The zone between knowing and not knowing
Part 2: Modelling and orchestrating
In part 1, we talked about slowing down the inquiry as the class enters the zone between knowing and not knowing. The slow down occurs so that students can get in contact with the aims of the inquiry (Alrø and Skovsmose) and the teacher can ensure her intent and the students' intent coincide (Zuckerman). In order to achieve contact and coincidence, the teacher has to take on a specific role during the phase of slowing down that might be different to the ones she adopts later in the inquiry. She aims to encourage students to suspend any doubts they may have about operating in “a twilight of shifting and unclear purposes” (O’Connor et al., p. 119) by creating an open zone in which questions and contributions are treated respectfully and seriously.
Modelling an inquiring mind
The teacher should aim to model the disposition required to be an inquirer and to learn through inquiry:
If we show students what being curious 'sounds like' by regularly and genuinely voicing our own wonderings, we also help teach the art of questioning in a more informal, natural way. The key to fostering an environment where students feel safe to ask questions is to be comfortable with uncertainty ourselves.... Students need to see and hear us in that space, to see and hear our fascinations and uncertainties and finally, to see and hear our willingness to find out when we don't know. (Kath Murdoch, The Power of Inquiry, p. 57)
Modelling an inquiry disposition involves publicly pondering a student’s observation about the prompt or reflecting out loud on the meaning and implication of a question. The inquiry teacher holds back from evaluative statements in favour of seeking clarification and extending ideas. Praise for the depth and mathematical validity of a question can be communicated by expressing interest and musing over a student's contribution during a class discussion. A comment such as “that’s an interesting idea” is preferable to giving overt praise which might interrupt the class discussion and reinforce the impression of the teacher as an authority figure. In this stage of the inquiry, the teacher aims to promote a symmetrical relationship in which she stands as a learner of students’ initial ideas and starting points.
The inquiry teacher orchestrates productive discussions by giving students time to construct responses to a prompt and then by expecting all students to have something to contribute. While allowing students to ‘pass’ when it comes to their turn, she is aware of the students who regularly opt out and gives them extra one-to-one support before the next whole-class session. The discussion develops in a sequence from basic contributions to sophisticated reasoning:
Questions about defining terms;
Descriptions of relevant procedures;
Explanation of underlying concepts;
Conjectures based on empirical features of the prompt (pattern spotting leading to generalisation);
Conjectures based on mathematical structure (specialisation in the prompt as representative of a class of objects).
As the students are called on for their turn, the teacher links up ideas and speculates about other connections.
During the class discussion, the teacher oversees the use of two forms of mathematical speech (O’Connor). Through exploratory speech, ideas are generated, conjectures tentatively proposed, and partially developed ideas discussed. In this inductive phase, the teacher might overlook imprecise speculations and might pass over errors in calculations or meet them, not with direct corrections, but with counter-examples that provoke fresh thinking: “When we are in the heavy lifting and framing stages of developing new ideas, stopping to correct every flaw is disruptive to the real work” (O’Connor, p. 177). However, the teacher asserts the deductive side of mathematics when reviewing and summarising the exploratory discussion or when fully formulating an idea that has emerged in the discussion. In summative speech, when the focus of the discussion is clearly defined and stable, the teacher “tightens the criterion levels for precision and correctness” (p. 178). She attends to students’ mistakes and re-casts their ideas using formal mathematical language.
At this point, the class is ready to move on to the next stage of the inquiry. Aims can be negotiated, inquiry pathways can be set out and resources can be gathered. The class is moving further into the zone between knowing and not knowing. Now students and teacher know what they want to find out and how they will go about finding it out.
Andrew Blair, May 2018