holder for these ideas to exist outside of Ellie. Moreover, because this language is dynamic--it can be "run"--it provided feedback to Ellie's ideas. This trialogue between Ellie's mental model, the expression of her mental model in encapsulated code and the running of that code, allowed Ellie to successively refine the creative structure of her thought. Although one might concede that it is theoretically possible for Ellie to have resolved her problem with a prebuilt model in which a randomizing parameter was modified, the learner-modeling approach is clearly significant in its outcome and arguably more practical in its implementation. This is because, for Ellie to have come to a similar set of insights, a model designer would have had to anticipate all of Ellie's concerns and built them into the model. Clearly, this is impossible to do in general--educational software designers cannot anticipate all the directions that a learner might want to investigate and incorporate them into a parameter model. Moreover, users of "parameter-twiddling" software realize that they are pursuing someone else's investigation. This realization decreases the motivation of discovery. Lastly, such closed environments reinforce a view of mathematics learning as a process of verifying already known mathematics, as opposed to seeing it as a personal odyssey of mathematics making. In designing computer-based environments for learning probability, we must remember that allowing users to create their own models is necessary for truly learner-owned investigations.
For many learners in the Connected Probability project, this experience of doing Connected Mathematics was so different from their experience in regular mathematics classrooms that they did not recognize their activity as being mathematics. Learners who had "always hated mathematics" and had been told that they were not "good at mathematics" were excitedly engaged in doing mathematics that could be easily recognized by mathematicians as "good mathematics." Having created a strong intuitive foundation for the conceptual domain, learners could also go on to engage the formal approaches and techniques with an appreciation for how they connect to core ideas of probability and statistics. Even more importantly, they now understood that mathematics is a living growing entity which they could literally make their own.
A version of this chapter appeared in 1995 in the Journal of Mathematical Behavior, 14 ( 2). The preparation of this chapter was supported by the National Science Foundation ( Grant 9153719-MDR), the LEGO Group, and Nintendo Inc. The ideas expressed here do not necessarily reflect the positions of the supporting agencies. I would like to thank Seymour Papert, Mitchel Resnick, and David Chen for extensive feedback about this research. I also thank Donna Woods, Paul Whitmore, Ken Ruthven, Walter Stroup, David Rosenthal, Richard Noss, Yasmin