Geoff Wake is Professor of Mathematics Education at Nottingham University.

Contact him at Geoffrey.Wake@nottingham.ac.uk.

Have you ever paused to consider what is it that students **really** think and understand about the work that we, as teachers, ask them to do? It is quite tricky to gain insight into what it is that they are thinking as they set out to tackle the problems we pose.

Below I’ll outline the approach to **Dialogic Teaching and Learning** in mathematics, and the research we are conducting to better understand its impact. Some key takeaways:

- Dialogic Teaching involves
**meaningful discussions between students and teachers**about what students currently understand, and**cumulatively (re-)constructing their understanding** - Getting this right requires the
**development of (new) skills by teachers**to foster such discussion, and also**careful design and/or use of tasks**that challenge students’ thinking in the right way - Our research – in the context of GCSE maths resit classes – is based around five key topics, for which we have carefully designed teaching sessions over a number of iterations
- Critical to making this approach work has been
**establishing ground rules and ways of working**for students which**promote collaborative discussion**to improve the students’ understanding - Early indications suggest that this approach can
**help uncover deep misconceptions**about a topic, and that its use ensures**students feel more confident**with their mathematics and are more likely to “have a go” at a problem

As part of the work we, at the Centre for Research in Mathematics Education at the University of Nottingham, have been doing in recent years, particularly in our project Maths-for-Life (M4L), funded by the Education Endowment Foundation, we have been designing for learning that probes and builds on student understanding. Central to our approach has been *dialogic teaching and learning*.

In this approach, based on the work of Robin Alexander and Neil Mercer, of the University of Cambridge we aim to provide students with tasks that prompt students to work with each other and with their teacher(s) to have meaningful discussions about what they understand and to (re-) construct their understanding in ways that gain insight into mathematical structure.

The context of our work is that of GCSE resits – all of the students involved have had some 11 years or more of learning mathematics and sadly there still seems a lot of which they only have a tenuous grasp. The approach taken in the project includes their teachers working in collaborative clusters to a modified lesson study model focused around five lessons that are concerned with five key topics (see the inner layer of Figure 1). Each lesson has an important pedagogy (middle layer) as a central concern as well as a particular facet of dialogic learning (outer layer).

**Figure 1. The conceptual framework of the Maths-for-Life project, including dialogic learning as the outer layer.**

Alexander (2008) identifies the five principles of dialogic talk as being talk that is:

**Collective**(there is joint learning and enquiry)**Reciprocal**(participants listen to each other, share ideas and consider alternative viewpoints)**Supportive**(participants feel able to express ideas freely, without risk of embarrassment over ‘wrong’ answers, and they help each other to reach common understandings)**Cumulative**(participants build on their own and each other’s contributions and chain them into coherent lines of thinking and understanding)**Purposeful**(classroom talk, though open and dialogic, is structured with specific learning goals in view).

Such talk can be developed in a range of settings: for example, in classrooms or when students are working individually with a teacher, in pairs, small or even larger groups.

A key question for teachers then, is how to best facilitate such talk. In our experience, this requires not only the development of a range of skills by teachers so that they can support students in ways that foster such discussion – but also tasks that are carefully designed to ensure that students’ thinking is challenged, that is tasks that introduce a certain cognitive conflict or dissonance (Swan, 2005) that needs to be resolved by collaborative discussion.

This is probably best illustrated by providing a specific example from a Maths-for-Life lesson. This lesson focuses on the mathematical topic of “parts of a whole”. See Figures 2 and 3 which aim to illustrate important features of our M4L design. This particular lesson and task were developed through a number of iterations. Initially, we made a minor redesign of resources available as part of the ‘Standards Unit Box’ (DfES, 2005).

In our first iteration we had an activity which asked students to match cards to indicate how money earned by two students, Ali and Blair, might be divided expressed as either a fraction or a ratio. As Figure 2 illustrates most students (working in pairs) have the misconception that a ratio of 1:2 corresponds to the fraction ½, the ratio 1:3 corresponds to the fraction 1/3 and so on. Having allowed students to do this without any input, the teacher then introduces a third set of cards (as in Figure 3) which students are asked to match to the cards already paired up so that all three cards are consistent in representing the same division of the money.

**Figure 2. Initial materials for lesson 1 of the Maths-for-Life programme.**

With the initial design we found that students often worked individually to match the cards with little effort to come to collective/shared decisions. Bringing about a change in the socio-mathematical norms (Yackel and Cobb, 1996) or the didactical contract (Brousseau, 1997) of the classroom needed a significant design modification of the initial card-matching activity. The lesson re-design as may be apparent in Figure 3 included:

- providing students with a template structure that signalled to them that on placing a card they should
**justify**to others their thinking. The template also allowed the teacher, as they moved around the classroom, to more easily and quickly identify student thinking, thus supporting their ongoing formative assessment; - signalling to students when they had ‘completed’ the task;
- facilitating the gradual hand out of cards ensuring that the task isn’t overwhelming in the early stages.

**Figure 3. Redesign of the materials for lesson 1 of the Maths-for-Life programme.**

Important in prompting more in the way of dialogic talk in the classroom is establishing ground rules to which everyone works. Figure 4 shows some overarching values that we attempt to promote in Maths-for-Life activities and in the Table below we set out in some detail some of the “rules” we discuss with students about how we can collaborate to improve our understanding of the maths of the lesson.

**Figure 4. The values that underpin the Maths-for-Life programme.**

1. Give everyone in your group a chance to speak |
“Lets take it in turns to say what we think”. “Claire, you haven’t said anything yet.” |

2. Listen to what people say |
“Don’t interrupt – let Sam finish”.”I think Sam means that ….” |

3. Check that everyone else listens |
“What did Ali just say?.” “I just made a deliberate mistake – did you spot it? |

4. Try to understand what is said |
“I don’t understand. Can you repeat that?” “Can you show me what you mean?” |

5. Build on what others have said |
“I agree with that because …” “Yes and I also think that ….” |

6. Demand good explanations |
“Why do you say that?” “Go on … convinced me.” |

7. Challenge what is said |
“That cannot be right, because…””This explanation isn’t good enough yet.” |

8. Treat opinions with respect |
“That is an interesting point.” “We all make mistakes!” |

9. Share responsibility |
“Let’s make sure that we are all able to understand this.” |

10. Reach agreement |
“We’ve got the general idea, but we need to make sure that we all understand it.” |

**Table of Maths-for-Life ways of working.**

Although the outcomes of the research project are yet to be published, anecdotal evidence suggests that students gain in confidence when using the approach and teachers report that students are much more likely “to have a go” at working on a problem. Spending time discussing with students their understanding of the mathematics they need to know has certainly uncovered deep seated misconceptions and we have found that the type of task illustrated here can prompt students to confront their misunderstanding by working with each other and with their teacher to go back to the structure of the mathematics at what is a basic but important level.

The change that the approach requires of both students and teachers takes some considerable time to bring about but it is very encouraging that we have found students really value being asked to talk about their maths in meaningful ways and having their thinking listened to and built upon.

**References**

Alexander, R. J. (2008)* Essays on Pedagogy. *Abingdon: Routledge*. *

Brousseau, G. (1997). *Theory of didactical situations in mathematics 1970–1990*. Dordrecht: Kluwer.

Swan, M. (2009) *Standards Unit –Improving learning in mathematics: challenges and strategie*s. London:Department for Education and Skills.

Yackel, E., & Cobb, P. (1996). *Sociomathematical norms, argumentation, and autonomy in mathematics*. Journal for Research in Mathematics Education, 27, 458-4