When to tell: the inquiry teacher's dilemma

Emily Wood who is studying for an MEd at the University of Cambridge (UK) contacted Inquiry Maths. As part of her research, Emily aims to promote a growth mindset by teaching mathematics through inquiry.

However, Emily was in a quandary. As the research class had low prior attainment in the subject and were new to inquiry, she had decided to plan a structured inquiry lesson. Emily’s concern was that the structure would invalidate the inquiry. How could the teacher, Emily asked, provide structure in a way that meant the students were still learning through inquiry?

This is a common concern in the inquiry classroom. Surely, the argument goes, teachers should not 'give the game away' by telling students what to do and how to do it.

At the same time, sticking to the role of 'guide on the side' can lead to rising levels of frustration when students do not know how to proceed.

This is the inquiry teacher's dilemma: tell and dilute the inquiry or do not tell and risk students' disengagement.

Research

Norwegian secondary school teachers involved in a research project about the introduction of inquiry-based learning faced the same dilemma. 

As the students in the research classes were unaccustomed to taking the initiative, the teachers foresaw problems in using the prompt x + y = 7 without guidance. They decided to prepare hints and suggestions on cards, but did not want to give “unnecessary cues” that led directly into graphing the equation.

The wording of one of the hints caused confusion in all the video-recorded lessons. Students failed to interpret the instruction to “draw the number pairs” as an invitation to plot x- and y-values as coordinates on a graph. When students in one of the lessons grew frustrated at not knowing what to do, the teacher was reluctant to intervene; instead, he hoped students would discover the alternative form of representation for themselves.

Should the teacher have just told the students to graph the equation or was it worth holding out? Certainly, in reviewing the lesson, the teacher felt that the period when students were stuck went on for too long.

Factors

In the Inquiry Maths model, the regulatory cards, through which the teacher and students can communicate, offer a means for resolving the dilemma. Students can use the cards to indicate they are stuck and ask the teacher to introduce a concept or point out a connection that opens a new pathway through the inquiry.

Even then, is it incumbent on the teacher to tell? The decision will rest on two factors:

• Knowledge and experience

If the knowledge required to make progress in the inquiry exists in the class, the teacher should aim to draw it out through discussion and questioning. By assessing students’ initial responses to the prompt in the question, notice, and wonder phase, the teacher can also determine whether students are able to derive the new knowledge from their current understanding.

If students are experienced inquirers, the teacher should draw upon their knowledge of directing inquiries to negotiate a way forward.

However, if the students possess neither the knowledge nor the experience, the teacher should tell them what they need to know to make progress.

• The learning intention

The learning intention of the inquiry is also an important factor in deciding whether to tell or not.

Was the point of the inquiry in the Norwegian research that students discover a connection between algebraic and graphical forms? If it was, the teacher was right to hold back.

However, discovery is never a good aim of inquiry, although it might be an extremely exciting and rewarding by-product. If students do not discover the intended knowledge (or only a few do), then the teacher is in an impossible situation. Telling becomes a sign of failure, which can lead to disappointment and even, as in the case of the research, to teachers blaming students’ for their lack of curiosity.

The intention of using the Inquiry Maths prompt y - x = 4 is not to discover an alternative form of representation, but rather to understand how the graph changes as the equation changes. The field of inquiry involves exploring the graphs of other equations. As a prerequisite for entering the field, students have to be told about the graphical representation if they cannot see the connection.

Open inquiry

Students do not learn to conduct open inquiries by being thrown into open inquiry. The results of such an approach are likely to be disheartening and bring learning through inquiry into disrepute with students and teachers alike.

Before open inquiry, students have to learn how to inquire. That will often involve the teacher telling students what to do and telling them about concepts relevant to the inquiry. However, the telling is not an end in itself as it is in traditional mathematics classrooms; rather, the telling opens up the field of inquiry for students to explore in a purposeful way.

The teacher does not tell as an ‘expert’ who aims simply to pass on knowledge. Rather, the teacher tells as an active participant in the inquiry and, in so doing, acts as a representative of the culture, history and practices of mathematics.

Andrew Blair, November 2024