This study explores the qualitative difference in performance between those who are more successful and those who are less successful in simple arithmetic. In the event that children are unable to retrieve a basic number combination the study identifies that there is a spectrum of performance between children who mainly use procedures, such as count-all in addition and take-away in subtraction, to those who handle simple arithmetic in a much more flexible way. Two independent studies are described The first contrasts the performances of children in simple arithmetic. It considers teacher selected pupils of different ability from within each year group from 7+ to 12+. It takes a series of snapshots of different groups of children and considers their responses to a series of simple number combinations. This first experiment shows qualitatively different thinking in which the less successful children are seen to focus more on the use of procedures and in the development of competence in utilising them. The more successful appear to have developed a flexible mode of thinking which is not only capable of stimulating their selection of more efficient procedures but, the procedures they select are then used in an efficient and competent way. However, the use of procedures amongst the more successful is seen to be only one of two alternative approaches that they use. The other approach involves the flexible use of mathematical objects, numbers, that are derived from encapsulated processes. The below-average children demonstrate little evidence of the flexible use of encapsulated processes. It is the ability of the more able children to demonstrate flexibility through the use of efficient procedures and/or the use of encapsulated processes that stimulates the development of the theory of procepts. This theory utilises the duality which is ambiguously inherent in arithmetical symbolism to establish a framework from which we may identify the notion of proceptual thinking. The second study considers the development of a group of children over a period of nearly a year. This study relates to aspects of the numerical component of the standardised tests in mathematics which form part of the National Curriculum. It provides the data which gives support to the theory and provides evidence to confirm the snap shots taken of children at the age of 7+ and 8+. It indicates that children who possess procedural competence may achieve the same level of attainment as those who display proceptual flexibility at one level of difficulty but they may not possess the appropriate mental tools to cope with the next. The evidence of the study supports the hypothesis that there is a qualitative difference in children's arithmetical thinking.

This thesis consists of a theoretical building of a cognitive approach to the calculus and an empirical testing of the theory in the classroom. A cognitive approach to the teaching of a knowledge domain is defined to be one that aims to make the material potentially meaningful at every stage (in the sense of Ausubel). As a resource in such an approach, the notion of a generic organiser is introduced (after Dienes), which is an environment enabling the learner to explore examples of mathematical processes and concepts, providing cognitive experience to assist in the abstraction of higher order concepts embodied by the organiser. This allows the learner to build and test concepts in a mode 1 environment (in the sense of Skemp) rather than the more abstract modes of thinking typical in higher mathematics. The major hypothesis of the thesis is that appropriately designed generic organisers, supported by an appropriate learning environment, are able to provide students with global gestalts for mathematical processes and concepts at an earlier stage than occurs with current teaching methods. The building of the theory involves an in-depth study of cognitive development, of the cultural growth and theoretical content of the mathematics, followed by the design and programming of appropriate organisers for the teaching of the calculus. Generic organisers were designed for differentiation (gradient of a graph), integration (area), and differential equations, to be coherent ends in themselves as well as laying foundations for the formal theories of both standard and non-standard analysis. The testing is concerned with the program GRADIENT, which is designed to give a global gestalt of the dynamic concept of the gradient of a graph. Three experimental classes (one taught by the researcher in conjunction with the regular class teacher) used the software as an adjunct to the normal study of the calculus and five other classes acted as controls. Matched pairs were selected on a pre-test for the purpose of statistical comparison of performance on the post-test. Data was also collected from a third school where the organisers functioned less well, and from university mathematics students who had not used a computer. The generic organiser GRADIENT, supported by appropriate teaching, enabled the experimental students to gain a global gestalt of the gradient concept. They were able to sketch derivatives. for given graphs significantly better than the controls on the post-test, at a level comparable with more able students reading mathematics at university. Their conceptualizations of gradient and tangent transferred to a new situation involving functions given by different formulae on either side of the point in question, performing significantly better than the control students and at least as well, or better, than those at university.

This research study investigates the concept of function developed by a sample of secondary and university students in England and Malaysia studying mathematics as one of their subjects. It shows that whilst students may be able to do the 'mechanical' parts of this concept, their grasp of the 'theoretical' nature of the function concept may be tenuous and inconsistent. The hypothesis is that students develop 'prototypes' for the function concept in much the same way as they develop prototypes for concepts in everyday life. The definition of the function concept, though given in the curriculum, proves to be inoperative, with their understanding of the concept reliant on properties of familiar prototype examples: those having regular shaped graphs, such as x2 or sinx, those often encountered (possibly erroneously), such as a circle, those in which y is defined as an explicit formula in x, and so on. The results of the study in England revealed that even when the function concept was taught through the formal definition, the experiences which followed led to various prototypical conceptions. Investigations also show significant misconceptions. For example, three-quarters of a sample of students starting a university mathematics course considered that a constant function was not a function in either its graphical or algebraic forms, and three quarters thought that a circle is a function. The extension of the study in Malaysia was made with the hypothesis that there is a significant difference between the concept as perceived to be taught and as actually learned by the students. Although the intended curriculum emphasises conceptual understanding, in the perceived curriculum (curriculum as understood by the teachers), only 45% of the teachers follow this approach. The tested curriculum as reflected in the public examination questions, only emphasises the procedural skills and the results of the learned curriculum show that learning of functions is more consistent with the theory of prototypical learning. Students in Malaysia develop their own idiosyncratic mental prototypes for the function concept in much the same way as those students in the UK.

The fragmentation of strategies that distinguishes the more successful elementary grade students from those least successful has been documented previously. This study investigated whether this phenomenon of divergence and fragmentation of strategies would occur among undergraduate students enrolled in a remedial algebra course. Twenty-six undergraduate students enrolled in a remedial algebra course used a reform curriculum, with the concept of function as an organizing lens and graphing calculators during the 1997 fall semester. These students could be characterized as "victims of the proceptual divide," constrained by inflexible strategies and by prior procedural learning and/or teaching. In addition to investigating whether divergence and fragmentation of strategies would occur among a population assumed to be relatively homogeneous, the other major focus of this study was to investigate whether students who are more successful construct, organize, and restructure knowledge in ways that are qualitatively different from the processes utilized by those who are least successful. It was assumed that, though these cognitive structures are not directly knowable, it would be possible to document the ways in which students construct knowledge and reorganize their existing cognitive structures. Data reported in this study were interpreted within a multi-dimensional framework based on cognitive, sociocultural, and biological theories of conceptual development, using selected insights representative of the overall results of the broad data collection. In an effort to minimize the extent of researcher inferences concerning cognitive processes and to support the validity of the findings, several types of triangulation were used, including data, method, and theoretical triangulation. Profiles of the students characterized as most successful and least successful were developed.Analyses of the triangulated data revealed a divergence in performance and qualitatively different strategies used by students who were most successful compared with students who were least successful. The most successful students demonstrated significant improvement and growth in their ability to think flexibly to interpret ambiguous notation, switch their train of thought from a direct process to the reverse process, and to translate among various representations. They also curtailed their reasoning in a relatively short Period of time. Students who were least successful showed little, if any, improvement during the semester. They demonstrated less flexible strategies, few changes in attitudes, and almost no difference in their choice of tools. Despite many opportunities for additional practice, the least successful were unable to reconstruct previously learned inappropriate schemas. Students' concept maps and schematic diagrams of those maps revealed that most successful students organized the bits and pieces of new knowledge into a basic cognitive structure that remained relatively stable over time. New knowledge was assimilated into or added onto this basic structure, which gradually increased in complexity and richness. Students who are least successful constructed cognitive structures which were subsequently replaced by new, differently organized structures which lacked complexity and essential linkages to other related concepts and procedures. The bits and pieces of knowledge previously assembled were generally discarded and replaced with new bits and pieces in a new, differently organized structure.

This thesis is concerned with students' understanding of the core concept of function which cannot be represented by what is commonly called the multiple representations of functions. The function topic is taught to be the central idea of the whole of mathematics. In that sense, it is a model of mathematical simplicity. At the same time it has a richness and has mathematical complexity. Because of this nature, for students it is so difficult to grasp. The complexity of the function concept reveals itself as cognitive complications for weak students. This thesis investigates why the function concept is so difficult for students. In the Turkish context, students in high school are introduced to a colloquial definition and are presented with four different aspects of functions, set-correspondence diagrams, sets of ordered pairs, graphs and expressions. The coherency in recognizing these different aspects of functions by focusing on the definitional properties is considered as an indication of an understanding of the core concept of function. Focusing on a sample of a hundred and fourteen students, their responses in the questionnaires are considered to select nine students for individual interviews. The responses from these nine students in the interviews are categorized as they deal with different aspects of functions. The data indicates that there is a spectrum of performance of students. In this spectrum, responses range from the responses which handle the flexibility of the mathematical simplicity and complexity to the responses which are cognitively complicated. Successful students could focus on the definitional properties by using the colloquial definition for all different aspects of functions. Less successful students could use the colloquial definition for only set-correspondence diagrams and sets of ordered pairs and gave complicated responses for the graphs and expressions. Weaker students could not focus on the definitional properties for any aspect of functions.

AIM To investigate mathematically able adolescents' conceptions of the basic notions behind the Calculus: infinity (including the infinitely large, the infinitely small and infinite aggregates); limits (of sequences, series and functions); and real numbers. To observe the effect, if any, on these conceptions, of a one year calculus course. EXPERIMENTS Pilot interviews and questionnaires helped identify areas on which to focus the study. A questionnaire was administered to Lower Sixth Form students with 0-level mathematics passes. The questionnaire was administered twice, once in September and again the following May. The A-level mathematicians had received instruction in most of the techniques of the Calculus by May. Interviews, to clarify ambiguities, elicit reasoning behind the responses and probe typicality and atypicality, were conducted in the month following each administration. A second questionnaire, an amended version of the first, was administered to a larger but similar audience. The responses were analysed in the light of hypotheses formulated in the analysis of data from the first 5ample. PRINCIPAL FINDINGS Subjects have a concept of infinity. It exists mainly as a process, anything that goes on and on. It may exist as an object, as a large number or the cardinality of a set, but in these forms it is a vague and indeterminate form. The concept of infinity is inherently contradictory and labile. Recurring decimals are perceived as dynamic, not static, entities and are not proper numbers. Similar attitudes exist towards infinitesimals when they are seen to exist. Subjects' conception of the continuum do not conform to classical or nonstandard paradigms. Convergence / divergence properties are generally noted with infinite sequences and functions. With infinite series, however, convergence / divergence properties, when observed, are seen as secondary to the fact that any infinite series goes on indefinitely and is thus similar to any other infinite series. The concept that the hut is the saue type of entitiy as the finite tens is strong in subjects' thoughts. We coin the term generic hiuit for this phenomenon. The generic limit of 0.9, 0.99, is 0.9, not 1. Similarly the reasoning scheme that whatever holds for the finite holds for the infinite has widespread application. We coin the term generic law for this scheme. Many of the phrases used in calculus courses (in particular hut, tends to, approaches and converges) have everyday meanings that conflict with their mathematical definitions. Numeric/geometric, counting/measuring and static/dynamic contextual influences were observed in some areas. The first year of a calculus course has a negligible effect on students conceptions of limits, infinity and real numbers. IMPLICATIONS FOR TEACHING On introducing limits teachers should encourage full class discussion to ensure that potential cognitive obstacles are brought out into the open. Teachers should take great care that their use of language is understood. A-level courses should devote more of their time to studying the continuum. Nonstandard analysis is an unsuitable tool for introducing elementary calculus.

This thesis has two integrated components, one theoretical and one investigative. The theoretical component considers human reason about categories of objects. First, it proposes that the standards of argumentation in everyday life are variable, with emphasis on direct generalisation, whereas standards in mathematics are more fixed and require abstraction of properties. Second, it accounts for the difficulty of the transition to university mathematics by considering the impact of choosing formal definitions upon the nature of categories and argumentation. Through this it unifies established theories and observations regarding student behaviours at this level. Finally, it addresses the question of why Analysis seems particularly difficult, by considering the relative accessibility of its visual representations and its formal definitions. The investigative component is centred on a qualitative study, the main element of which is a series of interviews with students attending two different first courses in Real Analysis. One of these courses is a standard lecture course, the other involves a classroom-based, problem-solving approach. Grounded theory data analysis methods are used to interpret the data, identifying behaviours exhibited when students reason about specific objects and whole categories. These behaviours are linked to types of understanding as distinguished in the mathematics education literature. The student's visual or nonvisual reasoning style and their sense of authority, whether "internal" or "external" are identified as causal factors in the types of understanding a student develops. The course attended appears as an intervening factor. A substantive theory is developed to explain the contributions of these factors. This leads to improvement of the theory developed in the theoretical component. Finally, the study is reviewed and the implications of its findings for the teaching and learning of mathematics at this level are considered.

Thesis (Ph. D.)--University of Toronto, 2005. / Includes bibliographical references (leaves 125-134).

Thesis (Ph.D.)--University of Tulsa, 1989. / Bibliography: leaves 103-107.

Thesis (Ph. D.)--Michigan State University. Dept. of Teacher Education, 2008. / Title from PDF t.p. (viewed xxx). Includes bibliographical references (p. 268-273). Also issued in print.