The teacher’s role in inquiry 
It is a common misconception that the inquiry teacher tries to do as little as possible in the classroom. For those who caricature inquiry as a discovery model, the teacher is obliged to let students develop concepts by themselves. For those who define inquiry as exclusively open learning, the teacher must refrain from intervening in order to allow students the freedom to find their own pathways. 
The first approach can lead to awkward interactions when the teacher refuses to give knowledge for fear of denying students the satisfaction of discovering it for themselves. The second approach often leads novice inquirers to complain that they “do not know what to do”. 
Even Dewey, who advocated experiential inquiry based on children’s lives in and outside school, did not encourage a passive role for the teacher. In Democracy and Education, he explains that the opposite of the teacher’s role in traditional teaching is not inaction, but rather participative activity
This does not mean that the teacher is to stand off and look on; the alternative to furnishing ready-made subject matter and listening to the accuracy with which it is reproduced is not quiescence, but participation, sharing, in an activity. In such shared activity, the teacher is a learner, and the learner is, without knowing it, a teacher and upon the whole, the less consciousness there is, on either side, of either giving or receiving instruction, the better. (p. 188)

Participation in the inquiry process requires more skills than in traditional classrooms. The teacher is a learner in the sense that she is continually assessing students’ understanding and taking on-the-spot decisions about whether to structure or guide the inquiry or encourage students to set off on their own. 
Although the Inquiry Maths model aims for participative activity in Dewey’s sense, it is also the case that participants adopt roles consciously and make them an object of reflection. The regulatory cards allow students to make suggestions about how the inquiry should proceed. This includes the activity they will embark upon at a particular stage of the inquiry and whether new knowledge is required to make progress. 
Importantly, the cards provide a mechanism for students to ask for an explanation from the teacher. At such a time, the inquiry classroom might take on the appearance of the traditional transfer of knowledge. However, as the 'transfer' is both meaningful in and relevant to an inquiry process partly directed by students, it is consistent with the teacher's democratic intent to give students control over their own learning.
The role of the teacher in conveying knowledge is the most misunderstood point of all when in comes to views of inquiry. The teacher, as the representative of the discipline of mathematics, introduces a new concept or procedure when it overcomes an obstacle to inquiry. 
Even the leaders of Dewey’s school, Mayhew and Edwards report, changed their model in recognition of the advantage of subject specialists over general teachers:
One of the reasons for this modification of the original plan was the difficulty of getting scientific facts presented that were facts and truths. It has been assumed that any phenomenon that interested a child was good enough, and that if he were aroused and made alert, that was all that could be expected. It is, however, just as necessary that what he gets should be truth and should not be subordinated to anything else. (pp. 35-36)
In this description, the teacher is the arbiter of what constitutes facts and truths. In the Inquiry Maths model, the teacher might also instruct students (although attempting to co-construct knowledge as much as possible) when participants in the inquiry, including the teacher, identify the need.

Andrew Blair
January 2017