, but those representations develop from grade 6 until grade 7. In the Same/Different task, participants only relied on componential strategies. From grade 6 on, pupils apply the same heuristics as adults in fraction magnitude comparison tasks. Moreover, we have shown that correlations between global distance effect and children’s general fraction achievement were significant.<p>Fractions are well known to represent a stumbling block for primary school children. In a third study, we tried to identify the difficulties encountered by primary school pupils. We observed that most 4th and 5th-graders had only a very limited notion of the meaning of fractions, basically referring to pieces of cakes or pizzas. The fraction as a notation for numbers appeared particularly hard to grasp. <p>Building upon these results, we designed an intervention programme. The intervention “From Pies to Numbers” aimed at improving children’s understanding of fractions as numbers. The intervention was based on various games in which children had to estimate, compare, and combine fractions represented either symbolically or as figures. 20 game sessions distributed over 3 months led to 15-20% improvement in tests assessing children's capacity to estimate and compare fractions; conversely, children in the control group who received traditional lessons improved more in procedural skills such as simplification of fractions and arithmetic operations with fractions. Thus, a short classroom intervention inducing children to play with fractions improved their conceptual understanding. <p>The results are discussed in light of recent research on the mental representation of the magnitude of fractions and educational theories. The importance of multidisciplinary approaches in psychology and education was also discussed. <p>In sum, by combining behavioural experiments in adults and children, and intervention studies, we hoped to have improved the understanding how the brain processes mathematical symbols, while helping teachers get a better grasp of pupils’ difficulties and develop classroom activities that suit the needs of learners.<p> / Doctorat en Sciences Psychologiques et de l'éducation / info:eu-repo/semantics/nonPublished