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.

This research aimed at measuring the extent to which a group of senior secondary pupils were attaining desirable cognitive objectives in mathematics. The summary of the design and procedures adopted in this study and the major findings which emerged is presented here. A scheme of objectives for mathematics learning at the senior secondary level was suggested in accordance with Bloom's Taxonomy of Educational Objectives and recent research relating to the Taxonomy and other classifications used in mathematics education. Multiple choice-type test items were constructed with reference to the above scheme of objectives and to content areas selected from the standard grade senior secondary mathematics syllabus. A pilot test was administered and analysed. The selection of items for the final form of the test was based on a consideration of item analysis data, distractors, reliability, validity, rating of items according to objectives and length of test. The final forms of the test and questionnaire were administered to a selected sample of 769 standard nine pupils from 14 Indian high schools in the Durban and District Area. The test was manually scored and the scores were subjected to statistical analyses by computerization. The findings suggest that: (i) it is possible to devise a reasonably reliable and valid test instrument to test at least two different levels of objectives in mathematics learning at senior secondary school level; (ii) the lower level objectives in mathematics are significantly easier to attain than the higher level objectives, which tends to support - in at least two levels - the assumption of hierarchical structure of a taxonomic classification of objectives; (iii) the performance in mathematics of the higher grade pupils tends to be adversely affected by being taught mathematics in mixed higher and standard grade classes. / Thesis (M.Ed.)-University of Durban-Westville, 1975.

In this study I examine the effects of visualization in the teaching of problem solving in grades 6 and 7 in a school south of Durban in KwaZulu Natal. One of the goals of mathematics instruction according to the Department of Education is to prepare learners to become proficient in solving problems (DoE, 2003). Whilst many studies have been conducted in the field of problem solving, using visualization as a strategy to solve problems has been a neglected area in mathematics teaching in some schools. A literature survey shows that the link between solving problems and visualization strategies is making finding solutions easier for learners. The literature suggests that visualization assists learners to develop their problem solving skills as it allows them an opportunity to show their interpretation of the problem and the understanding of mathematical concepts. Through the use of problem centred mathematics, problem centred learning, growth of mathematical understanding and realistic mathematics education, learners see the connection and employ appropriate strategies to solve problems. This study examines the strategies employed by educators in the teaching and learning of problem solving and the strategies used by learners when solving problems. Data was collected from educators using a questionnaire, observation of grade 6 and 7 learners in the classroom and semi structured interviews. The conclusions from the data analysis have shown that problem solving is been neglected and that visualization does assist learners in solving problems. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2009.

Too often, discussions about Mathematics express feelings of anguish and despair; and, indeed Mathematics results in general in South Africa can be described as dismal. The Department of Education (DoE) reported that in the 2010 National Senior Certificate examinations, 52.6% of learners obtained less than 30% in Mathematics and 69.1% of learners obtained less than 40% (DoE, 2010). This implies that a very small percentage of grade 12 learners would be eligible to further their studies in the fields of Mathematics and science at tertiary level, resulting in a depletion of science and Mathematics-oriented professionals. This study explored the mathematical attitudes and achievement strategies of successful Mathematics learners to overcome the factors that might impede achievement. This study has the potential to improve practice because the findings of the study and recommendations are made implicit in the discussion. In particular this study sought to investigate the following issues: (a) What are secondary school learners' attitudes towards Mathematics? (b) In what ways are these attitudes linked to factors to which the learners attribute their achievement in Mathematics? (c) What strategies do successful Mathematics learners use to overcome the factors that they identify as impeding their performance in Mathematics? This research involved a case study approach. The study solicited both quantitative and qualitative data from the participants. The participants comprised 95 Grade 10, 11 and 12 Mathematics learners. The Fennema-Sherman Mathematics Attitude Scales (FSMAS) questionnaire was used to collect data from participants. The data was analysed using Attribution Theory and Achievement Theory. Two learners, who obtained more than 60% in the 2011 half-year Mathematics examination, from grades 10, 11 and 12 respectively, constituted the focus group. The focus group interview enhanced the study by clarifying the responses to the questionnaire and providing answers to the second and third research questions. The findings of the research include the following: teachers play an important role in shaping learners’ attitudes toward Mathematics; learners are anxious when asked to solve mathematical problems; parents are very encouraging of their children learning Mathematics; the importance of Mathematics for future careers exerted a significant effect on mathematical achievement; and finally the various strategies that learners employ that positively impact on their achievement in mathematics include mastery experience, motivation, private tuition and peer group teaching-learning. The final section of this dissertation discusses the implications of this study for practising Mathematics teachers and suggestions for further research in the area of affect. / Thesis (M.Ed.)-University of KwaZulu-Natal, Edgewood, 2011.

This qualitative interpretive inquiry investigates how mathematical meaning is constructed in a sixth grade classroom during one academic year in an English medium suburban school. Mathematical meaning is situated within Piaget's constructivist theory and Vygotsky's socio-cultural theory of the development of higher mental functions. Halliday's social theory of language use provides another theoretical framework for interpreting the daily interactions between the teacher and learners and among learners. Particular focus is on modality and the use of modal auxiliaries in teacher interrogative modals across the teaching of two strands, fractions and geometry. Data collected and analyzed includes 107 audio taped mathematics classes, participant observations of the teacher and six focal children over one school year and school artifacts such as the mathematics textbook and paper handouts. / Findings indicate that the teacher used modal auxiliaries in both the fraction and geometry strands but more modals in the geometry strand. While there were similarities in the teacher's rejoinders across the two strands there were also some distinctions particular to each strand. Data sets suggest that mathematical meanings ultimately made by these learners are influenced by both personal and social factors. Teacher interrogative modals mediated understandings by encouraging multiple learner contributions which resulted in alternative strategies and thoughts being respected. Analysis of the task and activities, event flow of mathematical lessons influenced how the children perceived mathematics and how it should be accomplished Results indicate that these learners, understandings were unique and that through the power of questioning that encourages multiple learner perspectives educators may obtain insights into children's mathematical meaning making in classroom contexts.

The purpose of this study was to investigate the effectiveness of two different methods of teaching learning disabled middle school students (6th, 7th, and 8th grades) how to solve one-step addition and subtraction mathematics story problems. This study also compared the generalization of the two instructional methods to problems written in simple syntax which required the performance of two mathematics operations, addition and subtraction, in order to obtain the correct written solution.Teachers were randomly assigned to one of the two instructional methods. The students in the experimental and control classrooms were administered the The Mathematics Computation Screeninq Test, the One-step Story Problem-Solving Test of Mathematics Reasoninq and the Two-step Story Problem-Solving Test of Mathematics Reasoninq. Students who obtained above 80% mastery on the The Mathematics Computation Screening Test and at or below 67% mastery on the pretest of the One-step Story Problem-Solving Test of Mathematics Reasoning were included in the experimental and control groups. Students in the experimental and control groups who meet the above criteria and were at or below the 60% mastery level on the pretest of the Two-step Storv Problem Solving Test of, Mathematics Reasoning were included in the analysis of two-step problems. There were 70 students who meet these criteria. Following administration of the tests, students received 17 days of instruction in one of the two instructional methods.Previous research has shown that good problem-solvers initially have a mental representation of a story problem prior to solving the problem and that accurate performance may be increased by teaching students to generate diagrammatic representations of the problems. Based on this research it was hypothesized that learning disabled students who receive instruction in generating diagrammatic representations would have a higher mean performance on a linear composite of writing number sentences and solving one-step addition and subtraction story problems than learning disabled students who did not receive this instruction when pretest performance on one-step written solutions was held constant. It was also hypothezied that when presented with two-step addition and subtraction story problems learning disabled students who receive instruction in how to generate diagrammatic representations for various one-step addition and subtraction story problems would have a higher mean performance than learning disabled students who do not receive this instruction when pretest one-step and two-step written solutions were held constant.A 2 X 2 X 2 X 2 hierarchical multivariate analysis of covariance mixed effects design followed by examination of step down F ratios was used to test the one-step hypotheses. Analysis of the data indicated no significant difference between the groups on number sentence writing and on solving one-step addition and subtraction story problems varying in syntactic complexity and position of the unknown term. The data did indicate a significant interaction between the within subject factors of syntax, position of the unknown term, and mathematics operation.A 2 X 2 hierarchical analysis of covariance design was used to test the hypotheses regarding generalization of the two instructional methods to two-step story problems of addition and subtraction. Analysis of the data indicated no significant difference between the problem-solving performance of students taught with the diagrammatic instructional method and those taught in the control group. / Department of Special Education

The study examined the relationship between the sexrole stereotyping and the mathematic achievement of academicallytalented tenth grade females. The sample was drawn from two middle schools and one high school in the Lake Central School Corporation, St. John, Indiana. The sample included students in grades six and ten. The sixth grade sample was used forinstruments: the quantitative battery of the Houghton Mifflin Cognitive Abilities Test (CAT), the mathematics computation and mathematics concepts batteries of the CTB/McGraw-Hill Comprehensive Tests of Basic Skills (CTBS), and the Maferr Inventories of Feminine and Masculine Values. There was not a significant decline in the quantitative achievement of scores of academically-talented tenth grade females as determined by achievement scores from the females' seventh, eighth and tenth grade CTBS test. There were no significant differences in the seventh, eighth and tenth grade quantitative achievement scores of academically-talented tenth grade females as compared to male peers. A weak negative correlation was found between the non-significant changes of the tenth comparative purposes only. There werethree data gathering grade females' CTBS score means for seventh, eighth and tenth grade and the females' inclination toward family orientation, self orientation or a balance between the two as perceived the way the females actually were and, then, perceived the way the females' ideal woman would respond. Additional analysis resulted in the following: 1) tenth grade females indicated that the ideal female would be inclined toward self orientation; 2) sixth grade females indicated that the ideal female would be inclined toward a balance between family orientation and self orientation; and 3) sixth grade females were inclined to be self oriented, while sixth grade males were inclined toward an even distribution among the three - family orientation, balanced and self orientation. Recommendations for further research included: choosing an achievement test that would provide more differentiation (an out-of-level test might be considered); conducting a longitudinal study, following the sample through five years of school; finding a better method of gathering sex-role stereotyping data; increasingthe sample size; and locating a more representative geographic location for the study.

The purpose of this study was to test the predictive ability of the Personality Assessment System (PAS) with special emphasis placed on predicting the personality patterns of two groups of mathematics students. Two ancillary purposes were also identified. These included the possible contributions to academic/vocational counseling and the addition of the findings to a PAS validation data bank that has been established.The sample included 26 selected volunteers from Purdue University and 26 selected volunteers from Ball State University. These students were identified into two distinct groups. The Purdue group was composed of senior students who were majoring in pure mathematics, while the Ball State group was composed of students who were majoring in mathematics education. Each group contained an equal number of males and females.Each student was administered the Wechsler Adult Intelligence Scale (WAIS) from which a personality profile was derived. This profile was based on the theoretical constructs of the Personality Assessment System.The hypotheses in the study were in the form of predictions. The predictions attempted to identify the personality patterns of the two groups of mathematics students.The resultant data were analyzed using percentages, a t test for independent groups, and a chi square analysis. Trends in the data, rather than statistically significant outcomes, were the main focus of the analysis.The predictions generated by the researcher correctly identified eight of 26 or 30% of the specific expected PAS patterns of the Purdue group. When the original predictions for the Purdue group were expanded to include all possible mathematics patterns, 23 of 26 or 88% of the Purdue group was correctly identified. The predictions correctly identified eight of 26 or 30% of the specific expected PAS patterns of the Ball State group. When the original predictions for the Ball State group were expanded to include all possible mathematics patterns, 15 of 26 or 57% of the Ball State group was correctly identified.Trends were suggested by the data for other types of measurement. The most consistent trend indicated by the data was the preponderance of IRU Primitive personality patterns in the Purdue group as compared to the general population, and the preponderance of IRA Primitive personality patterns in the Ball State group as compared to the general population.Another trend indicated by the data was the effect of a low Digit Symbol score on academic achievement in the Purdue group. Members of the group with a low Digit Symbol score consistently obtained a Grade Point Average below the mean Grade Point Average of the group.A final trend in the data was found between the combined groups of mathematicians and the general population on the R-F dimension of the PAS. The general population tends to be F or flexible, while the mathematicians tend to be R or regulated.The findings indicated that the PAS could adequately identify the personality patterns of a group of mathematicians. Rather than one "mathematical personality" being identified, the trends in the data point to many mathematical personalities. Recommendations for further research were made.

This study was conducted to determine if differentiated instruction improved student growth. The overall effectiveness was studied as well as that of gender and the aptitude of average and above average students. The study was that of a quasi-experimental design using student subjects in the classrooms of three second-grade teachers. The school in the study was located in an affluent suburb outside of a major city in the Midwest. This quantitative study concluded that differentiated instruction did not have an overall effectiveness at a significant level. Students with a higher academic ability benefited significantly with opportunity to be challenged at a higher level while students of average ability did not. There was no significant difference between the achievement of males and females. / Department of Elementary Education

Recent literature on mathematics education, and more especially on the teaching and learning of geometry, indicates a need for further investigations into the possibility of devising new strategies, or even developing present methods, in order to avert what might seem to be a "problem" in mathematics education. Most educators and textbooks, it would seem, do not address the need (function and meaning) of proof at all, or those that do, only address it from the limited perspective that the only function of proof is verification. The theoretical part of this study, therefore, analyzed the various functions of proof, in order to identify possible alternate ways of presenting proof meaningfully to pupils. This work further attempted to build on existing research and tested these ideas in a teaching environment. This was done in order to evaluate the feasibility of introducing "proof" as a means of explanation rather than only verification, within the context of dynamic geometry. Pupils, who had not been exposed to proof as yet, were interviewed and their responses were analyzed. The research focused on a few aspects. It attempted to determine whether pupils were convinced about explored geometric statements and their level of conviction. It also attempted to establish whether pupils exhibited an independent desire for why the result, they obtained, is true and if they did, could they construct an explanation, albeit a guided one, on their own. Several useful implications have evolved from this work and may be able to influence, both the teaching and learning, of geometry in school. Perhaps the suggestions may be useful to pre-service and in-service educators. / Thesis (M.Ed.)-University of Durban-Westville, 1998.