An information processing approach to the investigation of mathematical problem solving at secondary and university levels
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.