Abstract | This thesis presents the findings of a research project which explored students’ learning during an activity-based mathematics programme. The research investigated what students learnt about solving linear equations and examined the role of activities in this learning.
The investigation of learning in the classroom was guided by the principles of naturalistic enquiry. A longitudinal study was used to investigate students’ learning during a unit of work that that made extensive use of activities and contexts. The longitudinal design of the study allowed the development of algebraic thinking to be investigated. The ideas of both Piaget and Vygotsky suggest that it is necessary to study the process of change in order to understand the thinking of students. A group of four students, two girls and two boys, were studied for twenty-seven lessons with each student interviewed individually within six days of each lesson, using the technique of stimulated recall. All lessons and interviews were recorded for subsequent transcription and analysis.
Learning to solve equations formally, using inverse operations, proved to be difficult for all the students. For two of them, their poor understandings of arithmetic structure and inverse operations were impediments that prevented them from doing more than attempt to follow procedures. Two of the students did succeed in using inverse operations to solve equations, but were still reasoning arithmetically. There was little evidence in the data that any of the students got to the point of regarding equations as objects to act on. They consistently focussed on the arithmetic procedures required for inverse operations. Even by the end of the topic the most able student, like the others, was still struggling to write algebraic statements. One of the most striking features of the results was the slow progress of the students. For at least two of the students, lack of prerequisite numeracy skills provided a good explanation of why this was so. However for the other two, poor numeracy did not appear to be a reason. The findings are, however, perhaps not too surprising. For children learning about arithmetic, the change from a process to an object view, from counting strategies to part/whole strategies, seems a particularly difficult transition to make. To move from a process to an object view of equations appears to be a similarly difficult transition.
The way in which the students made use of the contexts showed that the activities did not directly facilitate the students to develop an understanding of formal solution processes. The students did not usually make use of the contexts when solving equations, working at the abstract symbolic level instead. Although it was hoped that, by engaging students in meaningful activities, the students would construct understandings of formal solution processes, this did not occur. None of the activities used in the study provided a metaphor for the formal method of solving equations. It is suggested that, for a context to be of great value for teaching a mathematical concept, the physical activity should act as a metaphor for the intended mathematical activity. |