Children’s failure to re-use knowledge will continue to be problematic until processes that contribute to conceptual growth are better understood. The notion that conceptual knowledge, soundly constructed and reinforced, forms the basis of future learning, as the learner uses it unproblematically to make sense of new situations in related areas, is appealing. This thesis will show this to be an overly simplistic view of learning, failing to take sufficient account of fine-grained processes that contribute to the micro-evolution of knowledge and of connections between cognition and other factors. Much previous research focused on abstraction as key to learning. This thesis examines the role of abstraction in the development of mathematics concepts by children aged 8-9 years, using negative numbers as a window on their development of knowledge in a new domain. The assumption, prevalent in the literature, that abstraction is a requirement for transfer of knowledge is questioned. Three research questions are explored: 1. What resources shape the nature of transfer and the growth of knowledge about negative numbers? 2. What is the role of the interplay of resources in the microtransfer of knowledge about negative numbers? 3. What is the relationship between abstracting and transferring knowledge about negative numbers? Methodology is based on a case study approach, initially recording the work of 3 small groups of children throughout a series of tasks and using progressive focusing techniques to create two case studies which are analysed in depth. The thesis reports how the extent of conceptual development about negative numbers was influenced by interpersonal and intrapersonal learner characteristics, and describes a complex interplay between cognitive and affective factors. Micro-transfer and intermediate abstractions, and reinforcement of the connections that these construct, are found to be crucial for conceptual growth, though abstraction is not a condition for transfer at the micro-level.

The aim of this study was to generate a model (or theory) that explains students’ concerns as they tackle non-routine mathematical problems. This was achieved by using the grounded theory approach as suggested by Glaser and Strauss (1967) and further developed by Glaser (1978; 1992; 1998; 2001; 2003). The study took place in the context of a problem-solving course offered at the undergraduate level. As methods of data collection, the study made use of the problem-solving rubrics (or scripts) that students generated during the course. Other sources of data included interviews with the students and observations in class. The model generated as a result of this study suggests that problem solving can be seen as a four-stage process. The process was labelled ‘solutioning’ and is characterised by students trying to resolve the following concerns: Generating knowledge; Generating solutions; Validating the results, and Improving the results. The model also makes reference to pseudo-solutioning as an alternative approach to solutioning. During pseudo-solutioning, instead of trying to resolve the concerns listed above, students focus on trying to satisfy the academic requirement to submit an acceptable piece of work. Thus, pseudo-solutioning can be seen as an important variation to solutioning. After presenting the model of ‘solutioning’, the study provides an illustration of how it can be used to describe students’ processes. This is done in set of case studies in which three problem-solving processes are considered. The case studies provide a view of how the model developed fits the data and serves to highlight relevant patterns of behaviour observable as students solve problems. The case studies illustrate how the concepts suggested by the model can be used for explaining success and failure in the processes considered. This study contributes to the study of problem solving in mathematics education by providing a conceptualisation of what students do as they try to solve problems. The concepts that the model suggests are relevant for explaining how students resolve their main concerns as they tackle problems during the course. However, some of these concepts (e.g., ‘reducing complexity’, ‘blinding activities’, ‘transferring’) may also be of relevance to problem solving in other areas.

The first mathematics course for approximately 53 percent of U.S. community college students is a developmental algebra course. Many such students appear to be severely debilitated by their previous encounters with mathematics. Due to numerous misconceptions that dictate against a traditional course, a "reform" beginning algebra course, with function as the unifying concept, was designed. Since there is little research on this population to justify such a approach, the key research question for this thesis becomes: Can adult students who arrive at college having had debilitating prior experiences with algebra acquire at least a process level understanding of function through appropriate instructional treatment? Answering this question provides crucial information for future curricular design in the area of developmental mathematics at the college level. The theoretical framework considers different aspects that make up the function concept, taking critical account of several current theories of multiple representations and encapsulation of process as object to build a view of function in terms of different facets (representations) and different layers (of development via procedure, process, object, and procept). Ninety-two students at four community colleges completed written function surveys before and after a "reform" beginning algebra course. Twelve students, representing all four sites, participated in task-based interviews. Comparison of pre- and post-course surveys provided data indicating statistically significant improvement in student abilities to correctly interpret and manipulate function machines, two-variable equations, two-column tables, two-dimensional graphs, written definitions and function notation. The students were divided into three categories (highly capable, capable, and incapable) based on their demonstrated understanding of function. Using the interviews, visual profiles for students in each category were developed. The profiles indicate that the development of the concept image of function in such students is complex and uneven. The cognitive links between facets is sometimes nonexistent, sometimes tenuous, and often unidirectional. The highly capable demonstrated some understanding across all facets while the incapable indicated understanding of the more primitive facets, such as colloquial and numeric, only. Profound differences were noted particularly in the geometric, written, verbal, and notation facets. Overall, the target population appeared able to develop a process layer understanding of function, but this development was far from uniform across facets and across students.

Recognising the powerful role that technology plays in the lives of people, researchers are increasingly focusing on the most effective uses of technology to support learning and teaching. Technology Enhanced Learning (TEL) has the potential to support and transform student learning and provides the flexibility of when, where and how to learn. At the same time, it promises to be an effective educational method (Wei and Yan 2009). One of the hottest topics in this field is adaptive learning (Mylonas, Tzouveli and Kollias 2004). Today, with the ability of advanced technologies to capture, store and use student data, it is possible to deliver adaptive learning based on student preferences. TEL can also put students at the centre of the learning process, which allows them to take more responsibility for their own learning. However, this requires students to be metacognitive so they can manage and monitor their learning progress. This thesis investigates the impact of student metacognitive skills on their learning outcomes in terms of recalling and retaining information within a formally designed and TEL environment. The learning outcomes of students who study a subject consistent with their learning styles and another group of students who study the same subject in contrast to their learning styles are then compared to determine which group performs better. Based on this approach, a TEL environment is designed for undergraduate students to use for the purpose of collecting the required experimental data. The results of this study suggest that effective use of metacognitive skills by students has a direct bearing on their learning performance and ability to recall information. The outcomes reveal that successful students use effective metacognitive skills to complete their studies and achieve their learning goals in a TEL environment. Therefore, it clear that metacognition can play a critical role in successful learning, and, furthermore, this approach can assist educationalists in understanding the importance of metacognition in learning and in considering how technology can be used to better to allow students to apply metacognitive skills. The designed TEL environment for this study can be utilised as a precursor to implement TEL environments that can be adapted to individual learning styles, and to support the development of metacognitive skills.

This study investigates the relationship between teacher’s presentation and children’s understanding of the number line within an English primary school that follows the curricular guidance presented within the National Numeracy Strategy (DfEE, 1999a). Following an exploratory study, which guided the development of a questionnaire, the preparation of a pilot study, and the initial investigation with the trainee teachers, the study was re-conceptualised to consider the way in which teachers within each year group of a primary school used the number line and the ways in which their children conceptualized and interpreted it. Using a mixed methodology, the theoretical framework of the study draws upon methods associated with case study, action research and ethnography and involved the use of questionnaires, teacher observations and interviews with selected children. Analysis of the questionnaire data is mainly through the use of descriptive statistics that lead to discussion on children’s embodiments of the number line, their interpretations of what it is and their accuracy in estimating magnitudes. The results of the study suggest that conceptualising the number line as a continuous rather than discrete representation of the number system that evolves for the notion of a repeated unit was less important than carrying out actions on the number line. It is suggested that this emphasis caused ambiguity in the way teachers referred to the number line and restricted understanding amongst the children that focused upon the ordering of numbers and the actions that could be associated with this ordering. The results also suggest that children’s conceptions of magnitude on a segmented 0 to 100 number line neither meet objectives specified within the National Numeracy Strategy nor confirm hypothesised models that suggest a linear or logarithmic pattern of accuracy. The number line is seen to be a tool but its use as a tool becomes limited because teachers, and consequently children, display little if any awareness of its underlying structure and its strength as a representation of the number system.

Thesis (Ph. D.) -- University of Tennessee, Knoxville, 2006. / Title from title page screen (viewed on Feb. 2, 2007). Thesis advisor: Vena M. Long. Vita. Includes bibliographical references.

Thesis (Ed.D.)--East Carolina University, 2009. / Presented to the faculty of the Department of Educational Leadership. Advisor: Marjorie Ringler. Title from PDF t.p. (viewed Apr. 23, 2010). Includes bibliographical references.

In recent years within the field of Mathematics Education, the role of technology has been an area of intense interest. Surprisingly the impact of technology use on assessment has been less considered. This thesis explores the differences between two high-stakes examinations, one where the use of Computer Algebra Systems (CAS) is required, and one where CAS is prohibited. Key questions in this comparison explore the extent to which CAS would trivialize current assessments, whether CAS-required assessments necessitate more high-level thinking, and whether CAS has a more pronounced impact upon certain topic areas. To address these questions, a content analysis methodology was adopted. Texts for comparison were questions from two examination bodies; the Australian VCAA board, which has a CAS-required examination, and the English MEI group, which has a CASprohibited examination. Test items [n=370] from VCAA and MEI examination papers covering 2009-2011 were categorised according to two criteria. Firstly according to the level of impact of CAS-use using the categories: CAS-Proof, CAS-Optional, CAS-Trivial and CAS-Essential. Secondly according to the level of conceptual difficulty using three levels Mechanical, Interpretive and Constructive based on a variant of Bloom’s Taxonomy. When comparing these CAS and non-CAS examinations, a similar distribution of questions across the levels of impact and cognitive difficulty scales was found, with the exception of calculus questions where a significantly larger proportion of questions in non- CAS examination were of a mechanical nature and considered CAS-Trivial. CAS offers the potential to enable a radical rewrite of school mathematics and of assessment practice. However in this study the impact of assumed CAS-use on the test items studied appeared to be quite restricted. Given the critical place of assessment in school mathematics, understanding the differences CAS-required and CAS-prohibited assessments in similar syllabi makes a useful original contribution to researching use of this technology.

The aim of this study was to explore how teachers use language in Chichewa medium and English medium mathematics teaching in standards 3 and 4 of selected primary schools in Zomba, Malawi. Chichewa is a local and national language whereas English is a foreign language yet the official language in Malawi. Chichewa is a language of instruction in standards 1to 4 whereas English is used from standard 5. Both Chichewa and English are subjects of study from standard 1. Issues investigated included: teacher understanding of the use of Chichewa or English in mathematics teaching; teachers' knowledge and use of mathematics vocabulary in Chichewa and in English; and teacher use of language in mathematics lessons. In this thesis, I develop a sociolinguistic approach to a study of teachers' perceptions and uses of language in mathematics teaching. I demonstrate how we can represent these perceptual structures using sociolinguistic tools and principles, which I use to study how 40 mathematics teachers linguistically organise and structure their teaching of mathematics. I adopt the position that teaching is fundamentally a language activity based on classroom communication activities which are fundamentally sociolinguistic in character, that sociolinguistic structures are dynamic and rational, yet exhibit a level of stability which results in diverse teacher dispositions gelling into conflicting tensions. I develop a theoretical base and iteratively explore this, evolving a description of how we might model what I call the sociolinguistic orientation of mathematics teachers. I construct theoretical, conceptual and methodological frameworks to enable me to study some of the underlying relationships among the tensions, teacher predispositions and the sociolinguistic environment in the classroom. I draw on a constructivist approach to mathematics education founded in Piagetian and Vygotskian theories and in particular draw on the concepts of coping strategies (Edwards and Furlong, 1987) to deal with the dynamics of classroom communications (Hills, 1969) which result in tensions in the use of language in mathematics teaching Pimm, 1987; Adler, 2001}. I begin by educationally, professionally and linguistically locating myself before moving on to looking at how we can understand communication in the mathematics classroom, the role of language in mathematics education with emphasis on bilingual mathematics education. I examine theories for understanding the interplay and interrelationship among teaching, communication, language use, and mathematics and bilingual classroom. Thereafter I look at the sociolinguistic roots of mathematics education in the Malawi Education System, identifying those areas where the current language policy in education does not consider the role of language in mathematics education. I draw heavily on sequential focus group discussion, interviews, tests and classroom observations and construct a perceptual model for the sociolinguistic orientation of 40 mathematics teachers towards use of Chichewa or English, and explore how these perceptions relate to the actual use of language in bilingual mathematics classrooms. To increase the validity of the data and findings, I used methodological and data triangulation. The findings of the study suggest that the sociolinguistic orientation of mathematics teachers relates to the linguistic nature of mathematics (the desire to teach the technical language as opposed to the ordinary language that pupils will easily understand), mystifying language policy in education (the inconsistency of language policy), dynamic classroom discourse (the multi-functions of language in the classroom) and inconsistent source of language for use in mathematics teaching (different competencies in language for teaching and learning among teachers, pupils and instructional materials). In addition, I illustrate how the teacher sociolinguistic orientation depends on whether the language of instruction is L1 or L2 which rest ideologically on code switching between Chichewa and English as well as marked difference in the patterns of language use between Chichewa and English medium mathematics lessons. The findings of the study can increase our understanding of the dynamics of mathematics classroom discourse by not only identifying more tensions in the use of language hut also the sources of these tensions. These might pave the way to find remedies to reduce the linguistic tensions in mathematics education. These findings imply that teachers need to be trained and supported in the use of language if they are to improve the teaching of mathematics. It is recommended that a programme he developed to train and orient teachers in the use of language in mathematics teaching, and to produce appropriate instructional materials that would assist teachers and pupils to use language effectively in mathematics.

This thesis describes an investigation into university students' manipulation of symbols in solving calculus problems, and relates this to other aspects such as drawing and interpretation of graphs. It is concerned with identifying differences between students who are successful with symbol manipUlation and those who are less successful. It was initially expected that the more successful would have flexible and efficient symbolic methods whilst the less successful would tend to have single procedures which would be more likely to break down. Krutetskii (1976) noted that more successful problem-solvers curtail their solutions whilst the less able are less likely to acquire that ability even after a long practice. This suggested a possible correlation between success and curtailment. An initial pilot study with mathematics education students at a British University showed that in carrying out the algorithms of the calculus, successful students would often work steadily in great detail, however, they were more likely to have a variety of approaches available and were more likely to use conceptual ideas to simplify their task. However, the efficiency in handling symbolic manipulation may not be an indication that the students are able to relate their computational outcome to graphical ideas. A modified pilot test was trialed at the Universiti Teknologi Malaysia before a main study at the same university in which 36 second year students were investigated in three groups of twelve, having grades A, B, C respectively in their first year examination. The findings of this research indicate that there is no significant correlation between ability and curtailment, but ability correlates with conceptual preparation of procedures where there is an appropriate simplification to make the application of the algorithm simpler. The more able students may have several flexible strategies and meaningful symbolic mathematical representations but these may not always relate to visual and graphical ideas. On the other hand the less able students are less likely to break away from the security of a single procedure and liable to breakdown in getting the solutions for the calculus problems.