Triangulation in teaching probability: teaching materials for the theoretical foundations of probability in real-world applications

This paper proposes using the concept of triangulation with probabilistic models as a means to enhance theoretical inversion for deepening students’ understanding of the nature of probability in real-world contexts. Triangulation refers to the combined application of multiple methodologies to investigate the same phenomenon, particularly in the social sciences. Theoretical inversion refers to a shift in focus from surprising outcomes to the theoretical foundations of probability. The paper introduces three types of problem-solving tasks designed to enhance one of four types of triangulations: theory triangulation. Theoretical inversion is expected to emerge through engaging in these tasks. The characteristics of the problems are as follows. Problem 1 promotes students to compare different probabilistic models of events under similar procedures. Problem 2 provides students with an opportunity to simplify an experiment by omitting steps that add no new information. Problem 3 enhances students’ ability to recognise how subtle differences in the experimental setup can affect the resulting probability. These tasks are designed to encourage students to view probabilistic reasoning as a form of modelling and to appreciate the importance of assumptions, definitions of elementary events, and clarity in procedural descriptions.