# An Inquiry Maths lesson

# Seven components of mathematical inquiry

Mathematical inquiries that develop from the same prompt can follow very different pathways. That is because the teacher invites students to participate as much as possible in the direction of the inquiry.

Each class will bring different experiences and prior learning to the inquiry and will express an interest in different lines of inquiry. Moreover, the teacher, in deciding on the appropriate level of structure, treats each class differently.

Nevertheless, mathematical inquiries share common features and aim to develop the same habits of mind. So, while there is no 'correct' way to orchestrate an inquiry, the teacher should bear in mind seven components.

### 1. Question, notice and wonder

In the orientation phase of the inquiry, students attempt to understand the prompt by making an observation, posing a question or wondering about its truth. They think individually before pairing up to discuss ideas. Each pair feeds back a question or comment to the class. In the process, the teacher aims to draw out relevant knowledge. (See How to get started with Inquiry Maths.)

(More question stems to promote mathematical thinking.)

### 2. Establish aims and plan actions

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.

### 3. Explore and conjecture

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.

### 4. Construct understanding

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.

### 5. Reason and prove

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.

### 6. Present results

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.

### 7. 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 lesson structures

The lesson structure above was developed in a community of teachers from different subject areas who were studying inquiry approaches. The school had 100-minute lessons, which explains the scale on the left. After students' initial responses to the prompt, the teacher reviews the questions from a disciplinary perspective.

Satvia Bahia (a secondary school teacher of mathematics) designed the diagram for a presentation about Inquiry Maths that she was giving to trainee teachers at the University of Sussex (UK). After students have used the regulatory cards, the teacher decides on the structure of the inquiry and whether an explanation is required.

# Elements more than steps

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 George Polya's view of mathematics as a process in which deduction 'completes' induction. Polya's description suggests mathematical inquiry advances from inductive exploration to deductive reasoning.

While this might be the general trend, the relationship between the two is not necessarily linear.

Inquiries can zig-zag 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.

Would your students learn best through a structured, guided or open inquiry? How much structure should you give them? How do you decide?

What if you have not anticipated students' responses to the prompt? What if the responses are so diverse it is difficult to know where to start?

# Ten frequently asked questions

### 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 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, 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.

### 6. How often should you use inquiries with your classes?

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.

### 9. 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 on the website will not suitable for all classes and should not be simply 'taken off the shelf'. The prompts should be adapted to sit just above the understanding of the class, thereby promoting curiosity.

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.

### 10. 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.

# Other structures of mathematical inquiry

### The 4D-cycle of mathematical inquiry

The inquiry cycle was devised by Professor Katie Makar (University of Queensland). Each part of the cycle is described in more detail on this page from Thinking through Mathematics. Additional information appears in Professor Makar's 2012 chapter 'The Pedagogy of Mathematics Inquiry'* and on the IMPACT website.

* In Gillies, R. M. (Ed.). Pedagogy: New Developments in the Learning Sciences. New York: Nova Science Publishers, pp. 371-397.

### The International Baccalaureate's cycle of mathematical inquiry

One aim of the IB diploma programme for mathematics is to promote inquiry approaches in which students learn by experimentation, questioning and discovery.

Students are expected to be active participants in learning activities. Inquiry should stimulate students' critical reasoning and problem-solving skills.

In order to achieve the aim, teachers are encouraged to use the flow chart when planning inquiry lessons. (Read a critique of the IB model here.)

4E x 2 model

Samia Henaine has adapted the 4E x 2 instructional model for mathematical inquiry. The model (see diagram) combines four inquiry processes (engage, explore, explain, and extend) with formative assessment and metacognitive reflection.

Samia has created a template of questions to guide students to engage, explore, explain, and extend their thinking. She has also created inquiry prompts from which to launch classroom inquiry.

### Language acquisition as the first phase of inquiry

Sonya terBorg blogged about using Inquiry Maths prompts with her primary class in Idaho (US). Her post describes how the class carried out a preliminary inquiry into concepts and language related to angles. The pupils then conducted an open and collaborative inquiry into the prompts by applying the language acquired earlier.

# 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 Maths.

## Ten questions to evaluate Inquiry Maths

These questions were designed to facilitate a discussion among members of a school maths department who had collaborated in planning and running inquiries.

## Maths inquiry template

Amelia O'Brien, a grade 6 PYP teacher at the Luanda International School (Angola), has shared her Mathematics 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').

## Questionnaire

A brief questionnaire to collect students' feedback on the differences and similarities between inquiry and other lessons.