Supporting Mathematical Thinking

Anne Watson, Jenny Houssart, Caroline Roaf, Anne Watson, Jenny Houssart, Caroline Roaf

• Publisher: Routledge
• Year: 2012
• eBook ISBN: 9781135395865
• Edition: 1
• Topic: Education
• Subtopic: Education General

To what extent do curriculum subjects, particularly highly abstract ones such as Mathematics, have their own unique special needs?

This book celebrates the work done by subject specialists in mainstream classrooms to promote inclusive practice. It describes new and creative ways of developing mathematical thinking among pupils. Each chapter demonstrates reflective minds at work, close observation of learners, willingness to understand the students' thinking process and patient commitment to students over long periods of time.

Features of the book include:

• how low-attaining students can think mathematically
• numeracy recovery
• task refusal in primary mathematics
• progression in written calculation strategies for division
• using graphic calculators with low-attaining pupils
• generalising arithmetic: an alternative to algebra
• learning support assistants in mathematics lessons
• inclusion and entitlement, equality of opportunity and quality of curriculum provision.

CHAPTER 1
Is there a substitute for experience in learning?

AFZAL AHMED and HONOR WILLIAMS

It is a good thing to experience everything oneself, he thought. As a child I learned that pleasures of the world and riches were not good. I have known it for a long time, but I have only just experienced it. Now I know it not only with my intellect, but with my eyes, with my heart, with my stomach. It is a good thing that I know this.

(Hesse, 1973, p. 78)

It is now more than 15 years since the publication of Better Mathematics (Ahmed, 1987), the report of the Low Attainers in Mathematics Project (LAMP, 1983–6). At an early stage of LAMP, it became clear to all involved that low attainment was not only a problem for pupils in the ‘bottom 40%’ attainment range. Changes in teaching approaches were encouraging all pupils to become more involved in their mathematics and to surpass traditional expectations at every level. The problem became one of under-achievement across the age and ability range.

It is interesting to note that, as indicated in the original report, it is still not difficult to find classrooms where children:

• lack confidence in the subject
• spend the majority of their time reproducing their teachers’ examples with different numbers
• answer only other people’s questions
• ask, ‘What am I supposed to be doing then?’
• fail to connect their mathematics with other subjects or with their life outside school, even when they are successful in their mathematics lessons
• dislike mathematics, seeing it as irrelevant and boring
• spend most of their time mystified.

(Ahmed, 1987, p. 3)

Some may believe that, as considerable resources have been spent on national initiatives to improve the teaching of mathematics since the report, the situation outlined above may now only be observed infrequently. In this chapter we will attempt to re-examine the essence of ideas and approaches underpinning LAMP and Better Mathematics. We will focus on what we considered was at the heart of improving the mathematical experience and confidence of pupils, that is, ‘Teachers must be actively involved in the process of professional development in order to develop their confidence, motivation, autonomy and professionalism both in providing for pupils’ needs and in terms of their own mathematics’ (p. 38). Adrienne, a teacher of pupils with moderate learning difficulties, gained confidence in enabling her pupils to find their own solutions to mathematical problems, choose their own methods of working and become independent learners. She found that her pupils had confounded many stereotypes about children with moderate learning difficulties.

In the past two years my own views of mathematics, how people learn it, and how it should be taught have changed. The stimulus for these changes has been a combination of my own experiences of doing mathematics and the discovery that my pupils could do mathematics in ways that I had not appreciated before. As I have altered the way I teach mathematics, I have found pupils have been more highly motivated and have demonstrated skills that I had not suspected they possessed.

(Bennett and Williams, 1992, p. 63)

Perhaps one of the most significant shifts in climate from 15 years ago is that the main challenge for us then was to articulate the notion of teaching being a research activity and to embed relevant theory into practice. We feel now that a greater hurdle will be to unpack the rhetoric associated with effective teaching and learning of mathematics as well as continuing to develop and support confident and competent practitioners. Some of the characteristics for achieving this have been documented by the external evaluator of the Raising Achievement in Mathematics Project (RAMP):

In supporting teachers in their professional development RAMP has enabled them to take responsibility for:

• research
• curriculum development
• enhancement of pupils’ learning.

It has enabled teachers to realise that they are the prime agents of change and that the shape of classroom practice in the future can be revolutionised by them. As such the following statements from Better Mathematics are endorsed.

Statement 16

Where research is embedded in teachers’ own experiences it holds more meaning and credibility for them.

Statement 18

No imported curriculum development exercise can be effective without working commitment and teacher involvement.

Statement 12

Teacher enthusiasm for and personal engagement in the processes of mathematics will greatly enhance the mathematical experience of their pupils.

(Ahmed and Williams, 1992, p. 18)

The importance of professional development has assumed greater emphasis more recently since the National Numeracy Strategy (DfEE, 1998) has provided all schools with an abundance of materials to support the teaching of mathematics. In addition, in Qualifying to Teach: Professional Standards for Qualified Teacher Status and Requirements for Initial Teacher Training (DfES, 2002), the focus on continuing professional development is clearly articulated:

They are able to improve their own teaching, by evaluating it, learning from the effective practice of others and from evidence. They are motivated and able to take increasing responsibility for their own professional development.

(para. 1.7, 6)

LAMP and RAMP managed to enable teachers to minimise the conflict between the practical demands of classrooms and to take a wider educational perspective on teaching and learning, particularly of mathematics. However, with the escalation in both teacher and pupil materials, many teachers find themselves in a parallel situation to that of many pupils – ‘I can’t do all of this, Miss – just tell me if I need to add or multiply’; similarly for teachers – ‘There is too much to choose from, just tell me what activity/text will work best with my pupils.’ This led us to revisit the following description of teachers’ professional development from the RAMP Report, ‘How teachers have become more discerning with regard to in-service provision, published material and other resources’ (Ahmed and Williams, 1992).

It seems, therefore, that there is a need to revitalise within the current context how teachers are able to develop themselves, as individuals or in groups, so that both pupils and teachers can:

• become confident in their own mathematical resources and in their power to understand, use and generate mathematical ideas
• extend their own knowledge by developing their own skills of self-teaching and enquiry, and
• learn to value their own and others’ expertise.

Moore and Morrison (1998) have argued consistently in their book that:

The personal development of teachers is the most important factor in developing provision to meet special educational needs. This personal development will not take place unless teachers, as individuals, participate in the development activities of their colleagues. The inability of some teachers to ‘share’ experiences in this way is, we believe, the biggest stumbling block to progress.

(p. 63)

Reflections on the prevailing educational thinking and ‘political’ context during LAMP and RAMP

Both LAMP and RAMP involved teachers in:

• working together, discussing, sharing and reflecting on classroom experiences, both successful and unsuccessful
• working in groups, focusing on specific issues, including the examination of a number of previous relevant curriculum development and research projects, both in mathematics and in other curriculum areas
• developing strategies for in-service work in their schools and with other teachers
• exploring mathematical situations and evaluating possible outcomes
• evaluating the effectiveness of commercial resources in the classroom
• exploring the creative use of calculators and microcomputers
• working with other teachers on in-service courses
• working with other teachers in their classrooms.

LAMP was one of the post-Cockcroft projects charged specifically to exemplify, through practice, some of the major recommendations and principles outlined in the Cockcroft Report (1982). The Cockcroft Committee was set up in 1979 in response to the ‘Great Debate’ on standards in education following the then Prime Minister James Callaghan’s Ruskin College speech in October 1976. Following this speech the curriculum and the assessment system as well as teachers’ performance came under increasing public scrutiny and criticism. Teachers’ freedom to decide, within a broad framework, what to teach, when to teach a particular topic and how to teach it, was being challenged. McCulloch and McCaig (2002) exemplify this freedom by quoting Bonham Carter in 1947.

Teachers are free to frame their own curriculum and to choose their own text-books within a wide general framework. This freedom inevitably develops in them a sense of responsibility and an initiative which could not exist if they were robots, subject to a rigid, centralised control.

(p. 249)

The Cockcroft Report, instead of recommending a curb to this freedom, re-emphasised teacher responsibility, the widening of teaching approaches and the importance of continuing professional development opportunities for teachers to be able to exercise the above, both professionally and effectively.

In the spirit of the Cockcroft Report and the HMI publication, Mathematics from 5 to 16 (DES, 1985), the project’s philosophy was deeply rooted in the thinking of pragmatists such as Charles Sanders Peirce (1839–1914) and John Dewey (1859–1952). According to Magee (1998), Peirce’s central contention was that knowledge is an activity and that we acquire knowledge by participating and not by being spectators:

We are moved to enquire, to want to know, by some need or lack or doubt. This leads us to evaluate our problem-situation, to try to see what it is in the situation that is wrong, or missing, and ways in which that might be put right. This scheme applies even when our problem is a purely theoretical one.

(Magee, 1998, p. 186)

Dewey considered all acquisition of knowledge as a human activity. Confronting a difficulty of some kind leads us to formulate a problem. This is often far from straightforward, but it is vital and may require several stages before clarity is achieved, and is followed by seeking possible solutions and testing them in practice. This enables us to ascertain whether we have solved the problem and can move on or, if not solved, to try and think again. Dewey considered the process applicable to all situations, although the problems, procedures, evidence and testing would depend on the context and the field of inquiry. ‘Learning by doing’ became his slogan. He regarded critique as a vital ingredient in the process and hence it is inevitably a social activity. The following example from Better Mathematics (Ahmed, 1987) attempts to explain an aspect of this process in practice.

The ‘statements’ in the report have been used as generalisations to be refuted. For example, a Project member wrote about what happened when she used Statement 4.

Statement 4

Mathematics is effectively learned only by experimenting, questioning, reflecting, discovering, inventing and discussing. Thus, for children, mathematics should be a kind of learning which requires a minimum of factual knowledge and a great deal of experience in dealing with situations using particular kinds of thinking skills.

I offered the teachers the statement and asked them to pinpoint any areas of disagreement. They had to provide concrete, personal examples to back up their point of view. No second-hand stories or myths were allowed.

After small group discussion there was a feeling that the word ‘only’ in the statement was wrong. Many teachers gave their own schooling as examples. They said they had been taught without any experimentation in their grammar schools and yet had learned mathematics effectively.

After some probing and discussion they decided that there was more to ‘experimentation’ and ‘discovering’ than ‘scissors and paste’. They identified that this had often taken place outside class time, with friends, or on their own when experimentation and reflection were internalised.

By the end of the session many wanted the word ‘only’ underlined and everyone had benefited from the discussion.

(Ahmed, 1987, p. 9)

Maintaining the spirit of LAMP and RAMP

One of the conclusions of the RAMP report was that facts and skills taught in isolation from conceptual structures and general strategies can undermine pupils’ confidence and competence. We signalled (Ahmed, 1995) that the implementation of the National Curriculum following the Education Reform Act (1988) ‘had led teachers to concentrate mainly on planning and on improving teaching approaches, with second priority being given to the study of how pupils form elementary mathematical concepts’ (p. 146). We also suggested that the rich research and development culture on teaching and learning mathematics in the UK, which was well known internationally, had lost its vigour since the introduction of the Act.

From our experience, one way of minimising the narrowing influence of an increasingly centralised curriculum on classro...

Table of contents

• Cover Page
• Half Title page
• Title Page
• Copyright Page
• Contents
• Preface
• List of Contributors
• Introduction
• 1 Is there a substitute for experience in learning?
• 2 Low attaining students can think mathematically
• Individuals
• Possibilities
• Looking forward
• Index