<p>The aim of this study was to investigate the hypothetical view that the use of familiar social institutions in the formulation of mathematical problems by mathematics pre-service teachers will enable them to find solutions to problems by taking meaning, context and realities of a problem into consideration. The approach to investigate this hypothetical view was to describe the mathematical behaviour of pre-service teachers in an application and modelling context. This study, therefore, described the strategies used to arrive at solutions for problems from real life situations that are familiar to the participants.</p>

Thesis submitted in fulfilment of the requirements for the PhD degree of the University of the Witwatersrand Johannesburg, 2016. / This study compares school based assessments to standardised national assessments in South Africa. The purpose of this research was to understand widespread evidence of a discrepancy in South Africa between the judgements of mathematical achievement made within internal school-based assessments (SBA) and external standardised national assessments (SNA). The study considers the possibility of a mismatch between the design feature rules of SBA and SNA and explores the nature of this discrepancy at a Grade 6 level in three different school settings (suburban, township, and inner-city). The context of the study was characterised by the aims espoused for assessment in South African national policy interpreted in terms of an activity theory-based nested conceptualization of motives, goals and operational activities. Specific interest was directed at how operations or design features at national policy document level come to function as “rules” for SNA and SBA activity systems. The study was then directed towards explanations and findings on discrepancies between advocated rules and espoused design feature rules interpreted from an analysis of artefacts in SNA and SBA. The key artefacts analysed in the study were the Annual National Assessment (ANA) designed by the Department of Basic Education and the internal SBA assessments designed by Grade 6 teachers. The study shows that motives and goals that advocate design rules are inferred from a curriculum implementation discourse, with less emphasis on psychological needs and curriculum standards. The enacted activities are considerably varied across school settings. This is evidenced by lower levels of coverage and range and scope of questions in the township school in comparison to the suburban and inner city schools, even when the local district is a protagonist. Across activity systems, comparative analysis of SNA and SBA artefacts showed that there were critical disjunctures leading to deviations from suggested design features, with the state driven ANA and district mediation tools unable to close the gap. The study concludes with notable theoretical and policy implications for thinking about Grade 6 SNA and SBA activity systems in South Africa.

The Mathematics Curriculum and Assessment Policy (CAPS) document defines ‘mathematics [as] . . . a human activity’ (DBE, 2011a, p.8). This adoption of a realistic approach to the learning and teaching of mathematics appears to be partial, however, in that at the entry point of the Intermediate Phase, the recommendations of the policy makers are read as prescriptions by practitioners. In particular, the recommendation that ‘as the number range for doing calculations increases up to Grade 6, learners should develop more efficient techniques for calculations, including using columns’ (DBE, 2011b, p.13) is taken as a prescription to push the standard methods as the way to solving (often de-contextualized) problems from the very start of Grade 4, in disregard to the admonition that ‘these techniques should only be introduced and encouraged once learners have an adequate sense of place value and understanding of the properties of numbers and operations’ (DBE, 2011b, p.13). In the background of reports that place South African schools well below international standards with regard to mathematics, with only a third of the learners in grade 3 having attained the minimum standard required of learners at their level in 2011, this report focuses on an exploration into the purported catalytic role that the emergent model of an empty number line can play in shifting learners’ attention from counting (calculation by counting ) towards a focus on the structural properties of number (calculation by structuring). The use of emergent models is meant to support and improve upon learners’ informal solution strategies whilst seeking to reverse what Freudenthal referred to as the “anti-didactical” use of models in a ‘top-down instructional design strategy in which static models are derived from crystallized expert mathematical knowledge’ (Gravemeijer and Stephan, 2002, p.146). With a particular focus on poor performance in numeracy, the Wits Maths Connect-Primary (WMC-P) project was established with the overarching aim of improving the learning and teaching of primary school mathematics. My investigation is located within one Grade 4 class in one of the WMC-P project schools, and in this project, I act as both the teacher of six intervention lessons focused on additive relation problems, as well as researcher of the models and strategies that learners use prior to the intervention lessons, within these lessons, and subsequently. This report presents evidence to illustrate, firstly, that at the entry point of grade 4 level, learners are highly dependent on concrete strategies for solving addition and subtraction problems, and secondly that with proper intervention, learners can make significant shifts towards more abstract calculation. On the one hand, the key finding that the majority of the problems were tackled using tallies in the pre-test confirms what research has observed regarding the tendency for learners to remain highly dependent on concrete strategies at grade 2 (Venkat, 2011) and grade 3 (Ensor et al., 2009). Also, the results indicate a high proportion of incorrect answers resulting from the use of the column model across all questions in the pre-test and the post-test. On the other hand, the imposition of the use of the empty number line in the delayed-post-test points to the fact that improvements can be achieved in relatively short time frames, and importantly, that these improvements can be retained beyond their immediate coverage in class.

Thesis (M.A.Ed.)--Chinese University of Hong Kong. / Bibliogarphy: leaves 51-53.

Thesis (M.A.)--Chinese University of Hong Kong. / Bibliography: leaves 56-59.

本論文透過兩項研究，探討家長與幼兒在家進行數字棋盤遊戲時，應如何與幼兒互動，以培養幼兒的數字感及學習數學的興趣。 / 研究1旨在檢視香港幼兒數字感的發展趨勢，以及不同級別幼兒參與數字棋盤遊戲的能力。此外，研究1會考察家長與幼兒進行數字棋盤遊戲時的表現，並探究其與幼兒數字感及學習數學的興趣的關連。因此，研究1收集51名香港三至六歲幼兒在八項數字感測試的得分，以及其學習數學的興趣水平，並邀請這些幼兒與其家長進行一個數字棋盤遊戲。研究結果發現，在不少數字感測試項目中，各級別幼兒之間都存有顯著的差異。一般而言，低班幼兒已具備參與數字棋盤遊戲的能力。當家長與幼兒進行數字棋盤遊戲，並輪至他們移動棋子時，他們很少邀請幼兒參與其中。雖然「辨讀棋子移動後所停留方格中的數字」、「辨讀棋子移動時所經方格中的數字」及「辨識棋子移動後所停留方格中的數字與其他數字的關連」三項行為在家長與幼兒互動時甚少出現，但有關行為的出現程度與幼兒部份數字感元素有著正向的關連。 / 研究2旨在了解當家長接受親子活動策略訓練後，家長定期與幼兒進行數字棋盤遊戲，能否培養幼兒的數字感及學習數學的興趣。因此，研究2以實驗形式，邀請四組共91位家長與其低班幼兒在家進行不同的親子活動 (經訓練的遊戲組、遊戲組、紙筆練習組，以及控制組)，然後比較各組幼兒在前後測的表現。研究結果發現，對數學能力非特別優異的幼兒而言，數字棋盤遊戲與紙筆練習兩者均能提升幼兒數字序列背誦及兩位數字認讀的能力。當家長接受訓練後，家長定期與幼兒進行數字棋盤遊戲，更能提升幼兒加法運算的能力。不論家長接受訓練與否，定期與幼兒進行數字棋盤遊戲能培養幼兒學習數學的興趣。 / 歸納以上研究結果，本論文顯示家長可透過與幼兒在家進行有關數字的活動，培養幼兒的數字感及學習數學的興趣。然而，不少家長與幼兒進行有關活動時，往往未有充分把握教學機會。因此，家長有需要接受有關方面的訓練。 / Through two related studies, this dissertation investigated how parents should interact with their preschool children during home numeracy activities, such as number board game playing, so as to promote children’s number sense and interest in mathematics. / In Study 1, I examined Hong Kong preschool children’s number sense development and their competence in playing number board game. Furthermore, I investigated whether parents had utilized number board game playing experiences to foster their preschool children’s number sense and interest in mathematics. The relations of the prevalence of various numeracy-related events during number board game playing to children’s number sense and interest in mathematics were also explored. Fifty-one Hong Kong three- to six-year-old preschool children were first tested on eight number sense tasks (including object counting, rote counting, missing number, numeral identification, numeral writing, numerical magnitude comparison, addition, and subtraction). Their interest in mathematics was also measured. Then, parents of these children were asked to play a number board game with their children in dyads. The parent-child interactions during the game were coded and analyzed. Results showed that third year kindergarteners (K3) performed better than second year kindergarteners (K2) on all number sense tasks except object counting and rote counting. K2 children performed better than first year kindergarteners (K1) on all number sense tasks except addition and subtraction. Usually, by K2, children could start to play number board games. When parents played number board game with their preschool children, they seldom elicited children’s participation during their turns to move the token. The events “announcing the final number“, “announcing the numbers passed through“, and “stating the numerical relations of the final number to other numbers only rarely occurred, though the prevalence of these events was positively associated with some components of children’s number sense (e.g., numeral identification, numeral writing, addition, and subtraction). / In Study 2, I investigated the extent to which offering a training program to parents on strategies of interacting with preschool children during number board game playing, together with regular playing of number board games with children, could promote children’s number sense and interest in mathematics. Ninety-one K2 children and their parents were recruited as participants; and an experimental design that involved the pre-test-post-test comparisons of four conditions (i.e., the game with a training condition, the game condition, the exercise condition, and the control condition) was used. Results showed that among children who were not very skilled in mathematics, number board game playing shared some similar educational benefits with completing mathematics exercises from books, such as promoting children’s rote counting and two-digit numeral writing skills. When parents received training, number board game playing could even promote children’s addition skills. No matter whether parents received training, number board game playing could promote children’s interest in mathematics. / Findings of these two studies suggest that home numeracy activities, such as number board game playing, can promote preschool children’s number sense and interest in mathematics. Nevertheless, parents may not be aware of the opportunities to incorporate numeracy into these activities. Therefore, training of parents is needed. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Cheung, Sum Kwing. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 195-211). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Acknowledgements --- p.iii / List of Tables --- p.vi / Abstract --- p.x / Chapter Chapter One --- Introduction --- p.1 / Chapter Chapter Two --- Study 1 --- p.47 / Method --- p.51 / Results --- p.62 / Discussion --- p.105 / Chapter Chapter Three --- Study 2 --- p.120 / Method --- p.127 / Results --- p.141 / Discussion --- p.173 / Chapter Chapter Four --- General Discussion --- p.184 / References --- p.195 / Appendix --- p.212

The primary aim of this study was to investigate the effects of two video-based interventions, one guided, the other non-guided, on pre-service early childhood education teachers' understanding of students' mathematical thinking. Five web-based lessons on various topics in children's mathematical development were created for this study. Each contained a short reading introducing a videotaped clinical interview of a young child performing a mathematical task. The unguided group then watched a 2-minute video, while the guided group watched the same video segmented into short clips and then answered open-ended questions at each break. The main goal was to examine the effectiveness of the use of videotaped clinical interviews in professional development. More specifically, I was interested in the types of experiences offered by the guided and unguided versions, as compared to those of the control group. The results of this study showed that both the guided- and unguided-video experiences were successful in changing the way prospective teachers interpreted children's mathematical thinking. While the results show it was possible to use videos to improve prospective teachers' interpretive abilities, it was not possible to improve their ability to apply the interpretations to developing appropriate teaching activities.

Education plays a vital role in any society; so much so, that countries strive to have not only adequate, but excellent educators in their classrooms. The aim of this study was to understand how beginning secondary mathematics teachers define success and to what experiences they attribute that success. Specifically, the central research question addressed was, "To what degree were significant attributes or experiences, important to the success of the first year teaching, learned pre-teacher education program, during a program, or post-program?" The practical goal of filling classrooms with great educators needs to be informed by research on how best to recruit highly qualified candidates into the field of mathematics education and how best to facilitate the teacher preparation process. This study employed a mixed methodology, using a sample of beginning secondary mathematics teachers to gather both quantitative and qualitative data on when they reported gaining influential knowledge or experiences. In particular, input from those who have had some success as beginning mathematics teachers was desired. The interview protocol designed for these participants added depth to the survey responses. Emphasis was placed on the relative importance of the three stages, pre-, during, and post-program, in developing common attributes associated with good teaching. Two characteristics were generally discussed as developing pre-program: being a self-starting and hard-working individual, and holding a belief that every student can learn. Beginning teachers viewed these traits as important for their success. Participants also felt that they acquired both practical classroom tools and educational theory from their teacher education program; having program instructors model pedagogy and mathematical instruction, and having opportunities to practice incorporating theory into their teaching were also seen as important. These aspects distinguished particularly prominent roles that the teacher education program played in shaping its graduates. Classroom management and being flexible and adaptive to different contexts were the most notable qualities frequently reported as being learned post-program. The study's results have implications for informing the types of students a mathematics education program should try to attract or recruit and defining areas where practicum or internship components might be incorporated into the teacher education process.

What is good mathematics teaching? The answer depends on whom you are asking. Teachers, researchers, policymakers, administrators, and parents usually provide their own view on what they consider is good mathematics teaching and what is not. The purpose of this study was to determine how beginning teachers define good mathematics teaching and what they report as being the most important attributes at the secondary level. This research explored whether there was a relationship between the demographics of the participants and the attributes of good teaching. In addition, factors that influence the understanding of good mathematics teaching were explored. A mixed methodology was used to gather information from the research participants regarding their beliefs and classroom practices of good mathematics teaching. The two research instruments used in this study were the survey questionnaire and a semi-structured interview. Thirty-three respondents who had one to two years of classroom experience comprised the study sample. They had graduated from a school of education in an eastern state and had obtained their teacher certification upon completing their studies. The beginning mathematics teachers selected these four definitions of good teaching as their top choices: 1) have High Expectations that all students are capable of learning; 2) have strong content knowledge (Subject Matter Knowledge); 3) create a Learning Environment that fosters the development of mathematical power; and 4) bring Enthusiasm and excitement to classroom. The three most important attributes in good teaching were: Classroom Management, Motivation, and Strong in Content Knowledge. One interesting finding was the discovery of four groups of beginning teachers and how they were associated with specific attributes of good mathematics teaching according to their demographics. Beginning teachers selected Immediate Classroom Situation, Mathematical Beliefs, Pedagogical Content Knowledge, and Colleagues as the top four factors from the survey analysis that influenced their understanding of good mathematics teaching. The study's results have implications for informing the types of mathematical knowledge required for pre-service teachers that can be incorporated into teacher education programs and define important attributes of good mathematics teaching during practicum.

Over the last few decades school accountability for student performance has become an issue at the forefront of education. The federal No Child Left Behind Act of 2001 (NCLB) and various regulations by individual states have set standards for student performance at both the district and individual public and charter school levels, and certain consequences apply if the performance of students in an institution is deemed unsatisfactory. Conversely, rewards come to districts or schools that perform especially well or make a certain degree of improvement over their earlier results. Albeit with certain conditions, the federal government makes additional education money available to the states under NCLB. While testing is nothing new in American public education, the concept of district/school accountability for performance is at least relatively so. In New York City, where New York State Regents Examinations (NYSRE) have been a measure of student performance for many years, scores on these tests are low, often preventing students from receiving course credit, which in turn results in failure to graduate on schedule. In addition, rates of graduation from public high schools are low. The city and state have kept data on student performance broken out by a number of factors including socioeconomic status, ethnicity, attendance, and gender which point to an achievement gap among different groups. This study investigates a series of those factors associated with the mastery of high school Algebra, Geometry, and Trigonometry. This study concerns itself specifically with the effect that gender, socioeconomic status, attendance, and ethnicity may have on student achievement in a mathematics course and on standardized tests, specifically the NYSRE, an annual rite of passage for students in grades 9 through 11. This research considered and ran tests on data gathered from a single large New York City high school. In this study, a 12 two-way (between-groups) univariate analyses of variance (ANOVAs) were conducted to assess whether there were differences in students' mathematics achievement scores by gender, ethnicity, attendance, and family socio-economic status (SES). In addition, three Pearson correlation analyses were conducted to determine whether there was a correlation among Integrated Algebra, Geometry, and Algebra II/Trigonometry unit examination scores and Regents scores. Nine Pearson correlation analyses were conducted to determine whether there was a correlation between Regents scores and mathematics achievement unit examination scores. A correlation was run between each mathematics achievement score with the Regents score from each subject. Six two-way (between-groups) ANOVA were also conducted to assess whether there were difference in students' mathematics achievement among Black males, Black females, Hispanic males, and Hispanic females. Data were gathered, merged, and transferred into a Statistical Package for the Social Sciences (SPSS) 19.0 (IBM, 2010) for analysis. The findings indicate that attendance and family SES have a meaningful relationship to mathematics achievement in the New York City public high school which was the subject of this investigation. On the other hand, gender and ethnicity showed no relationship to students' mathematics achievement. As an implication of this research, school policies must focus more on the achievement gap of students from low-SES families and must encourage students to maintain good attendance. Students should have access to different forms of academic interventions that go beyond after-school or Saturday tutoring; academic intervention services; community counseling or mediation; or peer intervention or peer counseling through which students learn basic mathematics skills from each other to achieve college readiness.