Independence through structure
Independence has long been an aim of education. Ofsted (the inspectorate of English schools) describe outstanding maths teaching as “nurturing independence”. Yet, what exactly is meant by ‘independence’? Do we mean 30 students sitting in silence solving problems? If students can solve it on their own, then surely they must know it – with ‘it’ normally defined as a particular procedure.
In Inquiry Maths, independence is not defined like that. An independent inquirer decides how to learn. The student might indeed choose to inquire alone, but the situation might call for collaboration with peers or for an explanation from the teacher or another student. Independence is the ability to make and justify decisions about how to learn in different situations.
Of course, to demonstrate such independence, students must be involved in deciding the direction of the lesson. In the early development of Inquiry Maths, I would ask classes “What shall we do next?” Students were baffled by this question and would either view me with a look of complete incomprehension or would carry on exploring the prompt aimlessly. Surely, I, as the teacher, was there to tell them what to do.
When students do have an idea of how to continue, a class discussion on the direction of the inquiry lacks depth. Ann Macdonald (an Inquiry Maths teacher) told me this week that when she asked pairs for ways to proceed with an inquiry, students latched onto one early suggestion about needing to practice (the addition of fractions) and repeated it around the room whether they needed the practice or not. Another common response is for students to say whatever comes to mind at the moment without any forethought about the implications of the suggestion.
I have learnt through these experiences that to develop independence, students, paradoxically, first need structure. The freedom to express an opinion is insufficient; students must learn how to respond to that freedom before they can use it effectively. Inquiry Maths lessons now include a phase in which pairs of students are required to select a
regulatory card that regulates their learning.

Inquiry teachers tell me that the cards make a huge difference. When Ann tried another inquiry with the cards, the regulatory phase was far richer because, once chosen, the cards ‘lock in’ each suggestion. The cards reveal a great deal about students’ level of mathematical thinking. A pair that chooses the card Inquire with another student might be exhibiting an anxiety about the freedom of the inquiry; a student who selects Decide on the aim of the inquiry aspires to set a mathematical agenda.
The cards can also reveal students’ attitudes about maths. While observing an Inquiry Maths lesson, I was intrigued by one student's reply when asked for his card: “I don’t know what this is about. We’ve done it." I interpreted this response as the student saying: “Look the prompt says one thing, but we’ve found a counter-example, so it must be false.” For that student, maths problems had one answer. There was no ambiguity, and no room for conjectures or for circumstances in which the prompt could be true.
Finally, the cards help provide evidence for developing independence. Students might spontaneously start to select sequences of cards when one is insufficient to map out their projected path through an inquiry. They might, with the teacher's encouragement, make up their own cards or even decide what to do before the cards are given out. At this point, students do not need the structure provided by the cards; they have become independent mathematical inquirers.

Andrew Blair
October 2013