<p>This thesis investigates grade seven learners perceived experiences in learning algebra.Things that learners do and say during algebra lessons and about algebra were investigated. The study was done at one of the previously disadvantaged schools in Cape Town, South Africa.The data were collected through observations, a questionnaire and interviews. Observations were made from the day the topic was started in two grade seven classes. Two different teachers taught the two classes. Focus group interviews were conducted, two group of learners, ten learners from each of the two classes were interviewed. Learners devised a number of strategies for solving problems related to sums and differences. The principal learning difficulties experienced by learners in algebra related to the transition from arithmetic conventions to those of algebra, the meaning of literal symbols and the recoginition of structures. It became obvious then that developing algebraic thinking is not necessarily dependent upon algebraic notation and that the presence of algebraic notation says little about the level of problem solving.</p>

This study was designed to determine the effects of two different schedule types on mathematics achievement in public high school students. The instruments used included the Texas Assessment of Knowledge and Skills, given annually to all students in grades 3 through 11, the Texas Algebra I end-of-course examination, given as a district option to Algebra I students, and student final course grades as determined by classroom teachers. The study compared students' performance in these three areas during the 2004-2005 academic year in one suburban school district in North Texas. The study considers the type of schedule, either traditional or 8-block, between students in teachers' classes who teach the same course on both schedules concurrently. This study also investigates a qualitative aspect by including a short opinion survey of teachers' perceptions regarding student academic performance, teacher satisfaction and retention, and the ability to accomplish curricular goals. Findings from this research suggest course schedule does not have significant effects on student academic performance as measured using analyses of covariance comparisons with a 0.05 alpha-level, leading to the conclusion that a particular course schedule does not adversely impact student performance on academic measures. However, in some comparisons conducted within the course of the research, statistically significant results emerged. Qualitative data generated from a survey of teacher perceptions regarding the benefits of the two scheduling types, traditional 50-minute verses alternating day 8-block, suggested teachers preferred a traditional schedule over that of a block schedule design. Most teachers who responded to the survey instrument expressed the perception that traditional daily meeting classes allowed their students to be more successful. Additional research into the effects of scheduling types on students academic performance are suggested and would include examining larger population samples, a narrower study of specific courses within the field of mathematics, or an expansion of the content areas explored to fields such as science, languages, or non-academic core subjects, including the fine arts.

The purpose of this research study is to describe and analyze the self-reported experiences of exemplary high school mathematics teachers who underwent personal and professional transformations in order to develop and use a standards-based, constructivist (SBC) teaching paradigm in their classrooms. These teachers were all past recipients of the Presidential Award for Excellence in Mathematics and Science Teaching (PAEMST), an award that required them to demonstrate that their mathematics instruction was rigorous in the manner described by the NCTM standards. The following research questions are addressed: (a) What are the paths SBC secondary mathematics teachers who received the PAEMST pursued to become highly effective?, (b) What obstacles and challenges did they encounter and how were these obstacles overcome?, and (c) What sustained them on their journeys? The research methodology used to be a narrative inquiry. Following a wide survey of PAEMST recipients, five volunteer participants were chosen for the study. Data were collected from each participant using a one-to-one interview and the written section of each participant's PAEMST application. A narrative was written for each participant describing the path they had followed to become a highly effective high school mathematics teacher. The narrative was sent to each participant, and a follow-up interview was conducted via telephone amending the narrative to reflect the participant's additions and deletions. From the five amended narratives, eight themes were identified: (a) influences; (b) education; (c) professional development; (d) NCTM standards; (e) teaching style: beginning, current, or end of a career; (f) obstacles; (g) personality traits and personal beliefs; and (h) student influence. Several of the themes were supported by previous research. However, this research study discovered two new findings. First, the five participants had common characteristics and beliefs: (a) belief in their students, (b) persistence, (c) belief that professional development is vital for teacher growth, and (d) passion about mathematics and about conveying that passion to their students. The second research finding pertained to the influence that their own students had on all of the five participants. All the participants purposely sought out their students' thoughts about the classroom curriculum and about the instruction they received. The teachers considered their students part of the classroom learning community, and they honored and acted on their input. Finally, in addition to describing the trajectory of five PAEMST winning teachers, this study offers recommendations for students studying to become high school mathematics teachers, teacher educators, and educational researchers. For these students, their teaching preparation courses need to be taught adhering to the four principles of learning: activity, reflection, collaboration and community. According to this research, the model of teacher preparation courses that emphasize the teaching of the above four principles using a traditional teacher-directed method does not prepare future mathematics teachers for the use of SBC teaching in their classrooms. Suggestions about further research are addressed.

This study is an attempt to determine how a course in general mathematics can satisfy some of the needs of youth. The purpose of this study is to find out how the course in general mathematics can be made to contribute effectively to pupil growth, and how well the present course is contributing to pupil growth.

This study has two purposes. The first is to compare the academic skills achievement in reading and mathematics of students participating in a performance contracting program with the academic skills achievement of students not participating in the performance-based program. The second is to determine whether those students participating in the performance contracting program show a significantly different level of achievement (higher or lower) than the control group approximately four months after the conclusion of the program.

Complex systems abound on this planet, in the composition of the human body, in ecosystems, in social interaction, in political decision-making, and more. Analytical methods allowing us to better understand how these systems operate and, consequently, to have a chance to intervene and change the undesirable behavior of some of the more pernicious systems have developed and continue to be enhanced via quickly changing technology. Some of these analytical methods are accessible by pre-college students, but have not been widely used at that level of education. Jay Forrester, the founder of one of the methodologies, System Dynamics (SD), used to study complex system behavior involving feedback, laments the lack of understanding of complex systems evident in short-sited decisions made by legislators -- global climate change and fiscal policies being cases in point. In order to better prepare future decision makers with tools that could allow them to make more informed decisions about issues involving complex systems efforts have been underway to increase pre-college teacher understanding of the SD method. The research described in this dissertation introduces the mathematics education community to the value of System Dynamics modeling in pre-college algebra classes, indicates a path by which a traditional mathematics curriculum could be enhanced to include small SD models as a new representation for elementary functions studied in algebra classes, and provides an empirical study regarding conceptual understanding of functions by students. Chapter 2 indicates the numerous beneficial learning outcomes that empirical studies have shown accompany model-building activities. Chapter 3 indicates the need for students to become familiar with complex systems analysis, how SD modeling (one method of complex systems analysis) aligns with the Common Core State Standards in Mathematics, and the work that has transpired over the past two decades using SD in K-12. Chapter 4 focuses on the importance of the concept of function in high school mathematics, some limitations of exclusive reliance on the closed form equation representation for mathematizing problems and the SD stock/flow representations of some of the elementary functions that are studied in algebra classes. Chapter 5 looks at the issues affecting two traditional teachers and the challenges they faced when trying to reintroduce SD modeling into their algebra classes. Chapter 6 explains the student component of the classroom experiment that was conducted by the teachers who are highlighted in Chapter 5. The analysis of the results of student model-building activities in the two classroom studies that are part of the third paper did not indicate a statistical difference between the two experimental groups and the two control groups. Many environmental and scheduling issues conspired to adversely affect the experiment. However, positive outcomes were evident from the two pairs of students who were videotaped while they built the final multi-function drug model, the final student lesson in the experiment. Research focused on student outcomes is needed to further assess the strengths and weakness of the SD approach for student learning in mathematics.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Signed on 10 September 2016 in Austin, Texas, USA. / This dissertation investigated the possible connection between the two constructs of metacognition and the strands of mathematical proficiency. Using a small convenience sample of Grade 12 learners at a public all-girls high school in Johannesburg, South Africa, the teacher/researcher examined if training in metacognitive knowledge influenced the learners’ conceptual understanding, procedural fluency, strategic competence and heuristic strategy use. Two instruments (pre-test and post-test) developed by the researcher and completed by the students were on-line assessments of metacognition utilising written self-reports of strategy use that occurred simultaneously with mathematical problem solving. Qualitative analyses were applied to the data and learners demonstrated declarative, procedural and conditional metacognitive knowledge in their explicit heuristic and content-specific strategies. The learners also demonstrated strategies which could be inferred from their working. The study showed evidence of mixed transferability of metacognitive training to mathematical proficiency. Questions for further research are provided. / LG2017

Master of Science in Mathematics Education 2017 / This study focused on mathematical teachers’ beliefs about teaching critical thinking and how these beliefs informed teaching mathematics in practice. Two sets of interviews were conducted that investigated what conceptions of critical thinking, teaching strategies and tasks, and assessment methods, teachers believed promoted critical thinking in their mathematical classroom. Qualitative methods of inductive analysis were used to analyse teachers’ responses based on grounded theory where categories were constructed and re- organised into themes. The study found that teachers had a limited understanding of critical thinking and their beliefs about teaching critical thinking skills in mathematics did not appear to be reflected in practice. These newly qualified teachers demonstrated that they need assistance in developing and using teaching strategies and tasks that promote critical thinking skills in mathematics. / MT 2018

A research report submitted to tbe Faculty of Education, UnIversity 'of the Witwatersrand, in partial fulfilment of requirements for the degree of Master of Education by coursework and research report Johannesburg, November 1994 / This study investigates the relationships between small-group work and ownership of mathematical kownledge in order to explicate relationships, an account of the interactions of a group of three std 7 pupils working together for a week on mathematical tasks in their mathematics classroom is presented. The nation of ownership in used to encompass both the elements of democracy and the development of knowledge which are expected to be produced by small-group work. The notion of ownership is elaborated through the research process. The study finds both beneficial and problematic aspects in the interractions of the group. Ownership is seen to be established, but in different ways for each pupil, because of inequalities within the group. A serious mathematical mistake is made and : [Abbreviated Abstract. Open document to view full version]

A Thesis submitted to the Faculty of Education, University of the Witwatersrand, Johannesburg in fulfilment of the requirements for the degree of Doctor of Philosophy / This researcn investigates success and failure in mathematics at the standard seven level in middle schools in the Bafokeng region. There is great concern expressed by parents, teachers and students about the high failure rate in this subject in the external examinations. Mathematics is seen to be important for securing good jobs in the platinum mines which are situated on the Bafokeng tribal lands. in general, greater efforts by teachers and students have made little impact on the public examination results. The present study undertakes ar xn-de th analysis of this problem in the anthropolog^/cal paradigm using the illuminatxve approach. Interviews and questionnaires are used for gathering data and triangulation techniques are used for verification and checking. Statistical correlation is applied to the variables but the study is not limited to numerical relationships between factors associated with scholastic success and failure. Two separate criteria of "success" are defined, namely the standard 7 mathematics mark and the performance of a ten per cent sample in each of five picture problems. The population xs composed of one educational unit: the one high school in Phokeng, capital of the Bafokeng, and the three middle schools which make up the catchment area of this high school. The number of students involved is approximately three thousand. Language and attitudes show strong positive correlations with both criteria of success. Other variables, like socio-economic status and degree of westernization are significantly related to the picture problem scores but not to the mathematics mark. Surprisingly, self-concept and embedded figures scores show no statistically significant relationship with either of the two criteria of success. In order to succeed in school mathematics a student must attain a high level of abstraction. This research does not suggest that the case is any different for Bafokeng students, buJj it attempts to explore the modes of perception, methods of concept formation and the development of appropriate schemata by which abstraction is achieved. Finally the Implications for teaching and learning of mathematics in school are discussed in the light of the findings of this research. This is the first study of its type, as far as can be ascertained, for Bophuthatswana schools. / Andrew Chakane 2018