Abstract:

The teaching of probability often relies on traditional axiomatic and algorithmic approaches, which frequently fail to overcome students' intuitive misconceptions and cognitive biases regarding randomness. This study explores the epistemological obstacles students face when confronted with uncertainty without prior theoretical instruction, utilizing the classic "Problem of the Points" (the historical foundation of modern probability by Pascal and Fermat) as an active didactic tool within a higher education educational innovation project. An observational study based on Problem-Based Learning (PBL) was implemented with a sample of 283 students distributed across 9 classes. Students were presented with the "Interrupted Game" scenario (a coin-tossing game interrupted at a score of 5 heads to 3 tails, with a target of 6 wins, and a total prize of 24 euros). They were asked to propose and justify a fair distribution of the stakes. Deliberately, no formal instruction on expected value, sample space, or compound probability was provided beforehand, aiming to trigger and analyze their natural heuristics. The qualitative and quantitative analysis of the students' written solutions revealed four distinct cognitive categories that closely mirror the historical evolution of mathematical probability. Only 18.02% of the students successfully applied normative combinatorial reasoning to calculate the expected value (a 21/3 distribution ratio). The largest group, comprising 30.74% of the sample, exhibited a deterministic bias, applying a retrospective proportionality rule based solely on past results (15/9 ratio), akin to Luca Pacioli's 15th-century flawed solution. Furthermore, 24.03% applied additive compensation strategies based on linear distance to the goal, while 9.54% fell into an equiprobability bias (proposing an equal 12/12 split or nullifying the game). Finally, 11.31% of students suffered a representational block, leaving the problem blank. The findings empirically demonstrate that untrained human intuition naturally defaults to deterministic or linear heuristics when facing the multiplicative nature of conditional probabilities. Exposing students to an open-ended historical dilemma generates a profound cognitive conflict, which proves to be an irreplaceable catalyst for deep mathematical learning. Through this didactic situation, mathematical expectation emerges not as a memorized formula, but as the only logically fair solution to the problem. This study highlights the critical need for active learning methodologies, historical contextualization, and visual semiotic scaffolding to effectively transform stochastic reasoning in mathematics education.

Keywords:

Probability teaching, Problem-based learning, Educational innovation, Intuitive probability, Mathematics education.