Dialogue and Learning in Mathematics Education: Intention, Reflection, Critique
Is learning possible without talk? Is it possible to talk without learning? Are some kinds of talk better for learning than others? What kinds of talk happen in your classroom? Thinking about my own teaching, I can suggest various kinds of talk that might arise in a classroom:
- rehearsed or routine talk (three threes are? nine),
- organising talk (you’ll be working in groups of three...),
- chat (...did you see Concorde yesterday?),
- commentary (that’s interesting...),
- ‘thinking talk’.
The last kind of talk on my list is more difficult to exemplify in a couple of words, so let me try to describe it. Thinking talk has an absorbed quality. Those involved are absorbed in the work they are discussing together, such as a mathematical problem, for example.
Thinking talk also has a timeless quality. For the participants, there is no awareness of the passing of time. Thinking talk has an intense quality, perhaps indicated by careful listening. And thinking talk is punctuated by moments of silence, the silence of a group of people all thinking very hard about something.
Helle Alrø and Ole Skovsmose of Aalborg University in Denmark have also been thinking about the relationship between talk and learning. In their recently published book Dialogue and Learning in Mathematics Education, they draw on their research in secondary mathematics classrooms to develop and illustrate their ideas on talk and learning, as well as on the wider role of mathematics education in society. Their ideas are extensively illustrated with extended transcript data taken from Danish mathematics classrooms. The transcripts show students working on extended ‘investigations’ of several days (in the UK we might call them projects). These investigations include an exploration of egg trading, in which students have to grapple with the possibility of salmonella, the cost of having eggs tested and the need to make a profit. The work is driven primarily by the scenario (e.g. the egg business), rather than by learning objectives. The egg investigation leads, however, to significant mathematical inquiry into, for example, probability, in the context of the risk of salmonella occurring in a particular box. The investigation also leads students to consider mathematically influenced aspects of the society in which they live. How likely are they to catch salmonella from eggs? How reliable is the information given on boxes of eggs in the supermarket? What are the different conflicting interests involved? Do, for example, supermarkets tolerate some degree of risk in the interests of making a bigger profit? There is an interesting moment when a group of students are composing a slogan to put on the packaging of ‘their’ eggs. They realise that ‘salmonella free eggs’ would be false, but fear that ‘almost salmonella free eggs’ 'doesn’t sound very groovy'. Eventually, they adopt the more neutral slogan ‘free range’.
The authors begin by developing the idea of dialogue as a key feature of what I called ‘thinking talk’. The essence of their notion of dialogue is talk in which participants:
- are engaged in a shared process of inquiry,
- listen to each other and explore each other’s perspectives,
- are prepared to take risks, through, for example, sharing uncertainties, or trying out ideas they are not sure about
In discussing the egg slogan, for example, the students are working together to come up with an agreed form of words. In the process, they also negotiate criteria for the suitability of their slogan; they should not be dishonest, for example. They listen to each other’s ideas and make sense of the perspectives which inform those ideas. And the participants are prepared to make suggestions, such as ‘salmonella free’, which can be discussed and rejected, but which contribute to finding a way forward.
In the later part of the book, the authors address the relationship between dialogue and learning, which they relate to intention, reflection and critique. They argue that participants' intentions need to match within a group, and that participants need to be responsive to each other’s changing intentions. In the egg group, the students engage in a dialogue to find a slogan, for example. Learners' intentions do not always have much to do with mathematics, as they illustrate with an example of a group who are ‘resistant’, preferring to joke and show off instead work on their investigation. Reflection involves participants explicitly considering their thoughts, feelings and actions, questioning themselves and each other. When ‘salmonella free’ is suggested as a slogan, for example, the first response is ‘they are not salmonella free’, indicating that the student has reflected on what has been suggested. Finally, critique involves participants considering the assumptions, responsibilities and interests that underlie mathematical aspects of society. As the authors observe following the students decision to reject ‘almost salmonella free’ in favour of ‘free-range’: 'we have witnessed a breakdown between scientific interests and economic priorities.'
For Alrø and Skovsmose, critique is a crucial element of mathematical learning. They argue that mathematics education has a responsibility to develop critically aware citizens. Learning mathematics is about more than arithmetic procedures or algebraic manipulation. Learning mathematics concerns learning to analyse and critique the way society is organised, using increasingly sophisticated, but often hidden mathematics. Our daily life depends on insurance, banking, electronic security, government statistics, scientific tests, medical data and so on. Students need to be able to bring a critical awareness to the information they read in the news, on product packaging or on websites.
I agree with this analysis, which raises questions for teachers of mathematics, who, the authors note, play a key part in sparking dialogue between students. What is our role in supporting our students to develop a critical awareness of the society in which they live? How can we create opportunities for dialogue? What is it we do, that has a group of people thinking through talk?
Richard Barwell
University of Bristol’s Graduate School of Education.
Dialogue and Learning in Mathematics Education: Intention, Reflection, Critique
Helle Alrø and Ole Skovsmose
Kluwer, 2002
284pp
Review Sections
See also
