Levels of Inquiry Maths

The greatest danger for a teacher embarking on inquiry with a class 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.
  
Level of Inquiry Maths

   Profile of class*

Phases of inquiry

Questions and observations

Regulation

Inquiry pathways

Results of inquiry

Structured inquiry The class is new to inquiry. It is not easy to identify students who show curiosity or take initiative in maths lessons. Very few, if any, are prepared to take risks. The teacher supports students in asking questions and making observations about a prompt. (This phase, for me, is the principal feature of an inquiry lesson and, as such, is 'non-negotiable'.) 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.
Guided inquiry 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 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. For example, one pathway might involve students learning a procedure; another might involve students in exploring more examples; and a third might involve finding examples that support a conjecture or finding a counter-example. 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.
Open inquiry 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. 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. Multiple pathways are in evidence. Groups or individual students present results that the teacher could not have predicted at the start. These might include novel approaches and findings.
   
* The profiles of classes are not related to ability or prior attainment. In my experience, '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 open inquiry. They require structure or guidance just as much as bottom sets.
   

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.

Andrew Blair
January 2015