This study compares formative assessment practices in primary mathematics classrooms in Guangzhou, Hong Kong and Melbourne. Nine schools (three in each research location) were studied to examine the similarities in and differences between formative assessment practices for mathematics, and to identity underlying factors that may account for these similarities and differences. Videotaped classroom observations and face-to-face semi-structured teacher interview were the main data collection methods employed. The study identifies several similarities and differences in formative assessment practices by reviewing a total of 1140minutes of videotaped classroom observations (380 minutes from each city). Four similarities were noted: (a)a common structure of formative assessment practice; (b) providing feedback on what students had done well and what they needed to improve; (c) encouraging students to engage actively in the feedback process; and, (d) infrequent use of practical assessments. In addition, six differences were found: (a) interpreting, judging and suggesting on students’ work; (b) forms of assessment (including self and peer assessment); (c) assessment items; (d) effective feedback;(e) specific format for presenting mathematics; and, (f) ways of receiving feedback. Three key factors have been suggested to account for the similarities in and differences between formative assessment practices in primary mathematics classrooms in the three research locations: (a) cultural influences on mathematics learning and examinations; (b) assessment reform initiatives; and, (c) teachers’ conceptions about formative assessment. This study contributes to the understanding of formative assessment practices in the classrooms by proposing a theoretical framework for comparing formative assessment practices that takes into account cultural, social, school and classroom factors. Potential directions for future research are suggested, including further comparisons of mathematics formative assessment practices at other schools in Guangzhou, Hong Kong and Melbourne, and in other countries with similar cultural backgrounds. / published_or_final_version / Education / Doctoral / Doctor of Philosophy

The purpose of this study was to identify preferred learning styles and strategies of secondary school students and to examine the prevailing problems that restrict them to use their own preferences. The study was intended to highlight a number of issues that need to be revealed and addressed in the learning of mathematics. The types of preferred learning styles and strategies students need to employ in learning mathematics, the assistance students require from their teachers, the conduciveness of the design of mathematics curriculum and the challenges they might face to use their own preferred learning styles and strategies in the learning of mathematics were addressed as basic research questions. The study dealt with various elements that were related to environmental, emotional, sociological, physiological and psychological categories of learning in the identification of the types of learning styles and strategies. This study is believed to contribute a lot in addressing the problems of learning styles and strategies, provide feedback to the concerned government bodies to help them improve the teaching learning processes in secondary schools. It is also to reduce the bias or prejudice on mathematics by assisting students to use their own preferred learning styles and strategies, and contribute to further investigations to make the learning of mathematics more enjoyable, participatory and lifelong career. This study was conducted in four secondary schools in West Shoa Zone. A qualitative method that was descriptive in nature was employed in the study while the instruments of the study were questionnaires and an interview. The sample comprised of 249 (128 male and 121 female) secondary school students and 30 (25 male and 5 female) secondary school mathematics teachers selected randomly. The result has shown that students were not learning mathematics on the basis of their preferred learning styles and strategies and the teachers were practicing autocratic teaching styles. Most of the students did not prefer learning mathematics through plasma television; they required brief outlines and concrete presentations, and indicated that there is not enough time to check and recheck the answers they found for the problems. These imply that the organization of secondary school mathematics curriculum requires reform to accommodate the preferred learning styles and strategies of students. / Further Teacher Education / M. Ed. (Mathematics Education)

Piaget and Freudenthal advocated thinking levels. In the 1950's the van Hieles developed a five level model of geometric thought. Judith Land adapted the model in 1990, utilising four levels to teach the concept of functions. These four levels have been considered here in the formation of concepts of sequences and series. The origin and relevance of sequences and series have been studied and the importance of visualisation, patterning and generalisation in the instructional process investigated. A series of lessons on these topics was taught to a group of six higher grade matriculation students of mixed ability and gender. Questionnaires related to student progress through the various levels were answered, categorised, graphed and analysed. Despite the small number of students, results seem to indicate that emphasising visualisation, exploring patterns and generalisation and teaching the topics as a reinvention had made a positive contribution towards progress through the various thought levels. / Mathematics Education / M.A. (Mathematics Education)

This investigation emanates from the realization that Grade 3 children at schools in disadvantaged areas perform poorly in basic mathematics computations such as addition, subtraction, multiplication and division. The aim of the research was to establish the approaches teachers use when teaching mathematics computation. The qualitative approach, together with the research techniques commonly used with it, namely observation, interviews and document analysis was deemed appropriate for the investigation. The outcomes of the investigation revealed that the multilingual Grade 3 classes made it difficult to assist all children who experienced mathematics problems because teachers could not speak all the other languages that were not the language of learning (LoLT) of the school. Another obstacle that prohibited teachers from spending adequate time with children with mathematics problems was the time teachers were expected to spend on intervention programmes from the Department of Basic Education (DBE) aimed at improving schooling in general. Teachers could not make additional time that could afford children the opportunity of individual attention. With regard to the approach used for teaching mathematics, this study established that the teachers used the whole class teaching approach which is not a recommended approach because each child learns differently. It is recommended that teachers use a variety of teaching methods in order to accommodate all children and also encourage children to use concrete objects. It is also recommended that teachers involved in the SBSTs should consist only of members qualified in the subject and once these children are identified, remediation should take place promptly by their being enrolled (children) in the proposed programme. Finally, this study could benefit foundation Phase teachers in teaching mathematics based on the proposed strategy outlined after teachers’ challenges were identified. The outcome of the study could also be of value to the DBE, especially with curriculum designers. / Early Childhood Education and Development / D. Ed. (Early Childhood Education)

This study explores junior secondary students’ schemata on a line reflection construction task, the research of which was conducted in a secondary school in Hong Kong. The theories drawn on in this study come from the literature on theories of schemata and the corresponding knowledge embedded within, namely conceptual knowledge, manipulation and procedural knowledge. The research built on existing theories on schemata and attempted to categorize the different kinds of schemata as well as investigating the relationship between them among four junior secondary students in the construction of a line reflection task. The study also tried to find out how and why students manipulated in a line reflection construction task and the extent to which manipulation could lead learners to successfully tackle the task. This study researched on four junior secondary students, drawing mainly on qualitative data used in the analysis, including task-based interview with the employment of think aloud method in a designed line reflection construction task, as well as study of students’ drawings. The data analysis mainly focused on three areas. First, the analysis of each of the four cases was conducted by looking into the different kinds of schemata possessed by the student informants. Second, analysis of the different knowledge (conceptual knowledge, manipulation and procedural knowledge) embedded in the schema possessed by the student informants was done. Third, synthesis was drawn upon the analysis made in an attempt to answer the research questions posed in this study. Findings from the study confirmed the core role conceptual knowledge plays in the establishment of a learner’s schemata. Findings also revealed that different learners may possess different schemata towards the same concept such as the concept of same distance. When investigating the manipulative actions employed by student informants, it was found that there is a reciprocal relationship between a learner’s conceptual knowledge and his manipulation. This is also apparent in cases where there was a misconception in the learner’s schemata. The research also found that students exercised manipulation very differently and these manipulative actions were largely informed by their corresponding conceptual knowledge. With regard to why they manipulated, the research revealed reasons including manipulation for exploration, manipulation for representation and manipulation for verification. Based on the observation and analysis done in the four cases, it was found that manipulation helped students in the completion of the task to different extents. Learners with weaker conceptual knowledge in line reflection benefited more from the manipulation done in the construction task. These findings have implications for the teaching and learning of line reflection. Teachers are suggested to consider introducing using manipulative tools when approaching the teaching of line reflection, especially when they are dealing with students without rich conceptual knowledge in the area. The effectiveness of having hands-on experience implies that simply teaching definition and inviting learners to rote-learn does not necessarily lead to effective acquisition of knowledge in the Mathematics topic of line reflection. / published_or_final_version / Education / Doctoral / Doctor of Education

published_or_final_version / Education / Master / Master of Education

Mathematics presents a stumbling block to many students, particularly those majoring in scientific fields, business administration, or elementary education. Improvement of student achievement in mathematics at the lower division college level is needed. Seven instructional methods were investigated in terms of student achievement: programmed instruction (P.I.), individualized instruction (I.I.), computer based instruction (CBI), laboratory and discovery methods (Lab), television (TV), audio-tutorial (A-T), and tutoring. The research questions were: (1) What is the relative effectiveness of the innovative instructional methods as measured by student achievement and compared to the traditional lecture method? (2) What is the relative effectiveness of the innovative instructional methods on students of differing ability and course levels. (3) What is the effectiveness of combinations of the innovative instructional methods? A meta-analytical approach was used. Studies comparing an innovative method to the lecture or to another innovative method were located, and the summary data in each were used to calculate an "effect size"--a standardized measure of the effectiveness of the innovative method--to which statistical procedures were applied. The meta-analysis found that (1) Relative to the lecture method, six of the innovative methods produced a positive effect on student achievement. The ranking of the methods in order of decreasing effectiveness was: tutoring, CAI, A-T, I.I., P.I., Lab, TV. (2) The most effective methods by level of course were: (a) Precalculus level: CAI, A-T, and tutoring; (b) Calculus level: tutoring, I.I., P.I., and A-T; (c) Foundations of Mathematics (elementary education majors): P.I.; Descriptive Geometry: TV. The most effective methods by ability level of the student were: (a) High ability: CAI and Lab; (b) Middle ability: CAI, I.I., and P.I.; (c) Low ability: P.I. and A-T. (3) The lack of empirical studies prevent a determination of the relative effectiveness of combinations of the innovative methods. Recommendations include the following: (1) Variation of instructional methods; (2) Incorporation of specific, effective elements of innovative methods into the lower division college mathematics instructor's repertoire; and (3) Empirical investigation of the effectiveness of combinations of methods and of various instructional methods on students of different ability levels.

This study explores the relationship between university mathematics teachers' beliefs about the nature of reading mathematics and their practices regarding reading mathematics. It is a response to the calls for reform in mathematics education, particularly to the assertion made by the National Council of Teachers of Mathematics in 1989 that not all students can read mathematical exposition effectively and that all students need instruction in how to read mathematics textbooks. It presupposes a collaboration between reading and mathematics teachers to help students learn to read mathematics. The objectives were (1) to examine mathematics teachers' beliefs and practices regarding reading, mathematics, and thereby, reading mathematics; (2) to determine whether the theoretical perspectives implicit in those beliefs and practices could be characterized vis-a-vis the theoretical orientations that inform Siegel, Borasi, and Smith's (1989) synthesis of mathematics and reading; and (3) to determine the relationship, if any, that exists between mathematics teachers' beliefs about reading mathematics and their practices regarding reading mathematics. The synthesis presents dichotomous views of both mathematics and reading: Mathematics is characterized as either a body of facts and techniques or a way of knowing; reading, as either a set of skills for extracting information from text, or a mode of learning. The latter view, in each case, can be characterized as constructivist. The researcher was a participant observer in a university sumner program. The primary participants were fourteen mathematics instructors. Interviews were conducted using a heuristic elicitation technique (Black & Metzger, 1969). Field notes were taken during observations of classroom activities and other non-academic summer program activities. The data were coded using a constant comparative method (Glaser & Strauss, 1967) comparative method. Twelve instructors held conceptions of reading that were consistent with their conceptions of mathematics. Of those twelve, two held conceptions that could be characterized as constructivist; ten held conceptions that were not constructivist. Two instructors held conceptions of reading that were not consistent with their conceptions of mathematics. Of those two, one held a constructivist conception of reading but not of mathematics; one held a constructivist conception of mathematics but not of reading. Teachers' practices reflected their theoretical orientations. The study has implications for teacher education: If teachers' beliefs are related to their practices, then teacher education programs should (1) acknowledge the teachers' existing beliefs and (2) address the theoretical orientations implicit in various aspects of pedagogy.

A hierarchy of skills in the measurement topic of area was validated on three-hundred and six students between grades six and nine. The hierarchy of skills was based on the rules underlying the individual skills. When a rule for one skill was considered a component of a rule for another skill, then the two skills were hypothesized to be hierarchically ordered. In addition, if a simple rule for a particular skill was replaced by a more complex rule, resulting in a different skill, then these two skills were hypothesized to be hierarchically ordered. The physical representations of the area tasks, as well as the shapes of the area figures were hypothesized as influencing the skill orderings. The use of Latent-class analysis revealed that seven of the nine skill orderings analyzed were hierarchically ordered based on difficulty level and not prerequisiteness. The other two skill orderings indicated equaprobable partial mastery classes. In addition to Latent-class analysis, the incorrect processes used by the students were coded and tabulated. The results revealed that (1) nonstandard shaped area problems were the most difficult for this sample, (2) the most frequent process associated with incorrect responses involved the addition of numbers shown in area problem figures, (3) the second most frequent process involved some form of multiplication, without regard to the area concepts inherent in the task, and (4) students beyond the sixth grade made more errors involving multiplication processes than errors involving addition processes. The study revealed that the use of rules, representations and shapes as the basis for a hierarchy does appear to have merit. In addition, process analysis revealed that students respond in a large variety of ways when they do not know the correct process for area tasks.

<p>This research provided insights into the mathematical achievement of a cohort of tertiary mathematics students. The context for the study was an entry level mathematics course, set in an engineering programme at a tertiary institution, the Cape Peninsula University of Technology (CPUT). This study investigated the possibilities of providing a bridge between the assessment of students by means of tests scores and a taxonomy of mathematical objectives, on the one hand, and the critical analysis of student produced texts, on the other hand. This research revealed that even in cases of wrong solutions, participant members' responses were reasonable, meaningful, clear and logical.</p>