Development of student understanding of outcomes involving two or more dice

Data from 154 interviews with students in grades 3 to 13 were analyzed to suggest a developmental progression of conceptual understanding associated with the sample space for two ordinary six-sided dice tossed simultaneously. The model was then considered in the light of responses to an extension task involving three six-sided dice with four sides painted black and two white. Forty-five of the interviews were longitudinal interviews of students 3 or 4 years after the original interviews, allowing for analysis of change in levels of understanding across time. This study suggests some of the intuitions and understandings that form the intermediate steps to a complete understanding of outcomes for two dice, as well as documenting the conceptual regression likely to occur when a more difficult task is encountered. Links to previous research, including well-known misconceptions, and educational implications of the model are among the discussion points.