Thesis (M.Ed.)--University of Hong Kong, 1980. / Includes bibliographical references (leaf 171-176). Also available in print.

The general performance of learners in mathematics in most secondary schools in South Africa is not encouraging. Statistics reveal that there is a general poor performance in mathematics and related science subjects especially in those schools located in previously disadvantaged communities. There is just a need for appropriate interventions of the significance of mathematics in present society which is now characterised by high information and technology thrust. The objective of the study was to investigate the impact of the tutorial intervention programme on the performance of a grade eleven mathematics class with a view to improving the mathematics performance of these grade 11 learners. The selection of the action research based method was premised on the idea that it would help the researcher to improve his tutoring practice as well as contributing to better learner performance. Data was collected through the use of qualitative research methods. These involved observation, the utilisation of focus group discussions of learners, and in-depth interviews with randomly selected learners and other mathematics colleagues. A careful mix of learners of different learning abilities was made so that high performers would assist those whose performance was lower. In each group of five at least two high performers were included. Findings of the study point to the effectiveness of the tutorial process, justifying its adoption on a long term basis. In general, it can be concluded that the tutoring strategies used by the researcher had a positive effect on the knowledge retention of the learners. This conclusion was arrived at from an observation of improved test results over time, the learners' views in the questionnaires as well as those of peer observers. / Magister Educationis - MEd

The purpose of this study was to learn about the underlying factors that might help to explain differences in performance and engagement among middle school girls in mathematics. The study employed a qualitative approach to observe and listen directly to the voices of middle school girls and their parents and math teacher as they reflected on their experiences and thoughts about the girls‟ performance in and long-term goals related to mathematics. My goal was to hear what forces were working in and around the girls that might lead them to engagement or disengagement with mathematics. Through the use of journals, interviews, and classroom observations, I collected data on six adolescent girls attending a middle school in a small New England city. The data collected were viewed through several lenses including the triads created by parent-student-teacher and the triads of “high-performing” and “low-performing” girls. Six themes emerged: factoring in the algebra class; finding seats; relating to the teacher; social networking and engagement; untangling performance and engagement; and structuring class. These themes helped to explain some of the differences between the girls‟ performance in and engagement with mathematics. In addition, they suggested that the concept of engagement was contextual and somewhat elusive. The study raised questions about where engagement was taking place (in school or out, in math or another class), whether it was a solitary endeavor or a social creation, and the complex relationship between engagement and performance. Further longitudinal work with girls and young women as they progress through school will be important to the understanding of how, why and when girls engage or disengage from the study of mathematics.

As mathematical modeling gains popularity in K-12 classrooms, it is important to define what this entails for both students and teachers. The following study reviews various definitions of mathematical modeling and how these definitions are relevant for middle grades (5-9) teachers. Following a professional development workshop on mathematical modeling, four middle school teachers expressed their views about teaching mathematics through modeling tasks. This study documents the teachers' perceptions of what it means to model with mathematics, which tasks are most appropriate for their students, and why this is important in each of their classrooms. Although the teachers varied in their views depending on the context and circumstances surrounding each modeling task, they agreed that mathematical modeling helps students build critical thinking skills and provides an opportunity to align mathematics concepts with engaging, realistic phenomena.

Response to Intervention (RTI) has primarily been used as an early intervention in the elementary grades to improve the reading of all students; however, in recent years, mathematics has been added to the program and this addition has not been systematically evaluated. Guided by Deno's problem-solving model, the purpose of this qualitative case study was to gain insight on how middle school mathematics teachers use the problem-solving process to design interventions for struggling students and to understand the strategies they used to implement the plan. The research questions addressed how the problem-solving method is used when creating and implementing interventions, as well as the impact of the intervention on student achievement. The first phase of data collection was a focus group interview with 6 middle school RTI teachers. A convenience sample of participants described how the problem-solving method was used in planning the RTI process. The focus group interview was recorded, transcribed, and coded to find common themes among the responses. Data regarding the RTI implementation, as well as associated instructional strategies, benefits, and challenges were discussed. The second phase of data collection came from mean mathematics state test data from a cohort group of middle school students in 2012, 2013, and 2014. Scores were compared to determine if there was an increase in the percentage of students who scored at levels 3-4, as well as a decrease in the level 1 and 2 scores. Inconsistent data on the state test did not support the findings of the focus group. Social change can be achieved through this RTI mathematics study by providing teachers with instructional strategies that cultivate the growth of academic confidence and achievement of all students in the general education classroom.

This study examined the different types of mathematical errors exhibited by primary level pupils in Brunei when working with fractions. In addition, the study examined pupils' attitudes towards the learning of fractions and investigated if there were gender differences among Bruneian pupils' performances with fractions and with their attitudes towards fractions. The study was longitudinal in nature and its two phases involved a single cohort of Primary 5 pupils followed through a full year period in four government-funded primary schools in Brunei Darussalam. Pupils' mathematical errors were assessed by means of researcher-developed paper-and-pencil tests, while pupils' attitudes towards the learning of fractions were measured by means of an adapted version of attitude questionnaire that has been used previously with Bruneian pupils. Guided by six research questions, a number of statistical analyses were carried out to ensure the validity and reliability of the instruments used. These included piloting and revising the instruments, the use of Cronbach's alpha with the items in the attitude questionnaire, and the calculation of the Pearson Product Correlation Coefficient between scales of the questionnaire. The data was analysed by calculating the percentages and means of occurrences of each type of error. Paired and independent sample t-tests were carried out in order to investigate gender differences in pupils' errors and the impact of further instruction on fraction at the P6 level, while the GLM test was administered in order to investigate if there were significant change in pupils' attitudes towards fractions from the pre- to the posttests. Qualitative information obtained through pupils' interviews, field notes and lesson observations was used to support the quantitative data. / The study revealed that though pupils' achievement in the post-test improved, their performances on fraction work remained generally unsatisfactory. Many pupils in the study continued to have difficulty with the basic operations on fractions and resorted to the use of keyword strategies in dealing with word problems. Despite the pupils' unsatisfactory performance in the diagnostic tests, they generally held very positive attitudes towards the learning of fractions. No significant gender differences were observed either in pupils' performance in working with fractions tasks nor with their attitudes towards the learning of fractions. The findings of this study also highlight a number of issues for mathematics teachers to consider when dealing with fractions, and the findings also have implications for the quality of the instructional activities provided by the teachers, for the impact of language transfer in the medium of instruction - that is, from Bahasa Melayu to English at the pupils' Primary 4 level- and for the quality of the teacher training program in Brunei.

The purpose of this research was to examine the possible associations between the perceived classroom environment of high school students, the level of mathematics anxiety that they possess, and their attitudes towards mathematics. This marks the first time that these three fields of research have been simultaneously combined. Data were gathered from 745 high school mathematics students in 34 classes in high schools in the Southern California area using three instruments: the What is Happening In this Class? (WIHIC) learning environment survey created by Fraser, McRobbie, and Fisher (1996), an updated version of Plake and Parker's (1982) Revised Mathematics Anxiety Ratings Scale WRS), and a mathematics version of selected scales from Fraser's (1981) Test of Science-Related Attitudes (TOSRA). This revised attitude instrument was called the Test of Mathematics-Related Attitudes (TOMRA). Using statistical methods, the three instruments were checked for internal consistency reliability, factor structure, and discriminant validity. The RMARS and WIHIC were both found to exhibit good reliability and factorial validity in mathematics classrooms in Southern California, while the TOMRA yielded two scales of the four a priori scales, Enjoyment of Mathematics Lessons and Normality of Mathematicians, which met reliability and factorial validity standards. Within-class gender differences were analysed using paired t-tests combined with a modified Bonferroni procedure and effect sizes. Between- student gender difference were investigated using MANOVA. Simple correlation and multiple regression analyses were performed to identify possible associations between the learning environment and anxiety/attitudes scales. Qualitative data were collected from interviews and inductive analysis was performed in order to refute or corroborate the quantitative findings. / Significant within-class gender differences were found in four areas of the learning environment (Student Cohesiveness, Task Orientation, Cooperation, and Equity), but no gender differences in attitudes were found. All four learning environment areas were perceived in a more favourable light by females than by males. Individual gender differences were similar, with a significant difference also being found in Teacher Support, as well as both types of mathematics anxiety, namely, Learning Mathematics Anxiety and Mathematics Evaluation Anxiety. In order to carefully identify the relationships between the classroom learning environment and mathematics anxiety, analyses were conducted for both factors of mathematics anxiety. While no association between the learning environment and Mathematics Evaluation Anxiety was found, there were significant associations between Learning Mathematics Anxiety and three areas of the learning environment: Student Cohesiveness, Task Orientation, and Investigation. Significant associations between the Normality of Mathematicians attitude scale and the learning environment scales Equity and Involvement were identified, while three areas of the learning environment (Investigation, Task Orientation, and Cooperation) had a significant relationship with Enjoyment of Mathematics Lessons. Qualitative data analyses confirmed relationships between anxiety, attitudes, and classroom learning environments. The data also suggest that the structure of the mathematical content is linked with the level of anxiety that high school students feel.

Syftet med min studie är att belysa elevers uppfattningar om vad det innebär att ha matematik. Delar av studien ägnas åt att lyfta fram uppfattningar som elever i behov av särskilt stöd, d.v.s. elever med åtgärdsprogram, har. Åtta elever intervjuades i en fenomenografisk studie om vad de lägger i begreppet "att ha matematik". Studien visar att det finns en uppfattning att det individuella arbetet är grunden för att ha matematik och att kommunikation mellan elever ses som ett avsteg från matematiken. Det kvantitativa formella matematikarbetet med läroboken betonas av eleverna. Men samtidigt finns också en uppfattning att matematik kan has i samlärande. När eleverna hjälper varandra utvecklas lärandet hos alla förutsatt att det finns ett matematiskt tänkande enligt denna uppfattning. Det finns också en uppfattning om att det existerar ett visst samband mellan den informella vardagsmatematiken och den formella skolmatematiken. Men utanför matematiklektionerna upplevs matematiken finnas endast när eleverna rent påtagligt räknar och mäter. De intervjuade eleverna med åtgärdsprogram i matematik tenderar ha en uppfattning som starkare betonar att matematik has individuellt i läroboken, helst utan hjälp och ofta med fokus på algoritmer. Att räcka upp handen för att söka hjälp och att samarbeta tycks därför för dessa elever inte vara eftersträvansvärt. Den informella och den formella matematiken uppfattas ha få beröringspunkter. Utanför läroboken finns inte så mycket matematik enligt intervjuade elever med åtgärdsprogram. Risken är att det gemensamma matematikarbete som framhålles i Lpo94 av eleverna inte uppfattas som matematiklärande. Eleverna kan därmed missa de lärtillfällen som bygger på lärande i kommunikation. Att endast arbete i läroboken räknas som att ha matematik torde inte underlätta för elever i behov av särskilt stöd att förstå matematikens roll i vardagen. Så ska vi med dessa elever arbeta vidare med deras formella lärande i läroboken eller ska vi i stället börja arbeta för att eleverna ska förstå kopplingen mellan den informella och den formella matematiken för att öka den matematiska förståelsen?

THESIS Pupils' problem solving strategies at lower and upper secondary level ABSTRACT Thesis Pupils' problem solving strategies at lower and upper secondary level deals with pupils' strategies of solving mathematical problems that we can observe at primary and secondary school. The theoretical part summarizes basic concepts in the field of mathematical problems and pupils' strategies. The main aim of the experimental part of the thesis is finding new knowledge from this field at the second level of Czech education. I focused primarily on tasks involving data, addiction and statistics, including the concept of a mathematical function that is deeply linked to these educational contents. In the experimental part, there are selected tasks, expected or discovered strategies, statistics of chosen strategies and the success of the solutions and strategies found, in some cases even the transcription of pupils' errors. Keywords Problem, word problem, assignment, strategy, error, analysis, success rate, solving problems.

Magister Educationis - MEd / The research explores the effects of Community of Enquiry practices on the teaching and learning of Mathematics and Literacy in two local primary schools. After the 1994 elections, both the government and education system changed in South Africa. With the introduction of Outcomes Based Education (OBE), critical outcomes that emphasized thinking and collaboration became a vital part of the curriculum. Soon after, the Education system adopted the National Curriculum Statement (NCS) and thereafter the Revised National Curriculum Statement (RNCS), which maintained these outcomes. The Curriculum and Assessment Policy Statement (CAPS) was introduced to the Foundation Phase in 2012 and to the Intermediate Phase in 2013 with the Critical Outcomes, (which emphasizes thinking) now stated as the aims of CAPS. However, no guidelines are provided regarding classroom practice. The approach to teaching these aims is not made clear. Lipman’s Philosophy for Children (P4C) is one way of working towards these aims, and promoting thinking and is consistent with many of Vygotsky’s ideas. He initiated ideas about cognitive development in which he refers to the importance of dialogue in which one is able to talk and communicate with others. Vygotsky also emphasised scaffolding where the teacher provides the learner with clues and suggestions in order to develop better problem- solving techniques and thinking habits. His concept of the zone of proximal development (ZPD) refers to the individual’s ability to accomplish more or to perform a challenging task with the proper assistance. The development of language is considered important within his theory as Vygotsky believes that individuals are born only with lower mental processes and develop their thinking ability (higher mental processes) by acquiring the thinking tools developed in a particular culture, the most important of which is language. The research followed a qualitative research methodology. The study explored the perceptions of both educators and learners after an intervention based on Philosophy for Children. Qualitative data involved two group interviews with teachers, one with the Cognitive Education Co-ordinator and interviews with four focus groups of selected Grade 5 and 7 learners (12 per group) whose teachers implemented Lipman’s Community of Enquiry pedagogy in the classroom the previous year. Quantitative data included a learner self-rating scale. All the educators of the two schools, who were involved in the classroom Community of Enquiry training, were invited to participate in the study, as were selected learners from the two Grade 5 and 7 classes at each school. I made use of thematic analysis of the interview data from both learners and teachers. Themes within the interviews were identified. Themes pertaining to teacher perception of self-change, teacher perception of learner change, and learner perceptions of self change were identified. During thematic analysis, the three research sub-questions were underlined. These were: (1) What are the teachers’ perceptions of self-change? (2) What are the teachers’ perceptions of learner change? (3) What are the learners’ perceptions of self-change? The conclusion of the study was that P4C has the potential to affect the teachers professionally and to influence the learners positively in Mathematics and Literacy classrooms. Ongoing support in cognitive education is vital in order to reach the aims required for the new CAPS curriculum.