How to get started with Inquiry Maths
This week I received an enquiry from a teacher new to an international school: "I am just about to start using inquiry in my maths classroom. Your website is just what I need. However, it all seems a little overwhelming at the moment. Where is the best place to start in terms of resources?" Five years ago I received a similar e-mail from a maths teacher who had looked over the Inquiry Maths website. She liked the ideas and wanted to carry out an inquiry: "The prompts look interesting and I can see how they could develop into inquiries, but how do I start?"
The questions of how and where to start also arise frequently in workshops. As inquiry is so different from conventional maths lessons, it can seem difficult to visualise an Inquiry Maths classroom. On occasions, I have been advised to produce videos to show how it's done. However, to me, this is contrary to the spirit of inquiry that I would hope to engender not only in students, but in teachers as well. An inquiry disposition is essential if the teacher is to respond to students' ideas with empathy, curiosity and genuine interest. Inquiry is not a model to be copied; it is a way of being in the classroom - for student and teacher. I would encourage all teachers to find their own way without thinking there is one right way.
Having said that, I know it can seem daunting to conduct your first inquiry. One 
teacher in an Inquiry Maths session described using investigations, but felt that inquiry seems like "another step again". If students are participating in setting the agenda through their questions and responses to the prompt, then teachers new to inquiry can become concerned about meeting specific learning objectives. If students participate in directing the lesson, then they can be fearful of losing control of the lesson. Inquiry requires a complete cultural shift in which learning is co-constructed through promoting student initiative and agency.
Navigating the first few steps towards an inquiry classroom can, therefore, seem daunting. With that in mind, I think the following seven points are worth considering when getting started with Inquiry Maths. 
(1) Choose a prompt linked to your scheme of learning. 
The prompts on the website relate to standard mathematics curricula and can lead into pathways that cover content descriptors. If you are concerned that inquiry learning involves too much ambiguity around content coverage, then you can select process objectives as the aims of the lesson (see this lesson plan for objectives such as devising questions, making conjectures and justifying and reasoning). In my response to the enquiry five years ago, I recommended ignoring content objectives as a 'hindrance'. After all, the more the teacher attempts to address curriculum content, the less space there is for students to inquire. However, this is a luxury most teachers don't have. Now I would counsel teachers to balance the requirements from school authorities with the desire to promote student exploration. As long as students follow a pathway linked to the curriculum (however loosely tied to the specific content in the scheme of learning), then that, for me, is legitimate mathematical activity.
(2) Choose a prompt that you think will intrigue the class. 
The prompt should be set just above the prior attainment of the class in order to generate curiosity and encourage questioning and noticing with which to launch the inquiry. In one school, for example, the percentages prompt was adapted (see the box) for classes with different prior attainment in order to provide the right level of intrigue.
10% of 50 = 50% of 10
40% of 70 = 70% of 40
47% of 74 = 74% of 47
20% of 30% of 40 = 40% of 30% of 20
The percentages prompt was adapted to provide an appropriate level of intrigue to classes that were set on prior attainment (going from lower to higher in descending order).
(3) Decide if you are going to run a structured, guided or open inquiry.
Over the last five years, I have given the difference between levels of inquiry greater importance in Inquiry Maths workshops. Many teachers associate inquiry with free discovery and have a preconceived notion of classrooms in which students do what they want. This is not the case. Students must learn how to inquire alongside subject knowledge. For a class new to inquiry, I would advise teachers to plan for structured inquiries, even if some of their students show signs of independence and preparedness to take risks.
(4) Prepare resources related to the prompt.
Inquiry requires careful planning. Although Inquiry Maths prompts draw on a specific area of mathematics, they have the potential to develop along different pathways. If I were to restrict that potential within a structured inquiry, then I would prepare resources for a single main line of inquiry. In subsequent lessons, I might offer resources to support different pathways that develop out of students' questions and observations. When I talk of resources, I include stems to promote questioning and noticing (see here for examples), diagrams to support explanations, models of different types of cases that students could use when, for example, practising a procedure and further prompts that structure students' activity.
(5) Plan how students will communicate their questions and observations to the class
I take feedback about the prompt in the same way at the start of each inquiry. As individuals think about the prompt, I circulate around the classroom to see what they have come up with. Students pair up (in order to emphasise the collaborative nature of inquiry), agree on a question or observation and then one of them reads it out as we go round the class. I do not necessarily follow a set order, although I might aim to take lower level questions, such as requests for definitions, first and build up to more sophisticated ideas, such as conjectures or even generalisations. I write out all contributions on the board for public inspection and as a record of our initial thoughts. Other teachers have used post-it notes that are stuck up in the classroom for discussion or a Padlet wall that allows students to contribute to ideas on a virtual board.
(6) Review the questions and observations
Next I review the students' questions and observations by talking aloud. I might classify them into groups based on similar types of questions (such as what, how and why) or on similar subject matter. I might feel that the class can address questions immediately by giving a definition or demonstrating a procedure. If so, I would orchestrate a discussion with the aim of unearthing such knowledge.
(7) Use the regulatory cards t
o decide what to do next
You might decide not to use the cards in your first inquiry and direct the inquiry yourself instead. However, the aim of Inquiry Maths is to involve students in setting aims and deciding upon the direction of the inquiry. As the cards provide a mechanism for doing that, I would encourage you to try using a set in subsequent inquiries. I note down the cards chosen (by ticking them on the board) and draw lines between them or number cards when students start to select more than one card. If a pair of students propose a different card than others have done, I will ask them to justify their choice. Sometimes, the choice of cards allows you to run a unified class-wide inquiry; sometimes, the inquiry is diverse and multi-faceted. You decide based on the mathematical validity of the cards chosen.

Andrew Blair
December 2018