3. Exploration

Students might decide on a period of exploration when they aim to generate more examples, find a case that satisfies the condition in the prompt or test a conjecture. At the end of this period, students might have formed a generalisation through an inductive process of pattern-spotting.

4. Teacher or student explanation

Students might identify an impasse that can only be overcome with new conceptual or procedural knowledge. Their request for an explanation might lead to a whole-class episode when knowledge is shared or constructed collaboratively. Alternatively, the teacher might encourage small-group instruction (led by the teacher or a student).

5. Reason and prove

Students prove a generalisation they have made earlier in the inquiry. They reason deductively, perhaps with formal algebra or through a structural analysis of a mathematical model.

6. Present 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.

7. Reflect and evaluate

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.

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. 

1. Can you expect students to inquire without being given content knowledge beforehand?

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. 

2. Can you expect students in bottom sets to take part in inquiry?

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.

3. What happens if the students don't ask any questions at the start of the inquiry?

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.

4. Isn't learning  through inquiry too slow to cover the curriculum?

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.

5. How can you be sure that the students meet lesson objectives during inquiry 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, 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.

6. How often should you use inquiries?

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.  

7. Isn't inquiry too unpredictable for inexperienced teachers?

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.

8. Isn't it too difficult for individual teachers to use inquiry on their own?

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.