Analysis of the example spaces of mathematical concepts

Example space consists of a set of examples (more precisely: examples, counter-examples and non-examples) that an individual can access along with the links between them. An individual forms a mathematical concept through examples, and the set of examples forms one‘s personal example space of that concept. The personal example space is also influenced by examples in school textbooks - these examples form the conventional example space. In the theoretical part of the thesis, we have presented several classifications of examples and their importance at learning mathematics. We emphasized the importance of the teacher's choice of examples and the individual's own construction of them. We defined the concept and different types of example spaces and described in detail the conventional and personal example space. In the empirical part of the thesis, we studied the conventional example space of two mathematical concepts (parallelogram and trapezium) by analysing several mathematical textbooks. By analysing teachers' lesson plans and students’ notebooks, we examined the extent to which experienced and novice teachers use different types of examples and whether their examples are pre-planned or spontaneous. Using a test of knowledge we compared the students’ personal example spaces of experienced and novice teachers. The results of the research show that Slovenian school textbooks do not contain reference examples and non-examples, they do not encourage students to construct their own examples and do not contain tasks with which the students would explore the properties of quadrilateral. We found that among the analysed textbooks Stičišče 7 provides the best set of examples for quadrilaterals. The analysis of teachers’ lesson plans revealed that they only consider geometric shapes in traditional position and that they do not present other reference examples, nor do they encourage students to construct their own examples. They do not deal with non-examples, nor do they present specific cases of concepts. Most of teacher’s examples are pre-planned. Finally, we found that, on average, novice teachers’ students have a richer personal example space for parallelograms, while experienced teachers’ students have a better structured example space, especially for the trapezium and rectangle