Independence, initiative and teacher direction
Margaret Ashcroft
is a secondary school maths teacher at the Diocesan School for Girls in Epsom (Auckland, New Zealand), teaching students from years 9 to 13. She has responsibility for investigating and planning inquiry-based learning for the mathematics faculty. Margaret contacted the website with some questions about student independence and teacher direction during inquiry. Andrew Blair replies.

Q: How does one of your inquiry lessons typically run - I am familiar with the seven parts to a lesson detailed on your website and have tried to follow this approach but how do you keep the students persevering without too much teacher intervention? This seems to be my greatest weakness - still too keen to help and consequently "direct" which is not the essence of inquiry.
A: I view open inquiry as the outcome of a process that might take five years of secondary school. Over this time the teacher is continually testing the degree of independence she can allow the class. On some occasions, she will have to direct students when they have no clear idea of how to proceed. However, the inquiry maths teacher aims to create the opportunity for the students themselves to identify when that point has been reached and encourages them to call on the teacher by using the regulatory cards. As to the phases of the lesson, again I think the cards play an important role for novice inquirers. Typically, with inexperienced class, there will be a lengthy phase of exploration. During this phase, it is the responsibility of the teacher to keep the 'big picture' in focus. Linking the procedural tasks the students are undertaking to their original questions is important for maintaining levels of perseverance.
Q: I have the same problem when creating a prompt - I need to be less curriculum driven. How do you balance inquiry with curriculum content? Ideally they overlap but I feel if I'm not careful my prompt is more "discovery" than "inquiry". How do you get around this?
A: Yes, the distinction between discovery and inquiry is important. For me, the knowledge that is to be 'discovered' in one classroom is incorporated into the process of inquiry in the other. For example, a student could be required to discover Pythagoras' Theorem inductively by drawing squares on the sides of right-angled triangles and finding their areas. In inquiry, the theorem would
be used to solve students questions about a mathematical prompt (e.g. the right-angled triangles prompt) and introduced through teacher instruction (although, again, identified as necessary by students and co-constructed as much as possible). As to prompts, I design them (or adapt them from different sources) purely for their potential to generate inquiry. They must have "less to them and more in them" as an inquiry maths teacher once said to me. Only afterwards do I link them to the curriculum.

Q: I have a range of classes, from low ability year 10s through to two very able classes in years 11 and 12 - one being an IB class and the other NCEA (official NZ qualification). Do you see inquiry working in any classroom? How do you vary the approach for different levels of ability and engagement in the subject?
A: Yes, I think inquiry can work in any classroom. As I said before, inquiry is a process of continually evaluating how much control you can give to students. This will depend mainly on two factors: (1) the students' experience of inquiry and (2) the levels of initiative in the class (which is not necessarily linked to prior attainment). On the second point, I have taught a class with low prior attainment that found it difficult to generate one question about a prompt (and we spent some time on learning how to ask questions). Yet a similar class (on paper) I teach this year contains two students who take the initiative not only by asking questions, but also by making conjectures in public. They effectively lead their peers and I direct less.

June 2014