Children’s failure to re-use knowledge will continue to be problematic until processes that contribute to conceptual growth are better understood.\ud 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.\ud Three research questions are explored:\ud 1. What resources shape the nature of transfer and the\ud growth of knowledge about negative numbers?\ud 2. What is the role of the interplay of resources in the microtransfer of knowledge about negative numbers?\ud 3. What is the relationship between abstracting and\ud transferring knowledge about negative numbers?\ud 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.\ud 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
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