The idea of a playground connotes free form, amorphous recess activities, the antithesis of the relatively rigidly managed classroom. But if you think back to actual recess, you're probably aware of cliques, subplots, key personalities, plus the actual equipment (swings, teeter totters...) isn't exactly "free form," as in: there's a right way to swing, or at least a wrong way (getting hurt is no fun).
So a computer lab is like that too: it's possible to get in trouble, mess up, attract peer pressure, make friends, make enemies. The interactions are complex, especially if you factor in broadband and so many relationships with remote others, via IM, elists and the rest of it.
Minimally, we're providing tools of identity. Make yourself a home page, with your picture, let us know you're here. Update your records, synchronizing with the recordkeeping of others. Get your story going. Let the technology give you a boost. A world readable home page, for a few bits of HTML and a fraction of a kilowatt, is hardly a bad deal.
On top of that, it's about problem solving and developing a shared model within which collaborations are possible, and happen often. We share a need for general prosperity, yet differ in our fantasies of what that might look like.
Fortunately, this isn't about monoculture. Computers have the potential to amplify the indigenous, to anchor the key memes, whatever these might be. This is about sharing Eskimo culture, without making everyone Eskimo.
The play space metaphor works best in that it rhymes with workplace on many levels. Students show up at the community TuxLab, queuing for CPU time, we hope not waiting too long (maybe try that Cafe down the street?). But they're not all there for the same sequences. Some want more about cures for diseases, others want to read up on banking.
Alternating with these "open library" times should be the programmed events, the specific scheduled classes offering paced instruction in selected topics. More typical "school" in other words.
But the computer lab itself is used in both modes: lots of free "play time" when you're encouraged to explore your own favorite topics; other time when you've picked from a list, and now stick with it for awhile, learning the discipline (same as in sports, learning judo, kung fu, archery or whatever -- takes practice, teachers, and a dutiful frame of mind).