In the UK, as many as 20% of children in primary schools are more than two years behind their peers in mathematics. Research-based intervention for such disadvantaged children has been shown to be effective but not always sufficient, such that alternative approaches might sometimes be necessary. One alternative might involve digital games. This study used a design-based research approach to investigate a prototype digital game, that implements principles of an effective numeracy intervention and draws on insights from learning theory and the cognitive sciences, designed for children in primary schools who are low-attaining in mathematics. It comprised three cycles of design, intervention, analysis and reflection. The first research cycle involved the initial design of a prototype digital game, which was researched in one school. The second research cycle involved a second iteration of the game, designed in response to the feedback of teachers and children, which was researched in three schools. The third research cycle involved the design of a final iteration of the game, which to achieve theoretical saturation was researched online with twenty-four schools. The study has shown that a game that implements principles of an effective numeracy intervention and that draws on insights from learning theory and the cognitive sciences can be designed and can be useful in schools for children who are low attaining in mathematics. However, for it to be taken up by schools, the game has to be perceived by teachers to have achieved a quality threshold. In any case, such a game is of limited use in and of itself. Where the prototype game has been shown to be most useful is when it serves as a fulcrum for social interaction and educationally productive discussion between the children and teaching staff: when it becomes an artefact that both supports individual learning and stimulates, scaffolds and mediates dialogue-based collaborative learning.

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Response to intervention has become a widely implemented early intervention and pre-referral program in many schools due to the reauthorization of the 2004 Individuals with Disabilities Education Act. Limited studies exist that validate how teacher preparation programs are preparing the next generation of teachers to assess students, apply early academic interventions, monitor progress, and make educational decisions for students with learning difficulties as part of an RTI program. The purpose of this study was to examine elementary pre-service teachers’ perceptions and experiences in a mathematics intervention project (MIP), as part of a university mathematics methods course as related to RTI practices. Data were collected from multiple sources, including: Seidman’s three-step interview series with pre-service participants and course instructors, document analysis of the Mathematics Interactions Project (MIP) students’ responses, mathematics methods course syllabi, and observations of the mathematics methods course instruction. Haskell’s transfer theory was used as the framework from which to analyze the data. It was assumed that if a majority of the 11 principles of meaningful transfer were addressed, higher levels of transfer from university instruction to intervention instruction would be observed during the MIP. Findings indicate differences in RTI understanding according to elementary education degree plan. Candidates in the English as a Second Language (ESL) program did not demonstrate a strong foundational understanding of RTI, evidenced by a lower level of transfer about RTI. Alternately, pre-service teachers in the special education degree plan had a stronger foundational knowledge of RTI, discussed how RTI learning was supported, and had more experiences to implement RTI (principles 1, 7, and 9). Pre-service teachers in the Special Education (SPED) certification degree plan demonstrated a higher level of transfer since more of the principles were met; this was foundational in Haskell’s transfer theory. Implications are that elementary education programs, and particularly projects such as the MIP, should focus explicitly on RTI practices, as these are increasingly necessary in the field of elementary education practice.

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

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