# 12 Thinking Maths Teaching style

The cognitive agenda of a Thinking Maths lesson requires teaching that creatively combines elements of investigation and instruction. The focus is on pupils’ thinking, and how to orchestrate pupils ‘talk in the classroom flexibly in a trajectory towards higher order thinking. Since adolescents tend to develop their thinking by internalising some better understanding from a peer who is at a slightly higher level (see page 8,The Vygotskian contribution) each Thinking Maths lesson is designed to produce as many and as wide a variety as possible of these fresh insights and competencies.

**Act 1** whole class concrete preparation: the teacher’s role is to show the task to the pupils and then get them to describe or re-express what the task is and possible ways to achieve it. The peer-peer mediation is the insights pupils offer to the whole class. The teacher records these on the board and encourages questions. The pupils need to understand how they are to collaborate in Act 2, and what is required from them in Act 3.

**Act 2** small group work: pupils develop ideas that they can now show and explain to others. Worksheets are to focus pupils on the challenge (they are not given any value for assessment) and pupils may make jottings as reminders for the discussion in Act 3. Individual pupils may present a “completed skill’ that they have arrived at as a result of the group’s contribution. The teacher’s role is to observe groups as they work and remember what ideas they are generating, in order to invite groups to present their work in the final section in a logical order.

**Act 3** whole class sharing: each group quickly reports to the rest of the class the high points of their discussion, and others comment (`I don’t agree with what John said, because…’). Pupils should use their own language style, which is often closer to the thinking level of their peers than the teacher’ language.

Since there will be different levels of mental development in the class, and different depths of interpretation, the teacher should ask the less able children to present any solutions to the lower level aspects of the task first.Then each subsequent group only presents ideas that have not already come up, keeping a brisk pace. Ideally every idea or strategy from the groups’ discussions then becomes available to everyone, increasing by a large factor the number of opportunities for each to complete their skill in a concept.

**Developing collaborative learning**

Collaborative learning skills can be learned in all lessons and even illuminate drill and practice lessons. Thinking Maths lessons are more effective when this learning culture is already established in the class, but can also aid in its development.

Ensure that pupils take turns and listen to each other in small group work and whole class discussion. Record pupils’ ideas on the board as a basis for further thought. Group pupils carefully, Avoid either having pupils at the extremes of ability range working separately from the mainstream, or placing the extremes together. The best pairing in ability terms is ‘lower with middle’ and ‘middle with higher. However, thinking ability is often not the same as achievement in tests that are narrowly focused on skills and memorised procedures. It is often better to start with current natural pairing and grouping, with judicious switching of pupils to avoid ‘bad chemistry. Natural groups often combine manageable mixed ability with a complementary aspect, e.g. where a dextrous and quiet student works well with an imaginative but boisterous and less organised student.The class should accept these pairings and grouping for study purposes, and that there is as much value in changing partners as in settling in with a steady one.

**Classroom layout**

Physical arrangements have a bearing on interactions. For Thinking Maths lessons the three types of classroom interactions – student-student, student-class and student-teacher-at-the-board – all need to be accommodated within the constraints of the room. To facilitate peer-peer mediation the preferred layout is with pairs or groups of 3-6 pupils partly facing each other and partly facing the rest of the class when the teacher is at the board.

The diagrams show two possibilities in a classroom with tables. Where these layouts are not practicable the teacher may need to improvise to ensure that:

- pairs or small groups of pupils can work together
- pupils are not obstructed in their view and access to the board
- pupils can see and be seen by the rest of the class when volunteering ideas.

**Judging the success of a lesson**

We can never be sure of what actually goes on inside anybody’s mind. Our task is to increase the probability that some positive development occurs.

- Whenever parts of a lesson involve most of the class in meaningful discussion, with many pupils volunteering ideas to their peers or to the rest of the class, a collaborative learning culture is developing in that class.
- Where the pupils offer a wide range of ideas related to the same mathematical/logical strand a significant impact on pupils ‘thinking is likely to result.

Thinking Maths has a positive effect on pupils’ motivation — demonstrated through their engagement with new ways of working. Over the longer term they come to accept and appreciate the cognitive challenges and demands of the lessons the ongoing development of thinking powers. At the end of the two years, the effects on pupils’ mathematical achievement are the result of their generally enhanced capabilities.

**A new culture of learning**

The extent to which pupils benefit from Thinking Maths lessons is often related to the classroom culture — their expectations, habits and attitudes to learning acquired through prior experiences in mathematics lessons. Thinking Maths offers scope for the teacher to build up a new culture where enquiry, collaborative learning and the sharing of ideas become dominant themes and school mathematics is no longer seen as an individual activity, where the student expects to be trained in the application of formal rules and procedures. While initially the lessons must be implemented within the current culture their structure and the challenges to be resolved have been designed to contribute to a conscious change of that culture, providing a social framework that pupils gradually internalise.

The culture of learning emerging over time from such classroom interactions around an appropriately challenging mathematical cognitive agenda is an indefinable outcome of Thinking Maths, especially important since it counteracts prevailing negative attitudes to mathematics as a subject, and to what it means to learn it.

The CAME approach emphasises the role of the teacher in the development of mathematical reasoning in pupils, rather than as a technician providing the fragmentary teaching of skills. Two more recent research projects,3’4 using the CAME approach with younger children, have shown how non-specialist maths teachers in primary schools find the approach empowers themselves as much as their pupils. Many teachers rethink their mathematics from scratch, so it becomes better grounded in their intuitions for the first time.

The Thinking Maths approach, which gives pupils time to develop and combine their ideas with others, leads gradually to higher levels of thinking and lasting understanding. In the same way, the professional teaching skills needed for this approach mature in a process structured around the activities, developed and integrated in the curriculum content and the ways of teaching.Teachers reconstruct for themselves, on behalf of their pupils, the underlying notions of number, algebra, relations in shape and space, data handling, probability, and correlation. How they handle these through language is crucial. As they work with materials with each other, listening to what other groups of teachers have constructed, and reflecting on the underlying mathematics of the lessons, so too do they gain insight into how to manage their pupils’ learning.