Levels of Inquiry Maths
The greatest danger for a teacher embarking on inquiry for the first time is to aim to run an open inquiry. Unfortunately, few students in the age range ten to 16 have had the chance to develop independent inquiry skills through formal schooling. Your students might respond to open inquiry with frustration at "not knowing what to do" or develop approaches that are inconsistent with mathematical reasoning. By evaluating the profile of your class in terms of experience and initiative, you can select a level of inquiry that will offer an appropriate amount of support. Open inquiry is the end of a journey that can take many months.
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 structure in a more directed inquiry.
When deciding on the level of inquiry you will use, 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 in maths by successfully completing teacher-set exercises can become anxious and even dismissive when faced with the challenges of inquiry. They require structure or guidance just as much as (and sometimes even more) lower attaining classes.
Profile of class
The class is new to inquiry. It is not easy to identify students who show curiosity or take the initiative in mathematics lessons. Very few, if any, are prepared to take risks.
The teacher supports students in asking questions and making observations about the prompt
The teacher regulates the lesson, closing down the inquiry by requiring students to complete pre-determined tasks (based on predictions about the students’ questions and observations).
The teacher prescribes the (usually one) pathway that the class works on as a whole. The teacher decides if instruction is required to make progress.
The results are broadly predictable, although they should still be linked to the initial questions and observations. The teacher might ask groups to present their findings, but most students will have reached similar conclusions.
Profile of class
EITHER The class is relatively new to inquiry, but contains an identifiable “breakthrough group” whose members generate ideas, offer conjectures and show high levels of curiosity. Even though the group could form a minority in the class, it has the social influence to 'carry' their peers – at least, in the early phases of inquiry.
OR The class has carried out inquiries before and the students are starting to show higher levels of creativity by suggesting their own inquiry pathways.
Students ask questions and make observations about the prompt – and might make conjectures and generalisations.
Students are given a role in deciding the direction of the inquiry by choosing a regulatory card. The number of cards could be limited to three at first, then six, and so on. The teacher uses the students’ questions and comments to suggest approaches, but requests for instruction generally come from students.
The teacher might encourage students to follow one of three or four different pathways based on the questions and observations. For example, one pathway might involve students in learning a procedure; a second might encourage them to explore more examples; a third might require students to find (counter-) examples to support or reject a conjecture; and a fourth might involve them in explaining an observation.
Students report on or present different findings depending on the pathway they have followed. The results might include some that the teacher had not foreseen at the start of the inquiry.
Profile of class
Students are experienced in inquiry and can direct their own learning by using the regulatory cards creatively or without the support of the cards at all. They can take a prompt and inquire independently in order to create a mathematically-valid outcome.
Students ask questions and make observations about the prompt, including suggestions for changing the prompt to explore it further. They make conjectures and generalisations that could lead to attempts at proof.
Student-led (teacher validation)
Students combine regulatory cards into sequences of actions, or make their own suggestions instead of using the cards. They monitor their own activity, set their own goals, and are able to justify the activity and goals to the teacher.
Student-led (teacher instruction when requested)
Multiple pathways are in evidence as students attempt to answer their own questions.
Student-led (teacher assessed)
Groups or individual students present results that the teacher could not have predicted at the start. These might include novel approaches and findings.
The following articles have been influential in drawing up the level descriptors:
Harpaz, Y. (2005). Teaching and Learning in a Community of Thinking. Journal of Curriculum and Supervision, 20(2), 136-57.
Herron, M. D. (1971). The Nature of Scientific Enquiry. The School Review 79(2), 171-212.
The work of Galina Zuckerman, particularly her concept of the "breakthrough group", has been important when profiling classes: 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.