This study investigated young children's construction of meaning for number and explored ways to more comprehensively assess and portray the development of number sense in young children. Greeno's (1991) conceptualization of number sense as situated knowing in a conceptual domain was used to consider both the mathematical tools available to the child and the extent to which the child makes use of these tools. The data consisted of four videotaped interviews for each of sixteen children between the ages of six and eight. Each of the four interviews involved a different number context: doubling, finding missing parts, sharing, and working with money. Each context involved a task presented in a series of increasingly difficult items, with number size predominantly determining the difficulty level. A dynamic interview format was used to encourage children to work beyond their independent level, or "number comfort zone." Cues and scaffolds were provided to support children's construction of meaning within their "number construction zone" and towards the outer limits of their understanding. Analysis focused on the strategies children used to make sense of each item, and the cognitive, affective and contextual aspects which enhanced or constrained their mathematical activity within the number construction zone. Results were reported two ways. The first, specific task performance across children, provided a means of describing the diversity of developmentally appropriate ways children made sense of the different tasks and provided a frame of reference for considering individual performance. The second approach to reporting results considered individual children's performance across tasks, and provided a means of focusing on characteristics of emerging competence. Results of this study illustrate how the nature and use of children's reasoning strategies can provide an indication of developing competence. Results highlight specific conceptual, procedural, functional, and affective characteristics that most directly affected children's capacity to make sense of number situations. No single characteristic alone accounted for children's success or lack of success, rather the inter-relationships of the different characteristics was apparent, with strengths in one area compensating for weaknesses in another. Though conceptual and procedural abilities appeared to shape to a great extent the nature of the number knowledge available to children, affective considerations and functional competence played a major role in shaping the extent to which children drew on this existing knowledge. Issues of context influenced both aspects of number sense: available knowledge and the nature of its use. Number size, context of the tasks, and presentation of tasks influenced children's mathematical activity in important ways. Children's personal number contexts were considered in terms of how they influenced their approaches to tasks. Overall, dynamic assessment techniques proved to offer a viable alternative for exploring the limits of children's ability to make sense of number situations, and for considering children's construction of meaning for number in developmental terms.

This study investigated young children's construction of meaning for number and explored ways to more comprehensively assess and portray the development of number sense in young children. Greeno's (1991) conceptualization of number sense as situated knowing in a conceptual domain was used to consider both the mathematical tools available to the child and the extent to which the child makes use of these tools. The data consisted of four videotaped interviews for each of sixteen children between the ages of six and eight. Each of the four interviews involved a different number context: doubling, finding missing parts, sharing, and working with money. Each context involved a task presented in a series of increasingly difficult items, with number size predominantly determining the difficulty level. A dynamic interview format was used to encourage children to work beyond their independent level, or "number comfort zone." Cues and scaffolds were provided to support children's construction of meaning within their "number construction zone" and towards the outer limits of their understanding. Analysis focused on the strategies children used to make sense of each item, and the cognitive, affective and contextual aspects which enhanced or constrained their mathematical activity within the number construction zone. Results were reported two ways. The first, specific task performance across children, provided a means of describing the diversity of developmentally appropriate ways children made sense of the different tasks and provided a frame of reference for considering individual performance. The second approach to reporting results considered individual children's performance across tasks, and provided a means of focusing on characteristics of emerging competence. Results of this study illustrate how the nature and use of children's reasoning strategies can provide an indication of developing competence. Results highlight specific conceptual, procedural, functional, and affective characteristics that most directly affected children's capacity to make sense of number situations. No single characteristic alone accounted for children's success or lack of success, rather the inter-relationships of the different characteristics was apparent, with strengths in one area compensating for weaknesses in another. Though conceptual and procedural abilities appeared to shape to a great extent the nature of the number knowledge available to children, affective considerations and functional competence played a major role in shaping the extent to which children drew on this existing knowledge. Issues of context influenced both aspects of number sense: available knowledge and the nature of its use. Number size, context of the tasks, and presentation of tasks influenced children's mathematical activity in important ways. Children's personal number contexts were considered in terms of how they influenced their approaches to tasks. Overall, dynamic assessment techniques proved to offer a viable alternative for exploring the limits of children's ability to make sense of number situations, and for considering children's construction of meaning for number in developmental terms. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

This study was an exploration into the effects of four categories of extra-student variables: high school performance, demographic characteristics, Myers-Briggs personality preferences and mathematics attitudes on three measures of college mathematics achievement (a Problem-Solving Test, an Algebra Skills Final Examination and course grade for all seven classes of 175 undergraduate students taking Pre-Calculus I Fall semester 1993). High school performance explained the most variation for all measures of mathematics achievement. Demographic characteristics and mathematics attitudes do not significantly influence any measure of mathematics achievement. The Myers-Briggs Type Indicator (MBTI) preference Extravert versus Intraverts (E versus 1) was a significant predictor for the Problem-Solving Test; the Judging versus Perception a versus P) preference was a significant predictor for the Algebra Skills Final Examination, and both E versus I and J versus P were predictors for the course grade. An experimental design was used to explore four classroom variables--3 class times, 2 instructional settings, MBTI E versus I and J versus P-- in six classes. Students taking 8:00 classes averaged 9 points lower than students taking 10:00 classes and 11 points lower than students taking 1:00 classes for all measures of mathematics achievement. There was no significant difference for the two instructional settings--cooperative learning or traditional lecture--for any measure of mathematics achievement. Students who were Introverted averaged 8 points higher on the Problem-Solving Test. Students who had the Judging preference averaged 11 points higher on the Algebra Skills Final Examination and 5 points higher for the course grade. There was a significant interaction (p<.01) for the Problem-Solving Test of class Time x instructional setting caused by the poor performance of the 8:00 Cooperative Learning class. The interaction of E versus I x J versus P or the EIJP learning styles was significant (p<.05) for the Algebra Skills Final Examination and course grade. The students with the IJ learning style averaged 13 to 20 points higher for scores on the Algebra Skills Final Examination and 11 points higher for scores on the course grade than students with the other three learning styles--EP, EJ and IP. / Ph. D.

In this study, senior certificate (standard 10) pupils' attainment of the cognitive and affective aims and objectives of the senior certificate mathematics curriculum was investigated. With regard to the attainment of the cognitive objectives and aims, senior certificate pupils' performance in their mathematics examination, in terms of three broad categories of cognitive abilities (lower level, middle level and higher level mathematical abilities) was analysed and examined. As no norms (criteria) for mathematical attainment in respect of these three categories of cognitive abilities could be identified, these norms had to be firstly developed by the researcher. However, suitable standardised scales were identified and administered to determine senior certificate pupils' attainment of the affective aims and objectives (attitude towards and interest in mathematics). Besides the quantitative analysis, qualitative assessments of senior certificate pupils' attainment of the cognitive and affective aims and objectives were also made using information obtained, by way of a questionnaire, from teachers of senior certificate mathematics classes. The main findings that emerged from this investigation were: * The senior certificate pupils are attaining the desired proficiency levels in the cognitive objectives and aims of the senior certificate mathematics curriculum. However, these pupils are not adequately attaining the affective aims and objectives of the mathematics curriculum. * Qualitative information elicited from senior certificate teachers of mathematics tends to support the above findings which were obtained from the quantitative analysis. * There is a need for curriculum development in certain areas of the senior certificate mathematics curriculum, particularly in Euclidean geometry, for standard grade pupils. In terms of the general findings, certain recommendations were also formulated. In several ways, the present research is a pioneering effort in evaluating the efficacy of the cognitive and affective aims and objectives of the senior certificate mathematics curriculum. It is hoped that this study will serve as a catalyst for future research. / Curriculum and Instructional Studies / D. Ed. (Didactics)

In this study, senior certificate (standard 10) pupils' attainment of the cognitive and affective aims and objectives of the senior certificate mathematics curriculum was investigated. With regard to the attainment of the cognitive objectives and aims, senior certificate pupils' performance in their mathematics examination, in terms of three broad categories of cognitive abilities (lower level, middle level and higher level mathematical abilities) was analysed and examined. As no norms (criteria) for mathematical attainment in respect of these three categories of cognitive abilities could be identified, these norms had to be firstly developed by the researcher. However, suitable standardised scales were identified and administered to determine senior certificate pupils' attainment of the affective aims and objectives (attitude towards and interest in mathematics). Besides the quantitative analysis, qualitative assessments of senior certificate pupils' attainment of the cognitive and affective aims and objectives were also made using information obtained, by way of a questionnaire, from teachers of senior certificate mathematics classes. The main findings that emerged from this investigation were: * The senior certificate pupils are attaining the desired proficiency levels in the cognitive objectives and aims of the senior certificate mathematics curriculum. However, these pupils are not adequately attaining the affective aims and objectives of the mathematics curriculum. * Qualitative information elicited from senior certificate teachers of mathematics tends to support the above findings which were obtained from the quantitative analysis. * There is a need for curriculum development in certain areas of the senior certificate mathematics curriculum, particularly in Euclidean geometry, for standard grade pupils. In terms of the general findings, certain recommendations were also formulated. In several ways, the present research is a pioneering effort in evaluating the efficacy of the cognitive and affective aims and objectives of the senior certificate mathematics curriculum. It is hoped that this study will serve as a catalyst for future research. / Curriculum and Instructional Studies / D. Ed. (Didactics)