Problem solving or inquiry?
  
Recently Dan Meyer posted an article on his blog about provoking curiosity in order to teach the "boring bits" of the maths curriculum. The article addresses issues about the starting point and structure of a maths lesson. Although Dan argues for an open middle in which the teacher can "leave some of the important trip-planning" to students, he recommends that the teacher is "exceptionally clear about where your students are and where they're going." Thus, the beginning and end of the process are pre-determined. This approach might lead to students being temporarily interested in a particular problem, but not to the promotion and development of mathematical curiosity. For that, students must be given the opportunity to notice and taught how to act upon what they notice. Andrew Blair posted a response for Inquiry Maths on Dan's blog. The exchange that followed is reproduced here.

    
Andrew Blair
You write that to provoke curiosity requires “keeping several lines tight – not slack – but not so tight they snap.” This is an illuminating way to represent the dynamic nature of an open middle. However, the idea of lines is under-developed. When you say that the teacher has to be clear about the beginning and end points of an activity, I take you to mean that the ‘lines’ act like guy ropes anchoring a tent to the ground. In this analogy, the tent is already erected in the mind of the teacher and the students are left with some decisions about which pegs to bang into the ground first. Notice, the teacher decides where the tent goes and how it will look, and, most importantly, is the only one who knows why it is being erected. In my experience, all this dampens curiosity. Yes, we must offer students mechanisms for getting out of a frustrating impasse; yes, a prompt balances the novel with the familiar and the comprehensible with the confusing; but, no, curiosity does not flourish in classrooms in which pre-determined lines allow students little room to inquire.
  
Dan Meyer
Hi Andrew, I’m not sure where you’re finding my recommendation that teachers should be “the only one who knows why [the tent] is being erected” or that teachers should “allow students little room to inquire.” You’re applying an early metaphor in my post to a much later concept, which perhaps explains the confusion. The tensions I mentioned at the beginning are between the novel and the familiar and the comprehensible and the confusing.
 
Andrew Blair
Thanks, Dan. I am not sure I am any the wiser after your reply. I find the idea of productive tensions very appealing; perhaps, the metaphor of ‘lines’, rather than clarifying the idea, detracts from it. I understand you to be recommending the open middle as one way to maintain those tensions. If that is the case, then you yourself make the link from the lines metaphor to the later concept (i.e. the open middle). You go on to say that the teacher has to be “exceptionally clear about … where [your students are] going.” From this sentence, I don’t think it is at all fanciful to extend my analogy of erecting a tent to saying that the teacher is the “only one who knows why it is being erected," although, of course, the teacher might share her intentions on the way. To conclude, I return to a point I made in my first response: the idea of ‘lines’ is under-developed and that, perhaps, accounts for my confusion.
  
Dan Meyer
I’m just not sure extending the lines to tents has been all that productive here. The tension I’m trying to resolve with an open middle is between an underspecified and overspecified task. This tension (and one possible resolution) is nicely illustrated by Sudoku. Underspecification starts with a blank game board and the instructions to make a game board that satisfies the constraints of Sudoku. This can result in excessive cognitive load and a certain sense that one is flailing. It’s too open. Overspecification starts with an entirely full game board and the instructions to check its correctness. This can result in a lack of self-determination, a sense that you’re wandering down a narrow path prescribed by others. Sudoku instead offers a partially-completed game board and invites students to complete the rest. The initial state is clear. The goal state is clear. But the middle is calibrated so that players feel self-determined but not overwhelmed. That task clarity, rather than undermining a student’s curiosity or sense of agency, has enhanced it.
   
Commentary
Dan Meyer's comment gives a lucid rationale for the problem-solving approach using the example of Sudoku. In his argument, the tension between the over- and under-specification of a task is resolved by the ‘open middle’. Although students are given the problem and must reach the required end-point, they can determine some of the solution process. This contrasts sharply with the inquiry model in which the beginning, middle and end are all potentially open. Students are encouraged to co-construct (and learn how to co-construct) aims, methods and outcomes. The discussion continued on twitter with Dan saying that "openness is a spectrum, not a switch." This is true, so why consider only the middle to be on the spectrum?

Andrew Blair
February 2014