How could we improve this task?
Fuller, Rabin, and Harel (2011) [pdf] define "intellectual need," "problem-free activity," and offer several ways to improve that task in one of the best pieces I read last summer:
When students participate in mathematical activities that stimulate intellectual need, we say that they are engaged in problem-laden activity. Unfortunately, many students are engaged in problem-free activity, in which they are driven by factors other than intellectual need and, as a result, do not have a clear mental image of the problem that is being solved, or indeed an understanding that any intellectual problem is being solved.
The piece features:
- Dialog between teachers and their students that results in "problem-free behavior" and "social need." There's something in here for everybody. Everybody — myself included — will feel a twinge of recognition reading one or more of those exchanges.
- Great suggestions for how to mend those scenarios, for queueing up intellectual need and problem-laden behavior.
- Five categories of intellectual need. The need for certainty, causality, computation, communication, and connection. You can lean on any of those categories and watch several great lesson ideas fall out.
The recursive part in the original question is especially annoying in that it sends the message that math is used to take something that is totally obvious (two more brick in the next row) and somehow make it seem complicated.