Thesis (Ph. D.)--University of Wisconsin--Madison, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1954. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 34-35).

Thesis (Ph. D.)--Georgia State University, 2006. / David A. Washburn , committee chair; Claudio C. Cantalupo, Eric J. Vanman, Duane M. Rumbaugh, committee members. Electronic text (102 p. : col. ill.)) : digital, PDF file. Description based on contents viewed July 13, 2007. Includes bibliographical references (p. 79-96).

Magister Educationis - MEd / Systemic evaluation undertaken by the Department of Basic Education as part of the Literacy and Numeracy Strategy 2006 – 2016 posed a serious challenge in South African schools. The numeracy and mathematics results in 2009 stated that 35% of learners in Grade 3 achieved the required level of competence in Mathematics. This has, however, improved to 48.3% in 2010 but dropped to 47.6% in 2011. The development of early number concept in countries such as the Netherlands, Singapore and Helsinki has shown that early intervention is essential for reaching mathematical success in schooling. The Curriculum and Assessment Policy Statement (CAPS) integrates the three learning programmes: Literacy, Numeracy and Life Skills for Grade R into a daily programme of activities. Within this daily programme it specifies that 35% of each day must be used towards Numeracy. The Grade R method of teaching emphasizes the fact that teaching must take place informally but planned formally. The purpose of this study is to examine how early mathematics is taught in an integrated and informal setting to improve number concept. The theoretical framework underpinning this study is based on the constructivist views of Piaget and Vygotsky and how these theories lay the foundation for the development of number concept in Grade R. Number skills to develop number concept were identified in nine lessons to underpin the content area 1, Numbers, Operations and Relationships as determined by the Grade R Mathematics Curriculum and Assessment Policy Statement (CAPS). The methodology employed to answer the research question were video-recordings, observations and interviews. The findings identified number skills such as emergent number concepts: distinguishing numerosity, imitating resultative counting and symbolizing by using fingers as well as growing number concepts: discovering different meanings of numbers, oral counting, one- to- one correspondence, rote counting, perceptual subitising, resultative counting, representing and symbolizing numbers, ordinality, place value, emergent object-based counting and calculating and golden moments. The discussion of the findings focused on the CAPS content area and how these number skills were used to achieve the demands of the content area 1. The major findings of this study presented a case of the utilization of number skills to achieve the development of number concept in Grade R, how mathematics should be made fun, and how incidental learning, “golden moments” can be used to introduce key mathematical concepts informally. This study has implications for teachers of Grade R and for the training of pre-service Grade R teachers at tertiary level.

Jean Piaget describes three stages in the development of that aspect of quantitative thinking which he named conservation of number. In the first stage, a child is quite unaware that one-to-one pairing of two sets implies equivalence of the sets. He is unable to make a correct one-to-one correspondence, and if presented with two sets of objects which have been matched unit for unit, believes that one set has become greater if its units are spread out, or smaller if they have been compressed. In the second stage, the child is able to make a correct correspondence, but does not believe in the continued equivalence of the sets when one is spatially rearranged. In the third stage, the child maintains that the matched sets remain equivalent even though the units of one set have been rearranged, that is, the child conserves number. Piaget postulates that conservation is a necessary condition of mathematical understanding. In this study, it was hypothesized that children who are in Stage 1 at the beginning of their Grade 1 year, and who are still in Stage 1 at the beginning of the second term show low achievement in arithmetic at the end of the school year. It was further hypothesized that each stage in conservation is associated with corresponding levels in terminal achievement in Grade 1 arithmetic. One hundred fifty-six children received an individual conservation test in October of their first grade year, and were thereby classified as being in Stage 1, Stage 2, or Stage 3 in conservation development. In January, those classified as Stage 1 received a second conservation test, and were again classified according to their stage in conservation development at that time. In May, the arithmetic sub-test of the Stanford Achievement Test, Primary I Battery was administered to all groups. A significant proportion of the Stage 1 group selected by the January conservation test had achievement scores which fell below the median for all subjects, while a significant proportion of the Stage 3 group selected by the October test had above-median achievement scores. Mean achievement scores for the two Stage 3 groups did not differ significantly from each other, but were higher than mean achievement scores for the Stage 1 and Stage 2 groups. No significant differences were found among mean achievement scores of Stage 1 and Stage 2 groups. The results were interpreted as being consistent with Piaget's theory. The superiority of the mean terminal achievement of early conservers over that of children who had not developed conservation by January appeared great enough to be of educational importance. Some individual scores showed marked deviation from the pattern derived from the group data, however, and caution in use of the conservation test as a predictive instrument was recommended. It was proposed that the conservation test could be a useful diagnostic procedure for the teacher. / Education, Faculty of / Graduate

No description available.

The purpose of this research was to examine children's unmanipulated self-reports of their internalized representation of numbers and the relationship of the spatio-organizational patterns that are represented by the children's drawings to children's ability to solve basic addition problems. Also of interest were possible changes that occurred in children's spatio-organizational patterns as a result of age, mathematics achievement or gender. It was hypothesized that children whose drawings demonstrated more structured spatio-organizational patterns would achieve a higher number of correct answers on a timed test of basic addition problems. It was also hypothesized that the structure of the spatio-organizational patterns that children drew would be influenced by age, gender and mathematics achievement. The results of this exploratory study of children’s unmanipulated internalized constructs of number provided some interesting results. The children were asked to image specific numbers of dots for numerals from 4 through 13 and then to draw a representation of their images. The representations were categorized according to the structure of spatio-organizational patterns. The analyses revealed that the patterns had more structure for older children. Multiple regression analyses also indicated that the correctness of the cardinality of the number of dots imaged was the most frequently occurring variable that had a significant effect on the Imagery Scores. Less than five of more than 450 students expressed any difficulty with the imagery task and then only as it related to one of the ten numerals they were asked to image. The students were asked to image at the foundational level of imagery--reproductive imagery (Piaget & Inhelder, 1971). Because the research task developed for the students did not involve anticipatory images, those requiring transformations or movement, these imaging tasks were not influenced by the children's IQ or mathematics achievement. According to Piaget and Inhelder, children's ability to use anticipatory images indicates that children are developing an operational understanding and use of imagery. The children in this study were not asked to do anticipatory imaging. This may account for the negative relationship of the Imagery Scores to the fifth-grade students’ math percentile scores and the positive relationship between Imagery scores and mathematics percentile scores for the primary level students. The imagery tasks requested of the students were not of sufficient difficulty to relate to any mathematical operations or logio-mathematical thinking for older children. The ability of children to produce reproductive images which have varying degrees of spatio-organizational patterns was demonstrated by this study. Future studies need to address the higher level of anticipatory images. If students were asked to image a specific number of dots and then to image adding another quantity of dots to the original image, would the spatio-organizational patterns used by children in this transformation process change or transform the image? Are there specific spatio-organizational patterns that more easily allow children to develop anticipatory images that use mathematical operations? Are there children who have developed static reproductive images, and as a result, have created internalized constructs that inhibit their future understanding and development of higher level mathematical concepts? / Ed. D.

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

National and international studies show that the standard of mathematics teaching and learning in South Africa is very low compared to other countries. These statistics are worrying because mathematics is one of the ‘gatekeeper’ subjects that determine learners’ access to higher learning and a better future. My study, aimed at exploring two Foundation Phase teachers’ selection and use of examples and representations when teaching number, forms part of a longitudinal study currently underway within the Wits Maths Connect Primary (WMC-P) Project. One of the broad aims of the WMC-P Project is to improve primary teachers’ mathematics content knowledge and also to see this translated into improved pedagogy on the ground. This qualitative study was carried out within the WMC-P Project’s 20-Day in-service training course and one of the ten government schools participating in the broader study. My study aimed to build on research that has been carried out on teachers’ use of examples and representations with a focus on the South African terrain. The dataset comprised of two Foundation Phase teacher’s pre-tests, course-work tasks, field notes, and transcripts of observed lessons. Data was analysed using an analytical framework based on current literature related to examples and representations within mathematics teaching. Findings from my study show possible associations between a higher content knowledge score and the extent of a teacher’s example space and more coherent connections between different representational forms. More studies around this topic are needed because research shows that teachers’ examples and representations in mathematics teaching are important for good teaching and conceptual understanding.