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Inquiry structures

Language acquisition as the first phase of inquiry
Sonya terBorg
 blogged about using Inquiry Maths prompts with her primary class in Idaho (US). The post describes how the class carried out a preliminary inquiry into concepts and language related to angles. The pupils then conducted an open and collaborative inquiry into the prompts by applying the language acquired earlier.

You can follow Sonya on twitter @terSonya.
Structure of an Inquiry Maths lesson
The lesson structure (right) was developed in a community of teachers from different subject areas who were studying inquiry teaching. The school had 100-minute lessons, which explains the scale on the left. After the initial responses to the prompt, the teacher is expected to order the questions (possibly on the basis of complexity). As students become more experienced with inquiry, they could play a greater role in deciding the order in which the questions are addressed.
An alternative lesson structure
This lesson structure was devised by
Satvia Bahia (a secondary school maths teacher) for a presentation about Inquiry Maths that she was giving to trainee teachers. It shows the teacher deciding the level of structure (for individual students or the whole class) as the inquiry progresses, although the decision about the type of inquiry might have been taken before the lesson based on the profile of the class

The 4D-cycle of mathematical inquiry
This inquiry cycle was devised by Professor Katie Makar (University of Queensland). Each part of the cycle is described in more detail on this page from Thinking through Mathematics. Further elaboration appears in Professor Makar's 2012 chapter 'The Pedagogy of Mathematics Inquiry'*.

* In Gillies, R. M. (Ed.). Pedagogy: New Developments in the Learning Sciences. New York: Nova Science Publishers, pp. 371-397.
IB diploma programme
One aim of the IB diploma programme for mathematics is to promote inquiry approaches in which students "investigate unfamiliar situations, both abstract and real-world, involving organising and analysing information, making conjectures, drawing conclusions and testing their validity." In order to achieve the aim, teachers are encouraged to use the flow chart (right) when planning inquiry lessons.