The nature of pre-service secondary mathematics teachers' knowledge of mathematics for teaching of functions
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In this grounded theory study I explored the nature of knowledge of mathematics for teaching of five pre-service secondary mathematics teachers. Task based interviews were conducted which involved a variety of non-routine mathematical questions requiring the participants to identify and use different representations of various types of functions and connections among these representations. Through the interviews I explored these questions: What is the nature of knowledge of mathematics for teaching of functions among pre-service secondary mathematics teachers? How and how well can these participants use different representations of functions, and identify and use connections between these representations? How flexible and fluent are these participants in moving from one representation to another? How and how well can these participants deal with various types of functions and various properties of functions? What is the nature of the pedagogical and instructional decisions taken by these participants as they prepare to teach about functions? Although the participants differed in the nature of their knowledge structures, but the main theme that emerged out of this study is that the knowledge of these pre-service secondary mathematics teachers was incomplete. Many pieces of knowledge were missing for most of these participants, and the pieces of knowledge that they did possess were sometimes not integrated. Some of the participants in this study were unable to identify and use different representations of functions or move flexibly among different representations. These participants were not accustomed to dealing with unfamiliar and open-ended problems. They did not use formal language or precise mathematical terms while discussing the questions. Most of them seemed to favor informal language, and seemed uncomfortable in using mathematical language. They seemed to view mathematics as a collection of theories, concepts, rules, and procedures that is developed by the mathematicians, and they viewed themselves and their students as outsiders who have to learn and master these theories, concepts, rules, and procedures.