Inquiries that develop from the same prompt can follow very different pathways. They might last one lesson or extend over a series of lessons. There are seven component parts to a mathematical inquiry that the teacher should have in mind.
Orientation to the prompt: noticing and questioning
The teacher invites pairs of students to make an observation or pose a question about the prompt, providing the class with stems (examples below) if appropriate.
Establishing aims and planning actions
The teacher reviews the questions and statements (perhaps 'thinking aloud') and might take the opportunity to comment on possible directions the inquiry could take. Students select a regulatory card - a selection that is then justified in a class discussion.
Students might decide on a period of exploration when they aim to generate more examples or find a case that satisfies the condition in the prompt. At the end of this period, they might have formed a generalisation through induction.
Students request an explanation or identify an impasse that can only be overcome with new conceptual or procedural knowledge presented by the teacher. This might lead to an episode of whole-class teaching or to small-group instruction.
Students prove a conjecture or generalisation they have made earlier in the inquiry. They reason deductively with formal algebra or through a structural analysis of a mathematical model.
Students present their results in written or other forms. The teacher often calls on students to present their work in progress or suggest new ideas and directions to the class.
Reflecting and evaluating
The teacher leads students in reflecting on the course of the inquiry, and in evaluating how successfully the class has resolved the questions posed at the beginning.