# Seven components of mathematical inquiry

Mathematical inquiries that develop from the same prompt can follow 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.

Inquiry Maths habits of mind poster.pdf

### 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 whether it is true or not. 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 that students already hold. (See How to get started with Inquiry Maths.)

(More question stems to promote mathematical thinking.

### Establish aims and plan actions

The teacher reviews the questions and observations (perhaps 'thinking aloud') and takes 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 different lines of inquiry as students attempt to answer more than one of their questions.

### Explore and conjecture

During a period of exploration students generate more examples, search for cases that satisfy the condition in the prompt, or make and test conjectures. As they explore, students look for patterns and connections from which they can form a generalisation or decide if the prompt is true or false.

### 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.

### 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.

### Present results and findings

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

### 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. Students might write about how they - as a class or individually - conducted the inquiry. They consider the new mathematical concepts and procedures they have learnt through inquiry.

# 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.

Levels of Inquiry Maths

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

Stumbling blocks

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?

# Planning for inquiry

## An Inquiry Maths lesson plan

Audrey Stafford (a teacher in Niagara Falls, New York) contacted Inquiry Maths to request a lesson plan template. Audrey teaches 5th grade in upper elementary and reports that inquiry teaching is becoming more popular in the US.

The lesson plan contains questions teachers should consider before the inquiry, ways to support students in the orientation phase, advice on the regulatory cards to use depending on the level of inquiry, and guidance on the types of resources required to support each line of inquiry.

## Planning with the regulatory cards

The regulatory cards are tools that students use to participate in the planning of an inquiry.

Before the inquiry starts, the teacher decides on the cards to be used depending on the students' experience of inquiry and the learning intentions behind using the prompt.

Using the planning sheet for the set of 20 cards, the teacher can consider what student activity each card might lead to in the context of the inquiry and, in turn, the resources required for that activity.

## Planning to develop inquiry habits of mind

Hamdi Ahmed, a teacher in north London (UK), designed the Habits of Mind poster. The inner ring displays specifically mathematical habits of mind, while the outer ring shows more general habits of mind that are developed through inquiry.

When planning an Inquiry Maths lesson, Hamdi focuses on one of the habits to develop with her students.

She says that the most important is the management of impulsivity. When students notice something in a prompt, they often want to dive in and explore without first thinking of the best way to inquire. The regulatory cards help students learn how to direct their inquiries.

Inquiry Maths habits of mind poster.pdf

## Mathematics 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 'Question-driven inquiry' on the  4-by-3 rectangle inquiry page.)

December 2016

## Planning lines of inquiry

Michael Shkurka and Robert England, teachers of mathematics at the British International School of Bratislava, made the notes in the picture during a departmental training day about Inquiry Maths

Participants planned an inquiry in pairs with each teacher having a specific class in mind. The pairs chose a suitable prompt to explore, mapped out possible lines of inquiry, and selected regulatory cards that students could use to direct the inquiry. They decided what each card 'looked like' in classroom inquiry and, where necessary, planned the use of a relevant task or resource.

February 2020

## Planning with an organiser

When Richard Glennie, a teacher of mathematics at Levenmouth Academy (Fife, Scotland), started using Inquiry Maths prompts he devised a structured approach to classroom inquiry. Richard divided each inquiry into four stages: notice, wonder, explore, and found. He designed an organiser so that students could keep a record of their inquiries as they passed from one stage to the next (see examples of students' organisers on the right-angled triangles inquiry page.)

Richard also used the organiser to plan the inquiry, replacing the final stage with 'learn'.  His notes (pictured) cover the whole inquiry from which he planned separate lessons. Richard comments on what the students might notice, 'key questions' related to the inquiry and 'prompt questions' that might encourage further thinking, lines of inquiry, and the learning intentions behind the inquiry.

January 2024

Planning and unplanning inquiry

Andrew Blair's article in Mathematics Teaching 271 (April 2020) considers the detailed planning in preparation for inquiry and an openness to ‘unplanning’ in the moment of inquiry.

# Structures of mathematical inquiry

### 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.

### 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.

### A three-phased model for inquiry-based mathematics teaching

Morten Blomhøj, Per Øystein Haavold, and Ida Friestad Pedersen presented their three-phased model in a conference paper (February 2022). In the paper the authors describe a classroom inquiry called taxicab geometry. They conclude that the three-phased model provides a frame for the planning and implementation of inquiry activities, although teachers might require support in the execution of phases two and three.

# Other resources

## Questions to evaluate Inquiry Maths

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

## Questionnaire

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