The IB cycle of mathematical inquiry
In preparing to run a day-long Inquiry Maths workshop at an international school in Geneva, I reacquainted myself with the International Baccalaureate (IB). I was reminded of how the IB curriculum is, by some way, the most advanced and progressive in the world. This comes from the Mathematics Guide for the Diploma Programme:
"In the IB classroom, students should regularly learn mathematics by being active participants in learning activities. Teachers should therefore provide students with regular opportunities to learn through mathematical inquiry."
Inquiry processes are even explicit in the assessment criteria, which means that the common objection to inquiry - that it is not part of the final examination - cannot be levelled at the IB. The assessment criteria for the Middle Years Programme (MYP) require students to become become risk-takers, inquirers and critical thinkers; in the Diploma Programme (DP), inquiry approaches, such as making conjectures and testing their validity, make up one of the six assessment objectives.
Furthermore, the 10 attributes of the IB learner profile seem to constitute a comprehensive list that all inquiry teachers could embrace.
Mathematical inquiry according to the IB
Given its commitment to inquiry, the IB's treatment of mathematical forms of inquiry is brief. In the DP subject guide we find a short paragraph about the need for inquiry, problem solving and critical thinking. Teachers are encouraged to take approaches and use tools "intrinsically linked to the IB learner profile, which encourages learning by experimentation, questioning and discovery."
While there are common features of inquiry across disciplines, any generic recipe, such as the IB's learner profile, risks losing sight of the specific nature of inquiry in mathematics.
The guide addresses this issue by presenting a cycle of mathematical inquiry in the form of a flow chart (see above). The cycle's five steps - four processes (in rectangles) and one conditional operation that leads to a decision (in a rhombus) - are presented without explanation. We review each step below.
Explore the content
The first step combines a process (explore) with the field of inquiry (content). In its simplicity, the step masks a complex relationship. The source of the content (curriculum, teacher or student) and the students' first contact with the content (through a teacher's question or a prompt) are both important - one might even say crucial - in motivating the class to explore.
There is also the issue of what 'explore' means for different content. While the step suggests that the nature of 'explore' does not change, this is not the case. The form of exploration has to be worked out in conjunction with the content for each new inquiry. Exploration could mean, amongst others, generating more examples of the same type, testing different types of cases or delving into the structure of one case.
Make a conjecture
A conjecture normally arises in the classroom when students notice a result or property occurs more than once and speculate that the pattern will hold in the future. How is the conjecture related to the content of the first step? Does this step mean that the only legitimate content for inquiry is that which will yield a pattern and is, therefore, susceptible to a conjecture?
Test the conjecture
The process by which a student determines the conditions under which a conjecture is true or not has the potential to lead to deep mathematical learning. Unfortunately, the binary choice in the IB cycle - either accept or reject the conjecture - restricts that potential.
The either-or approach to conjectures is not how mathematicians think. They might accept the conjecture in forming a generalisation, but only within certain constraints. For example, a straight line is the shortest distance between two points in Euclidean geometry, but that is not necessarily the case in non-Euclidean geometry. When students reason that "conjecture A is true within constraint B, but is not true outside that constraint", their inquiry acquires greater depth.
'Justify' is a curious term as a discrete step in mathematical inquiry. A student might justify a decision, but the proof of a generalisation requires rigorous deductive reasoning. Whether such reasoning rests on structural analysis or employs algebraic tools, it amounts to far more than the term 'justify' suggests.
The final step implies that inquiry is never-ending. There are always new contexts in which students can test a generalisation; there are always connections that can be made to other fields of inquiry. While this is true, is it helpful to have 'extend' as a separate and necessary part of mathematical inquiry? After all, students also need to learn how to decide when an inquiry has reached a satisfactory conclusion.
Rigidity and regulation
In presenting its cycle of mathematical inquiry as a flow chart, the IB promotes the view that inquiry follows a single structure. However, any predetermined approach denies students the possibility of navigating the inquiry in other ways.
In particular, there does not seem to be any intent for students to develop their ability to regulate (plan, monitor and evaluate) inquiry and, thereby, become more sophisticated practitioners of mathematical inquiry. Such sophistication would see students learning to adapt their approach as new situations arise, deciding for themselves when to explore, test cases, make conjectures, develop generalisations or reason deductively.
Indeed, the implication of the IB's cycle is that, as students become more experienced inquirers, they continue to apply the same structure - only more fluently.
The absence of student regulation from the IB's model compounds the problem of how new concepts and procedures are introduced into the inquiry. With a mechanism such as the regulatory cards, students can draw on the teacher's knowledge if their inquiry is stuck.
However, for the IB, this is no problem at all. In its discovery model of inquiry, the IB expects students to come to an understanding of the content by following the steps in the flow chart.
Returning to the list of attributes of an IB learner, the omission of self-regulator - a student who is capable of directing the inquiry process - is consistent with the IB's fixed step-by-step cycle. Rather than learn to regulate their own mathematical activity, students are required to apply and internalise the order laid out in the flow chart.
In contrast, the Inquiry Maths model encourages students to direct their own mathematical inquiries flexibly using the processes unique to the discipline. They take decisions about their activity based on the position of the inquiry, not the next step of a flow chart. This makes them more conscious of themselves as learners and of the nature of mathematical inquiry.
Andrew Blair, August 2022