Towards a foundation theory of learning and teaching mathematics

The thesis reflects the changes of the 60's in the learning and teaching of school mathematics, and develops a way of describing and interpreting the reactions of teachers to these changes. Early exploration which is reported considers changes in school mathematics and problems of learning and teaching mathematics, and draws heavily on experience. This exploration leads, to contact with the School Mathematics Project textbooks, which provide the main framework for expressing these problems in a more specific way and for reflecting on past events. The unfolding of these events using the School Mathematics Project textbooks goes through different stages. The first stage involves a study of the published textbooks, first drafts and reports of the School Mathematics Project, with particular reference to the approach adopted for topology, proofs and geometry. A look at this approach, together with that in two other modern textbooks, leads to the recognition of specific difficulties in teaching these aspects. The second stage involves discussions and dialogues with the School Mathematics Project authors and teachers, to illuminate aspects studied in the first stage and more general aspects related to the evolution of the 0-level syllabus and textbooks. The method by which the discussions were conducted, information obtained and interpreted was essentially a phenomenological and hermeneutical approach in which every participant was an autonomous interpreter. The reflection with authors revealed various descriptions and interpretations of past experiences and events. Although these were conflicting and pointed to different aspects of the School Mathematics Project textbooks, they suggested beliefs held by practising teachers as a major factor to be used for interpreting and describing the activities of teachers. Models of these activities are proposed and these models are thought to improve the description of the stability of the structures involved.