Introduction

In this paper I will explain how the construct of lack of closure seems to be emerging as an important theme in relation to my research into CAME Professional Development. I will use lack of closure as a way of understanding issues around primary teachers’ beliefs, understandings and practices in relation to mathematics and the primary mathematics curriculum, mathematics teaching and learning and children as mathematicians. Specifically I will use this construct to relate the terms used by teachers to the terms used by CAME researchers.

Closure, lack of closure and open-ended-ness appear to be key ideas in thinking about mathematics for all those involved in Primary CAME, although the meanings of these terms appear not to be shared. Teachers refer to the opening up of mathematics, whilst the CAME researchers refer to CAME Thinking Maths (TM) lessons having a ‘lack of closure’. Lack of closure here refers to a lack of a mathematical end result or learning point which is common to all pupils. CAME lessons aim for development for all pupils, but do not have a common mathematical objective which all pupils are expected to achieve, although lessons may have an element of ‘closure’ for some individuals (Adhami et al, 1997, under review). At a more general level this lack of closure implies a ‘fallibist’ conception of mathematics as dynamic, enquiry-driven, unfinished and uncertain (Lerman, cited in Thompson, 1992) and of the mathematics teacher as mediator rather than director of learning.

Lack of closure here is somewhat analogous to the mathematical meaning of lack of closure (Collis, 1978). Collis refers to the acceptance of lack of closure as the ability to work at an abstract level algebraically with generalised number without the closure of a direct relation to particular value. Acceptance of lack of closure is a well-defined mathematical learning objective. In contrast, the construct of lack of closure described here is much harder to specify and define referring as it does to teachers’ beliefs, understandings and practices.

I will first give a brief introduction to the project and my research. I will then look at the idea of lack of closure in relation to CAME. I will contrast this with the teacher’s views on closure and openness.

The CAME project ‘aims to contribute to pupils’ achievements and teachers’ professional development by basing classroom practice on research and theory-applicable research’ (Adhami et al, under review, p. i). CAME draws upon three strands of research and theory: research into children’s mathematical understandings; Vygotskian and social constructivist theories of learning; and, Piagetian and neo-Piagetian theories on levels of thinking. A programme of Thinking Maths (TM) lessons has been developed for use in years 7 and 8. These TM lessons are intended as a supplement rather than a substitute for the standard mathematics curriculum and are delivered by the children’s teacher on a fortnightly basis. Whilst these TM lessons are highly structured and very explicit in the guidance given to teachers, the focus is on effective teaching. Adhami et al (under review) note that CAME research ‘is based on the premise that raising standards of achievement is an issue to be addressed through more attention given to teaching, not merely through yet another large scale curriculum development project’ (p.1). The CAME model of teacher professional development and adult learning closely parallels the CAME teaching and learning model for the classroom (Adhami et al, 1997b). Research findings indicate the success of the CAME secondary programme in terms of increased pupil achievement (Adhami et al, 1997c).

Primary CAME is exploring the feasibility of extending the CAME approach already developed for years 7 and 8 into years 5 and 6. During the first year CAME researchers have worked with four teacher-researchers and the LEA mathematics advisory teacher to explore the feasibility of the approach and develop lessons for delivery by the first main cohort of twenty teachers in the coming school year.

In this first year Primary CAME is not delivering formal, separate Continuing Professional Development (CPD) as such. However, the teacher-researchers are new to the project and necessarily Research Team Meetings and school-based lesson development involve elements of CPD. This development phase has been informed and guided by the CAME project’s developing thinking about CPD.

My research is examining what makes CPD in primary mathematics education effective in relation to Primary CAME. I am using this first exploratory phase of the project as an exploratory phase for my own research. This paper is based on my initial analysis of my first interviews with the four teacher-researchers involved in Primary CAME, my research field notes on Primary CAME Central Research Team Meetings together with CAME papers and draft research memos, notes from lesson observations and school visits. The draft research memos are the personal reflections of one member of the project team made on the basis of Central Research Team Meeting, other discussions and lesson observations. The Thinking Maths Teachers’ Guide referred to (Adhami et al, 1997a) These are shared with all researchers including the teacher-researchers. Primary CAME is in an exploratory phase and the theory and practice of CAME generally are in development. Although there is broad agreement, the King’s researchers do take differing positions on some aspects of CAME theory and practice. The draft research memos that I refer to at several points are, it should be noted, one researcher’s view. Moreover the teacher-researchers do not speak with one voice in approaching and thinking about CAME and mathematics teaching and learning more broadly. Each brings different experiences, beliefs and understandings to the project. Nevertheless, in the context of this brief paper describing the construct, I feel that the contrast between the CAME / King’s researchers and the teacher-researchers is a useful one.

The CAME Thinking Maths (TM) lessons are characterised in terms of their lack of closure. At the same time they are very specific in the mathematics given to children. At the first meeting with teacher-researchers this idea of lack of closure was outlined as follows.

CAME lessons limit very strictly the investigation (not ordinary open-ended investigations) but open-ended in the points children reach conceptually. The conceptual challenges for children are very specific. … The teacher doesn’t wait for all children to finish notesheet activities ... CAME lessons are primarily conceptual - intended to challenge and promote children’s mathematical thinking skills - may be ‘sowing seeds’ for (much) later work. CAME lessons are intended to promote very specific mathematical connections and generalisations. (Central Research Team Meeting Summary, 14 November 1997)

This idea of CAME being ‘open-ended at a conceptual level’ and of ‘ideas planted without closure’ is returned to frequently at Central Research Team Meetings (DCJ, Central Research Team Meeting, 30 January 1998). The Thinking Maths Teacher’s Guide refers to ‘the big ideas or organising conceptual strands in mathematics’, to ‘pupils … struggling on the way towards the concepts … gaining insights at different levels of complexity’ (Adahmi et al, 1997a, p3). At the initial Central Research Team Meeting these conceptual strands were specifically linked to earlier research at King’s into children’s errors, misconceptions and naïve strategies and this has been a recurrent theme at Central Research Team meetings. The teacher-researchers were each given a copy of ‘Children’s Mathematical Frameworks 8-13’, a report of research in this area at King’s (Johnson, 1989).

CAME sees that a teacher’s mathematical knowledge should enable her to place the children’s learning in relation to the big mathematical ideas,

to look ahead on behalf of the pupil: that is, the aim is a very long-term one in which the pupil cannot see more than the immediate road ahead, but the teacher frames the specifics of each task so that ‘the road ahead’ does lead in the right direction. (Adhami et al, 1997a, p.5)

Where the lessons have a clear end point, this is often beyond the reach of the majority of the class. Different classes are expected to reach different points in the lesson. TM lessons aim to challenge children’s misconceptions and naïve understandings by ‘disturbing’ their thinking (MA, Notes to meeting, 12 December 1997).

There will always be occasions where the child’s thinking structures are disturbed by an episode and she may not like that. But in fact that is the way cognitive development happen! We should judge effects only at a distance. … have the repercussions of the lesson settle down in the mind first, since they are not simple ideas to be added but new ways of looking at things that needs adjusting mental structure to. (Draft research memo: MA, P-CAME Notes 2, 12 December 1997, p.3)

Children expressed this discomfort in a discussion about what they liked about TM lessons.

[Thinking Maths] makes you think

Sometimes it’s quite annoying the way you end with lots of questions

I agree. I can’t stop thinking about them in other lessons

(Fieldnotes: 3 children’s comments after TM lesson, 5 June 1998)

Questioning is a key skill for the teacher in her role as ‘manager of pupil-talk’ and promoting a classroom culture in which ‘enquiry, collaborative learning and the sharing of ideas become dominant themes and the learning of school mathematics is no longer viewed as just an individual activity’ (Adhami et al, 1997, p.10) MA outlines the process of opening up questions as follows,

The difficulty of opening up a closed question is mainly here: How to get away from looking for the right answer to focussing on the different ways possible to solve the problem? To reach any answer? When these different ways are aired, the teacher would ask questions such as: Looking at these two ways, what is common between them? In what situation this or that method is best suited? What is the advantages and disadvantages of this or that method? Which of these methods / ideas can be used in all cases and which is useful in special cases? Such questions are very demanding, and requires hints and prodding from the teacher. The difficulty is that pupil’s thinking in answering them is focussed on ideas and methods rather than on the concrete experiences. (Draft research memo: MA, P-CAME Notes 3, 16 January 1998, p.3-4, emphasis in original)

CAME TM lessons are structured around a cycle of different phases of concrete preparation, construction which may include cognitive conflict, metacognition and bridging (Adhami et al, under review). The teacher-researchers were given a presentation on this theory fairly early on (Central Research Team Meeting Summary, 5 December 97). Teachers are provided with very tightly structured lessons which ensure ‘clarity of challenge points rather than allowing varied interpretation of the task’ (Adahmi et al, 1997, p3).

CAME is underpinned by theories about children learning mathematics by learning to be and act as mathematicians. This process of ‘cognitive apprenticeship’ is supported by social interaction and collaborative learning (Brown et al, 1989). Adhami et al (under review) refer to the negotiation of mathematical meaning (Voigt, 1994), the establishment of explicit sociomathematical norms (Yackel & Cobb, 1996), and the importance of reflective discourse mediated by the teacher (Cobb et al, 1997). The teacher-researchers were given a copy of Cobb et al’s paper with a verbal annotation ‘not necessarily to read all the way through … focus on the vignettes of classroom discussion Monkeys in the tree and Double Decker Bus’ (DCJ, Central Research Team Meeting, 5 June 1998, p.3 l.7) . Classroom culture is referred to often during the Central Research Team meetings. When reviewing lessons, for example, the teacher-researchers are asked to reflect on the classroom interactions and classroom culture (e.g. Central Research Team Meeting, 20 March 1998).

Primary CAME is in an initial exploratory phase. The initial four teacher-researchers are involved in amending existing CAME secondary lessons and developing new lessons specific to primary. This exploratory phase is challenging and changing the King’s researchers’ thinking on CAME. DCJ said that the teacher-researchers’ perspective has given them some insights into what we needed to do in order to clarify the presentation of the lesson and its key challenge points. (Central Research Team Meeting, 22 May 1998, p.1, l.24) MS sees a difference between Primary and Secondary CAME.

Secondary CAME looks like a course of thinking maths lessons, but in Primary we’re developing a particular kind of skills towards general maths teaching. (Central Research Team Meeting, 22 May 1998, p.2, l.21)

In relation to TM5: Length of Words, a lesson originally developed for secondary CAME, MA commented [I am] realising only now after five years how we need to change the activity … practice proceeds realisation. (Central Research Team Meeting, 22 May 1998, p.14, l.1) Some of the detail of individual lessons is changing in the course of Primary CAME. The following exchange took place during the lesson simulation for the same activity, TM5: Length of Words.

Ursula: What about the vocabulary? You’ve used a lot of vocabulary.

MA: Range and mode are important. Spread … range and spread are the same … Should we use bulge? … Shape, picture of the spread? … Curve?

Lisa: I’m worried about using curve.

(Central Research Team Meeting, 22 May 1998, p.12, l.17)

Lisa was worried about the use of the term curve to describe a discrete variable. In fact this was dropped after discussion. The teachers’ Guide further stresses the importance of the terms range and mode (Adhami et al, 1997a, p.40). So whilst the mathematical vocabulary of mode and range are key, the informal connection-making vocabulary to be used is not fixed. This intermediate vocabulary of spread, bulge, shape, picture of spread is negotiated with the teacher-researchers (and subsequently with children when the lesson is delivered). Moreover the development of new lessons is in itself open-ended. At the end of a discussion on the development of a new fraction lesson DCJ said that it was deliberately being left open so that the teacher-researchers could use bits of ideas when trying out the activity and MS commented that this is like the CAME approach. (Central Research Team Meeting, 30 January 1998)

The notion of lack of closure espoused by the CAME project is a complex one combining an idea of mathematics and specific concepts within mathematics as open-ended, of open-ended aims and end points to lessons, yet within a tightly defined and structured lesson. This openness is underpinned by very specific beliefs and knowledge about mathematics, mathematics teaching and learning and children as mathematicians in which teachers and researchers and teachers and children negotiate mathematical meaning. At the same time the Primary CAME Project is in an extremely fluid and exploratory phase.

Openness and closure is referred to again and again by the teacher-researchers. Rhoda, the LEA maths advisor, refers to it’s contested nature in a discussion about a new activity, you’ll be surprised to hear me say this [I think it’s] too open-ended (Central Research Team Meeting, 20 March 1998).

Ursula, a BEAM enthusiast, who said she works in an investigate and talk way, perceived a conflict between CAME’s apparent closure and her ordinary open-ended mathematics work,

it’s quite hard, because having, having spent a year trying to open up Mathematics, and doing my staff meetings trying to open up questions in Mathematics and seeing what’s available in, in a sheet. Em .. I find it really quite hard to try and close it in again which is what I feel like I’m doing with something like Tournaments. With Tournaments, well, it made sense, because I knew where I was going, but some of the others don’t quite (Interview: Ursula, 25 March 1998, p.7, l.16).

There is a tension between CAME’s lack of closure and Ursula’s understanding of ‘good’ mathematics teaching as open-ended. The tightly structured lesson agenda is creating a sense of ‘closure’ and restriction for her that is at odds with her beliefs about teaching and learning mathematics. Yet for Tournaments, an early TM lesson with an uncharacteristic end-point, this made sense, because she knew where she needed to get to. Indeed all the teacher-researchers talked about Tournaments and Roofs, the first TM lesson, as being successful, because there appeared to be a definite mathematical end result (i.e. some generalisations both at specific points in the lesson and at the end). Alexandra, for example, said,

[Roofs and Tournaments were] so different to things they’d done before that it was very easy and it was, I suppose it was quite a visual thing as well somehow. Em, it was quite easy to keep in .. in my mind to do somehow. It .. it had a nice easy containment about it somehow. (Interview: Alexandra, 27 March 1998, p.17, l.21)

 

Whereas another TM lesson, Number Operations, which relates children’s arithmetical methods to different forms of representing number, was

actually quite, quite hard, quite hard to deliver and I can’t, I can’t say why now. I can’t really think back, but I know we had some uncertainty about .. about how it should be approached (Interview: Alexandra, 27 March 1998, p.17, l.21).

Containment here seems to have a sense of closure to it. The activity was easy because of its apparently clear end result (even if not all pupils reached this end point) together with its easiness to keep in and stick tightly to the lesson framework. Being the first two TM lessons, the CAME agenda for both these lessons is slightly different from the majority of CAME lessons ‘to give an opportunity for as many groups as possible to have something worth saying’ and so begin to establish a classroom culture in TM lessons (Adhami et al, 1997, p18). However, the lack of closure is firmly established in ‘seed-sowing some ideas on counter examples and generalised number’ and comparing different forms of representation (Adhami et al, 1997a, p11, 17; Lesson simulation notes 14 November 1997, 21 November 1997). So the most apparently closed lessons appear to be the most initially attractive to the teacher-researchers and the the teacher-researchers’ interpretation of the closure of the lesson seems to be overshadowing the lack of closure in the TM agenda.

At the same time the lack of closure in the thinking agenda is referred to in positive terms. Lisa, for example, commented,

we are looking within a certain amount of time to moving those children’s thinking forward, to help them to understand that not all children are going to get to the same point at the end. That’s not what it’s about. That we are trying to develop this conflict, to raise awareness, so that when children come back to that problem again hopefully they’ll be able to pick up from the last session, perhaps even think about it in between session, be very nice. Em, and that they understand that we’re not expecting .. a result or some recording from that session. That’s not what it’s about (Interview: Lisa, 31 March 1998, p.29, l.19).

Whilst Ursula argued for this lack of closure in relation to a new lesson,

That’s what’s nice about Whiskey and Water. You can leave it in the air. It’s the talk (Field notes: Central Research Team Meeting, 30 January 1998)

Although lack of closure is perceived to be at odds with some beliefs about the value of open-ended mathematics teaching and learning, the idea seems nevertheless to be an appealing one.

This idea of children reaching different points in their thinking has a bearing on the notions of differentiation. Lisa said,

But that’s why we differentiate our work, because we want the levels 2s and 3s to be able to move their learning forward. That’s why we differentiating. We’re not differentiating in a CAME lesson (Interview: Lisa, 31 March 1998, p.18, l.14).

Lisa, who elsewhere in the interview refers to TM lessons as involving and valuing the contributions of children at all levels of achievement, is referring to a more traditional task orientated notion of differentiation as stated by Alexandra,

you can approach a new subject at a whole class level, introduce it at a whole class level, but .. you need a range of extension activities for the more able so that they can progress quite quickly, em, and a lot of consolidation activities for the kind of middle, sort of middle, bottom group. (Interview: Alexandra, 27 March 1998, p.24, l.4)

Notions of depth and richness are highlighted again and again at CAME Research Team meetings and in research notes, yet are hard to pin down. Alexandra later referred to the need for extending the more able ones, but not necessarily up or, you know kind of lateral extension … , just making them think a little bit deeper about the concept or the idea. (Interview: Alexandra, 27 March 1998, p.24, l.16). This depth has similarities with the CAME definition of differentiation.

Instead, say, of planning a lesson around one major learning point hopefully to be achieved by all, the teacher has to look at the underlying agenda in a developmental way, in which there may be, for the same agenda, three or more different levels of realisation of achievement, each one of which is valid for some subset of the class. (Adhami et al, under review, p.18)

Differentiation here is by development rather than by task. Lack of closure in the thinking agenda requires a notion of differentiation by thinking outcome is at odds with more accepted notions of differentiation by task and requires consideration of mathematical processes in addition to content.

Alongside CAME’s lack of closure teachers experience a sense of freedom. Lisa referred to the CAME freeing me up to develop the classroom culture (Field notes: Central Research Team Meeting, 5 December 1998), to CAME being

a release for the teacher and for the children that we’re not, em, basing the success of the lesson on what evidence we have at the end of it, ‘cos we’re talking about the build up of experience and a way of thinking over a period of time not on short-term results (Interview: Lisa, 31 March 1998, p.13, l.4)

and to CAME giving her

the opportunity to walk around and pick up what’s going on, but also to, em, not feel restricted in who I ask so much and use that as part of the .. the teaching that we can, we can listen to all different points of view as to what people have got to say (Interview: Lisa, 31 March 1998, p.16, l.16).

So CAME gives Lisa freedom from the constraints of day-to-day maths lessons and, because of this, gives her freedom to include the whole class in the mathematics. Henrietta expanded on this point,

I think it [CAME] gives you quite a lot of freedom. And there’s not that pressure as well of thinking that they have to, I mean you do want them to get to a right point or something, but there’s not that pressure of thinking they have to know this by that date or, you know, of not necessarily going to a strict deadline. It’s, it’s more a case of learning, what they can learn in their best way, of just learning. (Interview: Henrietta, 10 March 1998, p.10, l.14).

Henrietta also referred to the freedom to make mistakes,

it gives you a sense of security to knowing how, how it went with other classes to see that you’re, you’re not an isolated case that, you know, if something went wrong, let’s say, in your class and it went wrong in another class you don’t feel as bad. You know, that’s normal. Em, but I think it helps definitely to hear how it’s gone in the other classes … but also to, to think about again if you do it again how you would do it, you know. (Interview: Henrietta, 10 March 1998, p.12, l.10).

This freedom to experiment and make mistakes is reinforced by the King’s researchers. For example, DCJ said that, although a lesson can be a disaster, it can still be successful for developing lessons and teachers’ understandings of the lesson. (Central Research Team Meeting, 22 May 1998, p.5, l.1)

Lack of closure and its inherent focus on children’s thinking instead of children’s skills creates a sense of freedom from everyday constraints and an opportunity to focus on mathematics teaching and learning.

CAME’s lack of closure necessitates a considerable amount of uncertainty as Henrietta notes,

there’s a sense of, em … uncertainty, because you’re not sure .. what’s supposed to happen sometimes. And I, I think there’s, initially there’s a nervousness and I know I’ve certainly felt nervous at times in a lesson not being sure, you know, what’s supposed to come off. But I think that the more CAME lessons you do the more confident you are. (Interview: Henrietta, 10 March 1998, p.20, l.2)

This was echoed by Alexandra,

through doing this .. you are kind of setting yourself .. up to, not, em, what am I trying to say, you’re making yourself almost vulnerable in a sense, because you’re not asking closed questions and you don’t, you’re not necessarily knowing what you’re going to get from the children and it’s, it’s not the case of guess the teacher or anything like that (Interview: Alexandra, 27 March 1998, p.13, l.12)

Lisa’s experience of the lesson simulation for TM5: Length of Words demonstrates one way in which this sense of uncertainty can be overcome. MA defined the thinking agenda for the lesson as

about distribution, shape of distribution, median, mode, range … understand[ing] that data-handling gives a partial picture of the data … [reduction of the data is necessary to handle it, but] reduction of the data removes a lot of the richness of the data and later says that ‘the comparison of the two distributions - the small and the large - is the important two star thinking activity. The three star activity is thinking about possible reasons for the small sample being unrepresentative and for the particular distribution on the large sample’. (Central Research Team Meeting, 22 May 1998, p.11, l.4 & l.18 & p.12, l.11)

During a lesson simulation, Lisa highlighted a section in Teacher’s Guide relating to the lesson which states: ‘So this is not a Thinking Maths activity where the content itself is intended to stretch the pupils.’ (Adhami et al, 1997, p.40). She said My cognitive conflict points to the pupils’ thinking (Central Research Team Meeting, 22 May 1998, p.12, l.6). As the simulation progressed she sought several times to clarify that the mathematical content of the lesson was at a low level in National Curriculum terms. Although not involving Lisa, the following dialogue gives a sense of the lack of closure as discussed in the lesson simulation,

Ursula: It is acceptable that it’s not giving you definite answers.

MA: Yes.

Henrietta: Where would you go next? … Ask them the questions?

Ursula: Data-handling is posing the questions.

Henrietta: Yes.

(Central Research Team Meeting, 22 May 1998, p.13, l.2)

At the end of the simulation she said I wouldn’t look at this on the page [of the Teacher’s Guide] and come up with this activity and refers to her cognitive conflict earlier. She said that this is very accessible to a large range of children, but that it is important to realise that the thinking agenda is at different levels for different children, although the actual maths content is at a low level. (Central Research Team Meeting, 22 May 1998, p.14, l.1) In a sense here Lisa is negotiating around her uncertainty through the lesson simulation together with an explicit discussion concerning the lesson’s lack of closure. She is still left with an uncertainty about the actual lesson will develop, but has a greater understanding of the potential children’s thinking and its relationship to mathematical big ideas.

Henrietta argued for examples of children’s work in the lesson guidance,

if you’ve not taught something like this before for me I would like to see something that’s been done before and what the children did - you know (Central Research Team Meeting, 22 May 1998, p.33, l.1).

Alexandra sees a problem with this.

I mean the problem is that you can think of the things the children might .. come up with but you can’t think of everything, can you. (Interview: Alexandra, 27 March 1998, p.13, l.27)

MA expands on this point in a research memo,

Why is it so difficult to be clear about which things in a TM lesson are indispensable as opposed to those that are important but are better left to other lessons, be they instruction and practice, investigation, or discussion lessons? This difficulty applies to teachers trying a well-tested TM lesson for the first time, but also to researchers who are observing new lessons being developed and show promise in terms of cognitive challenge. [Alexandra’s] simple response was ‘because there are so many things involved anyway’ (Draft research memo: MA, P-CAME notes 8, 30 May 1998, p.2)

Ursula referred to the variety of children’s thinking that can occur and feels that she wants to discover this range of mathematical thinking with the children.

I don’t like to expect things of the kids I expect then the kids to produce things that other kids have produced and I’d rather go with finding what they can do (Central Research Team Meeting, 22 May 1998, p.32, l.15)

She expanded on this in a description of a good (non-CAME) maths lesson,

there was a big open discussion on what the difference of money would always have to be.. in. And they were trying to find some sort of percentage or fraction or ratio terminology to sort of express how much the difference of money would always have to be. Em .. for it to work. I mean they couldn’t, and they didn’t and they lost me and they were, they knew, they kind of knew what they were doing in an odd sort of way. They just carried it along and they felt like they were getting somewhere. …. And err, that to me was quite a good Maths lesson. They were using a lot of Maths that .. I hadn’t even thought about in the first place (Interview: Ursula, 25 March 1998, p.3, l.10).

The idea that the children lost her and used maths that she hadn’t even thought about in the first place appears to contribute to the quality of the maths lesson. She went on,

now I know how much you can get out of it. I might channel it down a bit more and, and know what I was expecting each group to do. And be a bit more prepared for it. Em, I knew what the bottoms would, would not be able to do or be able to do. I knew where I was going with those. I didn’t expect the top group to take that anywhere near as far. (Interview: Ursula, 25 March 1998, p.4, l.14)

Ursula is working with the children to help her discover about mathematics and will use this knowledge to improve her future delivery of the lesson. She feels confident enough not to know everything in maths and knows that even if the children lose her she will be able to deal appropriately with the maths they come up with. Indeed, she feels that this contextualised knowledge is a better way of understanding the children’s mathematics than is any material produced in a book. She said that this knowledge should be more formally recorded in lesson write-ups,

what we don’t really have built in is any reflection time on what would happen differently next time. Em .. when perhaps if we had some time for that and to actually make alterations, official alterations or suggested alterations and things that people would do differently and things that .. particularly worked well or didn’t work well in that lesson I think maybe .. that would be more useful. Because we talk it over and don’t seem to reach many conclusions about the lesson. Em .. and we get old ground again when we review lessons. We review and review the lessons, if you know what I mean. When, because we haven’t sorted anything out properly. Em, it’s like, when we come back down to doing Roofs and Tournaments and things again, with parallel classes, I haven’t got anything extra to add to that than I had the first time. (Interview: Ursula, 25 March 1998, p.17, l.18)

There is a sense of negotiation around Ursula’s conception of the process of making alterations to lessons based on review and reflection.

The teacher-researchers experience uncertainty associated with CAME’s lack of closure. However teachers can become comfortable with this uncertainty. CAME lessons require a degree of knowledge about the mathematical ‘big ideas’ and children’s errors, misconceptions and naïve strategies in relation to these mathematical strands as well as children’s consequent constructions and thinking. This knowledge cannot simply be didactically ‘given’ to teachers. This learning process is perhaps better thought of as a process of negotiation over knowledge and meaning paralleling the negotiation of mathematical meaning with children (Voigt, 1994).

All the teacher- researchers refer to the difficulty of CAME theory.

I’m still not sure what the CAME theory is or how I would explain it. (Interview: Henrietta, 10 March 1998, p.17, l.25)

Although Alexandra now feels more comfortable with it,

The Vygotskian, zone of proximal development thing .. I think with that sheet, the construction zone activity thing, at the time I thought my god, I mean I thought yeah, this is stuff I did years ago at college a bit and it, it’s kind of hard really [laughs] and I thought I’m just not going to get to grips with it, but I mean I think the interesting thing is that now I feel quite comfortable with most of that and, em .. now, I mean I, it has meaning for me, but I think then it didn’t have a lot of meaning for me and I was thinking it was really quite, quite complex, em, thinking and what’s it got to do with teaching Maths in the primary school in a sense. Em, I mean it has given me a lot of food for thought really. I mean a lot of the ideas are very challenging. (Interview: Alexandra, 27 March 1998, p.10, l.6)

She seems to refer to a lack of closure in what’s it got to do with teaching Maths in the primary school and is typical in referring positively to the ideas being challenging. Elsewhere she referred to the difficulty of the mathematical ‘big ideas’,

I find sometimes, em, when people start talking about the higher Maths which is involved in it, I just tune out. Em, I mean a lot I think because partially because of my own Maths experience. Em, I mean I’ve got O-level Maths. I learned it in a very different way to, to this. There was lots by the time I got to the end of it I just could not understand. Em .. and I mean I, I do struggle with some of the mathematical thinking behind it (Interview: Alexandra, 27 March 1998, p.12, l.13)

However, the teacher-researchers chose to present the CAME theory in a promotional presentation of the project to LEA primary maths co-ordinators, because I think you need to know it … I mean this is the key, isn’t it, to, to what a CAME lesson is about (Interview: Alexandra, 27 March 1998, p.14, l.20). Lisa echoed this point,

I mean you need to understand the framework for how the lesson is being put together, because if you haven’t got that understanding you can’t see where it’s going to. (Interview: Lisa, 31 March 1998, p.21, l.1)

So, although CAME theory is difficult and challenging, the teacher-researchers see it as an important and valuable part of the project.

The development of new lessons highlights issues associated with CAME’s lack of closure. Henrietta and Lisa both expressed dislike of this area. Although Alexandra and Ursula both like this area, it is unclear to them what direction they are going in for these lessons,

And more input from them into our new lessons. It seems like, em .. we were allowed to choose a title, go away and try something, come back with it and, yes, it was agreed that there’s a possible CAME lesson, but … well, not really, without really having it explained to us why it was going to be a CAME lesson or, em .. well, how to improve on it .. particularly .. there and then would have been quite nice or some immediate sort of like small group work on tightening it up there and then. Em .. there seems to have been a big gap between then and now when we’ve been asked to trial it again like next week. You kind of lose enthusiasm doing that. You know you think that you’re on to something, and then you are, and then you’re not, then you are and then you’re not. And then you go to a meeting and they argue where the maths is going, which you don’t understand and it kind of loses you a bit. But, em, I mean, no, I’m all for trying it. I always like a bit more .. direction. (Interview: Ursula, 25 March 1998, p.21, l.6, my emphasis)

Alexandra expressed this in terms of openness and closure,

what I find quite hard is whether we’re satisfying CAME aims. We’re still not quite clear about that. So I guess most of all what I would feel is we need to go back now to the experts and say, you know, is this fulfilling CAME aims. Is, you know, is this sufficiently, is this sufficiently open and closing it down? Have we done that sufficiently? Em, I suspect we might have led a little bit too much for it, so that might be an issue, I don’t know. Em .. and once that’s done, I suppose, I mean the ideal would be to take that transcript maybe and pick out the bits that were possibly moving the thinking forward and sort of crystallise that somehow into a lesson plan. (Interview: Alexandra, 27 March 1998, p.19, l.20)

Alexandra’s reference to a CAME lesson being open and closed expresses the tension between CAME’s lack of closure within a tightly constrained agenda. Developing new lessons at some point requires each new lesson to be written up for other teachers to deliver. Although the lessons are still fluid, this process has an apparent finality and closure to it. This apparent closure of a lesson plan seems to be a focus for Alexandra on her uncertainty about CAME’s lack of closure.

I have used the construct of lack of closure to describe how lack of closure, closure and openness are a key area of focus for contested ideas in relation to teachers’ beliefs, understandings and practices of mathematics and learning and children as mathematicians. I have attempted to demonstrate how this construct of lack of closure can relate and describe these contested ideas in similar terms.

CAME seeks to develop in teachers an acceptance of lack of closure in relation to the teaching and learning of mathematics. This idea of lack of closure goes beyond the lack of a common end result in lessons to encompass a conception of mathematics itself as lacking closure. For the teacher-researchers lack of closure is a complex and somewhat contradictory idea. The CAME concept of lack of closure in terms of children’s thinking is distinct from their ideas of openness and closure and is at odds with more traditional task-orientated ideas of differentiation. Nevertheless this lack of closure is an attractive idea to the teacher-researchers. CAME’s lack of closure provides teacher-researchers with a sense of freedom to explore and develop the teaching and learning of mathematics without the constraints of the everyday mathematics curriculum. At the same time acceptance of lack of closure requires a degree of knowledge of the ‘big ideas’ in maths together with understanding of theories of teaching and learning mathematics associated with CAME. I have argued that this knowledge is developed through a process of negotiation. Finally, the Primary CAME Project is itself in an early and exploratory phase. This accentuates uncertainties that teacher-researcher have in relation to lack of closure.

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This document was added to the Education-line database 15 January 1999