Abstract:

Teachers’ knowledge of the mathematical content is one of the most important factors influencing how mathematics is taught and consequently on what students learn. However, there is a lack of consensus on the critical knowledge a teacher must have that ensures the students learn mathematics effectively. Teachers’ knowledge has many domains and categories. There is a growing need to explore how the critical components of teachers' knowledge can be identified, and how these components affect what teachers do in classrooms.
In this paper we explore the nature of knowledge of mathematics for teaching of functions for pre-service secondary mathematics teachers. We conducted task based interviews which involved a variety of non-routine mathematical questions. These tasks were designed to identify how the participants use different representations for various types of functions and connections among these representations. We also explored the flexibility and fluency of the participants in moving from one representation to another. The interviews were audio taped and we used the Interclipper® software for the qualitative analysis of these interviews. This included checking the correctness of their solutions and analyzing the thought process involved in problem solving as well as their ability to explain the solution method.
One of the findings of our study was that the participants’ knowledge was lacking in many essential components. Even the components of knowledge they possessed were not well integrated. The study also indicated that they were unable to ascertain the extent of their knowledge; consequently they did not try to expand their knowledge. It is quite plausible that they would pass on this incomplete knowledge to their students as well. Some other themes that emerged out of this study were:
• The natural curiosity or inquisitiveness, to explore different avenues while attempting a problem, seemed to be lacking among the participants.
• They were not accustomed to dealing with unfamiliar and open-ended problems. This discomfort with non-routine problems stems from their segmented viewpoint of mathematical knowledge, as well as their inability to think globally.
• They did not use formal language or precise mathematical terms while discussing the questions. Also they did not realize the importance of writing or presenting a solution of a problem in a structured, coherent and stepwise way.
• These pre-service teachers did not have a sense of belonging to the world of mathematics. In their view, mathematics is a world of theories, concepts, rules, and procedures that has been conceived by the mathematicians, in which they and their students are just outsiders.

Keywords:

teacher? knowledge, teacher? education, pedagogy