Abstract
The present study revealed developmental processes in the operations used by students to quantify probability. Junior high school students (n=32) and university students (n=54) answered questions about the probability of getting particular numbers under various dice-rolling conditions. Developmental stages were identified as follows: Stage I A: Basic first-order quantification of probability Stage I B: First-order quantification of probability with additive composition Stage II A: Basic second-order quantification of probability Stage II B: Second-order quantification of probability with additive composition Stage III A: Quantification of basic conditional probability Stage III B: Quantification of Bayesian conditional probability According to the results, many junior high school students remained at stage I A, whereas many university students reached stage II A. It seemed difficult for university students to deal with any possible cases needed to reach stage II B. Stage III required further competence, reversibility of thought, although some students reached stage II B.
