Abstract

It is widely recognized that educational research and theory should be motivated by the desire to continually improve the practice of teaching. However, bridging the divide between theoretical research outcomes and the practical constraints of classroom-based teaching has proved somewhat challenging. The involvement of teachers as the 'bridge-builders' between theory and practice could provide an effective mechanism for achieving this integration. The purpose of this study is thus to investigate whether the involvement of teachers in developing and implementing a theory-based teaching module would improve teaching practice in the classroom. A teaching module was collaboratively developed by a group of teachers for Grade 9 linear functions using: the principles of mathematical proficiency postulated by Kilpatrick, Swafford and Findell, ; the teaching phases formulated by van Hiele ; and the cognitive classification of classroom activities developed by Stein and Smith . This module was then taught to six Grade 9 classes by four teachers in one school in the Eastern Cape, South Africa over a period of 5 weeks. The effectiveness of the module, and its application in the classroom, was assessed in terms of: the extent to which theory could be used to inform the design and development of teaching materials; the efficacy of this teaching material in promoting teaching for mathematical proficiency; and the effects of extraneous influences on the usefulness of the module in teaching for mathematical proficiency. While the theoretical framework provided a sound basis for developing the teaching module, it was found that collaboratively transforming this theory into a teaching module for practical use in the classroom is certainly possible, but it requires considerable time and effort that practising teachers do not have. Developing the depth of understanding required for mathematical proficiency also takes time - a commodity often in short supply as teachers grapple with the demands of the curriculum. Teaching for mathematical proficiency is a layered process. It starts with thinking about an idea that is developed out of a related concept that then has a set of characteristic algorithms and actions which are learnt and performed in sequence. Building understanding in this way ends with a student being able to visualize and conceive the graph as a structure that can be described as if it were an object . This development of understanding is important for mathematical proficiency but is not necessarily easy. When teaching with the module, it was necessary to create an extra opportunity for students to use procedural knowledge and repetition in order to provide enough examples to help them see the link: between linear number patterns and linear graphs. Extraneous influences on teaching for mathematical proficiency were grouped into two categories - endogenous and exogenous influences. Endogenous influences were teacher related and included the attitudes, decisions and disposition of the teacher. Exogenous influences were more contextual and included teaching time available, curriculum, external assessments etc. Both of these influences were seen to affect teaching for mathematical proficiency, either promoting or inhibiting it. This research affirmed the central role that teachers play in teaching for mathematical proficiency. It is considered critical that research actively involve teachers in the evolution of mathematical theory. The development of an enabling environment for teachers will further enhance their capacity to teach for mathematical proficiency