It is important to note that PIE’s original conception and design was best characterized as somewhere between the Learner Centered Design and Activity Centered Design models. It was during its implementation and subsequent analysis that our own theoretical perspective evolved into the Activity Centered framework presented above. This evolution creates some apparent
conflicts between PIE’s design and our theoretical framework. For example, Activity Theory approaches to education are often associated with apprenticeship, whereas PIE’s approach seems to assume that students learn probability by doing scientific investigations. However, as we attempt to show below, we believe the students were actually learning mathematical practices ways of perceiving and talking about probability that were accepted by the classroom community as successful methods for justifying claims (e.g. the claim that a given game was fair).

Mediating Probabilistic Reasoning

When we examined PIE from the perspective of how it mediated probabilistic reasoning, we found that by the end of the three weeks most students justified their claims about probability by calculating the probability of a compound event (Vahey, Enyedy & Gifford, under review). The students’ practice in most cases was structured around the probability tree as an ordered inscription of the fully enumerated sample space (i.e. all the possible outcomes). Elsewhere, we have outlined the trajectory of this particular cultural tool within the classroom (Enyedy et al., 1997) and the different social contexts in which the tool was used (Enyedy et al., 1998).

The Interdependencies of Mediation

However, there is more to an activity system and learning environment than the relationship between a tool and a method of reasoning. In our analysis of PIE, both the division of labor between students and the "rules" by which they interact effect the organization of activity and ultimately the students’ learning outcomes. Both of these mediating factors will be examined in turn.
 
 

We use Figure 5 to graphically show the reduced set of interdependencies we examine in our analysis of how the different configurations of who did what during the PIE activities effected the students’ learning. In the prediction phase of the students’ investigations we saw at least two distinct divisions of labor. Although the students worked in pairs, PIE was designed with only one text box to record their predictions. Dissent from their shared answer, was intended to be expressed through the use of "agreement bars." One way in which the students organized themselves to accomplish this task was to alternate who was responsible for that particular question. Alternatively, some students attempted to reach consensus on each and every question. These two ways of dividing the labor within the constraints of the mediating tool resulted in different patterns of interaction and different learning trajectories.

Under the alternating responsibility method, points of disagreement between student understanding often went undiscovered or ignored. Alternating responsibility compartmentalized their answers and eliminated the need for coherence across the questions. This presented a difficulty for a curriculum based on the ideal of students refining their ideas because of cognitive dissonance. It effectively eliminated the social accountability for their answers, and as a result students using this method did not often refine their ideas based on the input of others. On the other hand, trying to reach consensus on each question had its own strengths and weaknesses. While the process of collaboratively reaching consensus made differences between students visible to each other, it did not always lead to deep reflection about those ideas. It has been pointed out that high bandwidth systems, which immediately attempt to reach consensus, tend to settle on a solution nearest to the initial center of gravity regardless of what the evidence suggests as the "best" solution (Hutchins, 1995). In PIE this meant that the students who tried to reach consensus on their predictions, often agreed on the first explanation that seemed sensible to them without fully exploring it or its alternatives.

What is clear from this quick look at some of the ways that students answered the predictive questions of PIE is that the tool that mediated the articulation of their intuitions (i.e. a shared space for answers) was in turn mediated by the way the students divided their labor. While it is unrealistic to think that we can predict or completely determine how students will use a tool, this example shows that in designing CSCL systems it is important to consider how the larger context of the activity system will mediate the tools use and the students learning trajectory.
 
 

We also examined how the participation structures (i.e. rules) for student-to-student interaction mediated the way in which PIE was used and what the students learned from the activity. Figure 6 shows the set of interdependencies we examine in the analysis of two students as they answered a predictive question that asked about the probability distribution of two coin flips.
This interaction is shown in Figure 7.
 

 Figure 7: Rosa and Maria setting the frame for their interaction.

The first turn of this interaction shows Rosa attempting to establish a shared understanding of their current task by reading the Predictive Question into the public interactional space. Having a shared understanding of the task has enormous implications for what actually gets done and what the students eventually understand. What is interesting about this interaction is that the students do not read the entire question (In Figure 7, compare Turn 1 to the text on the top left of the computer display). The part of the question that they do not read aloud, is exactly the parameter of the task they end up ignoring. Even though the  teacher in Turn 8 reminds them that they need to consider the total number of points of their prediction, the students do not make any attempt to make their prediction add up to twenty. In fact, they do not seem to be attuned to quantity at all. Nowhere in this interaction to they mention the cardinal value of any outcome or class of outcomes. Rather, they use relative terms like "higher" and "highest" to talk about the ordinal relations of the classes of outcomes. For this interaction, then, their activity only partially corresponds to the intended activity, because they negotiated the task to include only the relative value of the histogram bars. Even so, in Turn 3 and 4 we see that the two girls collaborating to create a preliminary conjecture that is backed by a justification, that in turn incorporates one of the inscription systems of PIE. That is, even though their assertion is incorrect, its form reflects the desired participation structure of a well-formed argument.

The Sociogenetic, Ontogenetic and Microgenetic Context of PIE

Finally, examining how PIE is situated with respect to the possible socio- and ontogenetic trajectories reveals both some of the strengths and weakness of our design. At the sociogenetic level, we find that PIE takes a somewhat restricted view of the context of probability in relation to the larger domain of practice. In all of the PIE activities, the context for probabilistic reasoning was analyzing games of chance. This corresponds well to the historical roots of classical probability in which probabilistic situations, usually games, are analyzed in terms the number of favorable and non-favorable equiprobable outcomes. It does not, however, address the many real world and far less structured contexts where students might profit by leveraging probability, such as the assessment of risk, or understanding the reliability of a medical test. Our restriction of the activities to game playing is likely directly tied to the students’ limited success at probabilistic reasoning in contexts outside of gaming (see Vahey et al, under review). At the ontogenetic level, however, we believe our choice of games was justified. Games and fairness are authentic interests of students of this age. The gaming context leveraged this interest and helped motivate the students throughout the activities. Finally, we found that students’ microgenetic trajectories through PIE were fundamentally conversations anchored by the available material resources. In some cases, the inscriptions of PIE anchored these conversations in ways that helped them realize the relevance of the normative resources of probability which they previously ignored (e.g. the sample space). In other cases, the inscriptions conflicted with the students’ intuitive practices and led them to totally reorganize they way conceptualized the domain (Enyedy, in process).

Conclusion

There is still an enormous amount of research needed to develop our understanding of how the material, social and mental worlds interpenetrate in mediated activity. Activity Theory begins to lay out some of the dimensions of this task, but it is not yet clear how to apply the insights of Activity Theory to the design (rather than merely critique) of Computer Supported Collaborative Learning Environments. Activity Centered Design is an attempt to move us toward a more appropriate theoretical framework for CSCL environments that will lead to a number of concrete design principles, but this promise is as of yet largely unrealized. What ACD has accomplished to date is to identify and provide a unifying theoretical perspective on some of the major areas where design principles for CSCL are needed. The areas addressed in this article included: how cultural tools mediate cognition, how activity systems (and thus cognition) are mediated by social interactions and different participation structures, and how activity systems are situated in larger communities and their practices.