Levels of Inquiry Maths
Levels of Inquiry Maths
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The greatest danger for a teacher embarking on inquiry for the first time is to run an open inquiry. Unfortunately, few students in secondary schools will have had the chance to develop independent inquiry skills.
Your students might respond with frustration at "not knowing what to do" or blame you for "not giving them the work." Left to their own devices, they might follow lines of inquiry without mathematical content or develop hidden misconceptions.
By evaluating the profile of your class in terms of experience and initiative, you can select a level of inquiry - either structured or guided - that will offer an appropriate amount of support. You should view open inquiry as the end of a journey that can take many months.
Responsive teaching
Responsive teaching
As you gain experience in inquiry learning, you will often find yourself moving between the levels in the same lesson as you respond to students' questions and developing independence.
You might give some students responsibility for developing their own lines of inquiry, while, at the same time, providing others with structured tasks as part of a teacher-directed inquiry.
Prior attainment
Prior attainment
When deciding on the level of inquiry to use with your class, do not select one based on preconceived notions of ability or prior attainment. 'Top' sets can show less propensity to inquire than 'bottom' sets.
Indeed, students who have previously achieved top grades by successfully completing teacher-set exercises can become anxious and even dismissive when faced with the challenge of inquiry. They require structure or guidance just as much as, and sometimes even more than, classes with lower prior attainment.
More relevant questions you might consider include:
Do your students ask questions about mathematics? Do they notice properties independently?
Are they enthusiastic to contribute to discussions?
Are they prepared to explain their ideas to the class, especially when they are not sure about them?
Can your students use metacognitive strategies when they are stuck?
Do they connect different areas of mathematics when thinking about a problem?
Revised March 2026
Structured inquiry
Structured inquiry
Profile of class
Profile of class
The class is new to inquiry and has little or no experience of student-centred learning. The teacher cannot easily identify students who are curious or take initiative in lessons. Very few are prepared to contribute to class discussions. In general, students are averse to taking risks.
Questioning
Questioning
Co-constructed
The teacher uses sentence stems to support students in posing questions, noticing properties, and wondering about the prompt. The teacher attempts to draw out relevant knowledge from the class. All contributions are recorded on the board.
Regulation
Regulation
Restricted
The teacher regulates the lesson, closing down the inquiry by requiring students to complete pre-determined tasks. The tasks, from which each student might select the most appropriate, are based on the teacher's predictions about what the students’ would say in the first phase.
Lines of inquiry
Lines of inquiry
Restricted
The teacher decides on the aim of the inquiry with the whole class usually following one line of inquiry. Each task is linked to that line of inquiry. The teacher determines if instruction is required to make progress, drawing upon any knowledge students already hold.
Outcomes
Outcomes
Restricted
The results are predictable, although they should still be linked to the contributions in the question, notice, and wonder phase of the inquiry. The teacher might ask pairs of students to present their findings, but most students will have reached similar conclusions.
Guided inquiry
Guided inquiry
Profile of class
Profile of class
The class has carried out inquiries before and the students are starting to show higher levels of creativity. They are beginning to ask more sophisticated mathematical questions or suggest their own lines of inquiry.
It is also possible to run a guided inquiry with a class relatively new to inquiry if it contains an identifiable breakthrough group. The members of the group generate ideas, show high levels of curiosity, and are prepared to explain ideas to the class. Even if it forms a minority in the class, the group might have the social influence to 'carry' their peers – at least, in the early phases of inquiry.
Questioning
Questioning
Student-led
Students pose questions, notice properties, and wonder about the prompt without the teacher's support. They identify terms they do not understand and might make conjectures and generalisations that form the basis of lines of inquiry.
Regulation
Regulation
Co-constructed
Students are given a role in deciding the direction of the inquiry by selecting a regulatory card. The teacher offers three cards at first, increasing the number as students become more experienced. The teacher uses the students’ contributions in the first phase to suggest lines of inquiry, but requests for instruction generally come from students.
Lines of inquiry
Lines of inquiry
Co-constructed
The teacher guides students to follow one of three lines of inquiry that have arisen out of the students' contributions. Alternativley, the teacher uses pre-prepared lines of inquiry linked to the regulatory cards. For example, students might be involved in practising a procedure, exploring more examples, or finding (counter-) examples to support or reject a conjecture.
Outcomes
Outcomes
Co-constructed
Pairs of students or individuals report on or present different findings depending on the line of inquiry they have followed. The results are generally predictable but might include one or two that the teacher had not foreseen at the start of the inquiry.
Open inquiry
Open inquiry
Profile of class
Profile of class
Students are experienced inquirers who understand mathematical forms of inquiry. The majority of the class is curious and prepared to discuss ideas and conjectures. Students can identify when they are stuck and know how to acquire new conceptual or procedural knowledge that develops lines of inquiry.
Questioning
Questioning
Student-led
Students analyse the prompt mathematically without guidance from the teacher. They define terms and variables, identify gaps in their knowledge, make conjectures and generalisations that could lead to proof, and identify potential lines of inquiry.
Regulation
Regulation
Student-led
Either students use the full set of 20 regulatory cards to create sequences of actions or they plan the course of the inquiry without the support of the cards at all. They decide on the aims of the inquiry and monitor their own activity, and are able to justify aims and activity to the teacher.
Lines of inquiry
Lines of inquiry
Student-led
There might be multiple lines of inquiry in the class as students attempt to answer questions, test conjectures, or prove generalisations. The teacher makes students aware of others who are following a similar line of inquiry so they have the option of collaborating.
Outcomes
Outcomes
Student-led
Pairs of students or individuals present outcomes that the teacher could not necessarily have predicted at the start. These might include novel approaches and results that have interest for the wider mathematical community.
Further reading
Further reading
Structured inquiry: a contradiction in terms?
Independence through structure
For guided inquiry: The teacher's response to the students' responses to the prompt
Acknowledgement
Acknowledgement
The work of Professor Galina Zuckerman, particularly her concept of a breakthrough group, influenced the development of the levels.
See Zuckerman, G. A. (2001). How School Students Become Subjects of Cooperative Learning Activity. In Hedegaard, M. (Ed.) Learning in Classrooms: A Cultural-Historical Approach. Aarhus: Aarhus University Press, pp. 229-243.
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