The aim of this thesis is to examine the relative importance of working memory, counting ability, and additive reasoning in children's mathematics learning. One hundred and fifteen 6-year-old Chinese children in Hong Kong participated in two waves of assessments. At the first time point (T1 - first grade), they were assessed using non-verbal intelligence, working memory (central executive, phonological loop, and visuospatial sketchpad), counting ability (procedural counting and conceptual knowledge of counting), additive reasoning (knowledge of the commutativity and complement principles), and mathematical achievement (calculation and story problem solving). Approximately 10 months later (T2 - second grade), children's mathematical achievement in calculation and story problem solving were evaluated once again. The extent to which various cognitive factors longitudinally predicted children's mathematical achievement was evaluated in this study. Several key findings were identified through two sets of analyses - multiple regression models and latent profile analysis. The multiple regression analyses showed that counting ability accounted for a significant amount of variance in T1 and T2 calculation beyond the effects of age, IQ, and working memory, in which conceptual knowledge of counting, but not procedural counting, was a unique predictor. However, counting ability did not contribute significantly to story problem solving at both time points. When additive reasoning was also included in the regression model, counting ability made a unique contribution to T1 calculation only, but not T2 calculation. By contrast, additive reasoning and working memory appeared to be more stable and stronger predictors of children's performance in calculation and story problem solving at both time points than counting ability. Additive reasoning explained a substantial and significant amount of variance in calculation and story problem solving at both time points after the effects of age, IQ, working memory, and counting ability were controlled for - Both knowledge of the commutativity and complement principles were unique predictors. Similarly, working memory also accounted for a significant amount of variance in calculation and story problem solving at both time points beyond the influence of age, IQ, counting ability, and additive reasoning. Among the three components of working memory, only the central executive was a unique predictor for all measures of mathematical achievement. Autoregressive analyses provided strong evidence for the longitudinal predictive powers of additive reasoning and working memory. The analyses showed that both additive reasoning and working memory remained significant predictors of T2 mathematical achievement (calculation and story problem solving) even after the effects of children's previous performance were taken into account (i.e. T1 mathematical achievement). Overall, additive reasoning accounted for the greatest amount of variance in mathematical achievement both concurrently and longitudinally among all the other factors. This finding underscores the importance of additive reasoning in the teaching and learning of mathematics in young children. Because additive reasoning (as indicated by the knowledge of the commutativity and complement principles) is a critical variable in this thesis and relatively scarce research has examined this construct, particular concern was paid to the measurement of additive reasoning. It was measured in two ways in the present study: with the support of concrete materials (the concrete condition) and without the support of concrete materials (the abstract condition). Latent profile analysis showed that all children who performed well in the abstract conditions also did well in the concrete conditions, whereas it did not reveal a group of children who performed well in the abstract conitions, but not in the concrete conditions as well. Another interesting finding was that all children who obtained high scores on tasks that assessed their knowledge of the complement principle also obtained high scores in tasks that assessed their understanding of the commutativity principle. The overall pattern of profiles provides initial evidence suggesting that additive reasoning may develop from thinking in the context of specific quantities to thinking about more abstract symbols, and children acquire the knowledge of the commutativity principle in abstract tasks before they start to acquire the knowledge of the complement principle. This finding demonstrated that patterns of individual differences are present in the development of different aspects of additive reasoning. If teachers possess some knowledge about the particular strengths and weaknesses of each child, it would be easier for them to devise teaching strategies that are tailored to the needs of different children, which may relate to the developmental order of the commutativity and complement principles, and the role of concrete materials in this development. Thus, this study contributes to the literature by showing that assessing additive reasoning in different ways and identifying profiles with classification analyses may be useful for educators to understand more about the developmental stage where each child is placed. It appears that a more fine-grained assessment of additive reasoning can be achieved by incorporating both concrete materials and relatively abstract symbols in the assessment.

Thesis (MEd (Curriculum Studies)--Stellenbosch University, 2010. / ENGLISH ABSTRACT: The aim of this research is three-fold. Firstly I aimed to show the difficulty of the concept of proportional reasoning through empirical research. Several researchers have shown the degree of difficulty learners experience with proportional reasoning and have even indicated that many university students (and adults) do not have sound proportional reasoning skills. Piaget’s controversial developmental levels classify proportional reasoning as a higher order thinking skill in his highest level of development, formal operational thought, and claims that most people do not reach this level. The difficulty of proportional reasoning and the fact that it is a skill needed within all Learning Outcomes of Mathematical Literacy creates a predicament in terms of the difficulty of the subject in general. Is it then fair to classify Mathematical Literacy as an inferior subject in the way it has been done over the last few years if it is a subject that requires learners to operate at such a high level of thought through proportional reasoning? Secondly, I would like to confirm with the use of a baseline assessment that learners entering Grade 10 Mathematical Literacy have poor proportional reasoning skills and have emotional barriers to Mathematics and therefore Mathematical Literacy. The research will be done in three private schools located in the West Coast District of the Western Cape in South Africa. If learners in these educationally ideal environments demonstrate poor proportional reasoning skills even though they were privileged enough to have all the possible support since their formative years, then results from overcrowded government schools may be expected to be even worse. The learners in Mathematical Literacy classes often lack motivation, interest and enthusiasm when it comes to doing mathematics. Through the baseline assessment I confirm this and also suggest classroom norms and values that will help these learners to become involved in classroom activities and educational discourse. Thirdly and finally this research will focus on the design of activities that will aim to build on learners’ prior knowledge and further develop their proportional reasoning skills. I argue that activities to develop proportional reasoning should take equivalence of fractions as basis to work from. The activities will aim to help learners to set up questions in such a way that they can solve it with techniques with which they are familiar. Interconnectivity will form a vital part to this investigation. Not only do I indicate the interconnectivity between concepts in the Mathematical Literacy Learning Outcomes of the National Curriculum Statement, but I would like to make these links clear to learners when working through the proposed activities. Making links between concepts is seen as a higher order thinking skill and is part of meta-cognition which involves reflection on thoughts and processes. In short, this research can be summarised as the design of activities (with proposed activities) that aims to develop proportional reasoning by making connections between concepts and requires of learners to be active participants in their own learning. / AFRIKAANSE OPSOMMING: Die doel van hierdie navorsing is drieledig. Eerstens will ek die probleme met die konsep van proporsionele denke uitlig deur eksperimentele ontwerp navorsing. Verskeie navorsers verwys na die moeilikheidsgraad van probleme wat leerders ondervind met proporsionele denke. Sommige van hierdie navorsers het ook bevind dat verskeie universiteitstudente (en ander volwassenes) nie oor die vaardigheid van proporsionele denke beskik nie. Piaget se kontroversiële ontwikkelingsvlakke klassifiseer proporsionele denke as ‘n hoër orde denkvaardigheid in sy hoogste vlak van ontwikkeling, formele operasionele denke, en noem dat meeste mense nooit hierdie vlak bereik nie. Die hoë moeilikheidsgraad van proporsionele denke en die feit dat dit ‘n vaardigheid is wat binne al die Leeruitkomste van Wiskundige Geletterdheid benodig word veroorsaak ‘n dilemma as mens dit vergelyk met die moeilikheidsgraad van die vak oor die algemeen. Tweedens wil ek met behulp van ‘n grondfase assessering bewys dat leerders wat Graad 10 Wiskunde Geletterdheid betree swak proporsionele denkvaardighede het, gepaardgaande met emosionele weerstand teenoor Wiskunde en Wiskunde Geletterdheid. Die navorsing sal gedoen word in drie privaatskole in die Weskus distrik van die Wes-Kaap van Suid-Afrika. Indien leerders in hierdie ideale opvoedkundige omstandighede swak proporsionele denkvaardighede ten toon stel, ten spyte van die feit dat hulle bevoorreg was om sedert hulle vormingsjare alle moontlike opvoedkundige ondersteuning te geniet, dan kan verwag word dat resultate komende van oorvol staatskole selfs swakker mag wees. By leerders in Wiskunde Geletterdheid klasse kan daar gereeld ‘n gebrek aan motivering, belangstelling en entoesiasme ten opsigte van Wiskunde bespeur word. Deur gebruik van die grondfase assessering wil ek hierdie stelling bewys en ook voorstelle maak vir klaskamernorme en waardes wat sal help om die leerders meer betrokke te maak by klaskameraktiwiteite en opvoedkundige gesprekke.