Study of the development of students' ideas in probability

For several decades, the learning of probability concepts has been recognized as being very complex (Piaget & Inhelder, 1975; Fischbein, 1975). In order to better understand how students develop probabilistic and statistical concepts Garfield and Ahlgren (1988) recommended that longitudinal studies be conducted. This research is a response to the need for such studies and describes, in detail, how a group of students build their understanding of fundamental probabilistic ideas. It also investigates what probabilistic ideas do students build while working on carefully designed tasks involving games of chance, how students represent their ideas and whether student use mathematical justification and connections while working together on open ended investigations over several years.; Qualitative methodology was used in this study to analyze videotapes that were collected as a part of a longitudinal study funded by the National Science Foundation. The cohort group was videotaped working on probabilistic investigations from grades six, seven and twelve. Video portfolios include videotapes of students working on mathematics problems, students' written work, and researchers' notes.; Analysis of students work indicated that the students overcame their initial believes of luck and realized the need to decide what the events and outcomes were in the sample space. They determined that to make a game fair the number of outcomes and not the events had to be divided equally between the players assuming all outcomes were equally likely. The students used a variety of representations to find the outcomes of the sample space. They discovered the multiplication rule of counting to account for all possible outcomes and explained why the rule worked. The students also were able to find the number of outcomes for more complex problems by generalizing their results. They made connection to earlier problem solving experiences by building on their earlier models and applying previous strategies.; This study provides evidence that students can build an understanding of fundamental probabilistic concepts while working together on strands of problem tasks under conditions of investigating rich tasks, collaborating and justifying their solutions, and having the opportunity to think deeply about their ideas before they receive formal instruction in probability.