The mathematical knowledge of secondary pre-service teachers

The problem under investigation in this study is teachers' lack of competence to teach for conceptual understanding of mathematics. It was assumed in this study that pre-service teachers of mathematics go through their teacher education and training with certain insufficiencies in their mathematical understandings and that these insufficiencies will eventually affect the way they teach. The aim of this study, therefore, was to identify what these insufficiencies might be by examining mathematical knowledge that secondary preservice mathematics teachers bring with them to teacher education programs. The nature of these mathematical knowledge insufficiencies and how these insufficiencies would affect a person's competence to teach were of primary interest. To explore these knowledge insufficiencies, a multiple-case study design was used. The nineteen cases (secondary mathematics pre-service teachers) from four universities at two Australian states were selected according to their mathematical backgrounds. It was expected that the pre-service teacher participants (university graduates) who majored in mathematics or in other science related areas would show less evidence of knowledge insufficiencies than pre-service teachers who majored in other areas (e.g. economics). Furthermore, it was expected that the participants with mathematics major backgrounds would show more confidence to teach for conceptual understanding of mathematics than participants with mathematics minor backgrounds. The data collection instrument was a set of three mathematical stimulus items representing trigonometry, logarithm, and statistics. All three items were designed to elicit responses associated with the respondent's knowledge of the mathematics. Written and verbal responses to these items were collected in one-to-one interviews. Skemp's (1978) model of mathematical understanding was the instrument for data analysis. The results of this qualitative analysis indicated four types of mathematical knowledge deficiencies. In addition, the pre-service teachers' existing mathematical knowledge was highly representative of instrumental understanding of mathematics. These mathematical knowledge insufficiencies were suggested to be the outcomes of learning mathematical content which lacked in essential knowledge aspects, rather than outcomes of rote learning. Furthermore, these insufficiencies tended to reduce the pre-service teachers' confidence and likewise their potential to teach for conceptual understanding of mathematics.