Inquiries that develop from the same prompt can follow very different pathways. They might last one lesson or extend over a series of lessons and might involve students in working collaboratively on one line of inquiry or individually on multiple lines. Nevertheless, the teacher should bear in mind the following seven components of mathematical inquiry when navigating a lesson. 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. 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. Exploration 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. Instruction Students identify an impasse that can only be overcome with new conceptual or procedural knowledge. Their request for instruction might lead to an episode of wholeclass teaching or to smallgroup instruction (from the teacher or a student).
Explaining and proving Students prove a conjecture or generalisation they have made earlier in the inquiry. They reason deductively, perhaps 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 evaluatingThe 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. Resources
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 'Questiondriven inquiry').
10 questions to evaluate Inquiry Maths These questions were designed to facilitate a discussion among members of a school maths department. The team had collaborated in planning and running an inquiry in each of four year groups. 1. Did the students ask the questions and make the observations about the prompt that you expected? Give examples. 2. What did you do after the question and observation phase? 3. What types of cards did students choose to regulate the direction of the inquiry? 4. What suggestions did students make to change the prompt? Or did you direct the changes? 5. What conjectures and generalisations did the students make? 6. How did the students generalise the mathematical properties of the prompt? 7. What concepts and/or procedures did students learn about during the inquiry? 8. What was the most challenging part of the inquiry? For you? For the students? 9. What was most rewarding about the inquiry? For you? For the students? 10. Do you have any evidence that students worked more independently from you than usual?

Inquiry Maths lessons are responsive to students' questions and observations about the prompt. The seven components of mathematical inquiry are, therefore, not intended to be seen as 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." For example, 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.
However, the Inquiry Maths model is built on Polya's view of mathematics as a process in which deduction 'completes' induction.
Polya's description suggests mathematical inquiry is linear, advancing from inductive exploration to deductive reasoning. While this might be the general trend, the relationship between the two is not necessarily linear. Inquiries can zigzag between induction and deduction when, for example, students use empirical tests to amend deductive arguments. Students can also use algebraic or structural reasoning from the start and extend the inquiry by changing the properties of the prompt. 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 prompt  40% of 70 = 70% of 40  Alternatives  50% 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.
