This thesis is concerned with students' understanding of the core concept of function\ud which cannot be represented by what is commonly called the multiple representations of\ud functions. The function topic is taught to be the central idea of the whole of mathematics.\ud In that sense, it is a model of mathematical simplicity. At the same time it has a richness\ud and has mathematical complexity. Because of this nature, for students it is so difficult to\ud grasp. The complexity of the function concept reveals itself as cognitive complications for\ud weak students. This thesis investigates why the function concept is so difficult for students.\ud In the Turkish context, students in high school are introduced to a colloquial definition and\ud are presented with four different aspects of functions, set-correspondence diagrams, sets of\ud ordered pairs, graphs and expressions. The coherency in recognizing these different aspects\ud of functions by focusing on the definitional properties is considered as an indication of an\ud understanding of the core concept of function. Focusing on a sample of a hundred and\ud fourteen students, their responses in the questionnaires are considered to select nine\ud students for individual interviews. The responses from these nine students in the interviews\ud are categorized as they deal with different aspects of functions. The data indicates that\ud there is a spectrum of performance of students. In this spectrum, responses range from the\ud responses which handle the flexibility of the mathematical simplicity and complexity to the\ud responses which are cognitively complicated. Successful students could focus on the\ud definitional properties by using the colloquial definition for all different aspects of\ud functions. Less successful students could use the colloquial definition for only set-correspondence\ud diagrams and sets of ordered pairs and gave complicated responses for the\ud graphs and expressions. Weaker students could not focus on the definitional properties for\ud any aspect of functions
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