This research is devoted to focussing on the influence of different learning style on the performance of undergraduate students in various parts of calculus. In carrying out the study, calculus materials were classified into four main categories (Z4,Z5,Z6,Cals) and, for the Iranian students, the results of their mathematical performance in the university entrance examination is labelled (En) to identify their grounding in high school mathematics at the beginning of the calculus course in higher education. Also, in the present study, students' performance (weakness) in the manipulation of mathematical notation and logical discussion is called (Z1) category and (Cal) indicates students' total achievement in calculus examination which is, in fact, the students' performance on the combination of the categories (Z4,Z5,Z6). These calculus categories are described in Chapter 5. However in short term, multi-conceptual and procedural tasks are classified as (Z4). The (Z5) category is defined as the translation processes between mathematical abstraction (analytic/symbolic) and (pictorial/visual) forms of calculus materials. Moreover, multi-skilled, transferable and procedural skills are labelled as (Z6) category. It should be noted that these categories are interrelated in a scheme to exhibit activities in calculus. 572 students participated in the experimental part of this study and were selected from two Iranian universities (Sabzvar University and Mashhad University) and Glasgow University in Scotland, U.K. During the period of the study, the samples of students were subjected to some psychological tests in order to assign their Field-dependent/Field-independent and Convergent/Divergent learning styles. It was found throughout the study that the most effective combination of learning styles which emerged from the interacting picture of all the psychological factors used in the research, were field-independent/convergent (F1+Con) in Iran, and field-independent/divergent (FI+Div) in Scotland in performing on the calculus. On the other hand, the combination of field-dependent and convergent styles (FD+Con) could lessen achievement in calculus by mathematics/physics students, and field-dependent and divergent styles (FD+Div) would lessen attainment in calculus by engineering students. In addition, when the mean scores in calculus categories were calculated for various groups of students with different learning styles, the convergent thinkers (Con) were found to be best in (Z6), while divergent thinkers (Div) exhibited higher performance in (Z5). These findings demonstrate that the Con/Div way of thinking is the most effective in influencing performance in different areas of calculus, the FI/FD factor takes the second position. All these findings have been combined to form a model which emerges at the end of this thesis. Moreover, in Chapters 3 and 4, a comparison is made between calculus in secondary (high school) and higher education in Iran and Scotland, focussing on content, teaching order, learning objectives and teaching methods.

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 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.

This thesis contains ten chapters: three of them are background literature and five have resulted from practical work during the whole period of the research. Chapter 9 is an attempt to extend the idea of the demand of a task, while the last chapter contains conclusions and suggestions for further research. In Chapter 1, the theories of Piaget, Gagne and Ausubel are described and compared with each other. Piaget's stages of intellectual development and how learning processes take place are described and explained. The contribution of the theory in the domains of curriculum, teaching Piagetian tasks as subject matter and matching instruction to development stages is stressed. However, the serious challenges to the theory are (i) horizontal decalage phenomenon, (ii) relating stages with age, (iii) assessing competence and readiness. Gagne's model of an hierarchy of learning comes from theories of transfer. It is built from the top down. The conditions of learning are internal and external and ranged from signal learning to problem solving. The learning process is based on associational chains. The difficulty of the model comes from the nature of a learning hierarchy and its validation. Ausubel's theory of meaningful learning is based on what the learner already knows. It is built up from seven elements which range from meaningful learning to the advance organizer. Meaningful learning occurs as a result of interaction between new and existing knowledge and its variation is due to the growth of differentiation and integration of relevant items in cognitive structure. Failure in learning may occur in situations such as those of conflicting ideas and forgetting. In Chapter 2, Information Processing Theories of Learning are described and the justification of these theories as a fourth paradigm to guide thinking about research is stressed. A model of human memory is given and the components of memory and their features are listed. Stress is placed upon the memory processes and their levels, organization of knowledge, working memory and chunking as a remedy for overload. Two examples of these theories are given namely Neo-Piagetian Theory and the Predictive Model of Holding-Thinking Space. The main goal of the former is to make Piaget's theory functional not just structural. The latter relates performance to the amount of information to be processed in learning and problem solving. This model is applied in both University and Algerian samples. This can be found in Chapter 3. In Chapter 4, the field dependent-independent cognitive style is considered as an important factor affecting performance. The differences between field dependent-independent people may be related to the perceptual field, selected information and the level of guidance. The reason for these differences may be due to the way in which information is both analysed and represented in memory. The practical work has been done with both University and Algerian samples. In Chapter 5, some other factors are described. Most of them are concerned directly with the subject matter. The activities involved in learning mathematics are classified and attention is given to Polya's version of heuristic strategies. The concept of understanding is considered as a basic goal of education and its meaning is given in three different aspects. Most attention is given to the third one, which is known as alternative framework or misconception. The levels of understanding of Skemp are defined and their goals are stressed. The causes of learning difficulties in mathematics are listed, while the different forms of mathematical language are described and their effect on learning is noted. In Chapter 6, the analysis of Paper I (multiple-choice questions) has been done for preliminary Examination of four Scottish schools (a fifth school used only traditional questions). The experimental work is concerned with language, formulation and type of question.