This study identified factors related to solving mathematical word problems and then examined the differences in characteristics between students with low achievement in mathematics who were likely to have a learning disability and students with low achievement in mathematics who were unlikely to have a learning disability. Factoral analysis identified two significant factors: abstract thinking and long term retrieval from memory. Results indicated qualitative differences between sixth grade students with achievement in mathematics at or below the 25th percentile with indications of learning disabilities (MLD) and students with achievement in mathematics at or below the 25th percentile without an indication of a learning disability (Low Math/NLD). The Learning Disabilities Diagnostic Inventory, which measures intrinsic processing disorders indicative of learning disabilities, was used to differentiate between students with MLD (n = 13) and students with Low Math/NLD (n = 16). The Woodcock-Johnson III Tests of Achievement, Clinical Evaluation of Language Fundamentals-Fourth Edition, and the Informal Mathematics Assessment (IFA) were used to compare the two groups. In contrast to students with MLD, students with Low Math/NLD had a higher mathematical performance and had more difficulties with math fluency. When solving mathematics word problems on the IFA, a test composed of word problems, student interview, and error analysis, students with Low Math/NLD had more correct answers, more computational errors, and fewer translation errors than students with MLD did. Students with MLD had conceptual difficulties in the areas of analyzing, reasoning, and abstract thinking.

The research presented in this thesis is a case study located in the interpretive paradigm of qualitative research. The focus is on the use of van Hiele's theory to explore problems encountered in circle geometry by grade 11 learners and making some policy recommendations concerning the curriculum structure and teaching of the geometry at all grades. The interpretation is based to the learners' background in geometry i.e. their prior knowledge and experience of learning geometry. The study was carried out over a period of three years. The data collection process took a period of two months (April and May 2003) with a group of 21 grade 11 mathematics learners in a rural senior secondary school in the Eastern Cape. The researcher used document analysis, worksheets, participants' observation, van Hiele tests, a questionnaire and semi-structured interviews to collect data. The study showed that the structure of the South African geometry syllabus consists of a some what disorganized mixture of concepts. It is not sequential and hierarchical and it sequences concepts in a seemingly unrelated manner. The study revealed that the South African high school geometry curriculum is presented at a higher van Hiele level than what the learners can attain. The findings of the study showed that many of the grade 11 learners were under-prepared for the study of more sophisticated geometry concepts and proofs. Three categories of reasons could be ascribed to this: Firstly, there was insufficient preparation of learners during the primary and senior phases. Secondly the study indicated that there is overload of geometry at the high school level in the South African mathematics curriculum. Thirdly, the over-reliance on the traditional approach to teaching geometry, poor presentation of mathematical technical concepts and language problems, were identified as possible additional reasons for the poor learner understanding of geometry in general and circle geometry in particular. The study recommends that the structure of the South African geometry curriculum should be revisited and redesigned. Teachers should be empowered and developed to be more effective in teaching geometry through further studies in mathematics and in-service workshops. They should also be engaged in the process of implementing the van Hiele's theory in the teaching of geometry in their classrooms.

Bibliography: pages 75-82. / This dissertation is concerned with teachers' beliefs regarding boys, girls and mathematics. The present study is a partial replication of a study conducted by Fennema et al (1990) and the results are compared. The present study extended the work of Fennema et al (1990) through an exploration of the structure of the data. Forty female teachers in the Western Cape region were interviewed. They were asked to identify their two most and least successful boys and girls in mathematics and to attribute causation for success and failure. They _were also asked to respond to 20 characteristics on a "Likert type" response format. The results generated from the present study concluded that teachers believed their female students to be their more successful mathematics students. They attributed the most successful girls' achievement mainly to effort whereas with the most successful boys, achievement was attributed to ability and effort. Both the most successful boys and girls failures on mathematics tasks were attributed to the difficulty of the task. Achievement of the least successful girls was attributed mainly to teacher's help and for the boys it was attributed to teacher's help and task. For both these groups, ability and to a lesser extent, effort, are given as the main reasons for failure on mathematics tasks. Very little difference was found between teachers' responses regarding the characteristics of their best boy and best girl mathematics students. When exploratory factor-analysis was performed a difference was found in the factor-solutions for the boys and the girls. This study suggests that there might be a difference in teachers' beliefs regarding boys and girls achievement in mathematics that is worthy of further exploration.

The state of Texas passed the Student Success Initiative (SSI) in 1999 which requires all 3rd graders to pass the reading portion of the Texas Assessment of Knowledge and Skills (TAKS) test to be promoted to the 4th grade, and for 5th graders to pass the reading and math portions of the TAKS test to be promoted to the 6th grade. Beginning in spring 2008, 8th graders will also need to pass the reading and math portions of the TAKS test to be promoted to the 9th grade. The purpose of this study was to examine the academic performance of 3rd and 5th grade students who did not meet the passing standard on the TAKS test and were retained during the 2005-2006 school year. The population of this study included 33 3rd graders and 49 5th graders who were retained during the 2005-2006 school year due to not meeting the promotion requirements of the SSI. There was also a second population of 49 5th graders who were retained in 3rd grade during the 2003-2004 school year due to not meeting the promotion requirements of the SSI. These students were enrolled in the 5th grade for the first time during the 2005-2006 school year. Their TAKS scores were examined to see whether students were still benefiting from the year of retention in 3rd grade. Results for all populations were broken down by ethnicity and program codes. The results of the study showed a statistically significant gain in 3rd grade reading and 5th grade math scores. The 5th grade reading scores did have a statistically significant improvement even though the reading mean score was still below the minimum passing score even after a year of retention. A cross tabulation done on students who had been retained in 3rd grade due to SSI requirements and were enrolled in the 5th grade during the study showed a greater significant growth in math than in reading. A strong correlation between the ITBS and TAKS tests were found in both 3rd grade reading and 5th grade math. A weak correlation between the tests was found in 5th grade reading.

This study examined the relationship between levels of implementation of the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards and 5th Grade Louisiana Educational Assessment Program Math Scores with the effects of race of students accounted for. Secondary areas of interest were the relationship between LEAP mathematics scores with the effects of race of students accounted for and the teacher characteristics of years experience and educational attainment and of the relationship between level of implementation of the Standards and teacher characteristics. The population, from which a sample size of 250 was randomly drawn, was comprised of 1994-95 Louisiana public school teachers who taught in a regular 5th grade or departmentalized math class. Survey research was used to place the responding teachers at one of the five levels of implementation. Hierarchical Multiple Regression was used to analyze the question of primary interest. Race of the students was found to have accounted for nearly 9% of the variance in LEAP mathematics scores. This figure was statistically significant. The independent variable Level of Implementation of the Standards produced ambiguous results. Students of Level 1 (non-implementers) teachers were found to have statistically significantly higher LEAP scores than did students of Level 2 teachers. The Level 1 students had scores which were non-statistically significantly higher than did those of Level 3 and 5. Students of Level 4 teachers had scores which were significantly higher than those students whose teachers were at Level 2 and 5. No significant relationship was found to exist between student LEAP mathematics scores and teacher characteristics of years experience and educational attainment nor between levels of implementation of the Standards and the same two teacher characteristics. Despite these findings, in light of the amount of research pointing to their value, implementation of Standards is still highly recommended.

published_or_final_version / Education / Master / Master of Education

School-related mistakes and low pass rates have led this researcher to perceive that some students are not adequately prepared for Mathematics 1. To address the problem of under-preparedness overseas universities use placement or diagnostic tests. Diagnostic testing identifies areas of weakness and provides information to guide the development of appropriate remedial support. This researcher embarked on a study to identify the sub-domains (criteria) that should be included in a diagnostic Mathematics test battery at the NMMU. An analysis of first-year curricula was undertaken to determine the required Mathematical pre-knowledge and skills entry-level students should have. Thereafter, the required pre-knowledge and skills were reflected against the standard grade school syllabi. From this it was determined that the school learners should acquire the necessary pre-knowledge and skills for university success as part of the school syllabus. However, in reality this is not the case as the researcher and other Mathematics lecturers identified a number of basic errors that incoming students make. This suggests that they have not developed all the required knowledge and skills. Furthermore, their performance in the matriculation examinations does not provide an adequate measure of the requisite Mathematical pre-knowledge and skills necessary for success at university-level Mathematics. No suitable existing diagnostic Mathematics test could be found. By utilizing both an action research as well as a test development methodology, the researcher thus proceeded to delineate the sub-domains that should be included in a diagnostic Mathematics test battery. Thereafter, test specifications were developed for two pilot tests and items were developed or sourced. The constructed response item-type was chosen for the pilot tests as it was argued that this item-type was more useful to use in a diagnostic test than a multiple-choice item format, for example. The pilot test battery, which consisted of a pilot Arithmetic and Algebra and Calculus tests, was administered to a sample of first-year students at the NMMU in 2004 and their performance in Mathematics at the end of the first year was tracked. Tests were scored holistically and analytically to provide a rich source of information. Thereafter, the test results were analysed to obtain evidence on the content validity of the pilot tests, including the item difficulty values and the item-total correlations; to determine the predictive validity of performance on the pilot tests with respect to final first-year Mathematics marks; and their reliability was determined using the Cronbach’s Alpha statistic. These findings suggest that appropriate sub-domains (criteria) were delineated and the items appropriately covered these sub-domains (i.e. the content validity of the pilot tests is acceptable). Furthermore, the predictive validity of the pilot ix tests was found to be acceptable in that significant correlations were found between the pilot tests and performance in first-year Mathematics. Finally, the pilot tests were found to be reliable. Based on the results, suggestions are made regarding how to refine the diagnostic test battery and the research related to it. The final diagnostic Mathematics test battery holds much potential to be able to assist in the early identification of at-risk students who can be timeously placed in developmentally appropriate Mathematics modules or provided with appropriate remedial intervention.

The study aimed to explore misconceptions that Grade 9 learners at a school in Soweto had concerning the topic of fractions. The study was based on the ideas of constructivism in a bid to understand how learners build on existing knowledge as they venture deeper into the development of advanced constructions in the concept of fractions. A case study approach (qualitative) was employed to explore how Grade 9 learners describe the concept of fractions. The approach offered a platform to investigate how Grade 9 learners solve problems involving fractions, thereby enabling the researcher to discover the misconceptions that learners have/display when dealing with fractions. The research allowed the researcher to explore the root causes of the misconceptions held by learners concerning the concept of fractions. Forty Grade 9 participants from a township school were subjected to a written test from which eight were purposefully selected for an interview. The selection was based on learners’ responses to the written test. The researcher was looking for a learner script that showed application of similar but incorrect procedures under specific sections of operations of fractions, for example, multiplication of fractions. Both performance extremes were also considered, the good and the worst performers overall. The written test and the interviews were the primary sources of data in this study. The study revealed that learners have misconceptions about fractions. The learners’ definitions of what a fraction is were neither complete nor precise. For example, the equality of parts was not emphasised in their definitions. The gaps brought about by the learner conception of fractions were evident in the way problems on fractions were manipulated. The learners did not treat a fraction as signifying a specific point on the number system. Due to this, learners could not place fractions correctly on the number line. Components of the fraction were separated and manipulated as stand-alone whole numbers. Consequently, whole number knowledge was applied to work with fractions. A lack of conceptual understanding of equivalent fractions was evident as the common denominator principle was not applied. In the multiplication of fractions, procedural manipulations were evident. In mixed number operations, whole numbers were multiplied separately from the fractional parts of the mixed number. Fractional parts were also multiplied separately, and the two answers combined to yield the final solution. In the division of fractions, the learners displayed a lack of conceptual knowledge of division of fractions. Operations were made across the division sign numerators separate from the denominators. This reveals that a fraction was not taken as an outright number on its own by learners but viewed as one number put on top of the other which can be separated. Dividing across, learners rendered division commutative. A procedural attempt to apply the invert and multiply procedure was also evident in this study. Learners made procedural errors as they showed a lack of conceptual understanding of the keep-change-flip division algorithm. The study revealed that misconceptions in the concept of fraction were due to prior knowledge, over-generalisation and presentation of fractions during instruction. Constructivism values prior knowledge as the basis for the development of new knowledge. In this study, learners revealed that informal knowledge they possess may impact negatively on the development of the concept of fractions. For example, division by one-half was interpreted as dividing in half by learners. The prior elaboration on the part of a whole sub-construct also proved a barrier to finding solutions to problems that sought knowledge of fractions as other sub-constructs, namely, quotient, measure, ratio and fraction as an operator. Over generalisation by learners in this study led to misconceptions in which a procedure valid in a particular concept is used in another concept where it does not apply. Knowledge on whole numbers was used in manipulating fractions. For example, for whole numbers generally, multiplication makes bigger and division makes smaller. The presentation of fractions during instruction played a role in some misconceptions revealed by this study. Bias towards the part of a whole sub-construct might have limited conceptualisation in other sub-constructs. Preference for the procedural approach above the conceptual one by educators may limit the proper development of the fraction concept as it promotes the use of algorithms without understanding. The researcher recommends the use of manipulatives to promote the understanding of the fraction concept before inductively guiding learners to come up with the algorithm. Imposing the algorithm promotes the procedural approach, thereby depriving learners of an opportunity for conceptual understanding. Not all correct answers result from the correct line of thinking. Educators, therefore, should have a closer look at learners’ work, including those with correct solutions, as there may be concealed misconceptions. Educators should not take for granted what was covered before learners conceptualised fractions as it might be a source of misconceptions. It is therefore recommended to check prior knowledge before proceeding with new instruction. / Mathematics Education / M. Ed. (Mathematics Education)