Mathematics for Curriculum Leaders (Primary Inset)

PRIMARY INSET SERIES

MATHEMATICS FOR CURRICULUM LEADERS

PRIMARY INSET SERIES Series editor David Wray

Already published Language for Curriculum Leaders Henry Pearson and David Wray Forthcoming Science for Curriculum Leaders Elizabeth Clayden and Alan Peacock

PRIMARY INSET SERIES

MATHEMATICS FOR CURRICULUM LEADERS William B.Rawson

LONDON AND NEW YORK

First published 1994 by Routledge 11 New Fetter Lane, London EC4P 4EE This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” Simultaneously published in the USA and Canada by Routledge 29 West 35th Street, New York, NY 10001 © 1994 William B.Rawson All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including recording, or in any information storage or retrieval system, without permission in writing from the publishers. However, permission is hereby granted to reproduce the materials in this book for non-commercial classroom and other educational use. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Rawson, William B., 1939– Mathematics for curriculum leaders/William B.Rawson. p. cm.—(Primary INSET series) Includes bibliographical references. 1. Mathematics—Study and teaching—Great Britain. I. Title. II. Series. QA14.G7R39 1994 94–1933 372.7′044′09041–dc20 CIP ISBN 0-203-42556-1 Master e-book ISBN

ISBN 0-203-73380-0 (Adobe eReader Format) ISBN 0-415-10389-4 (Print Edition)

CONTENTS

Introduction to the Primary Inset Series

1

Introduction to Mathematics for Curriculum Leaders

5

Unit 1

TALKING MATHEMATICS

11

Unit 2

DEVELOPING MATHEMATICAL IDEAS

20

Unit 3

APPLYING MATHEMATICS

34

Unit 4

MATHEMATICAL PROBLEM SOLVING

48

Unit 5

CHILDREN’S UNDERSTANDING OF MATHEMATICS

65

Unit 6

CALCULATING IN MATHEMATICS

84

Unit 7

REPRESENTING MATHEMATICS

100

Introduction to the Primary Inset Series

Several factors have combined over the last few years to create a much greater emphasis upon in-service education in schools. First and foremost, national educational reforms have ensured that there is a great deal that teachers need to know in terms of the demands of particular curriculum areas. Stress upon the core curriculum subjects of science, mathematics and English has produced programmes of study with which teachers urgently need to familiarize themselves. Second, changes in the funding arrangements for inservice work have meant that schools have been given the responsibility for planning and organizing their own in-service programmes. While in theory this is a very positive move since it allows schools to tailor their in-service to their own precisely identified needs, rather than simply having to accept what is offered by outside agencies, for primary schools, in particular, it has created problems in practice.

There is a great deal that teachers need to know…

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SERIES INTRODUCTION

Planning your own in-service programme to cover the full range of curriculum areas is only practicable if you either have sufficient knowledgeable specialists already on the school staff or the funds available to buy in expertise from outside. For primary schools in neither of these happy positions, this series of packs offers a resource which will help their planning of a coherent and effective in-service programme. The packs are aimed at staff with responsibilities for curriculum leadership who may be asked to plan and run in-service sessions as part of a school’s development plans. The packs do not assume any particular knowledge on the part of the curriculum leader, other than the ability to recognize and negotiate the in-service needs of their schools, and are designed to be as self-contained and as flexible to use as possible. They provide, therefore, a resource upon which curriculum leaders can draw as they see fit, secure in the knowledge that the activities in which their school staffs will be involved have been carefully planned to take them into the crucial issues in the particular areas chosen. The following suggestions concerning the use of the packs are addressed directly to the curriculum leader who will be responsible for this use. Planning Effective In-Service Sessions Determining needs The first step in the planning of an in-service programme is to make an assessment of the exact needs of the people who will be involved. This is something which can only be done by you, in collaboration with others in the school, and not by outsiders, however knowledgeable. It is worth spending some time, in staff meetings or preliminary in-service sessions, working towards some joint statements of what these needs are. The better you can specify these needs, the more likely you are to be able to plan in-service sessions to meet them. A first step towards this might be to ask your colleagues to complete a short and fairly open-ended questionnaire asking for their views on what they need in your curriculum area. Collect these a few days before a meeting to discuss them, so that you can get a sense of the general feelings of the staff. You might present a summary of these to the staff meeting to initiate discussions on in-service planning. Do not forget to include your own views in this summary, and argue for them if you have to. As curriculum leader you are likely to have a clearer overall picture of just where you are as a staff and where you would like to go. Aim to leave this first meeting with a list of in-service priorities to guide your immediate planning. There are two important points to note about this, however. 1. Remember priorities can, and almost certainly will, be the result of negotiation between a range of different viewpoints. It is likely that everyone asked will have slightly different views on priorities for action and the final agreed list will have to recognize all of these if it is to command the commitment it will need to be the basis of a successful programme for in-service work. This may mean that you will have to agree to include items in your list which you either do not rate as priorities or which you simply do not agree with, in exchange for the inclusion of items you think are important which most of your colleagues do not. This is what negotiation is about, and it is vital to understand this process. It is all too easy to lose a staff’s commitment because they do not feel their views have been adequately taken account of. 2. Remember that your initial in-service plans are not set in tablets of stone. It is inevitable that, as you get under way with your plans, fresh and revised priorities will emerge which will influence your future planning. This is not undesirable, but in fact is the way in which real priorities do emerge. Needs which

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arise in the course of work in a particular area are usually those which are more deeply felt and important than those which are expressed as ‘top of the head’ reactions to initial questions. Running sessions Running effective in-service sessions can be a delicately balanced operation. To begin with, you, as leader, and your colleagues may have different perspectives on the enterprise which may demand some diplomacy in its handling. The participants need to feel that their existing expertise and knowledge is given adequate credit and that, as teachers with often wide experience and well-developed skills, they have something to offer to the joint enterprise of school development. ‘De-skilling’ them by assuming, albeit unconsciously, that they know nothing about the subjects under discussion will hinder, not help, the process of effecting beneficial change in general school practice. Yet, as a curriculum developer, you need to feel that you are actually making progress and that things are developing, which implies that your colleagues will recognize possible shortcomings or gaps in their practice and take steps to remedy these. Holding the balance between these two pressures may require a good deal of tact and may sometimes involve your restraining yourself from making statements, especially critical ones. Successful development is always the product of negotiation. Remember, if development were simply a question of one or two people having excellent ideas, and everyone else just adopting them, there would scarcely be a need for in-service in the first place! Another balance which needs to be achieved is that between pushing through the material you have planned and responding to the interests of the participants. Again, some tact will be required. Although it can be infuriating not to complete work you have planned because the group became side-tracked by a discussion which, to you, seemed rather a ‘red herring’, remember that, to others involved, it may have been a very necessary ingredient in their development. People are idiosyncratic in the way they deal with new ideas and, if there is to be any long-term effect, they have to work these ideas out in their own ways. ‘Red herrings’ may simply be their way of coming to terms with the ideas. They may also be, of course, nothing more than ‘red herrings’, in which case you will need, diplomatically, to push things on. But take care with this, and be prepared to sacrifice a little pace in your in-service programme for what may turn out to be a better quality end-product. In general, it is a good idea to set targets for particular sessions (although you may not meet them) which all the participants are aware of. Perhaps distributing agendas might help in this. Also, requiring a group to work towards the production of a written statement over a set number of sessions can concentrate minds, especially as the agreed deadline approaches. Monitoring and evaluating progress You will need to keep careful track of where you have got to and, if appropriate, the decisions you have made as a staff. Although you, as curriculum leader, will need to take responsibility for this monitoring process there are several points you should bear in mind. 1. Do not keep your monitoring secret from your colleagues. There can be little less conducive to selfevaluation than the feeling that someone else is secretly evaluating you at the same time. Opening up the monitoring process will improve the quality of your development work and also help in target setting which, as explained earlier, can be a very useful way of sustaining progress.

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2. The more you can involve your colleagues in evaluating their own progress the more likely you are to initiate sustained development. Take time in some in-service sessions to allow participants to think and talk about where they are and where they are going in the development of your curriculum area. 3. Everyone likes to feel they have achieved something, especially if they also feel they have worked hard. Make sure that the emphasis in your monitoring is upon achievements rather than deficiencies, and be positive and celebratory about what has been done. Design of the Packs The packs are organised into Units, each Unit dealing with a particular issue or area. A Unit contains five kinds of material, as described below. The Briefing Paper, Stimulus Activities, Classroom Activities and Review Session are designed so that they can be photocopied for the participants, or the curriculum leader can use or adapt their contents and photocopy whatever he or she wishes. 1. Leader’s Guidelines This provides a summary for the leader of the aims and activities involved in each Unit, with suggestions for timing and guidance on running the sessions. 2. Briefing Paper These are short papers giving participants some background to the issues dealt with in the Unit. The leader’s guidelines indicate how the Briefing Paper might be used, but the most common pattern will be to distribute copies of it to participants immediately following their work on the Stimulus Activities. 3. Stimulus Activities This is the introductory session for participants, consisting of activities designed to stimulate discussion and thought about particular issues raised in the Unit. 4. Classroom Activities These are activities for teachers to try out in their classrooms following their initial discussion of the issues involved. Sometimes they consist of particular work with the children and sometimes involve observing children engaged in their normal class work. 5. Review Session This is the final session for participants to review what has been observed and learnt during classroom work. The material consists of activities and discussion points. Although the packs include some guidance on possible ways of using them in in-service sessions, it should always be borne in mind that they are first and foremost a resource for in-service work rather than a prescribed course. You should feel free to use them when and how it seems appropriate in your situation. In particular, you may find that you wish to vary the time you spend on particular activities in the materials, and they are designed to have this flexibility. You should always remember that the most effective inservice work is that which is tailored to meet the needs of your staff, rather than planned by an outsider.

Introduction to Mathematics for Curriculum Leaders

The materials in this Pack are to do with sharing and celebrating— sharing what children can do in mathematics and celebrating their learning. Our own professional development requires this too. Getting better at teaching mathematics will happen when we challenge and refine our current classroom practice by talking to colleagues and sharing our discoveries. A study of mathematics could be thought of as one way of making sense of the real world as well as a means of exploring and creating the abstract world of mathematics itself. It is very difficult, however, to know exactly how children learn mathematics since it all takes place in their minds. These materials provide children with opportunities for expressing their mathematical thinking in a variety of ways. What I hope participants working on these Units will consider and share are their observations of the variety of methods children employ while solving problems. Undoubtedly, mistakes they make will also be highlighted. These should be examined to see whether or not they, too, reveal evidence of creativity in mathematics. It is the responsibility of the headteacher and members of staff to consider whether their existing schemes of work adequately cover the Attainment Targets and programmes of study of Mathematics in the National Curriculum. Examining the issues prepared in this INSET Pack relating to mathematics in the primary school should support this on-going course development. Teachers themselves are a vital mathematical resource. This can be shown by the manner in which they approach the subject. I feel, therefore, that every activity in this Pack should be worked on with the wellknown Paragraph 243 of the Cockcroft Report in mind. It speaks of teaching styles: Mathematics teaching at all levels should include opportunities for: * * * * * *

exposition by the teacher; discussion between teacher and pupils and between pupils themselves; appropriate practical work; consolidation and practice of fundamental skills and routines; problem solving, including the application of mathematics to everyday situations; investigative work. (DES 1982)

A repertoire of teacher skills includes the ability to tell, describe, demonstrate and explain. It also includes selecting from a variety of options in order to clarify a point through the use of simile, analogy or metaphor. This demands an increasing awareness of and responsiveness to the audience. The memorization of facts and the practice of skills, processes and routines in mathematics should not be seen as ends in themselves but as leading to the development of important mathematical concepts.

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Not only should you aim to demonstrate a number of teaching styles during the Stimulus and Review sessions, but the participants should also be aware of applying this model to the teaching and learning situations during the Classroom Activities. Each Stimulus Activity contains some mathematics to be attempted by the participants at their own level. This can be intimidating at times when a search for the answer may prevail. This is not the intended point of the exercise. Each activity should provide an opportunity for involvement, experimentation, observation, reflection and discussion. All participants should be able to work in an atmosphere where any conjecture is respected and carefully considered. Processes of mathematical thinking should be shared. You should consider the wide range of thinking and ways of working among the group of participants as well as the methods they apply in order to record their work. It is often stated that practical work is an essential feature in primary mathematics which encourages much talk to help form mathematical ideas. As learning begins with uncertain understanding which needs refining, work on practical activities is intended to provide a platform for this to happen. Challenge this kind of assumption. The Units are intended to be flexible resources. You can decide to select a particular unit or even a group of Units and then return to the Pack at a later date. The numbering of the Units does not indicate the importance of one Unit above another. Each Unit provides you with material taken from the primary curriculum in order to help participants make a particular enquiry into mathematics. Below is listed a number of issues and the relevant Units to suggest various routes through the Pack. Mathematics and Language The influence which language has upon mathematics could take the following route: Unit 1 , ‘ZZTalking Mathematics’; Unit 2, ‘Developing Mathematical Ideas’; Unit 3, ‘Applying Mathematics’; Unit 4, ‘Mathematical Problem Solving’; Unit 5, ‘Children’s Understanding of Mathematics’; and Unit 7, ‘Representing Mathematics’. Calculations If you wish to establish a policy for calculations, your string of Units could be: Unit 4, ‘Mathematical Problem Solving’; Unit 5, ‘Children’s Understanding of Mathematics’; and Unit 6, ‘Calculating in Mathematics’. Shape Work An analysis of shape work could take in: Unit 2, ‘Developing Mathematical Ideas’; Unit 3 , ‘Applying Mathematics’; and Unit 5, ‘Children’s Understanding of Mathematics’. Problem Solving A closer look at effective problem solving will be helped by working through: Unit 2, ‘Developing Mathematical Ideas’; Unit 3, ‘Applying Mathematics’; Unit 4, ‘Mathematical Problem Solving’; and Unit 6, ‘Calculating in Mathematics’.

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Symbolic Representation Symbolic representation is dealt with in: Unit 3, ‘Applying Mathematics’; Unit 4, ‘Mathematical Problem Solving’; Unit 6, ‘Calculating in Mathematics’; and Unit 7, ‘Representing Mathematics’. Conceptual Development A possible route for the conceptual development within mathematics could be: Unit 1, ‘Talking Mathematics’; Unit 2, ‘Developing Mathematical Ideas’; Unit , ‘Children’s Understanding of Mathematics’; Unit 6, ‘Calculating in Mathematics’; and Unit 7, ‘Representing Mathematics’. A number of the activities may be familiar to the participants. If this is the case, considering alternative methods of working on these will enable them to articulate and consolidate their understanding. Similarly, participants should be prepared to accept a child’s ‘alternative framework’—that is, an approach to problem solving that they may never have considered. Excellent computer programs are being developed, and you may wish to include these in order to supplement the activities. Implicit within each Unit is an awareness for a response to the particular needs of each school with respect to multilingual and special needs. Stimulus Activities will provide ample opportunities for alerting members of staff to these needs in preparation for the Classroom Activities. This means that none of the activities should be inserted into a scheme of work without considering the part it may play in enhancing the child’s understanding of new concepts, or the way it may help to consolidate their learning. You will notice as you read through the Units that I remind the participants to identify their strategies for problem solving as well as those applied by the children while solving mathematical problems. The Cockcroft Report identified a number of strategies the Committee believed were used in primary mathematics, yet these form only part of the processes of mathematical activity. A list of strategies can be found in Mathematics from 5 to 16 (DES 1987) and also Primary Mathematics Today and Tomorrow (Shuard 1986). You will see that Shuard’s compilation is reproduced on pages 9–10, and you could make sure that each participant has a copy of this during your first session together. The advice taken from Mathematics: Non-Statutory Guidance is also applicable to this Pack. Activities should be balanced between different modes of learning: doing, observing, talking and listening, discussing with other pupils, reflecting, drafting, reading and writing, etc. (NCC 1989: B9– 10) I hope that this Pack will help the participants towards stimulating and motivating children’s learning in mathematics. Writing for others to direct INSET sessions is always a difficult task as there is the obvious isolation from the actual session itself and the inability to experience the ‘feeling’ of the particular group. Having worked in similar situations around the country, there is one thing I do know—fellow participants are infected by a curriculum leader’s contagious enthusiasm for both teaching and mathematics. This, in turn, can be passed on to the children with whom we work. References Department of Education and Science (1982) Mathematics Counts: the Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the Chairmanship of Dr W.H.Cockcroft, London: HMSO.

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PACK INTRODUCTION

——(1987) Curriculum Matters: Mathematics from 5 to 16, London: HMSO. ——(1991) Mathematics in the National Curriculum, London: HMSO. National Curriculum Council (1989) Mathematics: Non-Statutory Guidance, York: NCC. Shuard, H. (1986) Primary Mathematics Today and Tomorrow, Harlow: SCDC Publications, Longman.

Strategies in Doing Primary Mathematics

Making a mathematical model of a problem: * * * * * *

exploring a problem; analysing a problem; representing a problem, using objects, diagrams symbols or mentally; working within the model; translating back to the problem situation; verifying the reasonableness of the solution.

Identifying or formulating a problem: * * * *

understanding that a problem exists and that mathematics might be relevant; organizing the information; defining terms and relationships; mathematical reading, including decoding, and the comprehension of words, diagrams and symbols.

Using known concepts and skills of number, measurement, or other parts of mathematics within the mathematical model: * identifying and carrying out the correct operation; * estimating the result; * using successive approximation and trial and error. Finding and using relationships: * * * *

relating different units; noticing quantities that increase or decrease together; noticing a decrease compensating an increase; systematically controlling variables.

Finding and examining mathematical patterns:

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STRATEGIES IN DOING PRIMARY MATHEMATICS

* exploring in a systematic way, leading to a pattern in number, in shape, in a graph or in a practical experiment; * describing a pattern and explaining the reason for the pattern; * making a hypothesis based on the pattern, and verifying or disproving the hypothesis. Systematizing: * * * * * *

searching for patterns; finding further examples; making hypotheses; trying out a hypothesis—accepting or rejecting it; using counter examples; explaining, verifying and proving.

Generalizing: * knowing when something will always work; * stating generalizations; * using symbolic notation in words or in algebra. Classifying problems: * noting similarities and differences with other problems; * constructing and trying out simpler related problems. Using appropriate ways of recording what has been done: * * * *

in talk (to the group, the teacher or the tape recorder) ; through models and other artefacts; in pictures, diagrams, graphs; in written language—words and symbols.

Setting up experiments: * to find something out; * to try out a new idea. Remembering: * identifying and remembering structural features of a problem. Reference Shuard, H. (1986) Primary Mathematics Today and Tomorrow, Harlow: SCDC Publications, Longman, pp. 103–4.

UNIT 1 TALKING MATHEMATICS

LEADER’S GUIDELINES This Unit identifies discussion as an important element of mathematics learning. It establishes the principle of encouraging talk about mathematics as a means of enhancing thinking about mathematical ideas. Implicit in each of the ensuing Units is the high profile given over to talking and listening. Briefing Paper You should distribute copies of this at the end of the Stimulus Activities. It contains recommendations for further reading on this subject. Stimulus Activities Three hours should give you sufficient time to work through the activities and consider the roles of both teacher and pupil during mathematical discussion. 1. Sharing recordings of mathematical discussions. Before this session, participants should make recordings of either a teacher-pupil or pupil-pupil discussion in mathematics. They should prepare a short transcript. Distribute copies of Stimulus Activity 1 well before the session in order to help each participant in their preparation and analysis of their transcripts. 2. Mathematical terms. Keep the group together for this activity and write down on a large sheet of paper the mathematical terms they use. 3. Fishbowl. Have ready a supply of pens and paper, geometric shapes, building blocks, measuring instruments. Participants work in groups of three. One observes the other two working on a problem they have chosen from the list. Depending on the time available, participants can repeat the process by swapping roles so that someone else can observe while the other two work on another problem. Classroom Activities Participants should be given around 2 weeks to arrange for small groups of children to work on some of the classroom challenges.

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UNIT 1 TALKING MATHEMATICS LEADER’S GUIDELINES

1. 2. 3. 4. 5. 6.

Number stories. Which is heavier? Triangles on a 3×3 pinboard. Behind the ‘wall’. Between 8 and 10. Plan a day. Review Session

About 3 hours should be assigned for this session. 1. Reporting on classroom activities and considering classroom roles. 2. Advantages and disadvantages of encouraging discussion. 3. Brainstorming further points and activities. BRIEFING PAPER Most teachers in primary schools would like to think that their classroom is a place where children are encouraged to handle, examine, explore, manipulate and talk. Indeed, HMI report that: The predominant way of organising effective mathematics work in the primary classrooms was within small groups. Working in this way enabled the teacher to promote discussion as an important element of mathematical learning. (DES 1989:20) Being encouraged to work collaboratively in this way on a shared mathematical task provides children with opportunities for posing questions, testing ideas, removing the fear of taking risks and supporting one another as they develop in understanding. However, conversations during mathematics lessons can easily turn to a form of exposition where the teacher instigates the question, prompts the child to elicit a response, and then supplies some comment which indicates an evaluation of that response. Often quite unintentionally, adopting this kind of teaching style can discourage children from contributing to classroom discussion. While working alongside groups of children, you may soon become aware of the great amount of off-task conversation. Does this indicate a need to examine both the task and the group composition? A closer examination of children’s talk during collaborative work often reveals the ‘cut and thrust’ of statements uttered, solutions offered, accepted, ignored and often left unchallenged. This can be disconcerting and can provide you with sufficient evidence to convince you to change your teaching style!

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Mathematics in the National Curriculum (DES 1991) identifies in Attainment Target 1 the important role of talk which is further supported by Mathematics: Non-Statutory Guidance (NCC 1989). This Unit aims to provide you with opportunities for analysing various classroom activities that involve mediating learning through conversation. Further reading on discussion-based teaching is recommended, and details will be found in the References at the end of this Briefing Paper. Children Talking Analysing conversations between children while working on mathematical problem solving reveals how they apply numerous ‘verbal strategies’. Some of these are identified in the following extracts taken from transcripts: 1. Here is an example of a breakthrough in understanding by a nine-year-old. No doubt she has worked on many activities dealing with ¼+¼=½ in her school life, yet in this specific context she now appears to have made a connection: Pupil : If I put half with this half, it makes one big square. This quarter joined to this. Eh, does a quarter and a quarter make a half? Two quarters make a half. 2. Examining the structure of a situation is an essential part of the problem-solving process. Questioning and mulling over the problem is a feature of this section of the dialogue: Pupil

: Was there six between the next stage?

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UNIT 1 TALKING MATHEMATICS LEADER’S GUIDELINES

3. Causal links between parts of an experience are justified as a child makes deductions. On the basis of these, the child becomes satisfied with the work that has been carried out so far: Pupil : Ten of these must equal forty-one because if you add four each time… 4. In order to ensure that successful and reasonable outcomes are possible, children apply a system of checking and modification. This is captured in the following extract: Pupil 1 : Let’s work it out again. Let’s work out eight. Four made seventeen, five equals twenty one… Pupil 2 : But that can’t be thirty-three. It must be thirty-two. You take another one off that. Remember? 5. Having to explain the results of your efforts to someone else is an excellent way of clarifying your own thinking: Pupil : We’ve just worked out four. If you times that by two, that’s eighteen. Take one, that’s seventeen. So you times that by two and take one. 6. As you observe groups working together on a problem, a wide range of abilities among those involved soon becomes apparent. Insights for one child may be of no value for another at that moment. On such occasions, these insights are often ignored. Four children were involved in identifying the shapes within shapes. Three were counting hexagons and triangles while the fourth noticed: Pupil : A hexagon looks like a box. When I am close, I see a box. When I am far away, I don’t see a box. This information was ignored by the other children. 7. Placing children in groups provides examples of interesting aspects of ‘collaborative discussion’. It appeared that these two boys were working independently, following their own lines of thought throughout the problem. A look over their comments, however, for the entire problem-solving session does show how their comments formed an interwoven whole and appear eventually to have influenced each other’s thinking: Pupil 1 : So that would equal eight. Pupil 2 : Would it work for odd numbers, though? Pupil 1 : Forty-five can’t be right. Pupil 2 : They’re all odd numbers, though. 8. The mathematical terms that are used can often be difficult to understand. Fortunately, on this occasion the child asked for an explanation: Teacher : We are going to do area today. I want you to cover this surface with these shapes. Pupil : What’s surface? Strategies Listening to oneself after recording a session often reveals how easy it is to slip into a teaching style that produces the familiar pattern: question by the teacher—response from the pupil—comment by the teacher— next question…. You need, therefore, to make a conscious effort, while in the ‘heat of the moment’ of discussion, to refrain from immediate intervention and think carefully about the kind of interaction that is taking place. Opportunities for mediating learning through structured conversation are characterized, for example, when a ‘Well done’ is changed to ‘Tell me how you worked that out’. Other modifications can be

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made in line with the technique of ‘open’ and ‘closed’ tasks discussed in Mathematics: Non-Statutory Guidance (NCC 1989: pp. D5–7). Genuine problems of communication occur quite naturally when children are given opportunities to talk about their mathematical understanding. Often they can be seen to engage in an intensive struggle to communicate meaningful thinking associated with problem solving. Their solution to a problem may not be immediately apparent to the teacher whose role is to help each child organize his or her thinking in a way that becomes accessible to others. Mathematical ideas that arise from children’s stories is an avenue worth exploring as a means of encouraging mathematical discussion. The ‘feely bag’ is a well-tried piece of equipment that gets the children to pass on information and respond to questions. A shift from the norm that provides teachers with an insight into a child’s understanding is as follows: first, give an answer; and then ask the child to come up with a suggestion about what might have been the question. Tasks completed in isolation and only presented in written form simply reinforce the notion that school mathematics bears no relation to real life. Ideas need to be explored, mathematics applied and principles shared with peers and others. The audience may be varied; and this requires children to be adaptable in their talk about work in progress, relating facts, explanations of completed tasks or representation of alternative points of view. HMI emphasize the vital role the teacher plays in this: It was the quality of the exposition and dialogue with the teacher that enabled the children to reflect upon and think through mathematical problems and ideas. This factor more than any other marked the difference between good and mediocre work. (DES 1989:27) References and Further Reading Baker, A. and Baker, J. (1990) Mathematics in Process, Portsmouth, NH: Heinemann. Bickmore-Brand, J. (ed.) (1990) Language in Mathematics, Brunswick, Vic. : Australian Reading Association. Brissenden, T. (1988) Talking about Mathematics: Mathematical Discussion in Primary Classrooms, Oxford: Basil Blackwell. Department of Education and Science (1982) Mathematics Counts: the Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the Chairmanship of Dr W.H.Cockcroft, London: HMSO. ——(1987) Mathematics from 5 to 16, London: HMSO. ——(1989) Aspects of Primary Education: the Teaching and Learning of Mathematics, London: HMSO. ——(1991) Mathematics in the National Curriculum, London, HMSO. Durkin, K. and Shire, B. (eds) (1991) Language in Mathematical Education, Guildford: Open University Press. Mathematical Association (1987) Maths Talk, Cheltenham: Stanley Thornes. National Curriculum Council (1989) Mathematics: Non-Statutory Guidance, York: NCC.

STIMULUS ACTIVITIES These activities are intended to help you consider the complexity of mathematical language by articulating ideas, providing valid descriptions, and offering alternative analyses and justifiable explanations.

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Activity 1 Record either a teacher-pupil or a pupil-pupil discussion in mathematics. The following questions and instructions, taken from Maths Talk, prepared by the Mathematical Association, may be helpful in providing a starting-point for the recording: * * * * * *

Do we have 1000 books in the library? Describe to a partner a design for them to draw. How many times does your heart beat in a day? What do ‘heavy’ and ‘light’ mean? How do you weigh a live goldfish? How high would a pile of 1000 sheets of paper be?

Prepare a short transcript (about 3 minutes) of aspects of this discussion that are particularly informative. Share excerpts from your transcripts of teacher-pupil or pupil-pupil discussions. You may wish to comment on your: * * * * *

instructions use of mathematical terms questions cueing to elicit response echoing of a child’s response.

Your recording may provide you with a fine example of collaborative work. Share any incidents of a child’s: * * * * *

use of mathematical language form of questioning way of explaining mathematical terms formation of ideas response to another’s suggestions.

Consider the following questions: * What actually has been learnt from this experience? * In what way will you change your teaching style, if at all, as a result of what you have heard? * What does this kind of work reveal of the child’s understanding of mathematics? Noelene Reeves, in Language in Mathematics (Bickmore-Brand 1990:95–7), suggests that mathematics ‘amounted to the mind dealing with information about space, time and quantity in three ways: grouping, ordering and changing’. She then provided examples of the kind of talk associated with these three categories; for example, * Grouping talk: box, sharp, long… * Ordering talk: in, smoother, equal to… * Changing talk: put in, add, repeat…

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Examine your transcripts once again in order to provide examples of the context within which such talk took place if at all. Activity 2 Produce a list of the terms and expressions that carry mathematical meaning different from the usual, everyday meanings; for example, volume. Discuss ways in which you would introduce some of these terms into your lessons. Activity 3 Materials needed: geometric shapes, building blocks, measuring instruments. Form groups of three. One participant takes notes of the mathematical terms the other two use while working on one of the following challenges. (If you are taking notes, you must restrain yourself and not join in the conversation!) * Draw a simple design or build a simple structure with Multilink. Describe your design or structure for your partner to draw or build. Discuss the final result. * Describe the movement of a shape for your partner to follow in the mind. Discuss what you visualized. * Explain the difference between ‘tall’ and ‘high’. * Decide on the total cost of books in your school library. Share your observation with your group and then draw some conclusions from what you have observed about talking mathematics. Present these to the other participants. CLASSROOM ACTIVITIES Observe small groups of children working on some of the following activities: Activity 1 Materials needed: a card with 26+15=41 written on it. Ask the children to make up number stories and word problems for this calculation. Get them to represent their stories in whatever way they choose. Activity 2 Materials needed: a balance, a paper clip and a matchstick (make sure you have more balances, more clips and matchsticks ready but out of sight). Challenge the children with the question: ‘Which is heavier, a match or a paper clip?’ Get them to work out a solution and write down how they tackled this problem. Activity 3 Materials needed: 3×3 pinboard, rubber bands.

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Ask the children to show you how many triangles they can make on a 3×3 pinboard. Get them to record their work. Activity 4 Materials needed: wooden blocks, Multilink, Unifix cubes and a large piece of cardboard. Sit two children opposite each other and place a cardboard dividing ‘wall’ between them. Make sure the ‘wall’ is high enough to prevent each child seeing what the other is building. Now ask one child to build a structure and then describe it so that the other child can make a copy. Remove the ‘wall’ when they have completed the task and listen to their conversation as they compare the structures. Make notes on the ways they describe their work. Activity 5 Ask the children to tell you what numbers there are between eight and ten. Make notes on the children’s responses. You may feel you have to probe further by, for example, drawing a number line. Activity 6 The timing of this Unit might coincide with an out-of-school visit. Get your children to plan the entire day. Make notes on the mathematical content of the planning session as well as characteristics of the interaction within the group. REVIEW SESSION 1. Report on the various Classroom Activities. Make comparisons with other participants’ experiences and identify any features that are common to each report. During the discussion, question your role in the classroom: * * * *

What kind of listener are you? What prompted you to interject? Which of your instructions were necessary, superfluous or inadequate? What circumstances make you hold back from taking on an instructional role?

2. Prepare a list of advantages and disadvantages of this kind of teaching that encourages a greater emphasis on mathematical discussion. You may find the following questions helpful: * What evidence can you provide that the collaborative work during the Classroom Activities has fostered effective mathematical learning among your children? * What criteria do you apply to identify that learning is taking place during collaborative activities? * What changes are necessary in order to adopt an approach to small-group, cooperative learning within your classroom? 3. Form groups and brainstorm:

BRIEFING PAPER TALKING MATHEMATICS UNIT 1

* Starting-points and challenges that can be used to stimulate mathematical discussion; * Activities that can help develop the use of particular mathematical terms.

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UNIT 2 DEVELOPING MATHEMATICAL IDEAS

LEADER’S GUIDELINES This Unit looks at how mathematical ideas may be effectively developed in the classroom through discussion, writing and practical work. The topic of ‘shape’ is taken merely to illustrate the arguments presented. It considers how ‘mathematical’ words can either clarify or confuse. Unit 1 suggests ways of encouraging talk in mathematics and this Unit builds on some of these ideas. Other Units, such as 3 and 4, provide opportunities for ideas to be developed and represented in problem-solving situations. Briefing Paper You should distribute copies of the Briefing Paper some days before the Stimulus Activities. Use the reference from the Cockcroft Report (DES 1982: para. 306) as a starting-point in preparation for the first Stimulus Activity. Stimulus Activities Three hours will be required. Participants should form groups of three or four. 1. Sorting shapes. Provide large hoops or skipping ropes, large sheets of paper, card, metal rulers, sharp knives and cutting boards. Participants are required to make their own supply of regular and irregular polygons. 2. Properties. Participants will need large sheets of paper, gummed paper, scissors, card, metal rulers, sharp knives and cutting boards. You may need to spend some time identifying the properties of various polygons and review the layout of Carroll diagrams before participants move into smaller groups to work on this activity. 3. Developing practical tasks for children. You will need to have on hand sets of published mathematics schemes you use in your school. Classroom Activities Allow a period of about 2 weeks for participants to work on some of these activities.

BRIEFING PAPER DEVELOPING MATHEMATICAL IDEAS UNIT 2

1. 2. 3. 4. 5.

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Sorting shapes. Provide participants with sufficient copies of Worksheet A (page 31). Describe a shape. Creating shapes. Dissection. Participants will need a number of copies of Worksheet B (page 33). Writing about shape. Review Session

Set aside a period of around 2 hours for participants to report on their pupils’ work and to respond to the questions related to the different kinds of demands made by ‘open’ and ‘closed’ tasks. Prepare copies of pages D6 and D7 from Mathematics: Non-Statutory Guidance (NCC 1989) to help participants identify the nature of these kinds of tasks. BRIEFING PAPER The Cockcroft Report, in its famous Paragraph 243, suggests that mathematics teachers should provide opportunities for (inter alia) discussion—not only between pupil and teacher, but also between pupils themselves. Further, in Paragraph 306 it states: There is a need for more talking time; ideas and findings are passed on through language and developed through discussion, for it is this discussion after the activity that finally sees the point home. (DES 1982) As well as directing your attention to recognizing the significance of discussion in helping children grasp mathematical ideas, you should also consider what specifically characterizes mathematical language. Indeed, your work on this Unit will highlight the complex nature of enabling your pupils to make connections between their use of mathematical language and the ideas themselves. Speaking You may have noticed how children often use non-explicit language when they attempt to express their ideas while working collaboratively. An understanding of the precise mathematical terminology and notation does not come quickly. Children need time to be able to absorb the mathematics and explore the mathematical language associated with that mathematics. For example, the idea of a ‘cube’ may begin with children using the word as a label for some specific object—an ice cube or a sugar cube; the word ‘cone’ may be used to stand for an ice cream or a traffic bollard; Volume’ may only be associated with the knob on the radio or TV set. Sometimes the mathematical definition of a term will conflict with the way the term is used in everyday language; for example, a ‘spiral staircase’ is really a ‘helix’. Mathematical terms have precise meanings among a restricted group of people who write mathematics for those who study mathematics. Does the language of mathematics hinder a child’s attempts at making inroads into the mathematics itself? A ‘cube’, for example, may be recognized as a three-dimensional shape comprising six congruent faces each of which is a square. This may be your idea of ‘cube’, but you have no control over the ideas that a child acquires in everyday life and brings into the classroom. How, therefore, can a child make inroads

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UNIT 2 DEVELOPING MATHEMATICAL IDEAS LEADER’S GUIDELINES

into this specialized knowledge? Your task is to provide opportunities for your pupils to make connections of their own and find out what these connections are. As a further example, consider the notion of a ‘rectangle’. Children sort from a collection of various shapes. Certain shapes appear to have in common the ‘rectangle property’, but the criteria applied by one child in assessing this quality can vary considerably from those of another. Is it only when a formal definition —‘A quadrilateral whose angles are all equal to one another is called a rectangle’—is provided that you can say that ‘rectangle’ has been clearly understood? A simplistic view of the process of refining a mathematical idea may look something like this: * some mathematical idea encountered through (1) observation, (2) manipulation of materials, (3) language; * these experiences discussed; * mathematical ideas thought about in the abstract. In essence, talking about the appropriate application of mathematics to the concrete assists the child to use and also explore the abstract nature of mathematics. Reading and Writing So far, a great deal of your attention has been directed towards the place of the spoken word in the learning of mathematics. In reality, you may find that more of the children’s classroom time is spent in contact with the written word through published schemes, workcards and other materials prepared for primary mathematics. If this is the case, you may consider what provision within the mathematics curriculum in your school is assigned to exploring ways in which you might help your children read, analyse and interpret mathematical texts. Shuard and Rothery (1984) have a great deal to say on this subject. They examine primary mathematics texts in detail by alerting readers to many problems of understanding mathematical terms, suggest ways of improving written materials and include activities to develop children’s reading of mathematical texts. You will find details of this book at the end of this Briefing Paper. Writing your thoughts is yet another way of communicating with others. It is also a good way of helping you clarify and refine your mathematics. We should not underestimate the amount of time spent on producing the mathematical texts children encounter in their classrooms. These are the result of adults agonizing over what they consider to be appropriate representations of mathematical ideas. However, very little is apparent to the reader of the process of clarification and refinement necessary to produce such finished articles. The suggestion in this Unit is that, among your repertoire of teaching strategies, you should give children the opportunity to experiment with writing. Azzolino (1990) shows how this can operate in class writing activities. The following notes and selected examples will give you something of the flavour of these methods. 1. Completion Start a sentence and have children complete it. This is appropriate for summarizing, comparing and analysing, and for the expression of feelings. E.g. Lines and line segments are different because______. 2. Lead sentence Make a statement and then get the children to support this with a further sentence or paragraph.

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E.g. The factors of 18 are 1, 2, 3, 6, 9. Factors are numbers that______. 3. Warm-ups Make a statement that requires little (?) thought to complete, and then make another statement on the same topic that requires more thought to answer. E.g. Shapes that are similar have the same______. State how ‘scale factor’ is related to the topic of similar shapes. 4. Rewording Take a statement, definition or procedure and ask the children to rewrite it, using their own words. E.g. The metre is the standard unit of length. 5. Word bank A list of words from which children can write a sentence or paragraph. E.g. Use the words ‘one’ and ‘is less than’ in a true sentence. 6. Debriefing Obtain feedback by asking children to list important ideas on completion of an explanation by the teacher or work on a mathematical task. E.g. These are the mistakes I made while working through exercise ______. Note with reference to work on shape In trying to grasp the meaning of certain terms associated with shapes and their properties, the idea of a ‘family of shapes’ may be helpful. For example, the family of rectangles belongs to the family of quadrilaterals. Within the family of rectangles, there are two families—the family of squares and the family of oblongs. Compare this with how we might view, for instance, a jackdaw as a member of the jackdaw family, which is part of the crow family, which in turn is part of the family of birds, which is part of… So, a picture of families nested inside one another can serve to clarify the terms themselves and the relationships between the different terms: * * * *

a cube is a special cuboid which is a special prism which is a special kind of 3-D shape.

Note regarding Carroll Diagrams A Carroll diagram is used to represent how a set of elements may be classified according to several criteria. For example, it could show which of two attributes are possessed by each element of the set. The diagram below shows how a set of numbers may be classified using the following criteria: ‘Is the number prime?’ ‘Is the number even?’ Set of numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

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UNIT 2 DEVELOPING MATHEMATICAL IDEAS LEADER’S GUIDELINES

References and Further Reading Azzolino, A. (1990) ‘Writing as a tool for teaching mathematics: the silent revolution’, in T.J.Cooney and C.R.Hirsch (eds) Teaching and Learning Mathematics in the 1990s, Virginia: The National Council of Teachers of Mathematics, Inc. Department of Education and Science (1982) Mathematics Counts: the Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the Chairmanship of Dr W.H.Cockcroft, London: HMSO. National Curriculum Coucil (1989) Mathematics: Non-Statutory Guidance, York: NCC. Shuard, H. and Rothery, A. (1984) Children Reading Mathematics, London: John Murray.

STIMULUS ACTIVITIES Activity 1 Materials needed: large hoops or skipping ropes, large sheets of paper, card, metal rulers, sharp knives and cutting boards. Form into groups of three or four and tackle the following: Construct an enlarged version of the Venn diagram below. (This will be used as the basis for the practical work.) Cut out of card a set of shapes, each of which is a polygon (i.e. a shape whose sides are all straight). Place each of the shapes in the appropriate region of the diagram. Design further shapes if necessary in order to ensure that, in the end, each of the eight regions has at least one shape inside it. When this is complete, consider how it might be possible to describe particular regions of the Venn diagram using mathematical terms such as: square, rectangle, oblong, rhombus, regular polygon.

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In formal mathematical language, there is a need for precision in the use of terms. Make notes of the terms which were not clear during the activity. Share these with the rest of the group during the discussion at the end of the session. Evaluate the discussion that occurred when you were deciding to which region a particular shape should belong. Think how language was used to clarify the way you categorized the shapes. Make a list of the decisions you made for two particular shapes.

Activity 2 Materials needed: large sheet of paper, gummed paper, scissors, card, metal rulers, sharp knives, cutting boards. Make several labels to mark the various partitions of a Carroll diagram. Each label is to correspond to some property a two-dimensional shape may possess. Choose properties of your own and/or from the following list: Properties of a 2-D shape: * * * * * * *

has no curved edge has reflective symmetry has rotational symmetry has one pair of sides parallel can tessellate is a convex polygon has an interior angle greater than 180 degrees

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UNIT 2 DEVELOPING MATHEMATICAL IDEAS LEADER’S GUIDELINES

As in Activity 1, create your own shapes to fit in, where possible, the eight regions of the diagram. Select one participant from the small group to stand back for some of the activity and take notes of the mathematical language used by the other participants in the group. Make sure each has an opportunity to act in this role of observer. Here is an example of a possible layout of the Carroll diagram:

Activity 3 Take a selection of mathematical ideas from those that were introduced in Activity 2 (for example, symmetry, parallel lines, tessellation). Write your own definitions for the mathematical terms and ideas you have selected. Experiment with some of the writing ideas suggested in the Briefing Paper by arranging them under the headings such as ‘Completion’, ‘Lead sentence’, ‘Warm-ups’ and ‘Re-wording’. Share your writing with other participants. For each idea, develop practical tasks aimed at giving primary school children experience of that idea at the appropriate level. Relate the children’s activity on a task to the processes identified within this model:

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In particular, explore the kind of language that the children might be encouraged to use in their discussion. Examine published texts from primary mathematics schemes that deal with the mathematical ideas you have selected for closer analysis. Pay particular attention to the demand these pages place upon reading. CLASSROOM ACTIVITIES Activity 1 Carry out the following sorting activities in a way that will allow for small-group discussion. Identify the mathematical language being used. You may find the following helpful: * * * * *

‘Find me a…’ ‘What makes it a…?’ ‘What is the difference between these two shapes…?’ ‘Find me a shape that isn’t a…’ Why isn’t it a…?’

Note the children’s responses and see if they are willing or reluctant to use mathematical terms. Paper Shapes Materials needed: copies of the worksheet (page 31). Ask the children to colour the squares red and the oblongs blue. Then let them choose a way of their own to sort the shapes. Templates Materials needed: card, scissors.

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UNIT 2 DEVELOPING MATHEMATICAL IDEAS LEADER’S GUIDELINES

Prepare strips of card about twice the size of the ones shown above right. Draw the shapes shown and cut them out to make templates. Ask the children to sort the shapes into two sets, ‘quadrilaterals’ and ‘not quadrilaterals’, by drawing around the cut-out pieces and the holes left in the template cards. Ask them to sort their own set of shapes from the templates and cut-out pieces.

Card shapes Materials needed: card, scissors. Prepare a large variety of regular and irregular 2-D card shapes, similar to those shown below. Ask the children to collect all the parallelograms and then choose ways of their own to sort the shapes.

BRIEFING PAPER DEVELOPING MATHEMATICAL IDEAS UNIT 2

Worksheet A

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UNIT 2 DEVELOPING MATHEMATICAL IDEAS LEADER’S GUIDELINES

Activity 2 Materials needed: card, scissors. Prepare various 2-D card polygons. (You could use the polygons prepared for Activity 1.) Place a variety of 2-D cardboard shapes in front of a small group of children and ask one child to choose a shape without picking it up. Let that child now describe that shape to the rest of the group. The other children are to find which shape has been chosen. Encourage them to ask questions about the attributes of that shape. Let other children take turns in choosing shapes. Activity 3 Materials needed: sticks, straws or card strips of length 3, 4, 5 units (use colour coding to identify the three different lengths), paper, scissors, 3×3 pinboards, ‘dotty’ paper. Give your children a starting-point that will enable them to create their own mathematical shapes. Encourage them to record in some way the shapes they construct. Ask the children to discuss their findings and, if appropriate, label their shapes and write about their discoveries. Examples of starting points 1. Provide strips of card, sticks or straws of three different lengths (3 units, 4 units and 5 units). Make different triangles using just three strips at a time. 2. Fold a square piece of paper in half. Now make one straight cut with scissors. What different shapes can you produce this way? 3. Work on a 3×3 pinboard. Join the points to make a quadrilateral. How many different quadrilaterals can they make? Ask the children to record their shapes on the ‘dotty’ paper. (You can modify these examples by varying the constraints—the number of units, the kinds of shape allowed or the types of material.) Activity 4 Materials needed: three card triangles, copies of the worksheet (page 33) scissors, glue, large sheets of paper. Prepare sets of three right-angled triangles in card the same size as the ones below. Get the children to talk about the shapes. How they are alike? Why they are different? Ask them to fit all three pieces together to make different shapes. They should then name and describe the new shapes they have made. The children should make a permanent record of their different shapes by using the shapes cut from the worksheet (page 33) and gluing them onto a larger piece of paper.

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Activity 5 1. Writing Select one writing technique (Completion, Lead sentence, Warm-up…) discussed in the Stimulus Activities. Prepare statements, definitions or questions on shape appropriate for the children with whom you are working. Ask the children to express their mathematical ideas on paper. 2. Explaining Ask a group of children to write down how to carry out a particular procedure in mathematics, such as adding two two-digit numbers together or calculating the area of a rectangle. Share this writing and get the children to cooperate in clarifying and refining what has been written.

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Worksheet B

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REVIEW SESSION 1. Report on your pupils’ responses to the Classroom Activities. 2. Compare the responses from children of different ages and provide examples of their use of mathematical terms. The following questions may help you focus your comments: * * * * * * *

What discoveries have you made? What discoveries have the children made? How far did the children engage in mathematical talk when they were describing shapes? How, through your questioning, were you able to encourage them to use mathematical terms? Describe how the children reacted to the more ‘open’ tasks. In what ways did they describe their constructed shapes? How far developed were the children’s ideas on scale (or similarity in shape—that is, the idea of ‘same shape/different size’? * Was there any reluctance among the children to accept irregular concave shapes during the dissection problem (Classroom Activity 4)? 3. Decide on three or four specific initiatives aimed at providing opportunities for talking and writing about mathematics in your school.

UNIT 3 APPLYING MATHEMATICS

LEADER’S GUIDELINES This Unit shows ways in which participants may encourage their pupils to use and apply mathematics in real contexts. It also looks at the place of games, puzzles and recreational activities in the curriculum as a means of providing insight into mathematics. For a deeper understanding into applications, Units 4 and 5 consider the underlying structure of problemsolving situations. Briefing Paper You should hand this to the participants at the end of the Stimulus Activities. Stimulus Activities A half day is likely to be needed for these activities. 1. Tiling a room. Have ready an ample supply of plastic squares. Quite some time will be devoted to looking at different ways of solving problems through the application of mathematical concepts and skills. Be careful to avoid Activity 1 being unnecessarily prolonged. 2. Mathematical model It is important that during the course of the discussion of this activity the participants consider the idea of what constitutes a ‘real’ problem or a ‘real-world’ problem or a ‘reallife’ problem, as opposed to a semi-realistic problem, a word problem or an abstract problem. 3. Three problems. Each group will need calculators, two identical vessels, a glass, water (clear and coloured), tape measure. Participants work in groups of three, and apply the ‘goldfish bowl’ technique of observation throughout the three tasks. 4. Children playing games. Two weeks before you begin work on the Unit, ask the participants to select one game they observe their pupils playing regularly, either in the classroom or in the playground during the period leading up to your first meeting. Give them the following questions to consider in preparation for their first meeting: * What game is popular with your children? * What do you identify as being the mathematical content of that game?

BRIEFING PAPER APPLYING MATHEMATICS UNIT 3

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Ask the participants also to interview the children they see playing the game: * Why do they play that particular game? * What do they think they are learning from playing the game? Participants will need card, scissors, blue and red crayon/pens, counters, dice, paper, 5×5 grid for the three games. ‘Backwards and forwards’ and ‘Fitting into place’ should be played in pairs; ‘Ladder’ can be played by the whole group; each participant needs to prepare their own ‘ladder’. Classroom Activities A period of about 2 weeks should be given for participants to complete the three activities. Note that the first involves the entire class working in groups on problems of their choice and then a period set aside for each group to report back. The second requires participants to focus on a small group of children in order to observe how they tackle a problem. The third is a game that needs plenty of space in which your pupils can move about. Review Session You should arrange for a meeting of around 3 hours’ duration for participants to report on their observations of classroom activities. Questions have been provided to focus attention on some of the issues relating to ‘real-world’ problems, their application in the classroom and the level of mathematics required to solve them. BRIEFING PAPER Your enthusiasm towards the game of snooker might be gauged by the speed at which you switch over to another TV channel! During that time, however, you might possibly catch a glimpse of an amazing demonstration of applied mathematics when a ball strikes the edge of the table three times before gently stroking another ball, which eventually disappears into a pocket you would never have dreamed of choosing. There may have been other ways of achieving the same goal, and so it is with mathematical problem solving. Having seen a problem approached in a manner you had never considered, you may become more inclined to appreciate that methods of solving mathematical problems can be many and varied. How far should you go towards advocating particular methods for pupils to use in dealing with particular types of problem? When tackling a real-life problem, what piece of mathematics is appropriate, and what ability do you have to perform the associated mathematical operation? Different Solutions The ‘tiling a room’ problem, Stimulus Activity 1, gives rise to several quite different forms of solution. Here are a few: 1.

There are 3×5=15 square metres

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UNIT 3 APPLYING MATHEMATICS LEADER’S GUIDELINES

Each square metre holds 16 tiles

So, number of tiles required=15×16 2. 20 tiles fit along one side. 12 tiles fit along the other side. This makes 20 rows of 12

3.

Area of room is 5×3=15 m2. Area of tile is 0.25×0.25=0.0625 m2 (by calculator). Number of tiles is 15÷0.0625

4.

Large area is 500×300 cm2

=240

=240

= 240

Small area is 25×25 cm2 Number of tiles is =240

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It is interesting to contrast different approaches. In (1) and (2) the solver is likely to be thinking about actually laying tiles in some imaginary room, whereas in (3) and (4) the method used is probably drawn from a standard procedure previously learnt. In the latter case, the solver may have been taught to be ready to recognize problems of this type—called ‘area’ problems. Standard Methods for Solving Problems Analysing the mathematics used in solving the ‘tiling a room’ problem, we can identify various kinds of operation, depending on the method adopted. Multiplication is common to all four methods; several multiplications are needed whichever method is employed. Division is also involved in each of the solutions (in (1) and (2) to determine how many 25 cm lengths make up a metre). Method (1) is not necessarily superior to method (2). You might argue, for example, that method (4) is more elegant than method (3) because it does not involve the use of a calculator, that method (2) is clearer than method (4) for someone who is trying hard to understand the stages of working. Surely what is important is that children are encouraged to develop their own ways of recognizing the mathematics in an unfamiliar problem situation and applying mathematical techniques they have previously learnt. If we accept that a true ability to apply mathematics is concerned with tackling unfamiliar problems, then there are obvious dangers attached to teaching through model examples. Often teachers will try to help their pupils by giving this kind of support: If the question asks you to find: how many more or fewer, how much greater or less, the result of decreasing or reducing an amount, the difference between…, then you will know it is a subtraction problem. The pupil can eventually be let down by this approach—when working with scale factors, for example. The Processes Involved The jump from the real-world situation to a corresponding mathematical description is often not an easy one. Before you can consider working on the mathematical routines, you need to: * draw out the essential elements of the problem; * get rid of irrelevancies; * introduce conciseness with the aid of symbols. You might then be ready to begin work on the solution. This transition from the real problem to the mathematical problem is part of what is termed ‘mathematical modelling’. The same mathematical model may correspond to quite different real-world problems. For example: 1. I have 34p and five pockets. The same amount is to be placed in each pocket. How is this done?

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2. I have 34p in my pocket. What is the greatest number of 5p coins I can have? Different aspects of division are embodied in each of these problems. In (1) we are sharing and in (2) grouping. Confidence in Applying Mathematics Confidence in your own ability to do abstract mathematics will not necessarily guarantee success in using mathematics to solve real-life problems. Mathematics: Non-Statutory Guidance (NCC 1989:D1–3) emphasizes the close relationship between acquiring knowledge, skills and understanding with using and applying mathematics. If a child lacks confidence in the understanding of mathematical operations and in the skill in performing these operations, there is the inevitable discouragement when asked to apply these to solving real problems. The calculator may help to boost confidence when a child finds that mental or pencil and paper arithmetical operations are too difficult. Fun with Mathematics You must have felt that embrace of a certain glow of satisfaction come over you when a child, quite spontaneously, shared a joke, placed a puzzle or trick on your desk and expected you to come up with a solution by morning playtime. These recreational challenges are further ways of creating opportunities for developing mathematical processes, provide you with valuable insights into children’s thinking and put flesh on the abstract nature of mathematics—and they are, above all, fun. Little by little, children achieve mastery over their mathematics, derive enjoyment from the freedom they have for experimentation and feel far more inclined to share their discoveries. Look out for books and articles in magazine or newspapers that contain excellent ideas for recreational mathematics. Begin to collect these and produce a school file. Most can be adapted for your particular class. Practical Experiences in Learning Mathematics It doesn’t take much effort to think of the names of one or two well-established board games. You may find it harder to identify games from much further back in time. An interesting fact about games is that they have been found in all cultures and they survive because people enjoy playing them. Bell and Cornelius (1988) provide a good source of games from around the world. These can be both enlightening and fun, which seems to be a good reason for playing games in the classroom. A word of warning, however, is called for at this point. Playing games in the classroom must be seen in a much wider context than simply to occupy children in an enjoyable activity that ‘fills’ free time after the completion of a task, in between topics or the last half-hour of school on Friday afternoon. Introducing games into the classroom simply to fulfil these objectives carries hidden messages of our attitudes about the nature and purpose of games. If you take time to analyse the mathematical content of many games your children play in the classroom, you may find that they incorporate counting, keeping score, developing strategies and practising routine tasks. Even though you can identify these aspects of mathematics, this may not have been the principal factor in their creation and design. Skemp (1989), on the other hand, has prepared games that are driven by rules and strategies that are largely mathematical. His claim is that these games give rise to mathematical discussion.

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While you work through this Unit, you should reflect upon the part games can play in your classroom as an essential component of learning and doing mathematics. Look out for games with clear mathematical objectives—those games that will help you identify changes in attainment as a result of playing them. References and Further Reading A bibliography of mathematical games features in Mathematics in School in the January 1988 edition. This journal also published between 1986 and 1989 a series of articles on mathematical games. Bell, R. and Cornelius, M. (1988) Board Games around the World, Cambridge: Cambridge University Press. Diagram Group (1980) The Book of Comparisons, London: Sidgwick & Jackson. McWhirter, Norris D. (any edition) Guinness Book of Records, London: Guinness Books. National Curriculum Council (1989) Mathematics: Non-Statutory Guidance, York: NCC. Skemp, R.R. (1989) Structured Activities for Primary Mathematics, vols 1 and 2, London: Routledge.

STIMULUS ACTIVITIES Activity 1 Try to work independently as far as you can on the following problem. Be prepared to describe to the group your method of obtaining a solution and all the detailed steps you covered. Materials needed: a supply of plastic squares. A rectangular room is 5 m long by 3 m wide. The floor is to be tiled. Each tile is square and measures 25 cm by 25 cm. How many tiles are required? Compare your method with those of the other participants. Is there a best method? Why would one method be preferable to another? What do you learn about the nature of the teaching and learning of mathematics from these methods? How do you respond when a child uses methods other than those you have taught?’ What do you think of an approach which encourages children to ‘spot’ problems, ‘Ah! Haven’t we seen this sort of problem before? It’s an area type problem, isn’t it?’ Activity 2 This illustration may help in showing the difference between a ‘real-world’ problem and a ‘mathematical’ problem: Real-World Problem Mathemathical Problem Kim’s grandma gives a bag of twenty-seven sweets to Kim to share among What is 27÷4? her friends Jan, Mel and Sal. How should they be shared among the four friends? What are the distinctions between the two kinds of problem?

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Think up quite different real-world problems which correspond to the same mathematical problem above.

In solving the real-world problem, it is transformed to a mathematical problem (the mathematical model), mathematics is used to solve it, and, finally, the solution is interpreted in terms of the original problem. Mathematical modelling may look like this:

Choose a real-world problem, typical of one that children might meet in their mathematics lessons. Identify the ‘mathematical problem’. What difficulties are the children likely to have in translating from the real-world to the mathematical problem? Analyse the way it may be solved (operations, routines). Explain what difficulties there could be with the interpretation of the solution in terms of the realworld problem.

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Activity 3 Arrange yourselves into groups of three. Work on these three problems using the ‘goldfish bowl’ technique. This technique is carried out as follows: two of you work together on a problem; the third member of your group will observe the way you attempt to solve the problem; take turns to be the observer. Problem 1 Materials needed: calculators. An article in a shop is priced for sale by adding 15% tax to the cost price. Later, the shopkeeper announces a ‘sale’ and prices are reduced by 5%. From the customer’s viewpoint, would it be better to have items reduced by 5% first then 15% tax added afterwards, or add the 15% tax first and then reduce by 5%? Problem 2 Materials needed: two identical vessels, a glass, water (clear and coloured). Two large identical vessels, A and B, are half-filled: A with wine and B with water. A glass full of wine is transferred from vessel A to vessel B. After thoroughly stirring this wine-water mixture in B, a glass full of this mixture is emptied into vessel A. At the completion of this exchange of liquids, is the percentage of water in vessel A greater or less than the percentage of wine in vessel B? Problem 3 Materials needed: tape measures. Because it was such a beautiful weekend, we decided to take the children to the seaside. Of course most of the time was spent swimming and building sand-castles. Above is an outline of the impression of a hand one of my children left in the wet sand. How tall is this child? When you are assigned to observe, you might like to consider the type of collaboration that goes on between partners. For example:

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* * * * * * * * * *

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What happened at the start of the task? What questions were posed by the pair? What statements were made? What plans were agreed? Were contributions accepted by both participants? Were lines of thought followed up? Were ideas ignored? How did the pair record their results? How did the pair encourage each other? How did the pair react to their final solution?

After working on the Stimulus Activities, share your thoughts with other participants on the following questions: * Can/did you confidently tackle these problems? * What reasons would you give for having confidence or lack of confidence? * How can you build children’s confidence in applying their own mathematics? Activity 4 Select one game you have seen your pupils play with enthusiasm during the last two weeks. Describe this game. Consider the mathematical content of this game.

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Discuss the appropriateness of this game for: * introducing a particular topic; * reinforcing a particular skill; * acting as a means of assessment.

Share the reasons your pupils give: * for why they play the game; * about what they think they are learning by playing the game. Now work through the following activities and note their: 1. mathematical objectives; 2. potential for variation; 3. investigational possibilities. (a) Backwards and forwards (for two players) Materials needed: strips of card, blue and red crayons/pens, counters, dice. Make a strip about twice the size of the one above. Divide it into thirteen equal parts. Colour one end blue and the other red. Place a counter on the middle of the strip. Decide who is blue and who is red. Blue throws a die and moves the counter towards the blue end the number of places shown on the die. Then red throws the die and moves the counter back towards the red end. The winner is the player whose counter moves off the coloured end first. (b) Ladder (for any number of players) Materials needed: paper, card. Each player draws a ladder with ten spaces. Make one pack of cards numbered from 1 to 20. Shuffle the pack and draw one card from it. Each player has to write that number in one of the spaces of their ladder. Draw out another card, and again players write that number in one of the remaining spaces on their ladder. Players who have guessed wrongly and have no space for a number that is drawn drop out of the game. Continue until all the ladder spaces are filled. The winner is the person with ten numbers in order, from the lowest at the bottom of the ladder to the highest at the top.

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(c) Fitting into place (for two players) Materials needed: card, 5×5 grid. Prepare a pack of cards with numbers 1 to 30. Prepare a 5×5 board and arrange around the columns and rows a selection of name cards marked, for example: Odd Numbers, Even Numbers, Factors of Two, Square Numbers. Cut the name cards to different lengths. (You could have the cards as long as one square, two squares or three.) These name cards can be varied according to levels of mathematical ability. Deal seven cards to each player. Take turns to place a card from your hand on one of the cells. When no more spaces can be filled or both players are unable to place any more cards, add the numbers on the remaining cards. The player with the lower total wins. Here is an example of a game in progress, showing the players arrangement of cards and their opening moves.

Player A was dealt number cards 1, 10, 12, 15, 16, 20, 21, and Player B number cards 3, 5, 6, 7, 9, 23, 29. Move Player A Player B

1 16 9

2 20 29

3 12 7

4 …? …?

5

6

7

Total left

CLASSROOM ACTIVITIES Activity 1 Materials needed: rulers, tape measures, calculators, large sheets of paper and felt-tip pens. You will need to set aside one of your lessons to have your class work in groups on one of the openended questions below. Let each group choose which question they would like to work on. Try to refrain from imposing your approach to solving the problems and let the children formulate their own ideas at the

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outset. Ask them to record their results so that they can explain to the rest of the class how they came to their solution. 1. 2. 3. 4. 5. 6. 7. 8.

Are there more boys than girls in the class? How many more? How many legs (human, table, chair…) in the classroom? Who will be next to have a birthday? If we all held hands, how big a ring could we make? Could we stretch around the classroom? Suppose everyone says ‘Hello’ to everyone else, how many ‘Hellos’ would there be altogether? Have you been in school for more than a thousand days? What fraction of your life have you spent in school? Plan your perfect school timetable.

Make notes on the manner in which they tackled the problems. Identify interesting features revealed during your children’s reporting-back session. Activity 2 Materials needed: paper, non-standard units for measure such as conkers and fir cones, rulers, balances, weights, lists of postal charges, a parcel already wrapped for (3) and access to children’s lunch boxes. Allocate one of the following tasks to a small group of children. Ask them to solve the problem, and tell them to record their work so that they can tell another group what they have done. 1. By bending a sheet of A4 paper so that the two opposite edges meet, you can make two different cylinders. How much space is there in each cylinder? 2. Select a shelf in your classroom and tell the children that you have been given enough money to buy books to fill that shelf. How much money have you been given? 3. You have been asked to send a parcel to a friend. How much will it cost to post? 4. How big should a shelf be to hold all the lunch boxes in your class? Collect samples of your children’s work in both activities. Prepare a report on what Activities 1 and 2 reveal of your children’s mathematical ability. Activity 3 Playground friends Materials needed: sheets of paper, safety pins, adhesive tape. Play the game with your entire class. Pin a piece of paper on each child with a number from 1 to whatever the number of children in the class on the day you play this game. Find an open space and call out the following instructions: ‘Join right hands with a number that is half your number’ or ‘Join right hands with a number that is double your number’ or ‘Join right hands with a number that is two more (three more…) than your number.’ Decide on other variations to the numbers and note your pupils’ reactions.

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REVIEW SESSION 1. Report on a sample of your children’s work. Pay particular attention to identifying the children’s approaches to transforming the real-world problem to a mathematical one from the Classroom Activities. The following questions might help you to focus the discussion: * * * * * *

Can you say that the children recognized the mathematics of the problem? Were your children able to develop a mathematical model? How much support did you give to your children in order to get them started? Are the objectives concerned with the application of mathematics difficult to teach? Is it difficult to think of real-life problems based upon the classroom environment? Do most of the problems you use tend to be contrived?

2. Now consider the way you frame questions. How a question is posed can often determine the level of application required. The following examples may help: * ‘How much higher than your chair is the teacher’s chair?’ is more direct than ‘Is your chair the right size for you?’ * ‘What is the area of the classroom floor?’ can be compared with ‘Which of these rooms is the bigger?’ or ‘Do we have room for another two tables?’ * ‘What is the height of this archway?’ is merely a task, in comparison to ‘Will a fire engine be able to pass through the archway?’ Many problems that are set can be pretty meaningless. It seems much more satisfactory to provide purpose to the activity. For example, instead of asking, ‘What are the dimensions of the cricket bat?’, you could find reasons why ‘this (adult’s) cricket bat is or is not the right size for you’. This kind of enquiry draws attention to the critical dimensions that have to be considered when making a decision.

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3. Comment upon the following statement: Children find less difficulty applying their ideas on number to ‘money’ than to other areas of work. Come to some conclusion about which of the contexts your pupils felt most at ease with. Brainstorm on developing problems where the application of mathematics can be fostered. 4. Work on this Unit has briefly touched upon the place of games in the classroom. Using games in mathematics may raise a number of eyebrows as well as questions. Discuss the following: * * * *

What is a game? What distinguishes a game from a puzzle? In which ways can games and puzzles contribute to the teaching of mathematics? What provision have you made in your curriculum for regular immersion in mathematical activity through games and puzzles?

5. Mathematical oddities, ‘magic’ numbers, tricks, puzzles and games provide that initial spark of interest which can lead towards the generation of an atmosphere of fulfilment, creativity and perseverance both inside and outside the classroom. Take, for example, this resource for mathematical discovery that fascinated children in my classroom. It is a question that appears on the jacket of The Book of Comparisons (The Diagram Group, 1980): If a flea were the size of a man, how far could it leap? This book brings together accounts of human endeavour and extremes in the physical world, and presents the comparisons we may encounter in everyday experience. Any edition of the Guinness Book of Records can stimulate mathematical activities in a similar way. Who, for example, is the champion left-arm-over-theshoulder-bean-bag-thrower (right arm if you are left-handed!)? Use these kinds of books as a resource from which to develop ideas for games and activities in order to enhance your pupils’ understanding of mathematical concepts.

UNIT 4 MATHEMATICAL PROBLEM SOLVING

LEADER’S GUIDELINES This Unit introduces teachers to some of the characteristics underlying problem solving as a mathematical activity. It follows on from aspects of problem solving dealt with in Unit 3. The methods of enquiry highlighted in this Unit are further applied in Unit 6. Engaging children in the strategies of mathematical thinking such as sorting, classifying and searching for pattern is a means of focusing upon helping children to recognize generalities and express these in words and symbols. Indeed, making sense in this way through an awareness of connections should be identified as a principal feature in each of the Units. Briefing Paper You should hand this paper to the participants after they have tackled Stimulus Activity 1. Allow them a considerable amount of time for reading and discussing the analysis and strategies described in the paper. Stimulus Activities Three hours should be sufficient time for participants to work on the following activities. Participants should work in pairs on Activity 1, and compare and review their solutions with the notes in the Briefing Paper. They should then work on Activity 2 and finally share their results with the other participants. The four tasks in Activity 3 could be worked as an entire group and Activity 4 in pairs again. 1. Frames. Have available plastic tiles and squared paper. This activity could be initiated by your posing the problem by arranging the plastic tiles as shown in the diagram on page 55. 2. Prisoners. Pairs will need sheets of large-squared paper and counters. You may find it necessary to suggest (after a while) that a simplification of the problem (to a 3-by-3 or 4-by-4 square) would help greatly. The arrangement of escaped prisoners outside the prison walls is crucial for the least number of moves. You may feel you should place restrictions on the problem by telling the participants that the prisoners should occupy a square formation outside the prison or that all prisoners must have final

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positions south of the door (which runs east to west). There are also possibilities for extending the problem, such as re-positioning the door, creating a second doorway or even introducing walls to restrict the movement of the prisoners outside the prison. 3. Testing statements. Small sheets of paper will be required as well as large sheets on which these can be glued when agreement has been reached on their distribution. Number sequences. Tile sequence. You will need sufficient plastic tiles or Unifix cubes for the participants to make at least four stages of their sequence. Continue the pattern. Objects such as wooden blocks, beakers, leaves, cones should be made available for pairs to devise their own linear arrays. 4. Building doorways. You will require Cuisenaire rods for the participants to build at least three stages of the sequence. A copy of the child’s written work should also be available. Classroom Activities These activities deal with shape and number. Participants should be given around 2 weeks before they report back on their pupils’ work. 1. 2. 3. 4.

Towers, squares and perimeter change. Consecutive numbers, adding pairs of numbers. Fold in half. Copy and continue.

It is important that participants make careful notes about the strategies children apply while working on these tasks as well as recording how they arrange the materials in the ‘copy and continue’ tasks in Activity 4. Get the participants to save examples of children’s methods of recording, which may be: * written generalizations; * tables of results; * diagrams or pictures. Review Session Allow 3 hours for this session. 1. Discussing the problem-solving attempts by the pupils. 2. Identifying situations where activities with pattern can be incorporated. 3. Other issues concerning problem solving. One outcome of considering the questions posed should be the collection of further ideas to supplement the present scheme of work for mathematics in your school. BRIEFING PAPER During a maths festival centred on a large building complex, groups of children faced up to challenges of ‘real-life’ problems. One of these problems was concerned with delivering bulky goods. What would be the

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maximum size of vehicle able to pass under the arch? Immediately, the children found the maximum height of the arch and claimed that this would be the height of the vehicle. They then realized that the vehicle would need to be wafer-thin to get through the arch! Parking facilities had also been a constant problem for this building complex, so the children set out to reorganize the position of the parking spaces. It eventually became apparent that packing cars tightly against one another, as they suggested in their initial plans, left drivers with the only option of climbing out of their sun roof. Those without this feature must have had to spend the rest of the day imprisoned in their cars! These are what you could call ‘everyday’ or ‘real-life’ problems. They need to be solved by directly applying a knowledge and understanding of mathematical techniques. Unit 3 deals with more aspects of this kind of problem. There are, however, problems that fall into another category. Fascinated by a card showing the twelve days of Christmas, some seven-year-olds decided to find out how many presents were actually sent altogether. As a result of this enthusiasm, their class teacher sent me ‘The twelve days of Christmas—a correspondence’, by John Julius Norwich, in G.Smith (ed.) The Christmas Reader, Penguin, 1986. Here are two excerpts: 25th December My dearest darling That partridge, in that lovely little pear tree! What an enchanting, romantic, poetic present! Bless you and thank you. Your deeply loving Emily. A few days later: 5th January Sir Our client, Miss Emily Wilbraham, instructs me to inform you that with the arrival on her premises at half-past seven this morning of the entire percussion section of the Liverpool Philharmonic Orchestra and several of their friends she has no course left open to her but to seek an injunction to prevent your importuning her further. I am making arrangements for the return of much assorted livestock. I am, Sir, Yours faithfully, G.CREEP Solicitor-at-Law Now, turning back from this diversion to the second category of problem solving, you may note that the main characteristic is that these problems challenge the problem solver to try out general strategies of working in what are, to a large extent, unfamiliar situations. This Unit concentrates on these kinds of problem-solving situations. The strategies that need to be adopted in solving problems are the subject of discussion in a number of books (Mason et al. 1982: Burton 1985). A list of general strategies to be treated as teaching objectives is also set out in the DES publication, Mathematics from 5 to 16 (1987). As there are a wide variety of strategies available to the problem solver, attention will be restricted in this Unit to those listed below: I

Internalizing the problem

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II III IV V VI VII VIII

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Simplifying a complex problem Approaches through ‘trial and error’ or ‘trial and improve’ Working systematically Looking for, identifying and explaining pattern Reasoning Hypothesizing or conjecturing Making generalizations Analysis of Problem-solving Activity

To illustrate the different types of strategies used in solving problems, one method of finding a solution for a particular problem is presented for you to work through. Certain crucial stages of the solution are highlighted (indicated by A, B, C…in the margin) and corresponding commentaries are given at the end. The comments make direct reference to the strategies listed above and marked by Roman numerals. The Problem The ‘frame’ in the diagram comprises sixteen squares on the outside. These surround nine squares inside. Find a frame which comprises the same number of squares on the outside as those inside.

A Solution

* A * B * C Frame comprises and encloses 16 squares

20 squares 25 squares

24 squares 4 squares

12 squares

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* D Table of results: When there is one square inside, we can say the length of its sides is 1 unit. The next inside square has to be 2 units long because it is made up of four tiles (whose sides are each 1 unit long) placed together in a square. These measurements are called ‘inside measurement’ in the table below. * E Inside measurement 1 2 3 4 5 6 7 8 No. of squares outside 8 12 16 20 24 28 32 36 No. of squares enclosed 1 4 9 16 25 36 49 64 * F In the table, you should be looking for two entries in the same column, in the second and third rows, which are the same value. Notice that in one set, the numbers increase by 4 each time, whereas in the other set they increase by growing amounts: 1, 3, 5, 7,…At the stage where the inside measurement is 5, the value for the number of squares enclosed ‘overtakes’ the value for the number of squares contained. At this point, the two sequences of numbers converge; after this, they diverge. This happens where the value for the inside measurement is between 4 and 5. * G Suppose the inside measurement is x.

The number of squares in the frame is: (4 times x) add 4 or The number of squares enclosed is: x times x or x2 You are looking for a value of x which fits this statement: You may be familiar with the technique of solving this equation. A graph may be of help too (see p. 52). If you look at the graph, you will see that the results show there is no whole-number value which does in fact fit. This agrees with what has already been said about a value being ‘between 4 and 5’. * H Suppose you start with an oblong frame… Comments on the Stages in the Solution 1. Here you become familiar with the problem by trying a few particular examples. At this stage, you may feel that you understand the problem— STRATEGY I.

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2. You decide to concentrate on square frames in order to keep the problem as simple as possible— STRATEGY II. 3. You now try several examples to see what happens—STRATEGY III. 4. You adopt a systematic approach and record your findings in a table— STRATEGY IV. 5. Now you begin to spot patterns—STRATEGY V. 6. You analyse the patterns in terms of the problem to be solved. The reasoning here depends on the argument that the sequences progress according to a rule. The rule at this stage is merely a conjecture— STRATEGIES V, VI, VII. 7. In order to search for a reasoned argument, you consider the general case. In fact, you idealize. The use of symbolism is important here. A diagram used to represent the ideal case is helpful when you try to form

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algebraic relations. From this, you can establish an equation which represents the problem in abstract form. You formulate a result by applying some standard mathematical techniques, and this is interpreted back in terms of the original problem—STRATEGY VIII. 8. Because of this success, you decide to go back to the initial problem and extend the class of ‘frames’ by considering oblongs. STRATEGY I Closed and Open-ended Problems

The wording of the problem that you have solved might suggest that, as soon as an answer is found (that is, a frame with particular dimensions satisfying a stated condition), then the solution is complete. However, at this stage you may be drawn into asking whether this particular solution is unique, in which case the problem is immediately extended. Perhaps other related questions come to mind, such as: ‘What happens if…’ If a problem lends itself to these possibilities and you take advantage of them, then the ensuing activity becomes investigational. You create your own problems and challenges based on a starting-point. Pupils working in this way are engaged in a truly mathematical activity. This is a valuable experience in itself, and it is where effective learning is likely to take place. Problem Posing

There is a great deal of difference between seeing, looking at and attending to what lies in our field of vision. Many signals compete for our attention. Making sense of these messages received from the world around us often takes place when similarities or differences are noted. Take, for example, the following anecdote of a purchase I made in a large store one Friday afternoon. I received the items I bought, along with the receipt which is shown below:

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These items were bought because: 1. I was due to travel by train; 2. I assumed it would be cheaper if I made the purchase before getting on the train. Already there are signs of making generalizations based upon my experience of rail travel. What followed, however, opened up a more interesting excursion into mathematical thinking. Before leaving the store, I decided to pick out one receipt from each waste-bin at the end of each checkout counter. These are reproduced below:

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On the basis of this evidence, I thought it reasonable to conjecture that only small numbers of items were bought at this time of day—every day or only Fridays? This was the beginning of an exercise in making an hypothesis, trying out an hypothesis and eventually stating a generalization. That afternoon, the train journey passed very quickly! References and Further Reading Burton, L. (1985) Thinking Things Through: Problem Solving in Mathematics , Oxford: Basil Blackwell. Department of Education and Science (1987) Curriculum Matters No. 3: Mathematics from 5 to 16, London: HMSO. Mason, J., Burton, L. and Stacey, K. (1982) Thinking Mathematically, Reading, MA: Addison Wesley. Moses, B., Bjork, E. and Goldenberg, E.P. (1990) ‘Beyond problem solving: problem posing’, in T.J.Cooney and C.R.Hirsch (eds) Teaching and Learning Mathematics in the 1990s, Virginia: National Council of Teachers of Mathematics.

STIMULUS ACTIVITIES Activity 1 Materials needed: plastic tiles, squared paper. This is a contrived problem which you could say is related to a practical situation such as laying paving stones around a grassed area. The ‘frame’ in the diagram comprises sixteen squares. The number of squares that will fit inside the frame is nine. Find a ‘frame’ which comprises the same number of squares as that which can be made to fit inside. Work in pairs on this problem and identify the stages through which you pass during the process. Make notes on the strategies you

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applied during the course of your attempted solution. Compare these notes with those the course leader hands to you when you decide you have worked on the problem as far as you can. Activity 2 Materials needed: large squared paper, counters. Work in pairs. An 8 by 8 square represents a prison. The diagram shows the wall and a door positioned at one corner. A prisoner (counter) occupies each of the 64 squares inside the prison. Now they start to escape! They move in turn to an adjacent square, to the left, to the right, up or down (not diagonally), continuing this way even outside the prison boundary. No two prisoners may occupy the same square. What is the least number of moves needed to get all the prisoners out of the prison? Identify the strategies you apply and share your solution with the other participants.

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Activity 3 Work as an entire group. 1. Testing statements Materials needed: paper, glue. Make statements about any topic and write each on small sheets of paper. Pool these statements and then sort them into those that are true, sometimes true and not true. Glue the statements in these categories on to three large sheets of paper. Identify the occasions when ‘counter examples’ are suggested that show how statements do not hold up. Example: Statement—All participants in this group are over thirty. Counter example—Mrs X is twenty-nine. 2. Number sequences Individually decide how you will extend the number sequence 1, 2, 3… Share this with the rest of the group who are to discover the rule you have chosen. Repeat this activity using other number sequences. Record the sequences devised by the participants. Note the constraints within these sequences on the number of stages to be revealed before each sequence can be confidently followed. 3. Tile sequence Materials needed: plastic tiles or Unifix cubes. Make up sequences with square tiles. Search for a variety of ways of continuing the sequence. At what stage do we know with certainty when we have identified the ‘rule’ that dictates the construction of the sequence?

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4. Continue the pattern Materials needed: a variety of objects e.g. wooden blocks, beakers, leaves, cones. Working in pairs, ‘copying, continuing and devising repeating patterns’. Use a variety of objects: natural materials and structured apparatus. Let one person in the pair arrange two stages of the pattern which has to be continued by the second person. Make a record of your patterns as well as the responses from the second person. Identify possible variations in interpretation of these patterns. Discuss with the whole group the implications of these activities in assessing a child’s understanding of pattern. Activity 4 Building Doorways Materials needed: black Cuisenaire rods, copies of the child’s written work. Build the first three stages of the chain of ‘doorways’ shown below:

Now discuss with your partner the meaning of the written response the child has made in answer to the following questions: How many rods would be needed to build (1) ten doorways (2) any number of doorways? (Remember, the child had sufficient rods to build three stages only.)

CLASSROOM ACTIVITIES Observe small groups of children working on the following problems. Ask them to keep a record of their work so that you can use this in your presentation during the Review Session.

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Activity 1 1. Towers Materials needed: a large supply of Multilink blocks. Build three towers, each of a different colour. Ask the children to find out how many different heights of towers they can make using all three blocks. They can stand them up or lay them on their side or fix them together end to end. Investigate other sets of blocks.

2. Squares Materials needed: a large supply of 3×3 grid paper, glue, large sheets of backing paper. Cut out one of the smaller squares from the 3× 3 grid and draw the children’s attention to the change this has made to the shape of the original grid. Challenge them to find out the different shapes that can be made by cutting out one square from the other 3×3 grids. Investigate by cutting out two, three,… squares.

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3. Perimeter Change Materials needed: a large supply of square plastic tiles. Place the tiles edge to edge in front of the group and count around the shape indicating the perimeter. Rearrange the same number of tiles and once again point out the perimeter. Let the children investigate the change in perimeter for a fixed number of tiles placed edge to edge.

Activity 2 Materials needed: a number line. 1. Place a number line in front of the children and identify sets of consecutive numbers. Select a pair of . Now add the previous three consecutive consecutive numbers and add them together; e.g. . numbers together; e.g. Challenge the children: ‘Does it always happen that the same number appears?’ 2. Tell the children that they can use +,×and − as many times as they like (along with=of course) with the digits 1, 2, 3 and 4. They are only allowed to use each digit once. Give them one example; e.g. or . Ask them to find as many sums as possible using different combinations. 3. Select any pair of numbers. They don’t have to be consecutive. Add these numbers together; e.g. use 2 and 6. Ask the children to find what numbers can and cannot be made by adding these two numbers together in different ways.

Now ask the children to investigate for other pairs of numbers. Activity 3 Fold in half. Materials needed: a large selection of sheets of paper of varying sizes (newspaper will do).

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Get the children to keep on folding a rectangular sheet of paper into half to find out what is the greatest number of possible folds. Let the children compare their results and note whether or not they generalize. Do they, for example, think the size of paper is an important factor? Activity 4 Copy and Continue (A). Materials needed: a wide selection of materials such as pencils, acorns, cups, beakers, scissors. Make a line of some of these materials according to your rule, and then ask the children to continue the pattern. Note what materials they select and how far they extend the pattern. Ask the children to explain the rule. Example: cup, beaker, beaker, cup, beaker, beaker… This is a 1, 2, 1, 2…pattern. Get the children to make their own linear patterns. Copy and Continue (B). Materials needed: tiles. Arrange two stages of tiles as follows:

Ask the children to lay the next stage, and then the next… Get them to talk about their rule. Copy and Continue (C). Write down a number sequence and get the children to continue the pattern. Ask the children to explain the rule. Get the children to make their own number sequences and explain their rules. REVIEW SESSION 1. Report on the outcome of your pupils’ investigations. You could use the following questions in order to focus your discussion. In relation to the outcome of the classroom activities and your own efforts at problem solving: * How difficult is it to identify the types of strategies used in solving problems? * Which strategies are the more readily adopted ones?

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* Should you teach children ways of approaching problem-solving situations and deciding upon appropriate strategies to be used? * How do you assess children’s ability to solve problems (i.e. problems set in unfamiliar situations)? 2. These questions are directed towards other issues of a general nature concerning problem solving. This opportunity of discussing them with other participants may help you establish a consensus about this aspect of mathematics teaching and learning in your school. * How can this model of teaching problem solving be justified to those who claim that basic computational skills are the most important part of mathematics? * If your school uses a published mathematics scheme, how much of the problem-solving model is supported by it? Which are the areas of deficit, and how can they be remedied? * How can you justify using a common starting-point for an investigation with a whole class of children when they must be at so many different levels of development in mathematics? 3. Consider different ways of posing problems. Take, for example, the frequent use of the format: a question with a unique answer. Apply ‘What happens if not…’ to this type of question. The following may help you get started: QUESTION: How many 2p coins does it take to make 10p? ANSWER: Five. Moses, Bjork and Goldenberg (1990) (see page 54) show how this kind of question can be presented in a far more challenging manner. They ask what kind of information the problem gives, what kind of information is unknown (and wanted), and what kinds of restrictions are placed on the answers. An analysis of the problem shows: (a) information ‘known’—the sum of money (10p); (b) information ‘unknown’—the number of coins (5); (c) restriction 1—the particular coin to be used (2p). What happens if ‘the sum of money’ is unknown and the number of coins is known? By making this reversal, a new question is generated: what sum of money can be made by 5 coins all of 2p value? What happens if the coin is not restricted to the value of only 2p? The problem might, therefore, look like this: (a) information ‘known’—the number of coins is 5; (b) information ‘unknown’—the sum of money; (c) restriction 1–2p must be used at least once. You may also wish to introduce further restrictions. Identify and change constraints to problems selected from the published scheme you use in your school. 4. Develop and resource your existing scheme of work in mathematics by ensuring that pattern becomes one of the major underlying themes in learning mathematics. Search for pattern while:

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counting arranging materials organizing data working with tables interpreting graphs engaging in daily routines.

UNIT 5 CHILDREN’S UNDERSTANDING OF MATHEMATICS

LEADER’S GUIDELINES This Unit explores what is meant by ‘understanding’ in mathematics and shows how you can help children develop this. Children provide ample evidence of their attempts at making sense of the world of mathematics. This Unit considers what creative thinkers children are by taking a closer look at the errors they make in this subject. Aspects of teaching and learning mathematics are explored in relation to the three elements—facts and skills; conceptual structures; and general strategies and appreciation—identified in Paragraph 240 of the Cockcroft Report. It provides a model by which the mathematical content of any of the activities in this and other Units can be analysed. Briefing Paper Let each participant have a copy of the Briefing Paper about one week before you begin this Unit. Under corresponding headings, there are brief accounts of various issues related to understanding in mathematics. Reference is made to ‘fractions’ to illustrate the particular issues, and it may be necessary to clarify some of the terminology associated with this topic. The notes on fractions at the end of the Briefing Paper should be useful in this event. Stimulus Activities Three hours will be required for these. Participants could work in small groups on each activity and then join together to discuss their work for the remaining 45 minutes of the session. 1. Parts of a whole. Groups will need copies of the worksheet (page 62) and scissors. 2. Computing fractions. Make available Mathematics in the National Curriculum (DES 1991) and copies of Paragraph 300 of the Cockcroft Report (DES 1982) ready to distribute among the participants when they have completed Activity 1. Structured apparatus should be available for those participants who want to demonstrate a point with concrete materials. 3. Decimal notation. Sufficient calculators and money tokens will be needed to share among pairs of participants. 4. Objectives. Make sure that participants have access to place-value apparatus as well as money tokens.

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Worksheet

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5. Placing mistakes in categories. Participants will need the Briefing Paper. Classroom Activities A period of 2 weeks should be sufficient time for participants to cover these activities with their children. The first activity continues with ‘fractions’, and then other topics are introduced to provide opportunities for children to demonstrate their understanding in a wider mathematical context. Participants should collect examples of errors that children have made in mathematics over a period of two weeks. In preparation for the Review Session, ask them to prepare a presentation that will: * identify and classify errors; * examine the mathematical context in which these errors occurred; * generate ideas for activities to help overcome these. 1. Finding half 2. Topic 3. Front page 4. Letter 5. Recording errors Review Session You should anticipate a number of important issues about the teaching and learning of mathematics arising from the participants’ reports. It is not necessary to have reports from every participant about each activity. You could ask a participant to identify one issue and then have the group discuss this with reference to a particular activity. It may become apparent from the informal discussions that participants identify certain children who are failing to use their knowledge in mathematics. If this is the case, you could get the participants to make suggestions why they think this might be. Remember that much more than an accumulation of isolated facts is involved in the teaching and learning of mathematics. You could get participants to consider: * * * *

whether or not they attempt to identify their children’s previous knowledge on a particular topic; what use they make of their children’s previous knowledge; the kind of classroom setting that is associated with mathematics; whether or not searching for alternative solutions and different points of view feature in classroom discussions. References and Further Reading

Department of Education and Science (1982) Mathematics Counts: the Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the Chairmanship of Dr W.H.Cockcroft, London: HMSO. —— (1991) Mathematics in the National Curriculum, London: HMSO.

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BRIEFING PAPER Serious challenges may come from those very close to you. They often arise out of desperation because of the time you are spending on preparing yet another topic. It runs something like this: ‘Didn’t you do that topic last year? Why can’t you repeat it? After all, they’re not the same children.’ It is hard to explain why you feel the need for a reappraisal. Attempting to do so often makes you begin to reason: ‘Why not repeat last year’s work?’ Deep down, though, you know why. There is something about revisiting a familiar topic with a ‘fresh pair of eyes’. You are a different person from last year, anyway. With all that knowledge gained by working on that topic, there is no chance of repeating it verbatim. You are pretty self-critical, and no doubt this drives you towards seeking to improve your own practice. Similarly, you may wonder about theories of mathematics teaching. Does ‘first concrete then abstract’ always apply? Is ‘practical work’ so important after all? Does ‘I do and I understand’ stand up to scrutiny? Not if it means repeating mundane tasks. Once you begin to reflect upon ‘understanding’ and, in particular, your children’s understanding, you naturally become critical of learning theories and the influence they have upon your own practice.

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As you work through this Unit, you will engage in: * * * * *

thinking about the nature of ‘understanding’ in mathematics; analysing different levels of mathematical understanding; considering how children develop their understanding of mathematical concepts; deciding upon the kind of teaching approach when aiming to promote learning with understanding; examining ways of assessing pupils’ mathematical understanding. The Nature of Understanding in Mathematics There is general agreement that understanding in mathematics implies an ability to recognise and to make use of a mathematical concept in a variety of settings, including some which are not immediately familiar. (DES 1982: Para. 231)

You may be tempted to say that a pupil has suddenly achieved understanding (the ‘Eureka!’ effect). But strictly speaking, understanding can only develop—more and more—as further appropriate experiences are gained. These experiences will provide the means for either drawing together concepts already learnt, or relating a newly encountered idea to some concept previously acquired. Levels of Understanding To say I understand how to do this piece of mathematics in this particular way. is quite different from I understand why I do this piece of mathematics in this particular way. ‘How to’ implies that my learning has been very much at an instrumental level. This means that I am at a level where I have learnt a method by rote which I can reproduce whenever I recognize the need to do so. ‘Why I’ implies that I am able to justify my way of working by relating it to ideas and processes with which I am already familiar. To illustrate this, take an example from the conversion of an improper fraction 7/3 to a mixed number 2⅓. Now, you may know how to carry out this conversion simply because you recognize a ‘top-heavy fraction’. The numerator 7 is larger than the denominator 3. Whenever you meet this situation, you remember that you can divide the numerator by the denominator and, in this case, will get 2 with ⅓ left over. This will therefore give the mixed number of 2 wholes and ⅓. Compare this approach with a way of working that results from a relational understanding built upon concrete experiences: I have seven thirds. Three thirds give a whole. Six thirds give me two wholes.

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And I’ve got one third left over. (Of course I need to understand all the relevant notation as well.) Often it is very much quicker to teach and learn instrumentally. But the pay-off with relational understanding is evident when, later, pupils have to deal with work which relates directly to this earlier work. Developing Understanding Mathematical understanding does take time to develop. It is generally accepted that children’s understanding of mathematical concepts will be enhanced if they frequently encounter the same concepts over a long period of time. In a gradual way, these are assimilated and allowed to grow. In addition, for children to grasp abstract concepts and handle them with confidence, it helps if they are allowed to meet them within a wide variety of concrete situations. For example, continuing with the theme on fractions, concrete experiences should ensure that fractions are embodied in real objects (chocolate bars, fish fingers, cakes, a class of children) or in classroom materials (plastic pieces, wooden strips, paper shapes that cover a wide range of possibilities—squares, oblongs, circles, triangles). Teaching Approaches Ideally the activities provided for children should involve them in constructing their own objects to work with, help them think and enhance their discussion. In this way, they should more readily relate their new learning to earlier experiences of a similar nature. Two points worth emphasizing here are: * the importance of organizing for practical work; * the freedom for children to express their thinking aloud. Actually, verbalizing your thought processes goes hand in hand with the development of understanding. Letting the children consolidate their thinking by also allowing them to prepare a classroom display about their practical work can also have true value. Assessing Understanding To determine a pupil’s level of attainment in some respect of mathematical understanding generally requires a good deal of teacher skill and time. Assessment tasks for this purpose will sometimes have to be of a practical nature, for it is the child’s response to a practical activity that often holds the clues as to whether or not there is an understanding of the mathematics involved. Think of your response to Kim as she shares something of the excitement she has experienced with ‘19’. (See page 67.) This was, however, written on 1 May 1990, and you can see that there is an opportunity for some teacher input concerning the numbering of centuries! Nevertheless, what Kim has written shows how important it is to examine carefully the errors children make. This Unit, therefore, should help you consider:

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some of the reasons why children make mistakes in mathematics; ways in which you can interpret children’s mistakes; children’s thinking that is revealed by their mistakes; ways in which children learn mathematics.

The Dilemma Tensions occur between teacher and pupil as a result of the beliefs and practices relating to the teaching and learning of mathematics. It may be associated with what Mason (1989) describes as a ‘didactic contract’: * The pupils know that the teacher is looking to them to behave in a particular way. * The teacher wishes the pupils to behave in a particular way as a result of their understanding of the concepts or the topic. * The more explicit the teacher is, the more readily the pupils can provide the sought-after behaviour. * The pupils are seeking this behaviour and expect the teacher to be explicit. * The teacher is concerned that, by being more explicit, teaching is less effective. There is an area of educational research which offers evidence that children learn mathematics only when they construct their own mathematical understanding. This means that you cannot do the learning for the child. When you emphasize the written format, the manipulative skills and technical terms in order to carry out set procedures, you have to remember that your children are also engaged in making sense of these factors by probing their past and fragmentary experiences. You will note that they are to enact for themselves verbs that permeate Mathematics in the National Curriculum (DES 1991): ‘select’, ‘describe’, ‘check’, ‘read’, ‘use’, ‘place’ and ‘solve’. Children engage in and impose their own interpretation on what is presented before them. Identifying Elements Recommendations in Mathematics: Non-Statutory Guidance (NCC 1989: D 6–7) show how ‘closed’ tasks can be changed to provide more challenging ‘open’ activities. These adjustments require pupils to employ a

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much wider range of skills and strategies. In this Unit, you will be working on activities that enable you to identify some of these, skills and strategies. This, in turn, should help you to formulate a teaching style appropriate to that particular piece of mathematics in which these skills and strategies are applied. Before you start work on the activities in the Unit, you should read the section in the Cockcroft Report dealing with teaching methods. Three elements in the teaching of mathematics have been distinguished by research, and these are described in Paragraphs 240 to 243. A wider discussion of these elements is found in Mathematics from 5 to 16 (DES 1987:7–25) under the section dealing with ‘objectives’ of mathematics teaching. Again, you should read these pages as preparation for this Unit. References and Further Reading Department of Education and Science (1982) Mathematics Counts: the Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the Chairmanship of Dr W.H.Cockcroft, London: HMSO. ——(1987) Curriculum Matters No.3: Mathematics from 5 to 16, London: HMSO. ——(1991) Mathematics in the National Curriculum, London: HMSO. Mason, J. (1989) ‘Teaching and assessing’, in Mathematics Teaching: the State of the Art, P.Ernest (ed.), Lewes: Falmer Press. National Curriculum Council (1989) Mathematics: Non-Statutory Guidance, York: NCC.

Notes on Fractions Common fraction (or vulgar fractions). A fraction expressed in the form:

and used to distinguish between fractions of this kind, decimal fractions and percentage fractions. Unit fraction. A common fraction which may be expressed with unit numerator. e.g. −unit Non-unit fraction. Three-quarters, for example, can mean either: three lots of or three wholes divided by four. Improper fraction. A common fraction expressed in a form where the numerator is greater than the denominator. e.g. Mixed number. A number which consists of two parts—a whole and a fraction. e.g. Equivalent fraction. Deals with the recognition that, e.g. has the same value as has the same value as has the same value as… NB The ideas associated with the terms ‘numerator’, ‘denominator’, ‘unit fraction’, ‘improper fraction’, ‘mixed number’ and ‘equivalence’ are important in learning about fractions, but the terms themselves need not necessarily be used by the children.

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STIMULUS ACTIVITIES Activity 1 Materials needed: copies of the worksheet (page 62), scissors. Here is a practical activity to get you going. Work together in small groups—don’t be afraid to discuss the ways you approach the activity. Cut out the shapes from the worksheet. Decide what fraction of the whole each shape represents. Reflect on the activity and decide what concepts you were dealing with. Prepare a statement in response to the following question: What significance do you attach to practical work and discussion in helping to promote a greater understanding of mathematical concepts? Share your statement with the rest of the participants. Activity 2 Materials needed: Mathematics in the National Curriculum, Cockcroft Report, Paragraph 300. Refresh your memories about what Mathematics in the National Curriculum has to say about fractions. Read this alongside Paragraph 300 from the Cockcroft Report. Write down your interpretation of what is expected of a child’s understanding of fractions at Key Stage 2. Now consider the different approaches you make in obtaining answers to these questions: (a) (b) Prepare a variety of different forms in which these questions may be presented to children, taking account of their level of understanding. How might children attempt these questions if their understanding is: (i) instrumental (ii) relational? Activity 3 Materials needed: calculators, money tokens. Consider the following concerning the teaching of decimal fractions based upon an understanding of concepts associated with common fractions. Make notes on your responses to the questions below. Traditionally, the teaching of decimal fractions was built on an understanding of common fractions; that is: 0.1 can also be written 0.01 can also be written 0.37 can also be written or

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If it is accepted that the concepts underlying common fractions are difficult for young children and should be developed gradually over a long period of time, then how does this affect the teaching of decimals? In explaining decimals to children, do we need always to talk in terms of tenths and hundredths? Are there other ways which will allow children to understand the decimal notation? For example: children experience decimal notation on calculator displays as well as money represented in written form. Consider how these might be used to develop curricular materials. Activity 4 Materials needed: place-value apparatus, money tokens. The list below deals with objectives in teaching mathematics that relate to a scheme of work on place value with a group of seven-year-olds. The content is concerned with analysing our base ten numeration system in a variety of ways. The work focuses attention on how it is possible to represent any whole number by combining a set of numerals in a particular way. The first objective has been identified by the code ‘A2’. The letter ‘A’ refers to FACTS and the ‘2’ refers to OBJECTIVE 2 in the FACTS section of Mathematics from 5 to 16 (DES 1987). ‘A2’, therefore, covers ‘remembering notation’. The coding for these objectives is according to the headings taken from this DES document as follows: A A1 A2 A3 A4 B B5 B6 B7 B8 B9 C C10 C11 C12 C13 C14 D D15 D16 D17 D18 D19 D20 D21 D22

FACTS Remembering terms Remembering notation Remembering conventions Remembering results SKILLS Performing basic operations Sensible use of calculator Simple practical skills in mathematics Ability to communicate mathematics The use of microcomputers in mathematical activities CONCEPTUAL STRUCTURES Understanding basic concepts The relationship between concepts Selecting appropriate data Using mathematics in context Interpreting results GENERAL STRATEGIES Ability to estimate Ability to approximate Trial and error methods Simplifying difficult tasks Looking for pattern Reasoning Making and testing hypotheses Proving and disproving

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E E23 E24

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PERSONAL QUALITIES Good work habits A positive attitude to mathematics

In pairs, (a) use this code to identify the objectives in this scheme of work (b) plan an activity for one of these objectives. (Remember to consider which teaching style(s) might best fit this activity.) Then compare and discuss the results of your choice of objectives with the other participants. Place Value Objectives 1. A2 Children will know that ‘U’ stands for the number of units, ‘T’ for the number of tens and ‘H’ for the number of hundreds. 2. – Children will be able to enter numbers up to three digits into a calculator. 3. – Children will develop their ability to estimate numbers (of centimetres) by visualizing groups of Hundreds, Tens and Units as part of an activity related to measurement of length. 4. – Children will remember that they cannot have a digit greater than nine in any column. 5. – Children will develop their understanding of place value in base ten. (The idea is that the value associated with a digit is governed by its place within the series of digits and the relationship between consecutive places is constant.) 6. – Children will increase their ability to apply place-value ideas when dealing with money. 7. – Children will learn how to say, in words, the number represented by, for example, ‘256’; and to describe how such a number may be decomposed. 8. – Children will become aware of the need to work in a systematic way as they group into Tens and then into Hundreds during their practical work. 9. – Children will appreciate pattern emerging from a set of numbers arranged in order. 10. – Children will remember that we read numbers from ‘left to right’. 11. – Children will gain confidence in handling numbers greater than one hundred. Activity 5 Materials needed: copies of the Briefing Paper. During this activity, you are asked to work with a model designed to place mistakes in categories. The model is described below and on pages 74–5. Spend a little time reading through this to prepare yourself for applying these ideas to a particular area of mathematics. Taking each category in the model, look at the topic ‘Angle’ and try to find specific examples corresponding to each category of mistakes children might make. What do children actually measure when they measure angle? * the space between the rays? * the distance between the rays?

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How do they manipulate the measuring instruments? * the straight edge on the protractor? * the centre point of the protractor? * reading the protractor? Angle as Amount of Turn The following starting-points could provide you with activities that expose children’s understanding of and misconceptions about ‘angle’. The pictures below show how rotations may occur in different objects. Look at the greatest rotation for each object. Place these angles of rotation in order of size.

Rotating yourself. Explain precisely how you should travel (in paces and turns) to get from the door of the room to your present position.

How should the bee move to get from A to B?

Decide which of the categories in these tasks are associated with mistakes that are: * the most commonly found in classrooms; * the most serious;

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* the most easily remedied. DIFFERENT KINDS OF MISTAKES This model is just one way of describing the wide variety of different kinds of mistakes children can make in their mathematics. We can start with three headings: A. Communication in mathematics B. Mental mathematics C. Practical mathematics Within each of these, we can establish several subcategories to describe particular types of error as follows: A1. Mis-reading mathematics A2. Mis-writing mathematics A3. Verbal slip B1. Mis-memorizing mathematics B2. Mis-understanding mathematics B3. Mis-applying mathematics B4. Mental slip C1. Mis-handling skills C2. Mis-using skills C3. Practical slip The following illustrations will help to describe the nature of each type of mistake. A1 Mis-reading mathematics The vocabulary used in a mathematical text may create problems for children reading it. Words used in a mathematical sense may have a different meaning from those used in an ordinary sense. Take, for example, words such as: ‘difference’, ‘product’, ‘parallel’, ‘mean’, ‘odd’ and ‘value’. A child reading: ‘What is the difference between 24 and 37?’ may answer: ‘One is bigger than the other’ or ‘One is odd and the other is even’. Mathematical symbols may also be read incorrectly. For example, does 10÷2 mean ‘2 into 10’or ‘10 into 2’? A2 Mis-writing mathematics A child may be able to competently communicate some mathematics orally but unable to set this down on paper in the conventional manner. A child can tell someone that: ‘Twenty three multiplied by ten is twohundred-and-thirty’ but then writes:

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A3 Verbal slip It is often difficult for children to express themselves verbally with mathematical precision. A child may understand what is meant by the term ‘prime number’ but, when asked to explain, replies with something like: ‘It’s a number that can only be multiplied by 1 and itself.’ B1 Mis-memorizing mathematics Children are expected to recall from memory a variety of different facts and skills, mathematical terms and symbols, number facts, formulae or routines. Memory aids are often used to help retention, but these may fail on occasions. TEACHER: PUPIL: TEACHER: PUPIL:

What are seven eights? Sixty-three. How many millimetres in a metre? A million. B2 Mis-understanding mathematics

A child will not readily recognize his or her own mis-understanding. (This is unlike non-understanding.) For example: (a) A child believes that, because the result of adding five to three is the same as adding three to five, then 5−3 will produce the same result as 3−5. (b) A child regards the operation of multiplication as one which always produces a result that is greater than each of the numbers to be multiplied. This is not so when dealing with fractions: 5×3 gives a result gives a result less than . greater than 5, but (c) A child regards the first shape as a pentagon but the other not.

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B3 Mis-applying mathematics A child is measured and found to be 128 cm tall. When the child is standing on a box, the measurement from the top of her head to the ground is 216 cm. What is the height of the box? Faced with this problem, the child realizes that the two measurements must be used in a calculation but selects the wrong operation, and writes:

344 cm is then given as the answer. B4 Mental slip A typical instance of a mental slip is included in this calculation:

We can assume that the algorithm has been recalled successfully from memory, and that a mistake has been made due to the lack of care in carrying out the routine procedure. C1 Mis-handling skills When a child is asked to use a piece of mathematical apparatus, a lack of skill may be considered the result of inadequate training. Inadequate training could mean: * the pupil has not been given the necessary instruction; * the proper instructions have not been followed; * the pupil has had insufficient practice. For example: (a) A child uses a balance for weighing without first calibrating it—ensuring that the pointer is initially at zero. (b) When a template is being used to draw a polygon, the pencil slips at the vertex of the template. C2 Mis-using skills Children may jump in quickly to use measuring instruments when a little careful thought is necessary. This can sometimes be seen when children use a ruler without checking whether or not the calibration starts at the end or just a short distance from the end.

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C3 Practical slip Slips can easily be made when children use calculators. The wrong button may be pressed in the middle of a long calculation or the child may forget which number was last entered. CLASSROOM ACTIVITIES Activity 1 In order to determine the extent to which a group of children in your class understand the meaning of ‘half you need to work with one pair at a time. Hold an informal discussion with the pair, asking them to tell you what they know about ‘half’ of something. Get them to describe situations where they have had to work with ‘half’. Ask them to give you instructions about finding ‘half’. It will be interesting to withhold what you mean by ‘something’ in order to see what objects and examples your children provide. Note what they say and do. You can compare this information with how they actually carry out the following tasks. Materials needed: Plasticine, counters, jug, water, cards of various shapes, scissors. Ask the children to make ‘half’ from: (a) a ball of Plasticine (b) a set of counters (c) a card (d) a jug of water How do they know they have ‘half’? Ask them to make ‘half’ again. Are the children always ready to refer to the ‘whole’ to which the ‘half’ belongs?

Are they anxious to obtain precisely two equal fractions?

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Do they accept that the equality of the two fractions can be thought in terms of length, area, weight, capacity or number? Make notes about their responses and record these on the Review Sheet for Classroom Activity 1, page 79. Activity 2 Materials needed: wooden blocks or counters, calculators, number lines, paper, scissors, squared paper. Focusing upon a small group of children, choose some relevant mathematical topic about which they will be able to display some kind of understanding. For example: * * * * * *

distinguishing between odd and even numbers expressing decimal fractions the meaning of negative numbers the effect of multiplying a whole number by ten or one hundred determining the average of a set of numbers finding the area of a rectangle.

Before the children work together, hold an informal discussion to find out what they know about the topic you have chosen. Make notes on what they say and then compare this information with what actually transpires when they demonstrate their understanding in written form or with concrete materials. Activity 3 Materials needed: front pages from different newspapers. Hold an informal discussion with the children and find out what they know about ‘area’. Make notes on their comments, and compare these ideas with your observations of how they worked on the following activity. Get different groups of children to work on different front pages and compare their answers to ‘How much of the page is taken up with pictures, headlines, adverts or unused surface?’ You may wish to take notes on: * * * * *

the materials they select for this task the mathematics they use how they record their work how they compare their results with other groups the conclusions they reach as a result of interpreting their results. Activity 4

Materials needed: a letter composed by you from an imaginary child due to join your class. Tell your children to pretend a new child will be joining your class next week, and read out a letter this child has sent to you asking about the mathematics you have been doing in class. Get your children to send a written reply, explaining the mathematics they think would be useful for this child.

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Prepare statements to be shared during the Review Session on what the results of this work tell you about: (a) your children’s perception of mathematics (b) planning and working methodically (c) supplying sufficient information (d) explaining and recording work systematically (e) clarity of presentation of results (f) understanding chosen pieces of mathematics. Activity 5 Over a series of lessons, you may come across a number of instances of children either making errors in mathematics or revealing their misconceptions. Make a collection of these, and organize them according to the classification model you worked on during the Stimulus Activities. Prepare a short presentation to share with the other participants. You should attempt to analyse both the mathematical content around which the error was made as well as the error itself. Having examined your data this way, consider what this tells you about the particular child and how you might provide support materials to help overcome the error. REVIEW SESSION 1. Share your observations and comments on any comparisons you may have found between your children’s statements and their actual performance within the first three classroom activities. What did the practical activities and discussion reveal about the children’s understanding? What were the difficulties you encountered in trying to assess a child’s level of understanding? The sheet ‘Review of Classroom Activity 1’ will provide you with some examples of children’s thinking that you also may have identified among the children in your class. What would be the consequences if a pupil progressed through later work on fractions but had certain misconceptions at a basic level? 2. Report on the information you have obtained about your children as a result of your informal discussions. You may wish to consider: * * * * * *

the knowledge children bring to a particular topic how your children view classroom tasks the effect of classroom practices on their performance strategies your children apply reasons your children give for knowing something connections your children make with their knowledge.

3. Share your statements on the items listed in Activity 4 of the Classroom Activities. 4. Report on your collection of children’s errors collected during the Classroom Activities. Break up into pairs and consider the following questions: * Was it possible to account for every category in your list?

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What kinds of errors are the most common? What kinds concern you most as a teacher? Why? What kinds are the most difficult for children to overcome? On which occasions might children’s mistakes be more/most informative/beneficial/useful? Must mistakes always be corrected? What ways are there of correcting pupils’ mistakes? What are the implications for the kinds of assessment tasks we set our pupils?

Prepare short statements arising from your discussion. Come together as a complete group and share these. Identify areas of your existing mathematics scheme that might be adjusted as a result of your work on this Unit. REVIEW OF CLASSROOM ACTIVITY 1 Did your pupils’ perception of ‘a half’ coincide with any of the following ideas, some of which are erroneous? A half is produced when: (a) the whole is broken into two pieces; (b) the whole is cut by a straight line through the middle; (c) the whole is ‘folded’ along a line of symmetry (making two halves), then again, making four halves; (d) the whole is split into two congruent pieces; (e) the whole is seen to comprise two pieces of the same size but not necessarily the same shape; (f) the whole is found to have an area twice that of apart; (g) water is poured into a container to a level midway along its height; (h) a shape is made from another by halving all its dimensions. This selection of ideas serves to illustrate the complex process associated with children's understanding of what might well be regarded as a simple notion. It points to the need for children to gain a wide variety of appropriate experiences, without which fundamental concepts may fail to develop sufficiently.

UNIT 6 CALCULATING IN MATHEMATICS

LEADER’S GUIDELINES This Unit will help participants to establish views about the nature of numerical calculation and to recognize the relative importance of the various aspects of this work. Inevitably, calculating in mathematics permeates throughout the Units. Unit 3, for example, considers the need to be able to perform arithmetical calculations as part of the process of applying mathematics. Unit 5 provides opportunities to look specifically at children’s errors in computation. Briefing Paper Five aspects related to computation are dealt with separately. You could hand out this paper at the beginning of the Stimulus Session in order to alert the participants to these five issues that you intend to cover. Stimulus Activities You could introduce this session by gathering ideas from the participants about the areas of computation they think are important for children. From this list, you should be able to identify the five that will be considered during the Stimulus and Classroom Activities as well as the Review Session. From this point, participants should break up into small groups, work through the activities, and then be given an opportunity to share their ideas during the final part of the session. 1. Different ways. Calculators will be needed. Groups could do their calculations on large sheets of paper in order to show the others their methods of calculating. 2. Mental skills. Provide card and scissors, copies of the Worksheet on page 83. 3. Pencil and paper. Have structural apparatus available for groups to demonstrate how the algorithm works. 4. Estimating. (a) Sharing understanding. Bring the group together to share each participant’s understanding of ‘estimation’ and ‘approximation’. (b) Collecting newspaper cuttings. Here participants apply their skills of estimation and approximation. The examples provided on page 91 can be copied for participants to start them off. They will need an ample supply of old newspapers and magazines, paper, glue and scissors.

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(c) Estimating the contents of collecting boxes. Try to have a sample of charity collecting boxes on display and ensure that you have at least 20p in small change (1p, 2p, 5p). Small containers, e.g. bottles, should also be on hand. (d) A thousand words of print. Participants need to bring a selection of children’s reading books. (e) Estimation from children’s stories. Plasticine, card, paper and building blocks are useful materials for this activity. Ensure that the participants have a selection of children’s reading books available. 5. Calculators. Each participant will need a calculator. Classroom Activities A period of two weeks should be sufficient time for participants to collect results of their pupils’ work on these activities. 1. 2. 3. 4.

Reviewing class number work. Number games. Children’s own way of working. (a) Pouring water; (b) Supermarket goods. It is essential that a wide selection of materials from the supermarket is on hand. Remind participants that they should prepare notes on their work with children and bring these to share with other participants during the Review Session. Look out for:

* the ways children describe their estimations; * their concern for accuracy; * situations where guessing was apparent. 5. Calculator challenges. Review Session You may choose to prepare a display of number activities—in particular, number games, puzzles and calculator activities—from each class in your school for this session. Around 3 hours should be made available for participants to report on their pupils’ work. As well as responding to the questions for this session, you may find that this is an ideal opportunity for participants to agree on a policy for calculators in school if one has not already been established.

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BRIEFING PAPER In this transcript, a student-teacher experiments with communication in mathematics: TEACHER: PUPIL: TEACHER: PUPIL: TEACHER: PUPIL: TEACHER: PUPIL: TEACHER: PUPIL: TEACHER:

How far can you count? To twelve. OK. Count up to twelve for me. 1, 2,.....12. Do you know what comes after twelve? No. Now think of a really big number. I can think up to five because I’m five. Now think of a very small number. I’ll choose one from here. It’s this one. But that’s a letter ‘a’.

Three seven-year-old children discuss number:

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TEACHER: N: A: R: N: A: R: N: A: N:

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What’s the biggest number you can think of? Miss, there isn’t a biggest number…. You can keep on counting. More than a million? No…the biggest number is one thousand and six hundred and… It’s not. A billion something. No. One thousand and three hundred and sixty nine. But…you can keep on counting…just keep on counting, my mum told me… My dad don’t know nothing about maths. Like one billion zillion trillion million gillion dillion.

These examples may be familiar to you. They begin this paper for a special reason. They act as a reminder, a warning in fact. Even though this Unit provides opportunities for you to put into perspective the different ways children are taught to do numerical calculations, you should never let go of those valuable experiences you have had of exposure to children’s perceptions of number. Five aspects of computation are dealt with in this Unit. These, in turn, are set out in this paper under separate headings and then through the five activities in the Stimulus and Classroom Activities as well as the Review Session. By focusing on them in this way it does not mean, however, that they are treated as isolated procedures to be introduced prior to other computing methods. 1. Different Ways of Calculating Mathematics: Non-Statutory Guidance (NCC 1989:E3–E4) gives examples of ‘pencil and paper methods’ encompassing a wide range of formal and informal techniques and methods. Children are expected to apply the four rules either mentally, using pencil and paper or with the aid of a calculator. Some teachers encourage their pupils to develop a flexible attitude towards the use of calculators. This attitude may be described as follows: If pupils are asked to perform a calculation, then they should first try to do it mentally. If this proves too difficult, then they should not be afraid to use the calculator. On occasions when the teacher stipulates that a calculator must not be used, then they employ a standard method or develop their own way of working—whichever they prefer. The flow chart below demonstrates this approach:

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2. Developing Mental Arithmetic Skills There is the need for children to be able to carry out calculations mentally. Exploring and inventing alternative strategies for mental computation requires time. The emphasis, therefore, should be on conceptualization rather than computational speed. Sadie has written down how she performed two calculations mentally. She also evaluates her own efforts in the process:

It is generally accepted that there are two main approaches to helping children in this: (a) Giving opportunities for lots of practice with number facts and rules. To do this, teachers might:

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(i) question their pupils orally on a regular basis, asking them to recall number bonds and multiplication facts together with simple arithmetical tasks; (ii) expect this kind of mental activity as the children do written exercises in their number work; (iii) involve their children in trying number puzzles and playing number games. (b) Encouraging work on seeing relationships between results in number. To do this, the nature of the activity is important. Work on pattern, often with the aid of some practical apparatus, helps to develop this ‘feel for number’. The work could be presented in the form of a creative activity, a number challenge, a puzzle or a game. 3. Paper-and-Pencil Methods Of course, mental methods of calculation are necessary when written methods are employed. Knowledge of number facts and simple numerical operations are built on to provide a means of dealing with some complicated computation. This involves a series of written recordings to keep track of the various steps in the procedure. It may be that a standard technique has been learnt (that is, an algorithm for dealing with a certain class of calculation), and it is a straightforward matter of reproducing this technique. On the other hand, pupils could be allowed to build up the calculation in their own way. For example, 17×25: First method—standard procedure:

Second method—pupil’s own way:

Third method—special ‘trick’ when multiplying by 25:

A child using the second method quite obviously knows what is happening. Teachers generally feel it is important that children should understand the arithmetical rules they are using. It is here that apparatus can be helpful. When apparatus is used, the process of the calculation may be demonstrated visually but, more importantly, the children can be allowed to construct the method for themselves. In this way, they are inclined to remember the abstract procedure and also they are likely to be helped towards a greater ‘feel’ for number.

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4. A Question of Practice Mathematics: Non-Statutory Guidance says: Whether using mental, pencil and paper or calculator methods, pupils must be able to estimate, approximate, interpret answers and check for reasonableness. (NCC 1989: E2) This is sound advice especially when, for example, confidence has to be developed in selecting correct key sequences while using calculators. An interesting issue can be raised, however, when dealing with the inexact nature of measurement. A well-tried practice is to estimate first, get the children to commit this to paper, then use an instrument to measure and compare results. Anyone having worked in primary school will have noticed how children feel uneasy about differences between these results. Rubbing out often accompanies such an activity. Prepare your responses to the following questions and share them during the Review Session: * Does estimating and then checking the estimate by actual measurement lead to the perception that estimating is not important? * Does this kind of activity seriously undermine the genuineness and potential value of the estimates? Very young children appear to have a high tolerance for extreme estimations, yet soon move towards the understanding that exact answers are the only ones acceptable. It could be that they see estimation as being in conflict with the assumed exactness of mathematics. However, an estimation is often the only reasonable choice we have in certain situations. Take an extreme example: what criteria are used by bank clerks as they estimate the height of a bank robber? There would have been no time to take out the tape measure and write down measurements! Under such stressful conditions, however, reference points around the bank might help to estimate by comparison. Other real-life situations place less emphasis upon the importance of accuracy. It would not matter too much if, in casual conversation, I told my friend that a certain number of people attended a rally and that that number fell short of the actual. Newspaper headlines often do this kind of thing. You may like to begin collecting headlines that show how numbers we encounter in real life are often estimates. A distinction is made between estimation and approximation in the HMI report, Mathematics from 5 to 16 (DES 1987:18–19). Mathematics Counts (DES 1982: Paras. 257–61) has more to say on this matter. You may also have experienced some people using these terms in a way that implies that they are synonymous. Think for a moment about the strategies you apply when carrying out a mental computation. You may find yourself intuitively estimating the size of the answer before actually performing the operation. On the other hand, you may ‘round the numbers’ first and then perform the operation upon these numbers in order to give an approximation. Estimation is one aspect of mathematical thinking which depends not only upon a good understanding of place value but also the meaning of ‘a unit of measure’. Approximation, however, can be forced upon us according to the instruments we use. The more refined the measuring instrument, the more accurate are the results. From this arises the concept of ‘degree of accuracy’ or the interval in which a certain measure falls. Such intervals are bounded by upper and lower limits, and an acceptable approximation will fall within these limits. You can produce an interval by purchasing 227 g boxes of chocolates, for example. (Don’t eat them, count them!) No doubt you will discover that each box contains a different number of chocolates because of

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the variation in the characteristics of the chocolates. From these results, you will be able to identify upper and lower limits. Armed with this information, you can quite confidently say that the number of chocolates within any 227 g box will fall within these limits. A further discussion of forms of measurement ‘within limits’ is to be found in Mathematics Counts (DES 1982: Paras 269–72). 5. Using Calculators Electronic calculators are part of life today and should not be ignored. Schools, therefore, should be a place where children can experiment with and make sense of this easily accessible piece of equipment. You can design specific activities to enable this to happen. For example, they may be used to check mental calculations and pencil-and-paper calculations. It is also important, however, that pupils regard the calculator as a tool for assisting in the solution of a realistic problem. Resorting to written mechanical arithmetic in the middle of working out a problem can often swamp the solution, thus defeating the object of the exercise. Further, the calculator can be used in a positive way to help children gain understanding of number properties. This may be approached in the following way: Repeat this on your calculator for lots of different numbers…. What do you see happening? Try putting different numbers in the calculator to find out which one works. Can you improve your guess each time? Children using calculators in this way surely improves their understanding of number. (See Shuard et al. 1991 if you wish to read more on this kind of approach carried out during the ‘Primary Initiatives in Mathematics Education’ project running from 1985 to 1989.) References and Further Reading Department of Education and Science (1982) Mathematics Counts: the Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the Chairmanship of Dr W.H.Cockcroft, London: HMSO. ——(1987) Curriculum Matters No. 3: Mathematics from 5 to 16, London: HMSO. National Curriculum Council (1989) Mathematics: Non-Statutory Guidance, York: NCC. Shuard, H., Walsh, A., Goodwin, J. and Worcester, V. (1991) Calculators, Children and Mathematics, Hemel Hempstead: Simon & Schuster.

STIMULUS ACTIVITIES Tackle the following in small groups of two or three. Activity 1 While working on the following two questions, try to find a number of different and contrasting methods of calculation which could be used. Question 1 A number when multiplied by itself and then by itself again gives 4913. What is the number?

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Question 2 The sum of two numbers is 1000. Their difference is 253. What are the numbers? After making your attempts, discuss the various possible methods with respect to the follo wing points: (a) To what extent was mental mathematics used? (b) Was any standard calculating technique used? (c) Was a routine procedure devised? (d) What part, if any, did the calculator play? What is your view on training children to be ready for using a variety of methods? Activity 2 Materials needed: card, scissors, copies of the Worksheet (page 83). Make a set of cards which will be used in a number game with your pupils. Complete each card by inserting different numbers and instructions; for example, I’ve got_____ Who has_____?

I’ve got_____ Who has_____?

I’ve got_____ Who has_____?

Now invent a set of rules for a game based on these cards suitable for a group of children to play. You could adapt this with your class in mind. What precise number skills would you say are being practised when playing this game? Share ideas of any number games that have been successfully used in your classroom. Activity 3 Here is a set procedure—an algorithm—for finding the difference between two numbers. The algorithm works for a pair of numbers the smaller of which is to be subtracted from the larger. Work through the steps of the procedure: Step 1. Think of two numbers. Step 2. Make sure the numbers have the same number of digits by introducing, if necessary, zeros on the left-hand side of the smaller number Step 3. Write down the smaller number. Step 4. Write down directly under each digit (of the number in Step 3) that digit’s complement to 9. If you have written 234, then each digit’s complement to 9 would be: 5 for the 4, 6 for the 3 and 7 for the 2. Step 5. Add 1 to the larger of the original pair of numbers and write this new number directly underneath the number arrived at in Step 4. Step 6. Add together the previous two numbers (i.e. those in Steps 4 and 5). Step 7. Cross out the left-most digit of the sum. The resulting number is the required difference. Find out if this procedure works for any pair of numbers? Try a few examples. It is possible to become very proficient at subtraction using this method. Yet, all you really need to know is how to add numbers less than 10 and how to form bonds to 9. With practice, it ought to be easier to use and more reliable than the standard algorithms that have traditionally been taught; namely, subtraction by decomposition or by equal addition. So why isn’t it taught?

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Can you understand why the algorithm works? Can you demonstrate the procedure with structural apparatus? Work together on this problem. Imagine junior-school pupils who have not been taught a subtraction algorithm. How might they tackle subtraction of a two-digit number from a two-digit number? Suppose they are directed towards using counters and bricks and then asked to record their working. What written methods could you see emerging? Activity 4 Materials needed: newspapers, magazines, glue, scissors, paper, charity collecting boxes, small change, small containers, children’s reading books, Plasticine, card, building blocks. (a) With the whole group, discuss the meaning of ‘estimation’ and ‘approximation’. Identify situations when and where you apply these strategies. (b) From a large selection of old newspapers and magazines, collect headlines or articles that incorporate estimation or approximation. (Examples are provided with this pack—page 91—to start you off.) Sort these into sets, such as estimations made by counting things or measuring. Discuss, mount and display. (c) Various charities ask the public to drop their spare coins into containers displayed in prominent places. Estimate, for example, how much a model lifeboat might contain. Brainstorm on estimation ideas involving containers and money. Fill a bottle with 2p coins. How much will the bottle hold? How many coins are there? If you place them end to end, how long would the line of coins be? Does this experience help if you repeat the activity with 5p coins or other containers? If so, why? (d) A picture is said to be worth 1000 words. Find pictures that are nearest in size to 1000 words of print from a children’s reading book. (e) Select a children’s story. Using materials of your own choice, provide an illustration of the number, geometry or measurement that it incorporates. You may ask yourself: ‘How tall was Baby Bear?’ Having decided on its height, you can then work on further questions: ‘How big was its chair, bed or bowl?’ For these activities in estimation, make notes on: * how you approached each task; * the methods you applied in solving the problems; * any difficulties you encountered. Activity 5 Materials needed: calculators. Work on the following calculator challenges: (a) Try the following kev sequence: Write down your answer. Clear the screen and key in another string of digits, e.g. 456456 and repeat the operation… . Suppose when you try to use the (b) Decide on a computation to be carried out, such as calculator again, you find the 6 is not working. Find other ways of obtaining the answer. Investigate for other operations and other damaged keys.

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(c) Show the key sequences you apply to display the numbers 1 to 12 on your screen when the only keys . that are working are: Sometimes calculators may serve as a device for checking one’s arithmetic. On other occasions they may present a quick way of obtaining the result of a calculation during the middle of a complicated problem. But how may they be used as a teaching aid as opposed to simply a calculating aid? Share your ideas with the whole group.

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CLASSROOM ACTIVITIES Activity 1 On various occasions, over a series of lessons, review the kind of number work in which your class has been involved. Assess the extent to which your pupils have engaged in each of the following aspects of calculation: * mental arithmetic; * using pencil and paper methods; * computation using a calculator. Focus your attention on just three children: (a) a mathematically able child; (b) a mathematically average child; (c) a mathematically less able child. Activity 2 Materials needed: game cards from Stimulus Activity 2, money or money tokens, number cards 1–100, a hat. Arrange for groups of children to tackle the following: (a) Play the number game I’ve got…Who has…?’ (b) Make 10p with just seven coins (or in general X with Y coins) (c) Draw three numbers out of a hat. Combining them with +, −, × and/or ÷ can they make 1? 2? 3?… Ask your pupils to tell you how they have carried out their tasks. What does this information tell you about their ability in computation? Activity 3 Teachers have reported on interesting results that have come out of giving children a calculation to do for which no general technique has been taught; for example, multiply two numbers without using the calculator. Try this out with your class and see how far they are able to develop ways of working that are their own? Activity 4 (a) Pouring water Materials needed: jug of water, pan, selection of other containers.

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Ask a child to pour water from a jug into a pan and to stop when they have poured out half the contents of the jug. Check the results. Discuss with the children how they knew when to stop pouring. Repeat with other containers. Set further challenges: * Pour water from a jug into a container until half the container has been filled. * Pour water from a jug into a container until a cupful has been poured. (b) Supermarket goods Materials needed: large boxes, selection of cartons, packages, bottles and cans from a supermarket (with prices clearly marked). Ask the children to fill a box carefully with a selection of goods from the supermarket and estimate how much they have spent. Get them to use a calculator to check their results. Allocate the children a certain amount of money, and ask them to decide what they can buy from the materials provided. Again let them use a calculator to find out how much they have actually spent. Activity 5 Materials needed: calculators. Observe small groups of children working on the following calculator challenges: (a) Ask the children to find the largest/smallest products when they use three digits and the multiplication sign; e.g.

Get them to investigate for other sets of digits. (b) Again using the multiplication key, let the children experiment with the following reversals. First, compute a pair of numbers; e.g.

Now ask them to reverse these and multiply once again; e.g.

Now ask them to investigate other pairs with their reversals. (c) The calculator opens up interesting avenues for investigation with division. Challenge the children to record those divisions which produce a repetition of single digits; e.g. or those that produce a two-digit repeat; e.g. (d) Allow time for the children to discover certain results; e.g. what happens when they enter:

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(number) and then (+0=) or (number) and then (×10=) or (number) and then (−number=). REVIEW SESSION 1. Share your responses to the issues raised in the Briefing Paper by making reference to your observations during the classroom activities. For example: What proportion of the time children spend doing arithmetic is devoted to: * mental calculation? * pencil and paper calculation? * the use of the calculator? The Briefing Paper also discussed the place of estimation and observation during the Stimulus and Classroom Activities, * * * * * *

what language did the pupils use in estimating? what processes and strategies were applied? how were judgements formulated? what indications were there of guessing? what concern was revealed for accuracy? what evidence was there of approximating?

Does estimating and then checking the estimate by actual measurement lead to the perception that estimating is not important? Does this kind of activity seriously undermine the genuineness and potential value of estimates? Identify ways in which the skills of estimation and approximation can be incorporated into your school’s mathematics scheme. 2. Here is an opportunity to share ideas about number games you have tried in the classroom. Often number games suggested are really versions of either bingo, dominoes (like Tve got… Who has…’), Snap, and Snakes and Ladders. The elements of these games are simple. How do you react to the statement: ‘Usually the simpler games are the most successful’? 3. Are the children in your class good at remembering how to perform a method of calculation and all the conventions associated with recording it? Do they understand why the calculating technique works? Will children readily tackle calculations without having been taught prescribed methods? Is there a case for training children in this from an early age? 4. Is there a need in your school for a policy regarding the role of the calculator in the classroom? If so, what are you going to do about it? How can we ensure that children will develop their skills with calculators and yet do not neglect their mental and written arithmetic skills? 5. What are your responses to the following statements: If children’s errors are not corrected, they will be reinforced. True? False? To what extent?

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Doing pages of sums is what got me where I am today. Getting sums right is all that is important. Agree? Disagree? Why?

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LEADER’S GUIDELINES This Unit is concerned with ways in which children represent their own mathematical ideas in written form. To become aware of something in common between objects or ideas is to perceive things as being ‘connected’. The identification of relations requires that some recognition of generalities (Unit 4) has taken place. This Unit considers an aspect of communication further to that of discussion already identifed in Units 1 and 2. The focus is upon the inventiveness of children as they use signs and symbols to convey meaning. A study of representation should not be isolated to one Unit; consequently, throughout each Unit there will be ideal opportunities for collecting examples of the ways children represent their mathematics. Briefing Paper You may wish to use issues raised in this paper as the basis for your introduction to the Stimulus session. This paper could, therefore, be handed out before or at the beginning of the session. Stimulus Activities Allow 3 hours for this session. Participants work in pairs for Activities 2, 3 and 4. 1. Symbols. This requires participants to brainstorm all the mathematical symbols they have encountered, explain them and sort them into types. Before they actually become involved in this activity, you could prepare an introduction based upon the examples of children’s work in the Briefing Paper. Asking participants to comment upon these will set the scene for the remaining activities. 2. Communication through graphs. 3. (a) Turning cups. At least five for each pair. (b) Folding stamps. Strips of postage stamps could be replaced by strips of paper with markings to represent the stamps. Have strips of four, five, six…‘stamps’ ready. (c) Envelopes. A large collection of used stamped envelopes.

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(d) Jugs. A set of jugs capable of holding 9, 5, 4, 2 units of water, and water is required for each pair. You could use a small plastic beaker as the 2-unit ‘jug’ and then cut large plastic lemonade bottles to the 4, 5, and 9-unit limit. 4. Relationships between variables.

Classroom Activities Allow 2 weeks for participants to work with their children on the five activities. Remind them of the resources that have to be prepared; for example, plastic lemonade bottles can be cut to fixed-unit sizes for Activity 5. 1. Signs and symbols. 2. Recording a race. 3. Shoe-box routes. 4. Turning instructions into symbols. 5. Describing an activity with symbols. Review Session About 3 hours will be needed for this session. The main aim is for participants to search for ways of providing for a classroom environment that encourages work on different forms of representation. 1. Participants report on their children’s work and share actual examples of the different forms of representation they applied. 2. Classify the samples of work to identify ways to widen the children’s repertoire. 3. Consider the quality of questioning used in the classroom activities. 4. Develop resources. 5. Brainstorm further activities for children.

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BRIEFING PAPER Why Teach Mathemathics?

This eight-year-old provides an answer. Mathematics is useful in that it has practical applications. She also reminds us that maths can be fun. As far as ‘utility’ is concerned, her statement agrees with the HMI report Mathematics 5–11 (DES 1979:4), which states the purpose of teaching mathematics in this way: Mathematics is useful, mathematics is part of our culture, and mathematics trains the mind. The Cockcroft Report (DES 1982: Para. 3) provides another answer to this question: We believe that it is the fact that mathematics can be used as a powerful means of communication which provides the principle reason for teaching mathematics to children. An important feature of this report is the specification of a broad range of learning outcomes as well as the provision of examples of a variety of ways in which mathematics can be used to present information powerfully, concisely and unambiguously. For ideas to be expressed and developed in mathematics, a considerable degree of dependence on formal symbolism takes place. It seems that learning mathematics means learning how to use symbols in order to convey to others messages of purpose and intention. You could, of course, approach mathematics as a ‘means of communication’ from another perspective. Enabling children to convey meaning by their own inventiveness of signs and symbols may help them later to adapt to formal mathematical symbolism. Mathematics can emanate from the intentions of the children themselves. This is shown in the following examples. A. Daniel (four years and one month) drew the map below after he had travelled on a section of the London Underground. During a discussion about his journey, he explained the meaning of the symbols he had used as follows: old stations stations that only District trains stop new stations stations yet to be built End of line are ‘bump’ stations

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B. One rising five-year-old modelled the teacher as she took the morning dinner register. During Kelly’s own planning time, she counted the number of packed lunches and school dinners for her class on that day.

C. The initial intention here was to identify right and left turns. The activity involved ‘driving’ toy cars up to a road junction set out on the classroom floor. Thomas (five years old) decided, however, to rearrange the three cars to find out the order they would take as they came up to the junction.

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D. Lizzie (six years old) is experimenting with the form of symbols that tell the story of ten.

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Symbolic Representation As you observe the real world, your attention may be drawn towards similarities within a situation or relationships between phenomena. You may even be moved to make inferences and predictions as a result of highlighting these experiences. Symbols can help in communicating your thoughts, actions and ideas as a result of being involved in this way. These are the outward expressions of your internal representations of the real world. They can take the form of charts, doodles, diagrams, reports or sketches, and open up opportunities for communication. The system of signs, letters and numerals used in mathematical texts today does not reveal anything of the struggle among mathematicians over the years. Accounts of the development of mathematical symbols and their eventual acceptance make fascinating reading. No wonder children encounter difficulty when they are quickly subjected to a diet of formal mathematical language. The arbitrary introduction of symbols as shorthand is often confusing and can produce insurmountable barriers to a child’s comprehension. The use of a form of specialized language in this way can easily prevent communication of the subject matter to anyone else. Members of the Association of Teachers of Mathematics put it this way: Language used when one is probing an idea is very different from that used in presenting an idea. An idea is presented to others already familiar with some elements of the idea. Children spend most of their time probing ideas, but many textbooks and teachers use language more appropriate to presentation. (ATM 1983:25) As well as the ‘punctuation’ and ‘alphabetic’ (a, b, c…) symbols that are used in mathematics, a form of representation is applied with symbols that closely relate to their meaning. These are mainly geometric. Gareth (nine years old) wrote in his journal a reminder about angles. His explanation uses this form of geometric representation.

Another category is made up of those specially invented signs for whole concepts such as % and £. Language as Experience The heading is taken from Brissenden (1988:217), where he draws the reader’s attention to a controversy dealt with in the journal Mathematics Teaching over whether or not ‘action and experience necessarily precede thinking’. One writer suggested that reasoning demanded reflection between signs without

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reference to what they signify. Another replied by submitting the following four-part model in favour of ‘experience’ as being the essential foundation for the learning of mathematics: ELPS: E(xperience with familiar objects), (spoken) L(anguage), P(ictures) and S(ymbols). (Brissenden 1988:218) Someone else argued that: With mathematics it is important to perceive language as experience as well as mediator between experience and the individual. (Ibid.) What bearing does this discussion have upon classroom practice? It once again invites teachers to consider their own perception of mathematics. Is it a ‘body of knowledge which pupils accept uncritically’ or ‘a creative activity—a way of knowing about ourselves and the world’? Does it mean that mathematics in primary school is simply to revisit those digits and letters that represent number and quantity normally covered in the primary curriculum or provide children with opportunities for inventing their own symbol system as models of the real world? The final outcome, surely, should be that all children are given the opportunity to become part of that group, if they so desire, among whom formal mathematics is a language of communication. References and Further Reading Association of Teachers of Mathematics (1983) Language and Mathematics, Derby: ATM. Brissenden, T. (1988) Talking about Mathematics: Mathematics Discussion in Primary Classrooms, Oxford: Basil Blackwell. Department of Education and Science (1979) Mathematics 5–11, London: HMSO. ——(1982) Mathematics Counts: the Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the Chairmanship of Dr W.H.Cockcroft, London: HMSO.

STIMULUS ACTIVITIES While you work through these activities, you will be able to explore situations that will help you to develop your own symbol system in order to communicate your mathematical ideas. You should note the different forms of representation you apply while you probe an idea and when you present your ideas for others to share. Also note the difficulties you encounter when you attempt to interpret another person’s representation. Activity 1 Brainstorm the symbols you have encountered in mathematics, and explain the meaning of the symbols you identify to the rest of the group. Sort these symbols into those that can be described as: * punctuation * alphabetic

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* specially invented * images of the object itself. Activity 2 Graphs form one of the key elements in the communication of information. Drawing graphs requires certain technical skills, such as the choice of scale, plotting of points and drawing of lines or curves. This activity emphasizes ‘interpretation’, which demonstrates the ways a graph can communicate a variety of situations. Work in pairs and interpret the following graphical representations. Provide reasons for your interpretation and share these with the rest of the participants at the end of the Stimulus Activities.

Activity 3 Materials needed: cups, strips of stamps or paper, used stamped envelopes, ‘jugs’, water. Work in pairs. Make a record of your progress while working through these activities, then share your methods of representation with other participants during the final part of the session.

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(a) Arrange a row of cups, with some facing up and some down. Get all the cups the same way up. Record your moves. (One move is turning two cups, one in each hand, at the same time.) (b) Record the number of ways a strip of five stamps can be folded. (c) You have a pile of stamped envelopes piled in random order. Taking each letter in turn, you may have to (i) flip it over, (ii) turn it a half turn. Record the moves to get them all facing stamp-upper-right-hand. (d) You have four jugs of capacity 9, 5, 4 and 2 units. The biggest jug is full of water. By completely filling or emptying a jug, divide the water into three equal parts in as few decantings as possible. Activity 4 One aspect of mathematical modelling is choosing possible relationships between variables. Decide on variables for the situations below and work out ways of representing these. Share your solutions with the other participants by getting them to interpret your representations. (a) Sucking a Polo mint. (You may select the relationship between time and the increase in diameter of the hole in the mint. There may be a constant yet small change in the diameter depending on the amount of ‘suck’. This is then followed by a sudden change—when the mint breaks apart.) (b) Running the marathon. (c) Hunger over a period of twenty-four hours. (d) Writing a story. (e) Burning off stubble in a cornfield. (f) School dinners. CLASSROOM ACTIVITIES These activities should provide you with examples of children * recording their own work * representing solutions for others to follow * interpreting another child’s representation. Activity 1 Get the children in your class to collect examples of signs and symbols used in the home, school and street.

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Let the children find different ways of sorting these into sets. Ask the children to write an explanation or meaning for each symbol. Activity 2 Materials needed: race card, two dice, squared paper. Play this game with two children. Prepare a ‘race card’ like the one below:

Ask the children to select four numbers each from the card. Each number represents an ‘athlete’. Place counters on each of the numbers. You take the three numbers that are left. Each person take in throwing two dice. Find the total score of the two dice at each throw. Move one square the counter that is that total. The counter that reaches the finishing line first is the winner. Play this game a number of times and choose different numbers each time. While the game is in progress, get the children to plot the game on a bar chart. Set out the bar chart with ‘athlete’s’ number along the horizontal axis and number of scores on the perpendicular axis. During the entire game, take notes on any comments children make on:

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* their understanding of the combination of the numbers on the dice; * the actual spatial array of counters on the board; * the final result of the frequency of totals plotted on the graph. Activity 3 Materials needed: shoe boxes. The cuboid shoe box represents a family home. Mark one upper-level corner as the bedroom and the diagonally opposite lower-level corner as the living room. Corridors are the horizontal edges and staircases the perpendicular edges.

Get one child to record the different ways of getting from the living room to the bedroom. Do not allow retracing of routes. Ask the child to choose one of these routes. Pass this route on to another child to interpret the information and follow the route. Get the first child to assess the second child’s efforts. Activity 4 Select an activity such as ‘making a cup of tea’. Get two children to work through the following stages: * Prepare a list explaining how to carry out the task. * Change the list of words to pictures. * Change the pictures to symbols. Pass this final set of instructions on to another pair for them to interpret. Let the children compare their results with the initial list of instructions. Activity 5 Materials needed: ‘squash’, three different-sized jugs holding 3, 5 and 8 units. (If you use a normal plastic drinking cup as the 3-unit jug, you could prepare the other two containers by cutting plastic lemonade bottles to hold 5 and 8 units respectively.) Give two children the three jugs. Make sure the largest jug is full; the other two jugs are empty. Get the children to pour the ‘squash’ from one jug to the other in order to finish up with the large and medium jugs each containing four cups of ‘squash’ each. Ask the children to record how they did this, and then ask them to refine their account by using symbols. Pass this account on to another pair for them to work through the instructions.

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REVIEW SESSION 1. Allow time for the participants to report on their children’s work. 2. Classify the samples of work so that you can identify areas that have not been covered during these activities. For example, if the form of representation used by the children is limited, have some children only used pictures or numbers? What ways can you suggest to widen your children’s repertoire, in working with scale drawings, various forms of graphical representation, tables or grids? 3. Consider the quality of questioning that was used while the children examined certain representations. * What information was surplus? * What information was lacking? * What parts were not understood? 4. What resources do you think need developing in order to encourage other forms of representation to be used in mathematics. 5. Brainstorm further activities in order to help children confidently apply different forms of representation.