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.

published_or_final_version / Education / Master / Master of Education

Young children are limited in their usage of comparative adjectives and ordinal numbers, typical ways of describing ordinal relationships. However, research in a number of areas suggests the possibility of a precursor level of ordinal concept. To facilitate the search for precursor ordinal skills, ordinal ability was defined in ordinal measurement terms. Only "greater than - less than," asymmetric judgements were required. Additionally, linguistic demands were reduced by using family-role terms as size designators. Experimental manipulations included variations in scale size and in the complexity level of ordinal conceptualization. Solution strategies based on "good form" and on "pairwise comparison" were precluded by using pictures of randomly placed objects which could not be manipulated by the child. Ninety-six 3-6 year old children pointed to "Daddy," "Mommy," "Big boy/girl," "Little boy/girl," and "Baby" when shown sets of 3 to 5 circles or squares which differed only in size. Tasks were of three types: Identification, mapping labels onto a single set of objects; Coordination, mapping labels onto two identical sets of objects in which corresponding "family members" are the same size; and Transposition, mapping labels onto two separate sets in which corresponding family members are of different sizes. Data were analyzed in an Age (3), by Scale Size (3), by Complexity Level (3), by Shape (2) mixed design ANOVA, and significant main effects were obtained for all variables. Tasks became more difficult with increases in scale size, and in complexity level. Square objects were slightly more difficult than circular, and older children were more proficient than younger ones. Post hoc tests generally supported the obtained main effects. Finer grained analysis using Latent Trait procedures supported the global ANOVA results, and supported the hypothesis that the end points of a scale are easier than the central positions. Response patterns indicated that errors were size-related, and suggested transitional levels of performance. The present study demonstrates that children as young as three can demonstrate a precursor ordinal concept when the task is framed in familiar terms and is placed in a context which is meaningful for them.

The purpose of the present research study was to produce a global, cumulative model of number concept development for children between the ages of two and eight years old. The theoretical and methodological orientation of this study was greatly influenced by Richard Young's production system analysis of seriation by young children (Young, 1971, 1976) and by Newell's (1973) seminal paper, ‘You can't play twenty questions with nature and win’. The methodology used in this investigation thus was as follows. A series of complex number tasks encompassing many aspects of the concept of number were developed. Five children aged between three and seven years then were videotaped while performing some of these complex number tasks. From a detailed protocol analysis of the video-recordings, computer simulation models written in the production system language PSS3 (Ohlsson, 1979) were produced. Specific production system models were produced for each of following aspects of the children's number knowledge: (i) sharing of discrete quantities; (ii) comparison of shares; and (iii) conservation/addition/subtraction of number. These domain-specific models were based on the converging experimental evidence obtained from each of the children’s responses to variants of the complex number tasks. Each child thus received a different set of problems which were chosen systematically in order to clarify particular features of the child's abilities. After a production system model for each child had been produced within a domain, these models were compared and contrasted. From this analysis, developmental trends within the domain were identified and discussed. The research and educational implications of these developmental trends then were discussed. In the concluding parts of this study, the children's domain-specific production system models were cumulated into global, comprehensive models which accurately represented their behaviour in a variety of number tasks. These comprehensive models were compared and contrasted and general developmental trends in young children's number knowledge were identified and discussed.

published_or_final_version / Educational Psychology / Master / Master of Social Sciences

Children’s counting strategies, such as counting from one or by groups of tens, reflect how much they understand the place-value structure of numbers. In a novel task for assessing place-value concept, namely the strategic counting task, children were asked to count small squares, which were arranged with or without correspondence to the base-ten number structure. The counting strategies of kindergarteners and first graders revealed that children developed from perceiving number as an undivided entity to seeing it as a collection of independent groups of tens, indicating a trend of increasing place-value understanding. First graders’ strategic counting task scores at the end of fall semester predicted their mathematical achievement at the end of spring semester, over and above age, intelligence, and measures of simple counting, number representation, place-value understanding, and arithmetic calculation. Based on item analysis, a brief version containing only five items was developed for more user-friendly classroom administration. First graders’ scores in the brief version uniquely predicted their mathematical achievement even at the end of second grade. Growth curve modeling revealed that children who were low mathematics achievers at the end of second grade had already shown poor performance in the brief version in early first grade and remained lagging behind their peers over the 18 months. Early poor understanding of place-value concept, then, seems to persist to upper grade and impede mathematical development. Implications for early support to children with difficulties in place-value concept were discussed. / published_or_final_version / Psychology / Doctoral / Doctor of Philosophy

Humans are born with a basic sense of number. This number sense, which is now called the Approximate Number System (ANS), allows us to represent numerosity without the use of symbols. There has been a debate on whether this nonsymbolic ANS contributes to our symbolic mathematics skills, and the recent findings are inclined to support the link between the two. However, what remains unclear is the mechanism underlying the relationship between the ANS and our mathematics skills, and whether children with Mathematics Learning Disabilities (MLD) suffer from a defective ANS. The present thesis aimed at addressing the above issues in two studies. Study 1 aimed at identifying the mechanism of how the ANS contributes to children’s mathematics skills. A group of 210 kindergarteners were tested on their ANS acuity, number-numerosity mapping skills (measured by counting and estimation tasks), and their arithmetic skills. They were then re-tested twice when they were in Grade 1.Using Structural Equation Modeling, it was found that children’s ANS acuity in kindergarten predicted their arithmetic skills one year later, and the relationship was mediated by their number-numerosity mapping skills. This suggested that ANS may contribute to mathematics learning by enabling more precise mapping between number symbols and the corresponding numerosity representation, hence making numbers meaningful. Studies 2A and 2B aimed at verifying whether children with MLD suffered from deficits in their ANS as well as their number-numerosity mapping skills. The same group of participants was followed one more time in Grade 2. Using the standard low-achievement method (Study 2A) and a more data-driven method known as the latent class growth analysis(Study 2B), two groups of children with MLD were identified. Both groups of children had deficits in both the ANS and their number-numerosity mapping skills as compared with their normally-achieving peers. Other groups of low-achieving children were also identified, and their difficulties seemed to be contributed by factors other than their ANS. While one of the low-achieving groups seemed to have deficit lying mainly on the number-numerosity mapping skills, the other low-achieving group did not show any cognitive deficits but had much lower SES compared to other groups. The relationship between the ANS and children’s mathematics achievement was supported and elaborated in the present study. The findings not only articulated a potential mechanism of how children learned about mathematics, but they also allowed educators to have better understanding of the cognitive profiles of children with MLD, thus facilitating early identification and intervention. The different profiles of the low-achieving groups also highlighted the need for differential intervention for different groups of low-achieving children. / published_or_final_version / Psychology / Doctoral / Doctor of Philosophy

Recent research on the mathematical achievement of young children prompts one to question the widely accepted views of Piaget in this regard. Researchers have begun to concentrate on assessing the development of mathematical concepts in appropriate contexts. Aubrey (1993), Hughes (1986) and Gelman and Gallistel (1978) examined the mathematical competencies of pre-school children and suggest how this knowledge could inform instruction and curriculum development. This study investigates the mathematical knowledge and competencies of 40 reception class children from English speaking, working class homes in Pietermaritzburg, Kwazulu-Natal. The assessment tasks were adapted from those of Aubrey (1993), Young-Loveridge (1989) and Wright (1991). These are compatible with the key number activities in the "Learning Through Activity Programme" used in the reception class in this province. The tasks were presented during individual interviews, using everyday objects and familiar activities. Tasks included rote counting, understanding the cardinality rule, numeral recognition, written representation of numbers, ordering numbers, addition and subtraction with concrete objects, social sharing and multiplication, estimation, patterning and an understanding of shape, space, measurement, time, and ordinal numbers. The results confirm the findings of previous studies: most children enter the pre-school year with considerable knowledge about number. Low-attaining children had some basic number knowledge but could not cope with higher numbers or more abstract tasks. Higher scoring children were already competent in most areas of the reception class mathematics curriculum. As the curriculum is suited to the low scorers, the majority of pupils are not provided with challenges to advance. Teachers may be unaware of the extent and range of children's mathematical knowledge, and the strategies used for manipulating numbers. Initial and ongoing assessment of each child's competence would enable teachers to develop and evaluate a meaningful curriculum. For every child to realise his/her potential implies instruction that is appropriate to the level and pace of learning. Further research should refine the assessment of children's mathematical knowledge and investigate the influences upon later mathematical achievements. / Thesis (M.Ed.)-University of Natal, Pietermaritzburg, 1996.

Thesis (Ph. D.)--University of Alabama at Birmingham, 2008. / Additional advisors: Jerry Aldridge, Lois Christensen, Lynn Kirkland, Maryann Manning. Description based on contents viewed Oct. 7, 2008; title from PDF t.p. Includes bibliographical references (p. 66-68).

Thesis (Ph.D.)--University of Central Florida, 2008. / Adviser: Juli K. Dixon. Includes bibliographical references (p. 167-173).