Section Three: How to use Let’s Think Maths
(Let’s Think Maths) is all about allowing children to work out what is going on for themselves but making sure that at the end of a session you, the teacher,
bring all those ideas together again, guiding but saying as little as possible yourself.
A primary teacher
Let’s Think Maths lessons are designed to help children develop the general thinking skills that are required in mathematics. Children are first asked to describe the task they are presented with in their everyday language, and to suggest strategies to carry it out. Then they are asked to work together in small groups, with the goal of presenting their solutions or problems to the rest of the class. In whole-class discussion, individuals and groups are invited to contribute their ideas and finally, the class is asked to summarise and reflect upon each of the important points.
The teacher’s role
As much as possible, the teacher’s role is just to guide the discussion rather than to lead it and say what the magic formula is.
You’re making yourself almost vulnerable in a sense, because you’re not asking closed questions and you don’t necessarily know what you’re going to get from the children.
Primary teachers talking about Let’s Think Maths
The role of the teacher in Let’s Think Maths lessons is threefold. Firstly, the teacher must be able to look ahead on behalf of the child. The mathematical and thinking aims within a conceptual strand are long term and the teacher frames the specific challenges so that children develop in the right direction. Of course, children can and do constantly surprise us with their inventive and imaginative mathematical constructions. Mathematics teaching is a learning experience for the teacher as well as the children.
Second, the teacher is a manager of children’s discussion. Those involved in the development of Primary CAME believe that much of the development of children’s thinking comes from internalizing, in a flash, some better understanding they see or hear from another child who is just beyond the level they have reached. Each lesson is structured to produce as many and as wide a variety of these fresh insights and competencies as possible. As well as reporting their own thinking, children must listen and respond to the reports of others’ thinking. Ideally, the teacher steps back from the discussion when children are confident enough to question and discuss their constructions with one-another directly. Thus, this role goes beyond good questioning to promoting good listening, questioning and dialogue by and between children.
Third, the teacher needs to encourage a classroom culture in which enquiry, collaborative learning and the sharing of ideas become dominant themes. They should discourage mathematics learning being viewed as just an individual activity where children expect to be trained in the application of formal rules and procedures. The development of this kind of classroom culture is facilitated through the creation of an environment in which the contribution of each child is valued and taken as valid for some purpose. The aim is to develop each child’s mathematical thinking to its fullest potential rather than simply to achieve ‘correct’ answers.Property of Let’s Think Forum – not to be copied or reproduced without permission
Lesson structure
Each lesson activity is centred around one or two mathematical ‘big ideas’, although the intention is for children to ‘struggle’ on the way towards these ideas rather than be taught specific knowledge or skills. Lessons last from 60 to 90 minutes.
Every lesson begins with an introduction, where both the language and context chosen should be within the grasp of the whole class, that is, it should be a familiar context that requires only early concrete operational thinking. An example might be recognition and single-step mental operations on familiar objects that may or may not be mathematical. The main purpose of this introductory activity, which is called the preparation phase, is to ensure that all children have sufficient grounding in the context, and confidence in the use of any technical terms introduced at this point, for the work that follows.
The preparation phase is followed by an activity that requires a higher level of thinking (mature concrete in most cases). The children know what the problem is about, know what the words mean, but need to shift their thinking to a higher level in order to cope with the new problem. This is typically done in small groups. This phase is known as the construction phase, and it is here that the children’s informal strategies or ideas often lead them to see some contradiction or cognitive conflict. This is followed by the children coming together in whole-class discussion where they share their constructions and reflect upon their thinking.
A cycle of the three phases – preparation, construction and sharing – is referred to as an episode and a typical episode consists of some introductory work with the whole class, followed by small group work and then returning to whole-class discussion to share ideas.
Each lesson consists of two or three episodes, with each episode designed to provide challenges at a higher level of cognitive demand than its predecessor. The last phase of the first episode generally provides the preparation for the second episode, and so on. The last episode should also include time for reflection.
Different classes will reach different points in a given Let’s Think Maths lesson. The aim is to promote children’s thinking and their reasoning skills, rather than to reach a particular common end point. The teacher should, however, ensure that at least one episode is completed, and time is reserved for a last reflection phase – this is crucial and needs to be included, regardless of how far the class has progressed.
Further descriptions of each of the three phases in each episode are given below. The concrete preparation phaseProperty of Let’s Think Forum – not to be copied or reproduced without permission
Most activities start with a problem set well within the children’s experience, which we call concrete preparation. The understanding barrier is set deliberately low for this initial phase, which often involves invoking links to previous ideas or previous Let’s Think Maths lessons. During this phase, the teacher needs to get the children involved in the problem by asking questions, sharing diagrams on the board, and so on.
The construction phase (and cognitive conflict)
Construction is the process by which children generate more powerful strategies or concepts. In Let’s Think Maths lessons, some construction may take place in the concrete preparation phase or later, during whole-class discussion, but most takes place when groups of children collaborate on tasks and together develop ideas to solve a problem.
Let’s Think Maths lessons often lead children to produce solutions that contradict their existing ideas. For example, in Lesson 12: ‘Roofs’, children may produce what appears to be a perfectly good rule for making a ‘roof’, only to find it disproved by a non-confirming example. This sort of surprise, or cognitive conflict, is an important element in cognitive development, promoting further and more sophisticated construction – it challenges the children to produce a higher-level strategy that does work.
Features inherent in school-level mathematics provoke cognitive conflict. Much of school mathematics is based on the idea of consistency, and recognising an inconsistency provides an opportunity for provoking higher-level thinking. These are the opportunities on which Let’s Think Maths capitalises.
The sharing and reflection phase (and bridging)
During this phase of the lesson, children reflect on their work and share the ideas they have constructed. Reflecting on one’s own actions or one’s own thinking is referred to as metacognition. It is important to encourage this reflection: without it children may easily lose the gains that they have made. In the Let’s Think Maths model, metacognition is primarily accommodated for in the class sharing of the many insights, difficulties and partially or completely successful strategies that have occurred in the construction phase(s) of the lesson. Metacognition leads naturally to, and is often helped by, the process of bridging.
Bridging is usually part of the reflection phase in the final class discussion, and normally takes place near the end of the discussion. The teacher asks the class to collaborate in finding ‘good names’ or phrases to describe what has been achieved and why or how it worked, and then records them on the board in the children’s own words The children are then asked to suggest other contexts where the reasoning pattern or the mathematical relation that underlies the Let’s Think Maths lesson might be applied. This leaves a ‘handle’ in children’s long-term memory with which the concept can be retrieved on another occasion, and applied to a new context.
Implementing the Let’s Think Maths approachProperty of Let’s Think Forum – not to be copied or reproduced without permission
Learning mathematics is a shared activity for adults as well as children. The more it is treated as an individual activity, the more difficult mathematics becomes. This makes it difficult to grasp the approach simply by reading the pages of a teacher’s guide. For this reason, the Primary Let’s Think Teachers’ Guide is only a small part of the professional development programme.
In the Let’s Think professional development programme, the teacher’s first experience of a lesson is in a simulation, during which they engage with the children’s possible constructions in the context of the mathematical ‘big ideas’.
They then begin to teach the lessons. After teaching the lessons, teachers reflect on the experience in the context of the Let’s Think theoretical basis for teaching and learning: increasing levels of cognitive demand and encouraging children to work together and share ideas.
Teachers have found it helpful to observe others teaching, to be observed by colleagues, and to team-teach lessons. They will often teach a lesson a second time to an older, younger or parallel group of children in order to better understand the lesson ‘flow’ and potential variations in differentiation between individuals and groups.
In summary, the Primary Let’s Think programme includes the teaching of the Let’s Think Maths lessons, teachers observing and team-teaching Let’s Think Maths with other teachers, and teachers talking and reflecting upon the lessons, together with the more formal Primary Let’s Think professional development sessions. Each of these aspects contributes to accelerating the learning of all those involved in the programme, children and teachers alike.
To find out how to access the full professional development programme, email [email protected]