Developing Algebraic Activity in a ‘Community of Inquirers’1

Laurinda Brown, Alf Coles, Rosamund Sutherland, Jan Winter

University of Bristol, Graduate School of Education

Paper presented at the British Educational Research Association Conference, Cardiff University, 7–10 September 2000

Abstract

We are working with teachers to develop the algebraic activity of pupils in four year 7 mathematics classrooms in three different secondary schools in the UK. Each of the researchers has responsibility for a particular strand of the work; Laurinda Brown, metacommenting, Alf Coles, teaching strategies, Rosamund Sutherland, algebraic activity, and Jan Winter, pupil perspectives. The teaching and learning styles, planning of activities and research methods are all based on looking for samenesses and differences within contexts and experiences to identify patterns which, when recognised, feedback into future action. In this paper each of us, separately, analyses for our strand the same transcript (Appendix 1, taken from a videotape of one teacher on the project) to illustrate patterns across the data. We then weave these strands into a story which describes the similarities between the different complex classroom cultures. Story: The teachers’ behaviours are contingent on their pupils’ actions and responses as they work on mathematical activities which allow the pupils’ to explore similarity and difference, often through classification. The pupils’ creativity is supported as they are encouraged in the asking of their own questions, the process of working on which leads to the articulation of complex structures and patterns supporting algebraic descriptions.

Background

In 1998/9 Alf and Laurinda worked with pupils aged 11-12 years old (year 7) on a Teacher Training Agency (TTA) project to explore Sutherland’s (1991) challenge:

Can we develop a school algebra culture in which pupils find a need for algebraic symbolism to express and explore their mathematical ideas? (p.46).

During that year they worked on ways to characterise pupils’ ‘needing to use algebra’ and saw this as linked to them being able to ask and answer their own questions. Following from this project they became interested in how to support pupils in asking their own questions, within contexts in the mathematics classroom, to create new insights into the structure of the problems they were working on. In reflection after lessons they became aware of a common strand of classification activities which seemed powerful in allowing pupils to ask questions e.g. when there were two examples to contrast or when pupils had a disagreement about what they saw or when they wrote an algebraic formula in many seemingly different ways. Alf and Laurinda started to refer both to what the pupils were doing and the activities which supported the behaviour as same/different. In turn this awareness of same/different started to inform planning for lessons. It seemed that investigating sameness led to the question ‘Why?’ of proof or ‘convince me?’ and difference led to a further exploration of structure.

Same/different

In observing the pupils’ work in response to comparing and contrasting two or more instances, what was striking was the motivation that they had for the tasks. The pupils were able to work from their own observations. St. Julien (1997) makes the connectionist claim that pattern recognition is the basic mental process and, as such, is the ground for all that we call learning (p. 275).If same/different is how we learn in the world then why is it that the model of teaching as transmission, where pupils have to rely on their memories, is standard? What would alternatives be like? The four of us were successful in a bid to the ESRC1to extend the TTA project into 3 more classrooms in the UK during the academic year 1999/2000. The aims are:

1) To create year 7 mathematics classroom cultures which provoke a need for algebra.

2) To investigate the similarities and differences developed in each of the teacher's classrooms.

3) To investigate the nature and extent of the support needed from the collaborative group of teachers to plan their classroom activities starting from the students' powers of discrimination.

4) To develop theories and methodologies to describe the complex process of teaching

and learning.

The influence of same/different on the aims for this bid is apparent and we believe that all individuals have powers of discrimination which allow them to learn, to make sense of the world they perceive, through an awareness of difference. What we experience through our actions is an interpretation based on all of our past. Therefore two people cannot see the same thing nor share the same awareness. We can communicate, however, because we can talk about the details of common experiences, exploring differences, and in doing so the gap between interpretations can be reduced. What would pupils do in response to their teachers’ planning and teaching using same/different? We were still investigating pupils’ ‘needing to use algebra’ and considered that teaching and research strategies would develop throughout the project. As researchers we would also be learning through same/different in a process described by Engeström and Cole (1997) as situated interventionism which calls attention to disturbances and discoordinations as indications of new possibilities in practice ... and is not satisfied with observing and analysing situated practices; it is engaged in creating new forms of practice (p.308).

Learning through the process of the doing of the research places us as enactivist (Reid, 1996, Brown and Coles, 1997, 1999, Hannula, 1998). In what follows we firstly describe how the ESRC project was constructed to give insights into the aims above, secondly link the four principles of our enactivist theoretical frame to methodology and methods and finally report the analysis of some of the data from the project.

Theoretical frame, methodology and methods

Over one academic year, September 1999 to July 2000 which is split into 3 terms, we (3 teachers, 1 teacher-researcher and 3 researchers) have been investigating the samenesses and differences in the developing algebraic activity in the classroom cultures of the 4 teachers through:

- working in a collaborative group, meeting once every half-term for a full day and corresponding through e-mail

- videotaping each teacher for one lesson in every half-term and researchers observing teachers in the classroom at most once a fortnight in teacher/researcher pairs

- every half term interviewing a) each teacher and b) 6 of each teachers’ pupils in pairs, selected to give a range of achievement within the class

- encouraging pupils to write a) in doing mathematics and b) at the end of an activity, about ‘what have I learnt?’: photocopies of all the ‘what have I learnt?’s are collected from each teacher as well as all the written work of the 6 pupils interviewed

- each researcher being responsible for viewing the data collected through one or more strands; metacommenting (Laurinda), teacher strategies (Alf), pupil perspectives (Jan), algebraic activity (Rosamund), and samenesses and differences in the classroom cultures, including teachers use of same/different in planning to teach through pupils using their powers of discrimination (Alf and Laurinda).

The classroom cultures were developed through the teachers sharing with their pupils that the year was about them ‘becoming mathematicians’. This structure is to support our looking at what pupils and teachers do in these classrooms. Schemes of work and organisational structures within the schools are different and it has not been our intention to change these. The content of the lessons has still to be decided by the teachers within those structures, but, during the day meetings there has been time to plan together, given those constraints, to allow pupils to use their powers of discrimination.

The first principle of enactivism is the recognition that we cannot take in the details of everything that is happening around us. We are naturally selective since there is a limited capacity to what we can attend to. What we notice and the connections we make guide our actions, often implicitly. It is in this sense that cognition is placed as being ‘perceptually guided action’ or ‘embodied actions’ (Varela, 1999, p.12, p.17).

The teachers have therefore given space within the classrooms for the pupils to work at making connections and for them also to communicate these to the whole class. The teachers make their decisions contingently upon what they perceive through their awareness of samenesses and differences within what the pupils are sharing. The teacher cannot be in control of the content nor hear and respond to everything that is happening in such classroom interaction. The teachers set up the possibility of the pupils making connections through common boards (see Fig 1.1) used for sharing questions, conjectures, homework etc.

The second principle of enactivism (adapted from Varela, 1999, p.10) is the belief that we are what we do. It is our actions and perceptions that make us who we are and these are dependent on the whole of our past experiences. Consequently the data collection on the project has been done over time. The teachers worked with pupils who had just started at a new school to establish new behaviours in their mathematics lessons related to ‘becoming mathematicians’.

The third principle we adopt is that we take multiple views of a wide range of data:

The aim here is not to come to some sort of ‘average’ interpretation that somehow captures the common essence of disparate situations, but rather to see the sense in a range of occurrences, and the sphere of possibilities involved (Reid, 1996 p.207).

Multiple views of the data are captured through the different strands which the researchers are investigating. We take one incident and interpret it through different strands and also tell stories of the changes that are happening over time. One powerful way of working with multiple views is through the use of the videotapes, using short extracts and talking through the details of what we see. What seems important is that overlapping themes emerge over time.

As a fourth principle, from these overlapping and interconnecting themes, theories which are ‘good enough for’ (Reid, 1996) a purpose emerge:

theories and models ... are not models of ... they do not purport to be representations of an existing reality. Rather they are theories for; they have a purpose, clarifying our understanding of the learning of mathematics for example, and it is their usefulness in terms of that purpose which determines their value (Reid, 1996 p.208).

There is no sense of there being a ‘best’ theory for our work. Our theories are ‘good-enough for’ our actions. The ideas that we continue to think about and use as teachers and researchers are those that inform our practice. We recognise what is useful for the practice of teaching by what is happening in the classrooms. What is not useful does not happen. We see ourresearch about learning as a form of learning (Reid, 1996 p.208) where our learning is gaining a more and more interconnected set of awarenesses about the teaching of mathematics. These are not the explicit products of this research, however, what is crucial here is the process of developing such theories and actions in the classrooms.

To illustrate this way of working each of us will analyse a video transcript (Appendix 1) from the point of view of our own strand ( Laurinda, metacommenting, Alf, teaching strategies, Jan, pupil perspectives, and Rosamund, algebraic activity) to show the richness of the multiple views of the data. Each of us looks through this common transcript to pick up themes running through all the data. In the conclusion we will weave together these interpretations to tell a story which not only seeks to describe the complexity of the four classrooms but also gives guiding principles for these teachers future practice, a theory which is currently ‘good-enough for’ our continuing work as teachers and as researchers.

Strand 1: Metacommenting - Laurinda

In setting up the cultures in the classroom we did not think it was possible to consider a community of mathematicians but it did seem possible to describe the practice as a ‘community of inquirers’. An aim of the project was to create a community of inquiry (Schoenfeld, 1996) within which the pupils participate in mathematical activity where: the real ‘authority’ is not the Professor - it’s a communally accepted standard for the quality of explanations and our sense of what’s right and the community is: dedicated to exploration and sense-making (p.16).

Bateson (1972) says that: human verbal communication can operate and always does operate at many contrasting levels of abstraction. (p.177). One of these levels he calls metacommunicative which is when: the subject of discourse is the relationship between the speakers. (p.178). We were interested in the teachers talking explicitly with the pupils about their relationships in the ‘community of inquirers into mathematics’. We took as evidence of the existence of a community of inquirers, the pupils asking their own questions and commenting themselves on the practice of the classroom and their practice as mathematicians. In establishing the culture in the classrom the teacher notices behaviours in the pupils which can be commented on as being mathematical and reports these to the class, because, although we believe they are all naturally inquirers, at this stage they do not know the culture of doing mathematics which they are entering (e.g. Pupil: How many different answers? Teacher D: That’s a very mathematical question (DO22, 29/9/99). We have come to call explicit metacommunicative messages, such as that of Teacher D, metacomments. To begin this process the teacher shares with the pupils, in their first lesson, a ‘purpose’ (Brown (with Coles), 1997) for them for the year as ‘becoming a mathematician’.

‘Another one’

Looking through the transcript (Appendix 1), however, there are no obvious meta-comments as there were at the start of the year. This is Video 4, Teacher A, 29/2/2000 and is therefore quite a late transcript. The culture has become established in which the pupils and teacher are used to collecting images on the board to discuss. Explicit metacomments are not apparent. The use of the phrase ‘another one’ is used in contributions 2, 29, 30, 77. We have noticed, as a research group when watching other videos of Teacher A, that he often asks ‘Another one?’. Here it is Pupil 2 who is offering using the same phrase (contribution 2), although it takes until contribution 29 for this to be heard. The discourse in the classroom is centred around ‘same/different’ which comes from the pupils (e.g. contributions 4 and 6) and, although not explicitly commented upon in this extract, ‘another one’ has a history behind it where it seems to be used to indicate ‘another, different, one’. In some cases, where a pupil offers a different one in their way of viewing the world, in discussion (e.g. contributions 4, 6, 10) there might be a decision to count this difference as the same (a x b is the same as b x a although the rectangles certainly look different (Fig. 1)). Classifying is a common activity in this classroom and questions arise both from the pupils (e.g. contribution 72) and the teacher (e.g contributions 13 and 61).

Getting organised

What did these metacomments look like at the start of the year? In an interview during the evening of the day when the first lesson of the year with the Year 7 class took place, Teacher D reported saying to them:

... about becoming a mathematician ... that what I mean by that is if you’re thinking mathematically then it’s about noticing things about what’s around you and it’s about writing things down about what you notice and often what you’ll be writing down will be a question about something which you’ve noticed - maybe you’ve seen a pattern and a question that mathematicians often ask is ‘why?’ so you might spot a pattern and think about ‘why does that pattern work?’. Make a prediction maybe based on that pattern. Say why you think that pattern will continue (Teacher D, First lesson interview, 9/9/99).

From the start of the year Teacher D metacommented on these behaviours, for instance, in the following extract he comments on the ‘very mathematical question’:

Teacher D: What were you doing last lesson?

Pupil: Carrying on from before, 4 digit numbers.

Teacher D: Were you working on a question?

Pupil: How many different answers?

Teacher D: That’s a very mathematical question. Any others?

Pupil: I’ve been looking at whether I can know which column any number goes in.

Pupil: I’ve been seeing whether the conjectures worked.

Teacher D: Try to be organised about how you try (DO2, 29/9/99).

What the pupils picked up on in their writing about ‘what have I learnt?’, however, was‘getting organised’ (here Teacher D suggest that pupils ‘try to be organised’) which itself became able to be commented upon and which the pupils themselves used to describe what they were doing:

Teacher D: Can someone explain what we were doing last time?

Pupil: Trying to get 3 down.

Teacher D: Can you expand on that?

Pupil: We were trying to be organised doing 6, 5, 4 ... down (DO2, 13/10/99).

Although Teacher D is initially commenting on the asking of questions (the pupil’s question ‘How many different answers?’ is described as a ‘very mathematical’ one) this question ‘How many different answers?’ is itself an example of one where the pupils have to ‘get organised’ or work systematically to answer it.