becomes essential because simply counting up outcomes will no longer correctly determine the expected probability distribution.
In the laboratory study, however, students exclusively relied on the counting strategy to decide if the game was fair and displayed no signs that they considered the frequency at which intermediate events occurred as relevant to the solution. Therefore, we took the static representation of the probability tree and added animation. (In the laboratory study, as an event happened, such as the first coin landing on heads, the probability tree would blink that branch of the tree). By animating a token along the tree, we hoped to highlight the connection between the outcome space and the probability distribution and help the students discuss the intermediate events that correspond to the constituent parts of the compound event. We predicted that this would help the students to perceive the process by which a compound event occurs and consider the frequency with which each individual constituent event occurs.
Preliminary analysis suggests that we had limited success. While the counting strategy was still the dominant way to use the tree and envision the outcome space, we did see some encouraging uses of the tree that may indicate that the students were more aware of the intermediate stages of the three coin flips and their relationship to the outcome space. For example in Figure 5, two adjacent pairs started to compare their ideas about what they thought was going to happen during the simulation. Student S points out that the HH intersection of the probability tree is important to their analysis, because if the first two flips are heads, it does not matter what the third coin lands on, team A wins either way (this state is re-created in Figure 1).
Do you guys think that A will|
win cause it has more?
Yep, because if it hits heads|
twice it automatically goes to
[in the simulation the 1st|
coin has just landed on heads]
If it hits heads, look, [S
points to the HH intersection
on the tree at the same time in
the simulation the second coin
lands on heads] Now A
automatically wins, see?
'Cause as soon as it gets two|
heads its only heads-heads-A A
[pointing to the outcomes HHH
FIGURE 5: Students discussing strategic intersections of the probability tree.
The students, in this example, used the tree as a resource in their discussion of why having more outcomes matters. That is, they used the tree to help them connect their new found concept of the outcome space to their existing ideas about the frequency of events (i.e. a probability distribution).
The progression in this case--from using the tree as a game board, to using the tree as a list, finally to using the tree as way to connect previously disconnected ideas--illustrates the importance of stable and persistent representations. The tree, although unchanged physically, changed function as the students' understanding progressed from isolated intuitions about specific aspects of probability to a coherent normative understanding of the outcome space and probability distribution.
Further, this data suggests that animation may play a role in extending a representation's utility across the different stages of a students learning trajectory in our environment. The value added in this case was that the animated tree supported a different type of conversation than the static probability tree. The animated probability tree supported students in their efforts to talk about the connections between their ideas.
This can be seen as a new take on an old story. Most studies about diagrammatic reasoning focus on individuals and how animated representation (and representations in general) can be used to suggest and constrain a student's interpretation. Here we have examined how animated representations support conversations among individuals. In addition, we have looked at how multiple interpretations of a single representation can actually contribute to the learning process.
This paper highlights the importance of considering the process of student-to-student, face-to- face interaction when designing educational software. In our introduction we stated that educational software should create an environment where students are active in their learning by reflecting on, modifying and articulating their understanding. It is our position that this can be accomplished successfully when the software is designed to foster and support domain specific conversations.
The two case studies of students using PIE begin to explore how the rich representational resources available in computer-mediated activities can be used to support collaboration. In the first case we showed how the computer can be an active participant in the interaction by using symbolic representations and text to focus the students attention and discussion on selected features of the activity. The second case study showed how graphical representations can be used as supportive resources for communication and problem solving. The dynamic probability tree helped to make the abstract structure of the outcome space visible to the students and allowed them to use