# An example in Mathematics

Introduction

Each LTTM! lesson uses a logical hierarchy within a topic and across topics. There is a logical and conceptual hierarchy of mathematical reasoning, with a range of thinking levels that is designed to match the typical ages in the group. But the hierarchy within a lesson is a segment of a much wider conceptual and reasoning hierarchy, other segments of which are addressed in other thinking lessons, using other contexts.

Working through this hierarchy of mathematical reasoning, often in fine and subtle grading in progressive formation of concepts and handling of misconceptions, is only possible in a collaborative classroom environment. For some experienced and innovative teachers this is part of effective normal teaching whether or not they are consciously aware of the underpinning theoretical background in social psychology. For many other teachers a developmental process of practice and reflection on practice is needed in a supportive school culture. We will explore this issue of professional development further in Chapter 8.
All Thinking Maths lessons (including LTTM! and PCAME) have two or more stages (or ‘episodes’) set within a context that is familiar to pupils in the given age range. Successive episodes demand gradually rising levels of thinking, starting from a relatively low ‘floor’ at the start of the lesson to a relatively high ‘ceiling’ at the end. The levels of difficulty of the ‘floor’ and ‘ceiling’ can be adjusted in different classes to maximise the amount of work done in the children’s Zone of Proximal Development, that is to ensure that they are challenged a little beyond their current capability. An episode can be seen as a self-contained thinking cycle of learning that takes from 10 to 30 minutes.

After setting the scene (concrete preparation), the cycle starts with a whole class discussion of a puzzle, a story or another type of challenge, often with some readily available materials or pictures for the pupils to handle. The aim here is to start to generate some cognitive conflict: an inconsistency to resolve or an intuitive pattern that seems difficult to describe. With younger children this is often done ‘on the carpet’, talking about the story or materials and agreeing what to call the objects on which they are to focus. This leads to pair or small group work to come up with further ideas in children’s own words. Here the challenge generates collaboration and social construction. The third act of the episode is the time for sharing, sifting and refining of ideas (whole class social construction and often metacognition), and possibly linking them to some formal mathematics (bridging). Where appropriate, this leads to some generalisation and focusing on a new challenge to be worked on in the same or subsequent session.

Depending on the responses in the particular class, the teacher may extend one of the episodes and leave the next episode for another day. Alternatively, the teacher may decide, because of the ability profile in the class, to condense an episode and go immediately to one with a higher-order thinking challenge appropriate to most of the class. Normally lessons with half the class at a time, e.g. with 15 pupils, end up with higher quality interactions to explore the thinking agenda in less time. Of course, you then need support to cater for the other half of the class. The teacher also benefits greatly from conducting the same lesson twice in succession, through recognising differences in the maths activity and the children. This is the teacher puzzling out reasons for differences and adjusting his or her expectations.