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The purpose of this study was twofold. First, to explore middle and high school students' intuitively-based (mis)conceptions in probability, particularly availability and representativeness heuristics. Second, to investigate teachers' awareness of these intuitively-based (mis)conceptions and the effectiveness of their instructional practices to support students' understanding of probability beyond heuristics. The participants were two middle school mathematics teachers, three high school mathematics teachers and their students. Data were collected through a diagnostic test administered to students as a pretest and posttest, interviews with the teachers and two students from each class and classroom observations. The findings indicated the existence of intuitively-based (mis)conceptions regarding availability and representativeness heuristics among middle and high school students. In general, the teachers did not consider the students' intuitions and difficulties in probability in their instructions. Not only did they rarely guide students to analyze and solve the tasks coherently and deliberately, but they also did not discuss students' intuitively-based (mis)conceptions. The results highlighted that if teachers do not attend to student thinking and change their instructional practices accordingly, their knowledge about students' difficulties will not necessarily help students overcome their intuitions and attain a probabilistic way of thinking.

This study investigated the sources of difficulties experienced by prospective teachers in mathematical modelling in the context of a semester-long course on mathematical modelling for teachers. The data reported in this study were collected from six prospective teachers through in-depth semi-structured interviews supported by audio and videotaped classroom observations and prospective teachers' written solutions for the modelling tasks. The results showed that the sources of difficulties experienced by prospective teachers in mathematical modelling could be subsumed under individual and contextual factors. The individual factors included four issues: lack of conceptual understanding of mathematics, the difficulty of connecting the real world and the mathematical world, the disposition to focus solely on reaching a result, and disorganized and unsystematic work. The contextual factors included two issues: insufficient previous experience with modelling tasks and time limitations. Some implications of these results are discussed.