Many Right AnswersLearning in mathematics through speaking and listening

Els De Geest

01—02

© The Basic Skills Agency, 2007Commonwealth House1–19 New Oxford Street London WC1A 1NU

All rights reserved. No part of this publication may be photocopied,recorded or otherwise reproduced, stored in a retrieval system ortransmitted in any form or by any electronic or mechanical meanswithout the prior permission of the copyright owner.

ISBN: 1 85890 501 3Design: elfen.co.ukPhotography: Media Trust ProductionPublished March 2007

Acknowledgements

Foreword

About this project

Part 1: generate

Beliefs and perceived issues about teaching low-attaining students

Why use speaking and listening in mathematics learning?

Collaborative work

Creating the desired classroom culture:

A. Building confidence and self-esteem

B. Developing new Selves/new behaviour

C. Many right answers

Part 2: experiment

Bev’s lesson: Dominoes

Judith’s lesson: Mystery

John’s lesson: 3D shapes

Corinne’s lesson: True or false?

Lynne’s lesson: How many cakes?

Brett’s lesson: Breadcake

Part 3: reflect

Speaking and listening develops understanding

Speaking and listening allows us to understand students’ understanding

Speaking and listening triggers reflection and meta-cognition

The role of disturbances in learning

Another argument for using speaking and listening

The importance of mathematics learning

Low attainment, intelligence and speaking and listening

Continuous Professional Development and speaking and listening

Revisiting issues of changing practice – possible effects

Some final reflections

Bibliography

Contents

2

3

4

8

11

12

14

22

22

23

23

24

24

28

28

32

33

34

36

38

39

40

42

44

The Basic Skills Agency would like to thank all the teachers, consultants, academics and others who participated in this project. It is their willingness to share their knowledge and ideas that allowed these materials to be developed. We hope these materials reflect the enthusiasm, professionalism, deep thinking and openness that we encountered.

The author is responsible for any interpretation and conclusion drawn from research, educational theories and from contributions made by the participating teachers in this project.

Acknowledgements

02—03

What particularly struck us during the making of this film was that some teachers told us in their ‘experimental’ lessons, with more speaking and listening activity, they were surprised by the increased engagement of some of their weaker pupils; the pupils that often did not seem to get a lot from their maths lessons. Others noticed that speaking and listening provided a quick and effective method of assessment for learning. Some noticed the special benefits of oral rehearsal for learners of English as an additional language.

We have produced this resource for teachers of mathematics in secondary schools to encourage them to embrace speaking and listening as a regular and integral part of their practice. Primary school teachers and teachers of adults, however, will find the film and booklet equally pertinent, providing a structure within which to analyse their own classroom practice and to seek to refresh their use of speaking and listening as a tool for learning.

We are thrilled that Professor Celia Hoyles OBE, the government Chief Adviser for Mathematics in schools agreed to become involved in the creation of the film that accompanies this booklet. Equally, the participation of the National Centre for Excellence in the Teaching of Mathematics and the Secondary National Strategy played a crucial role in creating and disseminating the materials.

Most of all though, it was the teachers of mathematics who made this piece of work really come alive. Their honesty, eagerness to learn and their professionalism were inspirational. To be willing to discuss what they saw as ‘areas for development’ in their own practice, to declare this to the world and then to be filmed ‘warts and all’ filled us with admiration.

Their lessons – inspirational as they may be – are not meant to be seen as perfect. They simply portray hard-working practitioners in a range of schools developing their practice. And then coming back together willing to discuss what went well and not so well.We hope that when the film is seen in secondary schools it will encourage other departments to engage in their own small piece of action research, following this model and using some of the tools that can be downloaded from our website to emulate the process.

Foreword by Carol Taylor Joint Director — The Basic Skills Agency

This project, ‘Many Right Answers: Learning in mathematics through speaking and listening’, was initiated to address two issues in the teaching of secondary mathematics:

1. speaking and listening and its relevance to engaging the disengaged; and

2. the use of speaking and listening by teachers of mathematics as part of their Continuous Professional Development (CPD).

Although aimed at teachers of secondary mathematics, many of the concepts, ideas and discussions are likely to apply equally to primary teachers.

We also aspired to offer something more. We wanted:

• to show the reality of teaching and learning: the ambivalence, the complexity, the jeopardy, the emotions;

• to explore the complexity of teaching and learning in mathematics;

• to portray and recognise teachers as highly trained and reflective professionals and to be informed by their thinking and practice;

• to encourage teachers to try out new ideas inspired by discussions with other practitioners;

• to be informed by research and educational theories; and

• to provide a focus on speaking and listening as a tool for learning mathematics and as a tool to make sense of complexity.

The passion, the professionalism, the commitment and exchange of ideas we wanted to capture are exemplified in the accompanying film; this booklet sets out to explore and develop them further.

04—05

About this project

What we mean by ‘speaking and listening’

There are two main types of spoken communication in classrooms: teacher/pupil talk and pupil/pupil talk. In this publication we focus mainly, but not wholly, on the latter, but the two are often inter-dependent.

Through the Secondary National Strategy, a range of publications has been made available to schools, including materials on the use of teacher talk and questioning techniques. See Teaching and Learning in Secondary School Pedagogy and Practice Pack (2004). Also, there are support materials and information in Literacy Across the Curriculum (2001), Literacy and Learning (2004) and Guidance for Senior Leaders (2004).

Approach

In these materials we offer descriptions and incidents from discussions and from the practice of participating teachers. We use research to show that these teachers are not unique in their thinking and in their responses. In this way, without being overbearingly academic, we are offering a pragmatic, though scholarly, evidence-based approach. We hope using literature in this way will support, challenge, expand or refine ideas and probe thinking.

This project worked in different dimensions. At one level it was about doing things to produce these materials. At another level it was more complex: the content of these materials would result from participating teachers’ beliefs, thinking and practice, and their development over a period of time. We believe teacher development is a continuum, not a one-off, and does not necessarily have a starting or a finishing point.

To reflect the organic nature of learning and of CPD we use the concept of ‘Story of the Teacher’ as a structure achieved by:

1. Bringing the teachers together for an initial day of discussions and voicing of ideas: the generate part of these materials.

2. Teachers trying out some of the ideas in their classrooms: experiment.

3. Another day of discussions and reflections: reflect.

We envisage that schools (or groups of schools) could come together in a similar way, using this three-part structure as a framework for CPD and to initiate discussions about speaking and listening and pedagogy. The questions and prompts we used during the discussion days to trigger reflection are available on our website: www.basic-skills.co.uk. You can also find information there about the supporting literature that was sent to participating teachers between the two discussion days.

This booklet has been designed to meet the needs of a wide range of readers. The page design draws attention to the more academic and theoretical information in grey and blue. In yellow, you will find quotations from the participating teachers. Summaries of the main points of each section are written in italics.

Part 1: generateThis section describes the discussions that took place when teachers came together before trying out ideas in their classrooms.

08—09

Part 1: generate

Beliefs and perceived issues about teaching low-attaining students:

The teachers discussed and reflected on their existing practice of teaching low-attaining students. They were asked: ‘Are there special requirements from the teacher?’ ‘In what way is it different from teaching high-attaining students?’ ‘Should it be different?’

The main strands of their contemplations were that:

1. Teaching low-attaining students is different. Teachers have to be more skilled to teach students who are struggling. What is required is a thorough understanding of the mathematics that is being taught, of pedagogy and an aptitude for empathy. Some teachers really enjoy it.

‘Teaching low-attaining students can be frustrating if they don’t “get it”.’ ‘You need, as a teacher, a good under-standing of what you are teaching.’ ‘With low-attaining students you have to think carefully.’

2. The belief and convention that mathematics and mathematics learning are hierarchical often influences the way mathematics is taught. An outcome of sticking to this belief with low-attaining students is going back to the basics, doing the same things year after year. In doing so, students are offered a poor diet of basic number work, practise simplified mathematics and experience little mathematical thinking. Progress is restricted. Watson, a mathematics educator whose research interests include working with low-attaining students, points out that low attainment thus may become the result, not only the reason, for teaching this way (2006, p103).

The teachers contemplated whether it would be possible to ‘skip’ levels, teach students something that is not the ‘the next curriculum step’ and come back to these lower levels later, possibly through subordination. Ernest, a philosopher in mathematics education, asks this same question: whether mathematics really has a unique and fixed hierarchical structure (1991, pp232, 237). He argues that there are strong theoretical reasons why this does not exist, one being that formal mathematics ‘is made up of a myriad of different theories and theory formulations, each with its own structure and hierarchy’ (Ernest, 1991, p233). He concludes that the mathematics curriculum must not represent mathematics as having a unique, fixed, hierarchical structure: ‘A fixed hierarchy cannot describe student learning’. (Ernest, 1991, p241).

‘Perhaps what we should do is allow low-attaining students to work across all sorts of other areas of mathematics that are considered ‘too challenging’. Maybe we will discover they can do all kinds of interesting thinking.’ ‘We should allow ourselves to branch out and not go back to basics all the time.’ ‘Low-attaining students do the samethings over and over again.’

‘It is common to hear teachers saying in staff rooms that “these learners cannot think for themselves” and “these learners have short memories” or “these learners cannot concentrate”. … It occurred to me that learners may not be being offered opportunities to think, remember and concentrate so get out of the habit of doing this in mathematics lessons, indeed they may not know how to do these things in mathematics lessons.’ Watson, 2006, pp102, 103

3. Students should learn to deal with the complexity of learning, of mathematics and of life. The teachers agreed that simplifying mathematics was not desirable from an educational perspective and that allowing complexity and making connections within that complexity is valuable, if not essential.

The benefit of teaching mathematics in its complexity is also something that is noted in the analysis of the TIMMS study, which compared mathematics learning in seven nations (Hiebert et al, 2003). The research showed that where learning was best, teachers in those countries preserved the complexity in the mathematical concepts and methods. The mathematics was not simplified.

‘We often make mathematics too simple.’ ‘We want to give students confidence and are in danger of over-structuring tasks. This takes so much of mathematics out of the question.’ ‘Students often do not understand how they got to that answer. We give them too much structured support and no opportunity for mathematical thinking.’

4. We have to think carefully about our roles as educators. For example, if students do not concentrate well, do we then give them tasks that allow for this short concentration span, or is it our role as educators to offer tasks that will help them to concentrate for longer? Similarly, if students do not ‘cope’ with complexity, do we offer them simplified mathematics that will cater for this, or is it our duty as educators to help them in developing ways of working with complexity, through developing ‘tools’?

Teaching low-attaining students requires careful and skilled thinking by teachers. Careful consideration has to be given to what mathematics is offered to low-attaining students. It is tempting to ‘help’ these students by making things clear-cut for them, by simplifying the mathematics. Sticking to the belief that mathematics and mathematics learning is hierarchical results in going over the basics time and time again. The danger of spoon-feeding students in an over-structured way is that they might never learn how to deal with the complexity of learning, of mathematics and of life.

‘Taking the complexity out of mathematics is like taking the mathematics out of mathematics.’Prestage et al, 2007, p88

10—11

Part 1: generate

Why use speaking and listening in mathematics learning?

The teachers discussed what teaching tools could be used to maintain complexity in the mathematics learning in the classroom. How to avoid the ‘Miss, Sir, I do not understand’ remarks? How to avoid the students’ expressions of bewilderment, and sometimes sheer panic? The teachers thought speaking and listening might be an effective tool because students learn more when they vocalise their thoughts and explain how they are working something out in their head. Students are more likely to be actively involved. Thus, speaking and listening helps students in their thinking, it encourages them to voice their opinion and to challenge something they think is wrong.

These ideas resonate in research findings. Swan (2006) reports how Vygotsky developed his theory that language is essential for developing intelligent learning and that speech is used to solve problems. Vygotsky observed that young children talk together much more when faced with challenging problems. Speaking allows the children to organise their thoughts and actions (Swan, 2006, p67).

‘Students are learning as much from each other as from me.’ ‘Speaking and listening is about interactive learning. It’s a two-way process.’ ‘With speaking and listening tasks there’s more opportunity for giving students challenging work because they – and you – have more chance to respond.’ ‘Using speaking and listening gives students time to respond.’ ‘Time allows students space to think and develop mathematically.’

Speaking and listening helps students in their learning because they have to vocalise what they are thinking. Students become more actively involved.

‘As with the use of multiple representations and multiple strategies, the features of communication and collaboration have been associated with meaningful learning.’ Silver et al. 1996, pp485-486

12—13

Part 1: generate

Collaborative work

The teachers noticed that learning through speaking and listening in a collaborative setting seemed to help the students in giving meaning to the mathematics, and in giving meaning to their learning of mathematics. Students act as teachers to each other. They can talk to each other in ‘student language’. They exchange ideas and information, which are then considered, supported, refined or rejected. Reasoning can be examined because it is said aloud.

Several pedagogical and psychological theories claim that learning is shaped by social interaction and that it is indeed this social element which is crucial in learning. Vygotsky coined the term ‘social constructivism’. Swan explains how Vygotsky came to this conclusion: ‘Vygotsky claims that people do not develop through repetition or discovery methods, but through social interaction. The child enters into relations with the situation not directly, but through the medium of another person (Swan, 2006, p68).

Other authors also support using collaborative learning. Silver et al. (1996, pp485-486) stated that thinking processes in collaborative activities differ greatly from those in more individual activities. Cuoco et al. (1996, p379) advocate a collaborative classroom culture where students can feel free to ask questions and comment on each others’ work because it is useful to formulate thoughts and exchange ideas.

‘Learning is a social activity. You learn by talking and explaining and making sense with the people around you.’ ‘When students speak to each other they can teach each other in a language that teachers don’t have. It breaks down a technical language barrier.’ ‘Speaking and listening helps learning because students do become involved – they are part of what is going on.’

Speaking and listening takes place in a collaborative setting. As a result, benefits associated with working collaboratively are also enjoyed.

14—15

Creating the desired classroom culture

How to go about creating a desired classroom culture? What are its elements? What should we be looking out for? The teachers highlighted three central features:

A. Building confidence and self-esteem

The instilling of confidence and self-esteem in the classroom is vital. This involves making students feel good about themselves as learners and as people. Feeling good about your own learning involves getting recognition that your thinking is being valued and considered worthwhile. Learning to accept and deal with not knowing what to do is also part of that.

To achieve this culture, teachers can do many things, at different levels.

• Stimulating and inviting contributions from all students by asking students to write their answers on wipe-boards and showing these; asking students to work in pairs and be ready to share with the class their partner’s answer; giving students time to think and to prepare an answer and then ask all students for their answers, even if some are repetitive. This makes students feel valued for their learning, for their opinions and thoughts.

• Offering students ‘get-out’ opportunities. This is very important for building trust in the teacher, trust in the supportive nature of the classroom culture and trust in the belief that being fallible is OK. This can be achieved by allowing students to say ‘pass’, ‘can I ask someone else?’ or ‘I need some more thinking time’ when they cannot answer a question.

• Making kind personal comments when students come into the classroom.

‘If you build up students’ confidence then they might be able to go back to the subjects they experienced failure in and have another go and feel that they actually can manage it this time.’ ‘Low-attaining students are poor at expressing themselves. I pick out elements of what they say and focus on those and inspire a bit of confidence that way.’ ‘Students feel good about themselves for giving a contribution.’

To instil confidence and self-esteem, students need to feel good about themselves as learners. Making positive comments, setting up systems that invite contributions from all students and offering students ‘get-out’ opportunities when they cannot answer a question can all help students to feel good about themselves as learners.

Part 1: generate

‘Behaviour can be altered, but it takes time, persistence and imaginative methods. Old habits have to be replaced by new ones over time. A training approach (with clear expectations and rewards) might be effective. Anything which disrupts old expectations (including expectations of the teacher’s behaviour) is worth trying. Time has to be given in lessons to establishing new habits, and time must be given over several weeks for them to become habits.’ Watson et al. 2003, p13

B. Developing new Selves/new behaviour

The second central feature for creating a desired classroom culture concerns the nurturing of new Selves or identities with new sets of behaviour and expectations. This is not easy and takes time. De Geest (2006) describes how Selves begin as a collection of practices appropriate for coping with certain situations. They develop into habits, with locked-in postures, gestures, tones, hierarchies, associated emotions and feelings. Selves are continually being created, developed and adjusted as life progresses, as life experiences are accumulated (De Geest, 2006, pp 190-192). Dennett, a contemporary philosopher with research interests in evolution and consciousness, writes that a Self originates from self-preservation, from ‘you are what you control and care for’ (Dennett, 1989, p4). This means that a Self originates from what Self cares for, what interests Self, what Self has affection for. This interest can be cultivated in the mathematics classroom.

For example, one of the recurring issues in the classroom identified by teachers is the discomfort of students when they are moved outside their ‘comfort zones’. This is often accompanied by less desirable behaviour.

In the creation of a desired classroom culture, an expectation of ‘it is OK, it is even exciting to work outside your comfort zone’ can be instilled as part of the new Selves. For instance, students can be offered a choice of tasks with different levels of comfort; rewards can be given for ‘getting in and out of a muddle’ for students working outside their comfort zones. To shift responsibility for learning to the student they can be told it is their responsibility to explain to each other, or use the mantra: ‘ask three before me’.

‘It’s not easy to build in new ways of doing things.’ ‘Tell the students it’s their responsibility to explain things to each other.’ ‘Give students a variety of tasks to let them move out of their comfort zones’ ‘It’s about clarifying expectations. Reward students for getting in and out of a muddle!’ ‘Shift the power and let students pose the questions. Answering questions that other students have asked is a lot less scary.’

To develop students, Selves with new sets of behaviour, expectations and interests have to be encouraged. This can include being prepared to work outside their comfort zone and taking responsibility for their own learning. Tasks can be devised to allow this change to happen.

16—17

Part 1: generate

‘The self is a virtual, imaginary place where past experiences are stored, formed by previous happenings in life. Self has thus cognitive, affective and enactive components amongst others, giving it a rich structure.’De Geest, 2006, p19

18—19

C. Many right answers

The third central feature for establishing a desired classroom culture is knowing how to deal with the belief that there is always a ‘best’ answer and ‘best’ way of working, that there is a ‘right answer’ in mathematics and in mathematics learning. Probing this deeply, the teachers felt there are many right answers. Acknowledging this is crucial when using speaking and listening in the classroom, as students vocalise their personal thoughts and workings out, which can result in many different answers.

Answers are a reflection of thinking and thinking grows organically. Individuals can even give differing answers to the same question from moment to moment. Lacan (1977), a psychoanalyst and psychiatrist, explains this inconsistency in his rather complex theory of ‘signifying chains’. He explores moments in life and in learning when thinking ‘Ah, this is what it means, this is the explanation. So x is like this because of y’ followed shortly after by ‘but if x is like this because of y, why is y like that? Is it because of z?’ Lacan argues there are answers, but never final answers and the thinking chain goes on and on. In Lacan’s language there is never a ‘signified’, only ‘signifiers’ (Lacan, 1977, pp153-154).

Does the absence of some final answers contradict mathematics as a subject? Is mathematics not rigid and factual? Lee (2006, p85) reports on the research of Evans and Tsatsaroni (1994) that argues this is not the case. They argue that mathematics is a dynamic tradition and system of knowledge thought about and developed by a community of people. It involves rediscoveries and reformulations. ‘Mathematising’ can thus be seen as a rather individual and organic experience that centres on making sense of the world and of the mathematical world. It does not therefore require discovering earth-shattering new things; reinventing the wheel. Discovering the meaning of current and old mathematical theories can be considered mathematising.

The teachers subscribed to the view that there are many right answers because thinking, learning and making sense are organic processes. This view not only offers a way to explain but also to invite a ‘yes, but‘ argument in the students’ thinking. Such debate should be stimulated and celebrated.

The teachers created a culture of openness by asking questions such as: ‘This is the answer, what is the question?’, ‘What if Libby’s answer is right, what would need to change about the question?’, ‘This question has more than one answer. Can you find them and what would be the difference in working?’

‘Change the problem stated. Start by saying “The answer is 13, what is the question?” There are many answers to this. The students then become used to thinking for themselves and see it is acceptable to have an answer different from the person next to them. The students gain confidence.’

Part 1: generate

Students vocalise their thinking when speaking and listening. Thinking is personal and can result in many different answers being offered. Creating a culture of openness that invites this diversity is crucial when using speaking and listening tasks.

Have the teachers’ comments in this section interested you? They were all collected during the course of the project and many of them are included in the film. When you watch the film, you will hear a more contextualised discussion. Viewed along with the comments made by Professor Celia Hoyles in her interview, this should provide you and your colleagues with starting points for discussion in your department.

‘Concepts are organic; that is, they are an individual’s attempt to make sense of the world and as such they constantly change and evolve.’Swan, 2006, p74

Part 2: experimentHere are further details about the six lessons that feature in the accompanying fi lm. All lessons included working on the Using and Applying strand (Ma1) of the National Curriculum.

22—23

1. Bev’s lesson: Dominoes

Description of the taskStudents work in pairs or small groups. Each group is given a set of dominoes and an A3-sized puzzle grid on which to place the dominoes. The teacher does not give much verbal instruction. There are written instructions and students have to work out together what to do. Students can get to the answer in their own way.

Aims and objectives • To use deduction and logic skills, trial

and error. To work on students taking on responsibility.

• To learn to concentrate, to think methodically and to realise it is acceptable to be wrong.

2. Judith’s lesson: Mystery

Description of the taskThis is a card activity for group work. On one of the cards the task is written, the others contain necessary information. They are not to show the cards to each other. The cards are dealt out within the group. A scribe is appointed, one pen and one big sheet of paper are given out. No further information is given by the teacher. Students have to collate information and make sense of the information and the task. In this instance, the task was to figure out the order of cars in a traffic jam.

Aims and objectivesThe students are encouraged to talk about what is on their card and to act on what they hear from other students. This offers opportunities to practise logic, reasoning and sequencing. Students also have to move from thinking, to speaking, to telling the scribe what to write, to the scribe writing. They have to collate and negotiate ideas.

This and similar tasks can be found in publications by Spooner et al. (2004) and Ginnis (2002).

Part 2: experiment

3. John’s lesson: 3D shapes

Description of the taskIn pairs, one student per pair comes to the front to look at a 3D drawing of a compound solid. The student then has to describe this to the partner, using only verbal language. The partner can ask questions and has to reconstruct this solid using multilink cubes. Some of the words that they might need to describe the shape may also be on the sheet or on the board to encourage use of proper vocabulary.

Aims and objectivesTo improve the use of accurate and mathematical vocabulary associated with 3D shapes. To learn to be more logical and more precise in the use of language by trying to describe shapes and by listening to others’ descriptions. To employ visualisation techniques to aid memory. To offer the opportunity to every student to describe a shape, even those who are sometimes reluctant to speak in lessons. To concentrate the students’ minds on exactly what they are saying and how they are saying it through the debate that will ensue. To improve listening skills, as the partner student has to deduce how to make the shape. To improve confidence. To learn how to react to their partner’s attempt to make the solid.

4. Corinne’s lesson: True or false?

Description of the taskStudents are given cards with six statements on, for example, ‘If you multiply 5 by a number the answer will always be larger than 5’. Students have to discuss these statements one by one in pairs or groups, decide whether the statement is true or false, cut out the statement and write next to it their conclusion and why they came to that conclusion.

Aims and objectivesBy being explicit, train the students to focus on how to work on this task. They have to understand that having lots of examples is not proof in mathematics. They have to start to think of ways to describe what it is they are trying to explain. They have to think mathematically to answer these statements. They have to work on coming to an agreement within the group.

24—25

Part 2: experiment

5. Lynne’s lesson: How many cakes?

Description of the taskWorking in pairs and groups, the students are in charge of packaging cakes for a supermarket. How will they package them? In fours? In sixes? Why? No further instructions are given.

Aims and objectivesExercise in division and fractions, for example, how would different-sized families divide up a six-pack of cakes. Focus on sharing ideas, sharing language and refining answers as a result. It is a small task with plenty of time to think and talk about responses. It gives the students a practical situation, time to think and opportunities to explore.

6. Brett’s lesson: Breadcake

Description of the taskThe task consists of three parts. In the first part students are asked to come up with different names for ‘breadcake’ (a term used in South Yorkshire for a burger bun) so students can see that something can be called different things but mean the same. This leads into the second part where students work on interchanging percentages, fractions and decimals, which builds on the same concept of equivalence. They are then offered the third activity to be completed in pairs: a hexagonal puzzle cut into triangles where the matching sides are equivalent to each other. Deliberate mistakes will be made by the teacher.

Aims and objectivesTo develop the concept of equivalence, sameness and synonyms. To work on equivalence of fractions, decimals, percentages. To offer opportunities for whole class discussion and paired discussion. To discover and air misconceptions.

One of the most important features of these lessons was that the teachers committed to experiment with something new. They were all prepared to try out a new activity using speaking and listening. If you are working on this with someone else, talk your ideas through with them at the planning stage. You might find that it is helpful to make a record of your intentions. We have included a ‘learning journal’, which you can download from our website: www.basic-skills.co.uk

Part 1: GenerateThis describes the discussions that took place when teachers came together before trying out ideas in their classrooms.

Part 3: refl ectThis part describes the discussions that took place when teachers came together after trying out ideas in their classrooms.

What effects of the tasks using speaking and listening as a tool for learning mathematics had the teachers observed during experimenting? What were their thoughts?

28—29

Speaking and listening develops understanding

The teachers found students in these tasks were encouraged to explain their answers, to verbalise and describe their thinking, to become more precise and explicit in their expressing of their thinking, to have to convince listeners and justify their arguments and to think about how to put speaking into writing. The tasks seem to trigger a focus on exploring, engaging and developing understanding.

Cuoco et al. (1996) describe similar aspects. The authors argue this is important because ‘describing what you do is an important step in understanding it. A great deal of what is called ‘mathematical sophistication’ comes from the ability to say what you mean’. (Cuoco et al. 1996, p379).

‘Is it when you let students discover that they understand?’ ‘Understanding seems to come from experiencing and growing over time.’ ‘As a teacher I rely on such activities to allow students to develop their understanding.’

What took the teachers by surprise was how these tasks had triggered active engagement and talking by learners of English as an additional language. They had wrongly assumed that students with a limited command of English would find it difficult to verbalise their thinking in a non-native language. Speaking and listening thus seems to present opportunities to all students, whatever their cultural background, language and attainment. Indeed, the opportunity for rehearsal through speaking and listening is often a vital element of learning for these students.

‘I was absolutely astounded that a student who is normally not confident and whose first language is not English was completely energised during this task.’

In their report on the QUASAR project in the USA, a study on the impact of teaching methods which focused on problem-solving, discussion, choice and learners’ ideas, Silver et al. (1996) found as well that ‘these features of instruction are quite likely to support the learning of culturally diverse students’ (Silver et al. 1996, pp485-486).

Speaking and listening tasks can be an effective tool for students to develop their conceptual understanding of mathematics, including for learners of English as an additional language.

Speaking and listening allows us to understand students’ understanding

The teachers became aware that they gained a better understanding of their students’ understanding especially in contrast with non-speaking tasks, which involve limited interaction or feedback. Students also showed more awareness of their understanding in these tasks. The teachers reflected that written questions and answers are often very structured due to the nature of communication in a written task. This results in ‘guiding’ the students’ thinking without having the opportunity to assess their thinking in that moment. The teachers felt written questions for low-attaining students therefore often offered simplified mathematics. They were of the opinion that the speaking and listening tasks offered greater engagement between pupils and between pupil and teacher, and this allowed for more accurate assessment of what the students can do.

Part 3: reflect

‘One result of encouraging the pupils to use mathematical language more in their learning is that the more teachers engage their pupils in conversations, the better they come to know the pupils and the better the pupils come to know their teacher. The relationship changes markedly. Lessons become joint enterprises in the struggle to know and understand more about mathematics.’ Lee, 2006, p95

30—31

Part 3: reflect

Sometimes teachers underestimated the mathematical prowess of the students, at other times they discovered that students did not understand a concept, that was considered mastered. Even when students gave a mathematically correct answer, this did not necessarily imply they understood the mathematical concept that should inform such an answer.

Speaking and listening allowed the students to show their understanding, and gaps in their understanding openly, for others to hear. Teachers were therefore better informed to formatively assess their students, and thus provide assessment for learning.

‘I found I did not have the right ideas about their understanding.’ ‘The feedback you get from the students is so much better in these tasks. My beliefs about my students’ understanding have changed.’

The question thus raised is: ‘Why is it so difficult to assess students?’ A possible explanation can be found in Reader Theory, which was developed in the realms of literary analysis with Jauss and Iser as main authors. They argue that what happens in the process of making sense depends on the individual and on their past experiences. It is therefore not observable, at least not by an external observer, and very often not by yourself either. Jauss described the process involved beautifully:

‘A literary work… does not present itself as something absolutely new in an informal vacuum. It awakes memories of that which was already read, brings the reader to a specific emotional attitude, and with its beginning arouses expectations for the “middle” and the “end”, which can be maintained intact or altered, re-orientated or even fulfilled ironically in the course of the reading’ (Jauss, 1982, p23).

What exactly happens in that period of making sense remains thus, inevitably, vague. Applying this theory to learning implies that we cannot control or know how students make sense. It explains why at times students come up with unexpected answers and interpretations, or why they might give a mathematically correct answer, but without the thinking we had intended or expected. It illustrates why students can find it hard to deal with word problems because the story that comes with such problems, and not the mathematics, becomes the focus of attention in their links with past experiences. It means that as educators we have to think carefully when devising tasks: if we want to restrict our students to making links only to certain past experiences, we will have to come up with tasks and questions that will have that restricting quality.

Speaking and listening can be a valuable tool for this because it allows for fine tuning of what is being said. The immediacy of the interpretations of others triggers rephrasing and re-explaining of thoughts. Vocabulary is developed out of the necessity of communication.

Reader Theory also supports the earlier conjecture that answers are individual articulations of a person’s ‘making sense’ process, resulting in many right answers. Speaking and listening activities can accept most contributions as valid and worthwhile. Opportunities are created for real personalised learning because students can build on their own level of mathematical understanding. Learning thus becomes full of meaning for the individual learner.

‘Students might give “correct” answers, but without understanding.’ ‘By having to talk and explain again and again in these tasks, students refine their ideas.’ ‘When using speaking and listening tasks students can go from their head, out through their mouth, and on to the paper.’

Thinking and making sense differs from student to student. Speaking and listening tasks allow teachers to better assess their students’ understanding of mathematics because thinking becomes audible and clarifying questions can be asked immediately. Personalised learning and assessment of and for learning thus happens naturally.

32—33

Part 3: reflect

Speaking and listening triggers reflection and meta-cognition

The teachers recognised that the process of thinking is more important to them than getting an answer. They wanted their students to be reflective, to give signs of serious thought about mathematics. They found that the speaking and listening tasks precipitated such activity.

Lee (2006) offers an explanation as to why speaking and listening works as an effective learning tool for reflective and meta-cognitive learning. She writes: ‘Asking pupils to articulate their ideas forces meta-cognitive activity and thus improves the clarity of their thinking… Discourse is important because making mathematics the subject of discussion in the classroom forces thoughts and ideas that may be tacit and latent to become the focus of attention. As the pupils formulate their own ideas in order to make them available for others, they make their thoughts overt and tangible for themselves’ (Lee, 2006, pp91-92).

Questions that can be used to trigger reflection and meta-cognition: • ‘How did you come to this answer?’ • ‘Why does it work?’ • ‘Tell me what you are thinking.’ • ‘Explain quickly to your friend how you

worked that out.’

‘The answer is not the important thing – it is the process of thinking.’ ‘By having to verbalise, you construct sensible thoughts about the answer you want to give.’

Speaking and listening can trigger reflection and meta-cognition because having to verbalise vague ideas focuses the attention on what is being thought..

‘Reflective discourse is “characterised” by repeated shifts such that what the students and teacher do in action subsequently becomes an explicit object of discussion.’Cobb et al. 1997, p258

The role of disturbances in learning

It became apparent that many of the tasks and the teachers’ existing practice included elements that would cognitively disturb the students’ thinking. The teachers reported using deliberate misconceptions and ‘incorrect’ answers to provoke confusion in their students. This they considered to be something positive to help students in their learning. They argued that such a disturbance forces students to work on their understanding, because it stops students in their train of thought, provokes re-evaluation and makes them rethink and reconsider the validity of their arguments. Disturbances indirectly let them see possible connections. They act as a trigger to make students reflect on their thinking.

This resonates with how Dewey (1859-1952), a pragmatic philosopher and educator, described what happens in a state of reflective thinking:

‘Reflective thinking, in distinction to other operations to which we apply the name of thought, involves (1) a state of doubt, hesitation, mental difficulty, in which thinking originates, and (2) an act of searching, hunting, enquiring, to find material that will resolve the doubt, settle and dispo se of the complexity’ (Dewey,1933, p12).

Should cognitive disturbances be part of lesson planning? Authors such as Festinger (1957, p3), Minsky (1988, p69) and Gattegno (1984, p5) argue that cognitive dissonance and disturbances are very important – even fundamental – experiences in the construction, negotiation and reconstruction of meaning and learning. They argue that disturbances initiate enquiry and create a tension and confusion within the

‘It’s mainly when our systems fail that consciousness becomes engaged. When we recognise that we are confused, we begin to reflect on how our mind solves problems and engages the little we know about our strategies of thought.’ Minsky, 1988, p69

34—35

Part 3: reflect

students that they want to dissolve by looking for an answer. Looking for an answer triggers them into reflecting.Bell (1986) wrote about the rationale that was used for incorporating disturbances in the tasks of the Shell Centre during the 1980s.

Bell (1986) wrote about the rationale that was used for incorporating disturbances in the tasks of the Shell Centre during the 1980s.

‘…a short start activity … would lead to the exposure of such misconceptions as are present in the pupils’ schemes and ideas. So it could be said that we deliberately gave them questions which at least some pupils would get wrong; and without forewarning them of possible hazards. The principle was that if there is an underlying misconception, then it’s “better out than in”; it needs to be seen and subjected to critical peer group discussion…. We saw the “conflict-discussion” as the main learning experience, and the written work as introductory, giving the opportunity for opening up the situation and allowing some mistakes to be made which led to conflicts and hence discussion’ (Bell, 1986, pp27-28).

‘To create cognitive conflict I give tasks with lots and lots of answers and students have to figure out what is different in those answers. I want to get them confused.’ ‘When students are confused I say ‘Good, it means you are thinking’.’ ‘I say to students all the time: ‘If we made no mistakes, we have learned nothing today’.’

Cognitive disturbances stop students in their thinking, make them reassess the validity of their arguments and trigger reflection. It is considered a crucial aspect of learning by many authors. Speaking and listening offers a supportive strategy for dealing with the effects of such disturbances.

Another argument for using speaking and listening

So far, the arguments for using speaking and listening have been that it is an effective tool for helping in learning because voicing thinking helps reflection. There is also an even more profound theory that says why the use of language, of which speaking and listening is a manifestation, is so crucial for learning. Lacan (1986) and Bruner (1990) argue that the process of making sense, of giving meaning, is placed in the narrative field: they argue that the mind is actually structured as language. Speaking and listening then acts as a mapping of how your mind makes sense.

Both Lacan’s and Bruner’s arguments are rather complex and are dealt with in more detail in other publications. In short, Bruner, a psychologist and educationalist, calls the process of making sense through the use of language the ‘biology of meaning’. He bases his view on theories from the field of semiotic realism (Bruner, 1990, p69). Lacan bases his conjecture on Freud’s theory and practice of bringing the content of the unconscious into the consciousness. Lacan observed that Freud’s dream analyses are mainly verbal as they depend on word-play, associations, etc. He deduces that the unconscious, which he sees as the ground for all being, with actions, thoughts, beliefs and making sense determined by it, is structured like a language (Lacan, 1986, p20 and pp24-28).

36—37

The importance of learning mathematics

Several elements of the tasks used in the ‘experiment’ stage depended on what the teachers consider the role of mathematics teaching to be. What is the importance of mathematics? What is it we want to teach in mathematics lessons and why do we actually want to teach mathematics? Looking at the aims and objectives of the tasks that were trialled, recurring themes are thinking, thinking methodically, using logic, learning to work in groups, asking questions and probing answers. These can all be considered mathematical ways of thinking and relate to the Using and Applying strand (Ma1) of the National Curriculum.

The more ‘functional’ view of preparing students for their professional life, for careers, jobs and ambitions, was also considered important. For this, students need to master certain mathematical skills, tools and techniques. We have to bear in mind, though, that the world is rapidly changing, and the jobs students are preparing for may be jobs that have not been invented yet, which might even rely on mathematics that has not been invented yet. How can we prepare students for such an unknown future? Cuoco et al. (1996) offer food for thought by arguing that mathematical thinking and reasoning, and mathematical decision-making actually form the utilitarian aspect of mathematics:

‘If we really want to empower our students for life after school, we need to prepare them to be able to use, understand, control, modify, and make decisions about a class of technology that does not yet exist. That means we have to help them develop genuinely mathematical ways of thinking’ (Cuoco et al. 1996, p376).

This echoes the aims and objectives of the speaking and listening tasks. It also fits in with Claxton’s view of lifelong learning. Claxton (1999) uses the expression ‘learning power’ to describe what is required for lifelong learning. This ‘learning power’ contains:

1. resilience, which he describes as ‘the ability to tolerate a degree of strangeness. Without the willingness to stay engaged with things that are not currently within our sphere of confident comprehension and control, we tend to revert prematurely into a defensive mode: a way of operating that maintains our security, but does not increase our mastery’ (Claxton, 1999, pp331-332);

2. resourcefulness, which he defines as ‘the range of learning tools and strategies that people develop and employ’ (Claxton, 1999, p334);

3. reflectiveness, which he explains as ‘the inclination to stand back from learning and take a strategic view, combined with the awareness and self-awareness to do so accurately and successfully’ (Claxton, 1999, p338).

Part 3: reflect

‘Children can rely on each other toomuch. Sometimes answers are given too easily. We have to let children experience “failing” from time to time so they develop coping strategies and learn to persevere.’ ‘My objective is to give opportunities for thinking and speaking logically and clearly.’ ‘The task pushes the students into taking responsibility for their mathematics learning.’

It is important to reflect on the role of mathematics teaching and learning. To support students in becoming lifelong learners and to prepare them for future jobs in this rapidly changing world, appropriate skills have to be taught. Speaking and listening can be a valuable tool for this.

Low attainment, intelligence and speaking and listening

One of the fundamental questions the teachers asked themselves was whether students were low attaining because of low intelligence. Is intelligence fixed from birth? If so, does that mean low-attaining students will always be low attaining? Is offering more challenging mathematics then futile? What if we subscribe to the notion that ‘intelligence is a malleable quality... that can be cultivated’?

Dweck’s research in the United States has shown that when teachers assume that intelligence can be cultivated and act upon that assumption in the classroom, the intelligence of low-attaining students improves.

Accepting this implies we really have to rethink the potential of our low-attaining students and their assessment. It also means we have to think very carefully about the mathematics and the tasks that we offer them, which should not put a ceiling on their potential learning.

Speaking and listening can again prove to be a very effective tool for students’ learning, as it helps them in their understanding, in organising their thoughts and in their engagement with mathematics. It also helps both student and teacher in the assessment and awareness of that understanding.

38—39

Part 3: reflect

‘Intelligence is a malleable quality, a potential that can be cultivated. Psychologists who study creative geniuses point out that the single most important factor in creative achievement is willingness to put in a tremendous amount of effort and to sustain this effort in the face of obstacles.’ Dweck, 2002, p16

Mathematics educators have a very important role to play in students’ learning by providing suitable and challenging tasks. It requires teachers to stay open to change, to try out new things, to listen to students. To ‘listen to’ as described by Davis (1997) ‘…listening to insights in their thinking, in their sense-making. For example, when you listen to their misconceptions and pick these up to construct a meaningful dialogue’ (Davis, 1997, p359).

‘If you are good at mathematics you are considered intelligent, but if you are not...’

Speaking and listening has the potential to heighten attainment effectively, which is even more important when intelligence is considered a malleable quality.

Continuous ProfessionalDevelopment and speakingand listening

As we learnt earlier in this booklet, teachers consider speaking and listening an effective tool for triggering thinking and reflection in their students. Teachers are also learners. The question is, can the same arguments be generalised and transferred to the realm of the CPD of any mathematics educator?

The teachers did feel that the speaking and listening that had taken place during this project had made them reflect quite intensively. They thought carefully about their existing practice. They became aware of subtleties they had not noticed before. They were inspired by the views and ideas of others. They voiced their fears and concerns. They asked and answered questions and raised questions anew. They experimented in their classrooms. They supported each other.Although they found this collaborative practice very effective, the teachers also pointed out that becoming reflective and developing your practice can also be achieved by other means. Thinking back to moments when they were triggered into reflecting on their practice in the past they remembered marking books, reading an article, something a student had said. These, as well as speaking and listening, are communication processes. It seems it is discourse, the communal name for these processes, that can trigger reflective thinking in CPD.

‘Teachers themselves need “permission” to take risks. If you can use departmental time to discuss changing practice, you give the go-ahead to try it.’ ‘For me, the “best” practice is when I can share ideas and try things out and discuss things like we are doing now.’

Discourse, which includes speaking and listening, can trigger reflective thinking in CPD. The teachers found that the added benefits of speaking and listening included the immediacy to respond in the moment and the supportive aspect of collaborative practice.

Part 3: reflect

40—41

Revisiting issues of changingpractice – possible effects

Both before and after trying out new things in their classrooms, some of the teachers expressed feeling uncomfortable, tense, even a fear of changing their practice. Other teachers looked forward to it. This can be explained in terms of having to build new teacher-Selves. As previously stated, speaking and listening tasks, and in particular group work, change many dynamics and relationships in the classroom. The hierarchy becomes different because students get more control and responsibility for their own learning. Teachers have to assess students differently. The role of the teacher as ‘the person who knows the answer’ and ‘the person who has to be relied on to explain’ changes. This can all be unsettling, even painful. An initial successful change does not automatically mean that the change is long term, that the new teacher-Self is fully established. It takes time for habits to develop.

The reassuring news is that such reactions are natural. De Geest (2006) writes about how the process of changing Selves can be described by Atherton’s (2005) theory of the psychological cost of supplantive learning. Atherton, an educationalist with a background of education in social work, states that supplantive learning happens when previous ways of acting or prior knowledge are called into question. He identifies three stages in this process of learning.1. Destabilisation: in which the previous

way of thinking or acting is upset. This disturbance unbalances the existing Self and can come from being required, demanded, or forced, or from creeping up into awareness. In the context of this project, this happened when a teacher realised his existing teacher-Self would not work for conducting group work.

2 Disorientation: the ‘trough’ in which loss of competence and morale combine to make the learning difficult, and there is a considerable temptation to return to the ‘old way’. In the context of this project, this happened when that teacher, who had the intention of using group work in his experimental lesson, did not do so at the last minute.

3. Reorientation: the gradual climb out of the trough. It concerns the acquisition of the new Self. In the context of this project, this happened to the teachers who have now incorporated speaking and listening tasks in their daily practice.

‘I chickened out of doing group work.’ ‘I do want to be in control and this task passes the control over to the students.’ ‘Trying out new activities – it could be absolute chaos.’

‘Border crossings [between Selves] are thus either moments of anxiety, or, in a familiar reversal, something to be especially enjoyed’ Dennett, 1989, p4.

It is also worth pointing out that changing existing Selves and practice does not necessarily have to be painful. One way of dealing with it is by using humour, by laughing it off; a strategy used explicitly by some teachers. Bakhtin (1986) argues that humour dissolves hierarchical tensions. He writes:

‘It is precisely laughter that destroys the epic, and in general destroys any hierarchical (distancing and valorised) distance. As a distanced image a subject cannot be comical; to be made comical, it must be brought close. Everything that makes us laugh is close at hand, all comical activity works in a zone of maximal proximity. Laughter has the remarkable power of making an object come up close, of drawing it into a zone of crude contact. Laughter demolishes fear and piety before an object, before a world, making it an object of familiar contact. Laughter is a vital factor in laying down that prerequisite for fearlessness without which it would be impossible to approach the world realistically. The object is broken apart, laid bare (its hierarchical ornamentation is removed). One ridicules to forget’ (Bahtkin 1986, pp23-24).

‘I’m a risk taker. I regularly make mistakes.’ ‘I will try letting pupils take on more responsibility. I am extremely uncomfortable with that, but I will do it.’ ‘I’ll explain to the students, “We are trying something different today. It may work, it may not, but we are all going to do our best.”’ ‘When I try out something different in my classroom it is going to be quite scary and it may not work the first time round.’ ‘If I teach the students to persevere, then I have to persevere as well.’

Changing your existing teaching practice and teacher-Self can be uncomfortable, even painful. Using humour as a strategy for change can help.

Part 3: reflect

42—43

Some final reflections

One of the teachers summarised using speaking and listening as a tool for learning mathematics as follows:

‘Once you try to incorporate speaking and listening, you will see your students grow. They are more likely to get the instant “feel good” factor when they contribute than if they are, for example writing. There will be some healthy competition – who can answer first? Who can give the most answers? As long as you value every contribution (sensibly!) they will love it.

‘You will get instant feedback on what they know and understand, and you will probably find you can raise the challenge more than you thought.’

‘Take risks – plan them first and make yourself go outside of your comfort zone now and again. Variety is the key!’

‘I think the whole aspect of speaking and listening is crucial, whatever subject or age group you are teaching.’ ‘The most important thing is to be open.’ ‘This all has to be gradual. A department has to move forward in an organic and gradual way.’

What are your responses to the teachers’ reflections? When you have progressed together through ‘generate’ and ‘experiment’, how will you manage, record and share the reflective process?

Bibliography

44—45

Atherton, J.S. 2005. Learning and Teaching: Resistance to Learning [On-line]: UK: Available: www.learningandteaching.info/learning/resistan.htm. Accessed: 28 December 2006

Bell, A. 1986. Diagnostic teaching 2: developing conflict: discussion lesson. Mathematics Teaching, 116, 26-9

Bakhtin, M.M. 1986 (4th edition). The Dialogic Imagination: Four Essays By M.M. Bakhtin. University of Texas Press, Austin. First Edition 1981

Black, P., Harrison, C., Lee, C., Marshall, B. and William, D. 2002. Working Inside the Black Box – Assessment for Learning in the Classroom. London: NfER

Bruner, J. 1990. Acts of Meaning. Harvard University Press, Cambridge, Massachusetts, USA

Claxton, G. 1999. Wise Up: The Challenge of Lifelong Learning. Bloomsbury, London

Cobb, P., Bouffi, A., McClain, K. and Whitenack, J. 1997. Reflective discourse and collective reflection, Journal for Research in Mathematics Education, 28, 3, 258-277

Cuoco, A., Goldenberg, E.P., Mark, J. 1996. Habits of Mind: An Organizing Principle for Mathematics Curricula. Journal of Mathematical Behaviour 15, 375-402

DfES 2001. Literacy Across the Curriculum

DfES, 2004. Literacy and Learning

DfES. 2004. Teaching and Learning in Secondary School Pedagogy and Practice Pack

DfES. 2004. Guidance for Senior Leaders

Davis, B. 1997. Listening for Differences: An Evolving Conception of Mathematics Teaching, Journal for Research in Mathematics Education, 28, 3, 355-376

De Geest, E. 2006. Using Energy Dynamics to Explore the Process of Making Sense and the Role of Multiple Selves in the Teaching and Learning of Mathematics. Open University, Milton Keynes. Unpublished doctoral thesis. Available on request from els.degeest@edstud.ox.ac.uk

de Saussure, F. 1974. Course in General Linguistics. Peter Owen, London. (edited by Bally, C. and Sechehaye, A.; translated by Baskin, W.). First published 1960

Dennett, D. C. 1989. The Origins of Selves. Cogit, 3, (Autumn 1989), 163-173

Dewey, J. 1933. How We Think: A Restatement of the Relation of Reflective Thinking to the Educative Process. D. C. Heath and Co., London

Dweck, C. 2002. Self-theories of Intelligence. Stanford University. Paper adapted from: Messages That Motivate: How Praise Molds Students’ Beliefs, Motivation, and Performance (In Surprising Ways). In J. Aronson (Ed.) 2002, Improving academic achievement New York: Academic Press

Ernest, P. 1991. The Philosophy of Mathematics Education. The Falmer Press, Basingstoke

Evans, J. and Tsatsaroni, A. 1994. Language and Subjectivity in the Mathematics Classroom. Cultural Perspectives on the Mathematics Classroom, edited by S. Lerman. Kluwer, The Netherlands

Festinger, L. 1957. A Theory of Cognitive Dissonance. Stanford University Press, Stanford

Gattegno, C. 1984. Understanding Disagreement. Newsletter, Vol XIV Number 1 (September 1984), Educational Solutions, New York

Ginnis, P. 2002. The Teacher’s Toolkit: Raise Classroom Achievement with Strategies for Every Learner. Crowne House Publishing, Carmarthen

Ingarden, R. 1973. The Cognition of the Literary Work of Art. Northwestern University Press, Evanston, IL

Iser, W. 1978. The Act of Reading. Routledge and Kegan Paul, London and Henley. First published in German in 1976

Jauss, H. 1982. Towards an Aesthetic of Reception. The Harvester Press, Brighton

Lacan, J. 1977. Ecrits. Tavistock Publications, London. (translated by Sheridan, A.). Originally published in French, 1966

Lacan, J. 1986. The Four Fundamental Concepts of Psycho-Analysis. Edited by Miller, J. (translated by Sheridan, A). Penguin Books, London. First published 1979

Lee, C. 2006. Language for Learning Mathematics: Assessment for Learning in Practice Open University Press, McGraw-Hill Education, Maidenhead

Minsky, M. 1988. The Society of Mind. Touchstone, Simon and Schuster, New York

Peirce, C. S. 1960. Collected Papers of Charles Sanders Peirce. Mass., Vol.2, Harvard University Press, Cambridge

Piaget, J. 1971. Biology and Knowledge: An Essay on the Relations between Organic Regulations and Cognitive Processes. University of Chicago Press, Chicago. (Originally published in French, 1963)

Prestage, S., De Geest, E. and Watson, A. 2007. Pocket PAL: Building Learning in Mathematics. Network Continuum, London

Silver, E. A., and Stein, M.K. 1996. The QUASAR Project: The Revolution of the Possible. Mathematics Instructional Reform in Urban Middle Schools. Urban Education 30: 476-521

Spooner, M. and Vickery, A. 2004. We Can Work it Out: Collaborative Problem Solving in the Mathematics Classroom. Association of Teachers of Mathematics, Derby

Swan, M. 2006. Collaborative Learning in Mathematics: A Challenge to Our Beliefs and Practices. National Institute of Adult Continuing Education, Leicester

Vygotsky, L.S. 1978. Mind in Society The Development of Higher Psychological Processes. Harvard University Press, Cambridge, MA

Watson, A., De Geest, E. and Prestage, S. 2003. Deep Progress in Mathematics: The Improving Attainment in Mathematics Project. University of Oxford, Department of Educational Studies, Oxford

Watson, A. 2006. Raising Achievement in Secondary Mathematics. Open University Press, McGraw-Hill Education, Maidenhead

DVD

Dr Els De Geest taught mathematics in secondary (11-18) schools and is a researcher in mathematics education at the University of Oxford. She also works as a mathematics consultant for Slough Local Authority and as a tutor for the flexible PGCE course at the Open University. Els’ passion is to put research into classroom practice and vice versa, and to value, cherish and build on the existing expertise and professionalism of teachers. She has worked on curriculum development with groups of teachers, consultants and teacher-educators. She is co-author of Deep Progress in Mathematics: The Improving Attainment in Mathematics Project (2003) and Pocket PAL: Building Learning in Mathematics (2007). Other publications include both research and research-based practical papers.

For further information cantact:The Basic Skills AgencyCommonwealth House1–19 New Oxford StreetLondon WC1A 1NU

Tel: 020 7405 4017Fax: 020 7440 770email: enquiries@basic-skills.co.ukwww.basic-skills.co.uk

For further copies contact:The Basic Skills AgencyPO Box 5050Sherwood ParkAnnesleyNottingham NG15 ODL

Tel: 0870 600 2400Fax: 0870 600 2401A2208