Composition of functions and inverse function of a function: Main ideas, as perceived by teachers and preservice teachers

The concept of function is one of the central concepts in mathematics; as such it has received considerable attention in mathematics education research. However, the research that focuses explicitly on the subtopics of composition of functions and the inverse function of a function is rather slim. This study investigated what teachers and prospective teachers consider to be the main ideas behind these two subtopics underlying the concept of a function and what they attend to when teaching or planning to teach these topics. The study also investigated connections between the three components of content knowledge: subject matter knowledge, pedagogical content knowledge, and curricular knowledge. Furthermore, the study examined the influence of the teaching experience on content knowledge. Ten prospective teachers and eight experienced teachers participated in the study. The data were collected through clinical interviews. In addition to this, the prospective teachers wrote a set of lesson plans for teaching the two considered topics. The data were analyzed in detail using a theoretical framework that was modified and refined based on prior research on teacher's content knowledge related to the concept of function. The framework contained five facets: essential features and the knowledge and understanding of the mathematical concept, different representations and alternative ways of approaching the topic, basic repertoire, knowledge of the mathematics curriculum, and knowledge of general mathematics. The results of the study suggest that the two groups of participants were not remarkably different in terms of their subject matter knowledge and their pedagogical content knowledge. Majority of participants presented a mainly procedural approach to both topics and disregarded some essential components, such as conditions for existence of inverse. However, the experienced teachers presented a higher competency in the area of curricular knowledge. With regards to the relation between the three components of content knowledge, subject matter knowledge and pedagogical knowledge appear to be interconnected, whereas curricular knowledge appeared to be independent of the previous two.