An important aim of Inquiry Maths is to develop students' ability to direct, monitor and regulate their own learning.
The model encourages students to make decisions on the approaches the class will follow, the goals it will set itself and on the selection of mathematical tools it will employ. A key question for students to answer is "What shall we do next?"
When the teacher first asks the question, school students often find it baffling. "Surely, that’s what you're here for – to tell us what to do," they respond. The idea that they have a role in directing learning can seem strange.
In Inquiry Maths, students learn to regulate their activity by justifying the choice of a card to their peers and, in turn, engaging with their peers’ justifications. They also benefit from the teacher’s rejection of non-mathematical suggestions and arbitration between contending legitimate ideas.
Regulatory cards help students structure their thinking. Stopping the lesson and asking the class (in pairs) to discuss the selection of a card has the advantage of making students conscious of how they are regulating their learning.
Using the cards in the classroom
In feedback about orchestrating inquiry lessons, teachers often report that the most difficult part is the regulatory phase. How do the cards work in practice?
The recommended way to use the cards is as follows:
(1) After questions and observations about the prompt, the teacher asks: “What shall we do now?” Pairs of students choose one card and try to justify their choice. The cards are physical objects for students to look through and manipulate.
(2) The cards are also displayed on the board. As students read out their choice and give a justification, the teacher ticks the card on the board. If the students have hand-held devices, they might interact directly with the board.
(3) The teacher's response to the the choice can take different forms:
If the choice is clear - for example, 'inquire in a group' to 'find more examples' - then the teacher would agree to that. If the questions have revealed that the majority of students lack procedural knowledge to make progress, she might choose to instruct the class first or ask a student to explain.
If the class is split into two or three distinct groups - for example, 'find more examples', 'practise a procedure', and 'test different types of cases' - then the teacher might divide the class along those lines and build in time at the end for each group to report back.
If students select cards about how to inquire only, which is typical of classes new to inquiry, then the teacher has to decide on the line of inquiry. This should be done publicly by 'talking-aloud' about students' questions and observations. Furthermore, the teacher should make students aware of the type of cards they have chosen and next time require them to choose two cards - one about how to inquire and one on what to do next.
If students select a variety of different cards, then the teacher might suggest a sequence of actions involving as many of the choices as possible or permit individuals to follow their own paths, especially if the class is experienced in inquiry.
(4) The teacher could use the cards more than once during an inquiry, particularly if it runs over two or more lessons. Using the cards at the end of a lesson has proved effective in structuring the next lesson.
(5) As students becomes more experienced in inquiry, a class might add cards to create their own unique set. The teacher could also remove cards if the class resorts to picking one card automatically without giving thought to the specific situation.
What regulatory cards should a teacher use?
A teacher should use cards that reflect the profile of her or his class and serve the aim of the inquiry. If the students are novice inquirers, the focus might be more on using social cards that focus on how to inquire. If students find the development of inquiry pathways challenging, then the teacher might use a set of cards in which inductive regulatory statements predominate. These cover the processes of exploration, identifying patterns, developing conjectures and forming generalisations.
Finally, to promote explanation and proof, the teacher might include some deductive cards that emphasise reasoning about mathematical properties and structure. The number of cards might increase as students become more experienced, perhaps using the six 'starter' cards with a class new to inquiry and moving on to the 20 mixed cards when the class has become more experienced.
How do you know when the students no longer need the cards?
The cards are not meant as a permanent feature of inquiry lessons. The aim is for students to become more conscious of how to monitor and regulate their learning and, thereby, develop into self-directing inquirers. There are clear signs when the class is becoming independent:
Students choose different cards by reflecting on their prior learning. In effect, students differentiate tasks in the inquiry for themselves, rather than defer to the teacher's direction.
Students are able to select two cards, combining an activity with how they will carry it out.
Students are able to select a number of cards that constitute multiple steps or a sequenced pathway through the inquiry.
Students make up their own cards. Teachers might encourage this by putting blank cards in the sets of cards.
Students make suggestions about structuring the lesson without waiting to be asked.
Developing independence and initiative
The regulatory cards are designed to support students in learning both what constitutes legitimate mathematical activity and how to direct mathematical inquiry. Ultimately, the teacher hopes the cards become surplus to requirements as students plan, monitor and reflect without them. When they take the initiative and regulate inquiry spontaneously, students demonstrate independence from the teacher and their peers. The first step in developing such independence is often taken by filling out one of the blank cards in a set of regulatory cards. The student is thereby indicating a wish to follow a different direction to the choices offered by the teacher. If a new suggestion has relevance for other inquiries, the teacher can add a card to create a bespoke set for each class.
The three examples that follow come from year 7 students at Haverstock School (Camden, UK) who had been offered the basic set of six cards. The first comes from the steps inquiry. During the question and observation phase, there was a claim that the difference between the outputs would always be the same. Aisha and Keana, in proposing Trial and error until you find a pattern!, wanted to change the operations in order to see if the difference was the same in other cases. Their regulatory statement, in pointing towards the mathematical concept of generalisation, had a wider relevance to other inquiries and was added to the class set of cards.
The second card appeared during the area inquiry. In wanting to Work out all three prompts, Somaya’s immediate intention was to measure the dimensions and work out the areas of the three shapes on the prompt sheet (although she later used the card Ask the teacher or student to explain because she did not know how to calculate the area of the circle). The teacher did not add Somaya’s statement to the class set because it only had meaning in that specific moment.
The third card records Jabril's desire to change the prompt in the number line inquiry. It has both a general application to other inquiries (in the regulatory statement Change the diagram) and an immediate relevance to the particular inquiry. The regulatory statement can be added to the class set, while the diagrams propose a new direction at that specific moment in the inquiry - that is, linking the numbers in a different way to the prompt.
However, the diagrams have an even more immediate importance as the foundation of generalisation and proof. In creating a learning episode around the card (by, for example, displaying it on the board), the teacher can point out the inconsistency in the diagrams. The last one decreases in size as it goes up, while the first two follow the prompt by increasing in size. This has implications when the class attempts to generalise and use algebra to prove an observation. When a variable n stands for the lowest number, students might become confused about how to arrange expressions (n + 1, n + 2 and n + 3) if they have not created their diagrams in a consistent way.