(More question stems to promote mathematical thinking.) 

The teacher reviews the questions and observations (perhaps 'thinking aloud') and might take the opportunity to comment on possible lines of inquiry. Students, still in their pairs, participate in directing the inquiry by selecting a regulatory card - a selection that is then justified to the class. 

The class might, under the teacher's guidance, decide to collaborate on one class-wide line of inquiry.  Alternatively, there might be multiple lines of inquiry as students explore individually.

Students decide on a period of exploration when they aim to generate more examples, find a case that satisfies the condition in the prompt or make and test a conjecture. At the end of the exploration, students form a generalisation by spotting a pattern or by identifying when the prompt is true or false.

When there is an impasse in the inquiry - perhaps due to the lack of conceptual or procedural knowledge or because of a commonly-held misconception - the teacher decides on how to intervene. That intervention might entail a whole-class episode when knowledge is shared or constructed collaboratively. In this phase the teacher aims to draw upon and develop students' existing knowledge. If the issue is isolated to a small group, the teacher might encourage a student  to explain to her or his peers.

Students explain why a conjecture is true or prove a generalisation they have made earlier in the inquiry. They reason deductively, perhaps with formal algebra or through an analysis of mathematical structure. 

As the inquiry develops, the teacher often calls on students to present their findings. Students report on lines of inquiry and mathematical breakthroughs, and suggest new ideas and directions to the other students.

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.

Yes. Inquiry provides a meaningful context to learn content and empowers students to make decisions about how they use that content. Inquiry lessons do not preclude the 'transfer' of knowledge. If students identify a need for new conceptual or procedural knowledge to make progress during an inquiry, the teacher is responsible for making it available. Moreover, if students request an explanation, they are more likely to be motivated to listen and engage actively with what the teacher or another student says. 

Yes. All students deserve the opportunity to experience the excitement of inquiry. 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. Moreover, it is not the case that attainment in mathematics can be used to predict an inquiry disposition. Students with higher prior attainment in the subject can be more anxious about inquiry because they are likely to have achieved their 'success' in traditional classrooms by answering repetitive exercises.

Three steps make this highly unlikely: firstly, set the prompt just above the understanding of the class to engage students' natural curiosity; secondly, structure the questions and observation phase by providing appropriate stems; and, thirdly, praise all mathematical contributions and return to them as they arise during the inquiry, acknowledging the author as you do so. In the event of not receiving any questions, probe students' understanding of the prompt and proceed with a teacher-directed, structured inquiry.

Inquiries might seem to start slowly, but the construction of a shared understanding in the first phase leads to a deeper understanding of procedures and concepts later in the inquiry. Indeed, it is essential that inquiries start slowly to ensure the involvement of everybody in the inquiry. Students are often more motivated to learn when answering their own questions and, consequently, their learning is faster and more memorable than in normal lessons.

The inquiries on the Inquiry Maths are linked to standard curricular objectives. As the teacher monitors the mathematical validity of students’ aims during inquiry, lesson objectives will be met even if they are not in the order prescribed in a scheme of learning. Moreover, inquiries integrate concepts from different areas of mathematics, making the subject more connected and meaningful (as opposed to being viewed as a list of discrete objectives). It is also often the case in inquiry lessons that students will challenge themselves to meet objectives at a higher level than expected in pursuit of answers to their own questions.

The frequency of inquiries depends on your national or state curriculum. In England, for example, inquiry encompasses one of the three aims of the National Curriculum. So a third of lessons, it could be argued, should be based on the processes of mathematical inquiry. Furthermore, students will become fluent in applying procedures (another aim of the curriculum) during inquiry, so the fraction could justifiably be more than a third. It is our contention that the whole curriculum could be taught through inquiry, but that might not be possible because of departmental, school or curricular restrictions.  

The potentially unpredictable nature of inquiry can be a concern for all teachers, not just for those who are newly qualified.  Students need to be trained to be inquirers. Teachers new to inquiry should take small steps, building up to open inquiry over months and years, rather than weeks. You could open up the start of the lesson for students’ questions and observations about the prompt, then use a pre-planned structure for the rest of the inquiry. In subsequent inquiries, you could give students the choice of more than one pathway to follow before encouraging them to devise and pursue their own ideas.

Sometimes it is hard to go it alone in a department that promotes teacher transmission and student performance. However, when you try inquiry, you might find unexpected interest from colleagues who themselves are looking for ways out of the sterile traditional model of teaching. Students who are used to repetitive practice are likely to find the thinking processes associated with inquiry challenging at first. Use a structured approach in your first attempts.

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 and would have been too challenging for the lowest. So the teachers adapted the prompt for their own classes as shown in the table.

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.