Lack of Closure in Mathematics Teaching and Learning: Understanding Primary Teachers Beliefs

Lack of Closure in Mathematics Teaching and Learning: Understanding Primary Teachers’ Beliefs and Understandings in relation to the Primary CAME Project

Jeremy Hodgen

Paper presented at the British Educational Research Association Annual Conference, Queen's University of Belfast, Northern Ireland, August 27th to August 30th

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

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

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.

CAME and Lack of Closure

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.

The Teacher-Researchers and Lack of Closure

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