The research documented in this report had a twofold purpose. Firstly, it was to design and implement an intervention based on the theory of Realistic Mathematics Education (RME) aimed at improving the mathematical understanding of learners in two Grade 8 remedial mathematics classes, by revisiting the key number concepts of place value, fractions and decimals. In doing so, a second purpose was to investigate the viability and emerging characteristics of an intervention based on the theory of RME in such a setting (i.e. with low attainers to revisit key number concepts). Pending the realisation of these immediate outcomes, more distant outcomes in subsequent research would be: that learners' understanding and academic performance in mathematics improves and to develop a local instruction theory in using the RME theory to revisit the concepts of place value, fractions and decimals with low attaining learners in order to improve their understanding in this regard. Grade 8 low attainers were selected as the target group for this research as a result of the pending implementation of Mathematical Literacy as a compulsory subject for all learners, possibly from 2006. Currently in South Africa, learners who are not meeting the required standard by the end of their Grade 9 year are able to elect not to take mathematics through Grades 10, 11 and 12. When the new Further Education and Training (FET) policy is implemented, this will no longer be the case. All learners, who do not elect to take mathematics as a subject, will have to take Mathematical Literacy as a compulsory subject throughout Grades 10, 11 and 12. Although less detailed and abstract than the subject mathematics, the Mathematical Literacy curriculum still requires learners to have an understanding of key number concepts and also contains a substantial amount of algebra. As Grade 8 is when learners start working with algebra more formally, and is also their first year at secondary school, it was decided that this would be an appropriate year to try and diagnose and remediate problems in learners' understanding of the key number concepts, if and where possible. The intention was that this would then equip learners with a more appropriate structure of conceptualised knowledge of the above-mentioned concepts on which they could further construct their understanding of algebra. The study was carried out at a local urban high school in South Africa and the research design of this study was informed by two development research approaches (van den Akker&Plomp, 1993; Gravemeijer, 1994). Also, the study was only implemented with a small number of participants, within a bounded setting and without the intention to generalise the results. It was therefore regarded as a development case study. The results appear to indicate that it is viable to apply the theory of RME with low attaining Grade 8 learners in order to revisit the key number concepts of place value, fractions and decimals. Copyright 2004, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. Please cite as follows: Barnes, HE 2004, A developmental case study : implementing the theory of realistic mathematics education with low attainers, MEd dissertation, University of Pretoria, Pretoria, viewed yymmdd < http://upetd.up.ac.za/thesis/available/etd-12032004-103122 / > / Dissertation (MEd (Curriculum design))--University of Pretoria, 2005. / Curriculum Studies / unrestricted

Thesis (PhD)--Stellenbosch University, 2012. / Includes bibliography / ENGLISH ABSTRACT: The rationale for this study emerged from a realization that conventional instructional design approaches for introducing Calculus concepts, based on the logical sequencing and structuring of the concepts, did not adequately attend to or address students’ ways of thinking. This was particularly important in a distance education environment where learners depend on instructional texts to make sense of what is being presented, often without support from tutors. The instructional design theory of Realistic Mathematics Education (RME) offered a promising approach for designing learning sequences based on actual investigations of the ways in which students think. This study’s focus was on trialling the process of RME theory-based design using the Fundamental Theorem of Calculus as an example. Curve sketching was prominent in this exercise. Applying RME required developing a hypothetical learning trajectory (HLT) while attempting to adhere to methodological guidelines of design research. In this project, the instructional designer’s conceptualization and interpretation of the derivative-integral construct has had the most immediate implications for the study. The line of inquiry has been largely didactic, in that it was framed by a need to establish ways of introducing the teaching of a mathematical concept following instructional design principles. Throughout the project, the instructional design space has been contested, broken down, rebuilt and, ultimately, enriched by the contributions of the expert teachers and the engagement of participating students. The series of design experiments have revealed knowledge about student reasoning in this learning domain in relation to four main areas of quantifying change, curve sketching, general mathematical reasoning and symbol use. The primary contribution of this research has been a deeper understanding of the extent to which RME can be used as an instruction design theory for planning and introducing a distance teaching Calculus unit. From the study, it is clear that successful adoption of the RME theory is influenced and facilitated by a number of factors, including: careful selection of the concepts and mathematical structures to be presented; a team of experts (mathematicians and mathematics subject didacticians) to research, test and develop the learning activities; opportunities for student interactions; and time and resources for effective RME adoption. More involved research is required to get to the stage of the evolution of a local instructional theory around introducing the derivative-integral relationship as expressed in the Fundamental Theorem of Calculus. / AFRIKAANSE OPSOMMING: Die rasionaal van hierdie studie het uit die besef ontstaan dat konvensionele onderrigontwerpbenaderings vir die bekendstelling van Calculus konsepte, gebaseer op die logiese ordening en strukturering van die konsepte, nie voldoende beantwoord aan die eise van hoe studente dink nie. Dit was van spesifieke belang in die geval van afstandonderwys waar hierdie studente sin moet maak van wat aangebied word, dikwels sonder die ondersteuning van tutors. Die onderrigontwerpteorie van Realistiese Wiskundeonderwys (RWO) bied belowende moontlikhede om leertrajekte te ontwerp wat gebaseer is op werklike ondersoeke van hoe studente dink. Hierdie studie se fokus was om die RWO-gebaseerde teoretiese ontwerp se proses wat die Fundamentele Stelling van Calculus as voorbeeld gebruik, uit te toets. Krommesketsing was prominent in hierdie oefening. Die toepassing van RWO het vereis dat 'n leertrajek ontwikkel moet word terwyl aan die metodologiese vereistes van die ontwikkelingsondersoekbenadering getrou gebly word. In hierdie projek het die onderrigontwerper se konseptualisering en interpretasie van die afgeleide-integraalkonstruk onmiddellike implikasies gehad vir die studie. Die lyn van ondersoek was grootliks didakties van aard. Desnieteenstaande was die instruksionele ontwerpruimte voortdurend beding, afgebreek, herbou en uiteindelik verryk deur die bydraes van die bedrewe onderwysers en die betrokkenheid van die deelnemende studente. Die reeks ontwerpeksperimente het kennis blootgelê van hoe studente in hierdie veld redeneer met betrekking tot die volgende vier hoof areas: kwantifisering van verandering, krommesketsing, algemene wiskundige beredenering en die gebruik van simbole. Die primêre bydrae van hierdie navorsing is die dieper verstaan van die mate waarin RWO gebruik kan word as 'n instruksionele ontwerpteorie vir die beplanning en bekendstelling van 'n Calculus eenheid in afstandsonderrig.Dit is duidelik vanuit die studie dat suksesvolle aanneming van die RWO teorie afhanklik is van 'n aantal faktore: 'n noukeurige seleksie van die konsepte en wiskundige strukture wat aangebied moet word; 'n span van bedrewe wiskundiges en wiskunde vakdidaktici om die leeraktiwiteite na te vors, uit te toets en te ontwikkel; geleenthede vir studente-interaksies, en tyd en bronne vir effektiewe RWO aanpassing. Verdere toegespitsde navorsing hierop is nodig om die fase te bereik van die ontluiking van 'n lokale onderrigteorie oor die bekendstelling van die afgeleide-integraal verwantskap soos uitgedruk in terme van die Fundamentele Stelling van Calculus.