Cross modal representation and co-construction: what is the learning?

Two 5th graders came to the studio to work on a real-life representation of a technique they use during math time -the ladder method. I've been thinking about reasons for older children to engage in inquiry in the studio, and it seems to me that there are 2 main ones, one more obvious than the other.

Showing Comprehension
When children represent ideas across modes (in more than one medium), they show their understanding of that idea. For instance, in the 3rd and 4th grade classrooms, students have been composing music to illustrate scenes in the books they are reading. These compositions serve to make the students' learning visible (and hearable), by letting other people see the students complex understandings and thoughts about the reading.








Co-construction of Understanding
Possibly an even more subtle and important reason for students to engage in this type of project comes from co-construction. While building the real-life math/ladder, the students have to reach a common understanding of the thing they are trying to represent (long division). As they show their individual understanding  to each other, little problems and glitches in what they each know come to light. By discussing these little 'knotty problems' together, the students refine their own and the other's understanding.

As they represent this knowing with paper, tape and a video camera, they are in effect testing that common understanding again, this time against the media and materials they are representing with.

So, beyond showing comprehension, students actually increase their understanding throughout the process of collaboration and representation. It reminds me of the dialectic (thesis-antithesis-synthesis), spiraling on to the next idea or thesis each time.
At least, that's the way it seems to me right now.
What do you think?

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