Here the key question is whether and how the collaborative interactions of the "studio model" helped generate positive results in this computer- supported application of design pedagogy in learning mathematics. The data from the Escher's World project shows that collaboration played an important role in students' learning. Moreover, the fact that the studio setting gave students control over their collaborative interactions was a key part of the success of the studio.
There were several ways in which students talked about feeling as if they were in control of their collaborations during the workshops, but the most prevalent comments were about students' control of the timing and extent of their collaborative activity. Students talked about their ability to decide for themselves when to work alone, when to work with a peer, and when to consult with an adult. One student said simply: "[In the workshop] if I don't know something, I just ask you or other friends to sit by me. In class [at school] you can't talk." Similar sentiments were echoed in two-thirds of the comments where students talked about both control and collaboration. In almost all of the comments about working with peers (16/19) and about getting help from adults (16/18), students talked about the fact that in the workshop they were in control of how and when these interactions took place.
A look at the relative frequency of student comments overall about Control and Collaboration make it clear that these were critical issues in students' experiences of the Escher's World mathematics studio (see Figure 1). Referring to the theoretical work described above in the introduction, excerpts from interviews in the Escher's World project were coded for Control when students referred to freedom of physical as well as intellectual movement, when they talked about making their own choices, judgments, or decisions--in short, when they described in a positive or negative way the effects of their own control (or lack of it) in their learning experience. Similarly, excerpts were coded for Collaboration when students referred to ways in which their learning experience was affected by the active participation of others (or lack thereof). This included descriptions of help given to or received from adults or peers, joint work with others, public presentations and feedback on ideas or work, and conversations or other "purely social" interactions--in short, Collaboration refers to the range of students' relations to other people as it connects to their learning experiences.
By these criteria, more than half of the excerpts from interviews about the studio as a learning environment (73/123 or 59%) were about either students' feelings of control over their learning experience or students' collaborative interactions with others. Perhaps more interesting, students' comments in these areas show significant overlaps. Students referred to both control and collaboration in 36 excerpts--that is, in almost 75% of the comments about collaboration students referred to the importance of feeling in control of their learning experience. Overall, student comments about collaboration were correlated with comments about control with r=0.79.
These two projects thus show that the design studio, with its combination of loose schedule and structured desk crits, provides a useful model for thinking about collaborative activity in an open learning environment. The design studio provides a framework for collaborative activity that preserves student autonomy in the learning process and provides a model for collaborative interactions. This work also shows that the design studio model can be used successfully in other disciplines with the help of computer