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An Inquiry Maths lesson

Inquiries that develop from the same prompt can follow very different pathways. They might last one lesson or extend over a series of lessons. The seven components of a mathematical inquiry (right) combine inductive and deductive reasoning.
Although deduction 'completes' induction, mathematical inquiry is not a linear process in which each component follows on from the one before in strict order. Rather, as Kath Murdoch says in The Power of Inquiry, the parts are "phases more than they are stages, elements more than they are steps."
Two examples of elements, not steps
1. The teacher should promote questioning throughout the inquiry, not just at the beginning. In this way, students deepen their initial questions and generate more lines of inquiry.
2. While deductive reasoning normally follows a period of inductive exploration, the relationship between the two is not always linear. Indeed, inquiries can zig-zag between induction and deduction when, for example, students use empirical tests to amend deductive arguments.

An Inquiry Maths lesson plan
Audrey Stafford (a teacher in Niagara Falls, New York) contacted Inquiry Maths to request a blank lesson plan template. Audrey teaches 5th grade in upper elementary and reports that inquiry teaching is becoming more popular in the US. Click here for a generic lesson plan with questions to help teachers prepare for inquiries and consider the resources required for different levels of inquiry.

Click here for a brief questionnaire to collect students' feedback on the differences and similarities between inquiry and other lessons.
Maths Inquiry Template
Amelia O'Brien, a grade 6 PYP teacher at the Luanda International School (Angola), has shared her Maths Inquiry Template with Inquiry Maths. The template helps students think about concepts relevant to the prompt and plan the inquiry. In their most recent inquiry, Amelia's pupils posed generative questions that opened up new pathways for inquiry (see a report here under the title 'Question-driven inquiry').

You can follow Amelia on twitter @_AmeliaOBrien.

Structures and cycles of inquiry in mathematics classrooms. 
Seven components of a mathematical inquiry
Orientation to the prompt: questioning and noticing
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.
Find more question stems to promote mathematical thinking here.

Jhahida Miah, a teacher of mathematics at Haverstock School (Camden, UK), designed this slide to support students' questioning and noticing. Students form groups of three and take on the following roles: scribe, speaker and manager.
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.
Teacher explanation
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.
Proving a conjecture
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.
Presenting results
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.

Frequently asked questions
(1) Can you expect students to inquire into topics without being given content knowledge beforehand?
Yes. Inquiry lessons do not preclude the 'transfer' of knowledge. They are not discovery lessons in which students are expected to discover a concept independently. If students identify a need for new conceptual or procedural knowledge to make progress during an inquiry, the teacher should give instruction. Moreover, if students request an explanation, they are more likely to be motivated to listen and engage actively with what the teacher says.
(2) Can you expect students in bottom sets to take part in inquiry lessons?
Yes. Often students are in bottom sets because they do not have the higher order skills required to regulate learning. Inquiry Maths gives all students the opportunity to develop those skills. To introduce inquiry (to any class), a teacher would require students to pose questions or make observations about the prompt. The inquiry could then be closed down, with the teacher structuring the rest of the lesson. (For advice on the type of inquiry to run, see levels of inquiry.)
(3) What prompt should I choose to get started?
Inquiry Maths prompts are designed around concepts in the school curriculum. You might start by choosing a prompt linked to the topic in your scheme of learning. However, the prompts as presented on the website are not suitable for all classes. A prompt should be set just above the understanding of the class to promote curiosity (see this post). Thus, you might adapt the prompt. An example comes from a secondary school maths department that was using the percentages prompt. The prompt on the website would have provided little intrigue for the highest set, but would have been too challenging for the lowest. So the teachers adapted the prompt for their own classes as shown in the table.
Main prompt40% of 70 = 70% of 40
Alternatives50% of 10 = 10% of 50
47% of 74 = 74% of 47
40% of 30% of 20 = 20% of 30% of 40 
(4) How can you make inquiry more accessible? 
This question comes from Alex 
Zisfein, a secondary teacher of mathematics in New York City, who felt the prompts are more suitable for advanced classrooms, rather than for general education groups. There are two ways to make the prompts more accessible  Firstly, the teacher can take more responsibility for structuring the inquiry by, for example, preparing a pathway for students to follow in the first lesson and then planning subsequent lessons that respond to the students’ questions and observations. Secondly, prompts can be adapted to ensure they are both familiar and unfamiliar. Familiarity gives students confidence to analyse and transform the prompt; unfamiliarity generates curiosity to understand the prompt more deeply.