The study investigated the effectiveness of Problem-based learning (PBL) in the Further Mathematics classrooms in Nigeria within the blueprint of pre-test-post-test non-equivalent control group quasi-experimental design. The target population consisted of all Further Mathematics students in the Senior Secondary School year one in Ijebu division of Ogun State, Nigeria. Using purposive and simple random sampling techniques, two schools were selected from eight schools that were taking Further Mathematics. One school was randomly assigned as the experimental while the other as the control school. Intact classes were used and in all, 96 students participated in the study (42 in the experimental group taught by the researcher with the PBL and 54 in the control group taught by the regular Further Mathematics teacher using the Traditional Method (TM)). Four research questions and four research hypotheses were raised, answered, and tested in the study. Four research instruments namely pre-test manipulated at two levels: Researcher-Designed Test (RDT) (r = 0.87) and Teacher- Made Test (TMT) (r = 0.88); post-test manipulated at two levels: RDT and TMT; pre-treatment survey of Students Beliefs about Further Mathematics Questionnaire (SBFMQ) (r = 0.86); and post-treatment survey of SBFMQ were developed for the study. The study lasted thirteen weeks (three weeks for pilot study and ten weeks for main study) and data collected were analysed using Mean, Standard deviation, Independent Samples t-test statistic, and Analysis of Variance. Results showed that there were statistically significant differences in the mean post-test achievement scores on TMT (t=-3.58, p<0.05), mean post-test achievement scores on RDT (t=-5.92, p<0.05) and mean post-treatment scores on SBFMQ (t=-6.22, p<0.05) between students exposed to the PBL and those exposed to the TM, all in favour of the PBL group. Results also revealed that there was statistically significant difference in the post-test achievement scores on TMT at knowledge (t= -23.97, p<0.05) and application (t= -11.41, p<0.05) but not at comprehension (t= -0.50, p>0.05, ns) levels of cognition between students exposed to the PBL and the TM. Based on the results, the study recommended that the PBL should be adopted as alternative instructional strategy to the TM in enhancing meaningful learning in Further Mathematics classrooms and efforts should be made to integrate the philosophy of PBL into the pre-service teachers’ curriculum at the teacher-preparation institutions in Nigeria. / Science and Technology Education / D. Phil. (Mathematics, Science and Technology Education)

M. Tech. Education / There are two systems of secondary education in South Africa. Some secondary schools are composed of five grades: 8 to 12 (Secondary System A, SS-A), while others start at Grade 10-12 (Secondary System B, SS-B). As can be anticipated, these two systems are compositionally different, with schools in SS-B being dependent on feeder schools around them for enrolment of Grade 10 learners every year. This means that the composition of Grade 10 learners in these two systems is not the same. This study acknowledges that there are important differences in Grade 10 learners belonging to these two systems.

Learning Mathematics in English is proved to have posed considerable challenges for many Hong Kong students who enroll in an EMI (English as the medium of instruction) curriculum. Solving Math word problems in English, still less, poses even greater challenges due to the high language demand involved for ESL students to understand the verbal text and conduct problem solving. Studies have shown that students generally perform worse in solving Math word problems than in solving problems with purely mathematical form. And this difficulty is even more pronounced for English learners. To help Hong Kong ESL students enhance their Math problem-solving skill in English, the current study conducts a genre analysis of the Math word problems chosen from four EMI mathematics textbooks most widely used in junior secondary classrooms. The analysis is a corpus-based one consisting of a linguistic analysis of the language moves and linguistic features in verbal text and a visual analysis of the Math visual presentations as to how they work with verbal text in meaning making. In the study, the linguistic analysis identifies 6 language moves and move patterns commonly used in the word problems, with move-specific linguistic features also examined to help students better distinguish each move. Since linguistic features vary greatly among content areas, linguistic features associated with each content area are also examined to cater to content-based difference. On the other hand, the visual analysis identifies 5 types of visual presentations in the word problems examined. By quantitatively calculating the number of each visual type, the study identifies the dominant visual type(s) in different content areas and further investigates each visual type’s visual-linguistic relationship according to the extent to which these visual take on the role of verbal text in meaning making. With the above findings of the linguistic and visual analysis of Math word problems, the study generates important pedagogical implication for the EMI Math learning and teaching in the junior secondary context and contributes to the development of language across curriculum in Hong Kong schools. Limitations of the study are discussed at the end of dissertation, followed by a list of implications for future research. / published_or_final_version / Education / Master / Master of Education

The twenty-first century society demands a high level of mathematical literacy. This drove Korean educators to evaluate their students using international mathematics tests such as TIMSS, PISA and IMO. In these tests, Korean students ranked highly among the participating countries. Korean students, however, had done poorly in the application of mathematics in daily life situations as well as in their interest in mathematics in comparison to those of other countries. Based on these observations, the present study was an investigation on the contexts which Korean grade 8 to 10 students would prefer to deal with mathematics, in order to improve these weak points and thus increase their mathematical power. The aim of the study was to investigate mathematical literacy in connection with the relevance of mathematics and mathematical modelling. The study paid more attention to mathematics education in real life situations.

The problem of this study was to investigate and compare the current status of mathematics programs in large, medium, and small high schools in North Central Texas in relation to selected factors.

The problem of the study was to examine the effectiveness of using supplementary materials in the teaching of eighth grade mathematics. The study considered the effect that supplementary materials had on students in terms of the following variables--attitude toward mathematics, mathematical concepts, problem solving ability, understanding common fractions, computation of common fractions, decimal fractions and per cent, concepts of numbers and numerals (number system and operations), and concepts of numbers and numerals (decimal place value).

M.Ed. / It has become quite clear in recent times that media is expected to play an active role in improving the learning - teaching process in school. When the media is intergrated into a school lesson, the expectation is created that it would bring about a renewal and improvement in the teaching - learning process. A geometrical problem can be solved in a logical manner, helping to expand and enrich the pupils point of reference in Geometry. In the learning process the teacher, pupil and learning content are inextricably linked. It is in this process where media can, playing a very important role, act as intermediate link between the elementary and the fundamental. Furthermore, the media can be used to solve various geometrical theorems and problems making it accessible, visual and concrete. This would greatly benefit the Standard eight pupil. There are specific problems which arise in Geometry. These problems relate to certain skills, for example, discovery method, observation, direct experience and verification. It is in this regard that the teacher needs to be very selective in the choice of media, so that it encourages the mastering of the content. Media promotes learning by involving both hemispheres of the brain. By virtue of the subject, visual presentation in Geometry makes learning easier. It is in this regard that models are the most suitable medium due to the characteristics they posses. A number of guidelines for planning a Geometry lesson for Standard eight are suggested in this study. Recommendations are made regarding the content of a lesson, the production of suitable models to provide for the needs of teachers and pupils as well as the active involvement of pupils in the production of models for Geometry. The empirical reseach has shown that models can serve as a medium to unlock the learning content of Geometry to the pupil. This can be done by concretising and presenting Geometry on a level which enables the pupil to experience it's simplicity with optimal learning. The important role of models in the daily life of every person makes further reseach into the nature of symbol systems and how they influence learning essential, so that every pupil may fully benefit from the potential of models.

The National Curriculum statement, or NCS for short, proposes significant shifts in the way that teachers carry out their work. Strategies, such as investigation and collaborative work were promoted as a reform model for effective teaching and learning. Thus, the intention of this research project is to determine how mathematics teachers are implementing the new reform in South Africa. Based on Sfard’s theory of commognitive development, a theoretical framework has been established and the focus specified in the following research questions: 1. How does a teacher mediate instruction during object-level & meta-level learning? 2. What enables and constrains her/his facilitative mediation in the case of Congruency in Grade 9? 3. What can we learn about the practical efficacy of Sfard’s discourse theory? In order to explore the actual teaching process the research project presents a case study constituted from two teaching practices on one topic, Congruency, at a College in Johannesburg. The purpose of observing and interviewing two teachers on the same lessons is to get a greater variety of conversation on object-level and meta-level learning. At the same time analysing their teaching process in-depth creates an opportunity to have different possibilities of mediating collaborative learning. The study addresses the three research questions through two related activities – non-participant observation and semi-structural interviews with teachers (in order to provide an opportunity for teachers to express their opinion). Two main findings can be summarized: Firstly, the way the teacher manages instruction originates from her/his teaching style. The data analysis clearly confirms that mediation of the two teachers on the topic Congruency does not differ according object-level and meta-level learning, but according to the teachers. The second finding is related with Sfard’s theoretical perspective: on the one hand the Department recommends investigative activities, whilst, on the other hand, Sfards’ theory states that reinvention by the learner is highly unlikely. Therefore the practical efficacy of Sfard’s theory is that in meta-level learning investigative activities are not appropriate and the role of the teacher should be dominant, not necessarily as facilitator. This research study is an empirical proof of the validity of Sfard’s theory and unspecified requirements of the Department of Education. KEYWORDS: object-level learning, meta-level learning, Congruency, Commognitive theory.

A thesis submitted to the Wits School of Education, Faculty of Humanities, University of the Witwatersrand in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016. / This study explores the potential for Web-based diagnostic assessments in the classroom, with specific focus on certain common challenges experienced by learners in the development of their rational number knowledge. Two schools were used in this study, both having adequate facilities for this study, comprising a well-equipped computer room with one-computer-per-learner and a fast, reliable broadband connection. Prior research on misconceptions in the rational numbers has been surveyed to identify a small set of problem types with proven effectiveness in eliciting evidence of misconceptions in learners. In addition to the problem types found from prior studies, other problem types have been included to examine how the approach can be extended. For each problem type a small item bank was created and these items were presented to the learners in test batteries of between four and ten questions. A multiple-choice format was used, with distractor choices included to elicit misconceptions, including those previously reported in prior research. The test batteries were presented in dedicated lessons to learners over four consecutive weeks to Grade 7 (school one) and Grade 8 (school two) classes from the participating schools. A number of test batteries were presented in each weekly session and, following the learners’ completion of each battery, feedback was provided to the learner with notes to help them reflect on their performance. The focus of this study has been on diagnosis alone, rather than remediation, with the intention of building a base for producing valid evidence of the fine-grained thinking of learners. This evidence can serve a variety of purposes, most significantly to inform the teacher on each learners’ stage of development in the specific micro-domains. Each micro-domain is a fine-grained area of knowledge that is the basis for lesson-sized teaching and learning, and which is highly suited to diagnostic assessment. A fine-grained theory of constructivist learning is introduced for positioning learners at a development stage in each micro-domain. This theory of development stages is the foundation I have used to explore the role of diagnostic assessment as it may be used in future classroom activity. To achieve successful implementation into time-constrained mathematics classrooms requires that diagnostic assessments are conducted as effectively and efficiently as possible. To meet this requirement, the following elements of diagnostic assessments were investigated: (1) Why are some questions better than others for diagnostic purposes? (2) How many questions need to be asked to produce valid conclusions? (3) To what extent is learner self-knowledge of item difficulty useful to identify learner thinking? A Rasch modeling approach was used for analyzing the data, and this was applied in a novel way by measuring the construct of the learners’ propensity to select a distractor for a misconception, as distinct from the common application of Rasch to measure learner ability. To accommodate multiple possible misconceptions used by a learner, parallel Rasch analyses were performed to determine the likely causes of learner mistakes. These analyses were used to then identify which questions appeared to be better for diagnosis. The results produced clear evidence that some questions are far better diagnostic discriminators than others for specific misconceptions, but failed to identify the detailed rules which govern this behavior, with the conclusion that to determine these would require a far larger research population. The results also determined that the number of such good diagnostic questions needed is often surprisingly low, and in some cases a single question and response is sufficient to infer learner thinking. The results show promise for a future in which Web-based diagnostic assessments are a daily part of classroom practice. However, there appears to be no additional benefit in gathering subjective self-knowledge from the learners, over using the objective test item results alone. Keywords: diagnostic assessment; rational numbers; common fractions; decimal numbers; decimal fractions; misconceptions; Rasch models; World-Wide Web; Web-based assessment; computer-based assessments; formative assessment; development stages; learning trajectories.

A thesis submitted to the Faculty of Humanities, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy September 2015 / This study explores the mathematical discourse of Grade 11 learners on the topic function through their routines. From a commognitive perspective, it describes routines in terms of exploration and ritual. Data was collected through in-depth interviews with 18 pairs of learners, from six South African secondary schools, capturing a landscape of public schooling, where poor performance in Mathematics predominates. The questions pursued became: why does poor performance persist and what might a commognitive lens bring into view? With the discursive turn in education research, commognition provides an alternate view of learning mathematics. With the emphasis on participation and not on constraints from inherited mental ability, the study explored the nature of learner discourse on the object, function. Function was chosen as it holds significant time and weight in the secondary school curriculum. Examining learners’ mathematical routines with the object was a way to look at their discourse development: what were the signifiers related to the object and what these made possible for learners to realise. Within learners’ routines, I was able to characterise these realisations, which were described and categorised. This enabled a description of learner thinking over three signifiers of function in school Mathematics: the algebraic expression, table and graph. In each school, Grade 11 learners were separated into three groups according to the levels at which they were performing, from summative scores of grade 11 assessments, so as to enable a description of discourse related to performance. Interviews were conducted in pairs, and designed to provoke discussion on aspects of function and its signifiers between learners in each pair. This communication between learners and with the interviewer provided data for description and analysis of rituals and explorations. Zooming in and out again on these routines made a characterisation of the discourse of failure possible, which is seldom done. It became apparent early in the study that learners talked of the object function, without a formal mathematical narrative, a definition in other words, of the object. The object was thus vested in its signifiers. The absence of an individualised formal narrative of the object impacts directly what is made possible for learners to realise, hence to learn. The study makes the following contributions: first, it describes learners’ discursive routines as they work with the object function. Second, it characterises the discourse of learners at different levels of performance. Third, it starts exploration of commognition as an alternate means to look at poor performance. The strengths and limitations of the theory as it pertains to this study, are discussed later in the concluding chapter. Keywords commognition, discourse, communication, participation, routines, exploration, ritual, learners, learning, narratives, endorsed narratives, visual mediators.