children learing mathematics readings

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EDMA 2213 (ED22M)Children Learning

Mathematics

EDMA 2213 (ED22M)Children Learning

Mathematics

B. Ed. Secondary (Online)

(READINGS)

UWI OPEN CAMPUS


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The Language of Mathematics do you speak it? Camille Bell-Hutchinson ..................................................................................................15 3

Communicating with symbolsCamille Bell-Hutchinson ..................................................................................................15 5

Mathematics Through Problem SolvingMargaret Taplin ..............................................................................................................15 7

Caribbean Examinations Council Regional Workshop MathematicsCamille Bell-Hutchinson ..................................................................................................16 5


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Learning to Teach Mathematicsin the Secondary School

Sue Johnston-Wilder, et al.

Task 1.2 Your personal vision for mathematics education

It is important to be clear about your opinions about mathematics and mathematicseducation, because they will influence your practice. In order to articulate your viewsclearly, it can be helpful to try writing them down in a concise form. Write a statementoutlining your personal vision for mathematics education. This could be kept in yourreflective journal and referred to at different times during your course For instance, youmight consider reviewing it prior to any job interview and also at the end of yourcourse.

Limit yourself to a maximum of 250 words. Include answers to the following questions:

Why is mathematics education important? What is mathematics? How is mathematics best taught and learnt?

Having finished your statement, you might like to ask other students on your coursehow it compares with their own views. How will the differences among you have animpact on the pupils you teach?

MATHEMATICS AND EDUCATION

While you are learning to teach mathematics you may feel that you have moreimportant priorities than considering the somewhat abstract issues of the nature of mathematics and the purpose of education. You may quite reasonably wish to focusinstead on surviving in a classroom full of pupils. However, it is important tounderstand at this early stage the way in which these philosophical issues doimpinge significantly on classroom practice.

Is the nature of mathematics a controversial issue? Surely everyone has a fairly clearidea of what the subject is and consequently what should be taught in schools? If you have had a chance to talk to other students about the issues raised in either Task1.1 or 1.2, you may already have encountered varying perspectives, if not conflictingviews on what is important about mathematics.

The two quotations that follow, written by professional mathematicians andpublished in the same year, illustrate that mathematics can mean different things todifferent people.

Reprinted from: Sue Johnston-Wilder, Peter Johnston-Wilder, David Pimm and John Westwell, Eds. Learning to

Teach Mathematics in the Secondary School . pp. 819.


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It is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematicsthe part of human knowledge that I call mathematics, as one thing one great gloriousthing.

(Halmos, quoted in Albers, 1986, p. 127)

Mathematics does have subject matter, and its statements are meaningful. Themeaning, however, is to be found in the shared understanding of human beings notin an eternal non-human reality. In this respect, mathematics is similar to anideology, a religion, or an art form it deals with human meaning and is intelligibleonly within the context of culture. In other words, mathematics is a humanisticstudy. It is one of the humanities.

(Davis and Hersh 1986, p. 410)

You might conclude that although the responses of these people are different, theyare nonetheless still talking about the same thing. However, there is significantdifference between these two views, and it is rooted in different understandings of

human knowledge. For Paul Halmos, mathematics seems a fixed, objective body of true ideas, independent of culture. Developing your standpoint on the nature of mathematics is important because it will influence the values about mathematicsthat you convey to your pupils.

The Aims of Education

Aims express intentions of individuals or groups; they are not just abstract ideas. It isimportant then, in order to understand educational aims fully, to ask whose aimsare being expressed. As with the nature of mathematics, there is no universal

agreement or happy consensus. This is because different groups can and do havedifferent sets of values which are in turn rooted in their different world views orbelief systems.

There are, however, some broad categories that help to illuminate the area of educational aims. These are indicated in Table 1.1, which presents a simplifiedcategorisation of the possible purposes for education. Each of these areas of development has, at some time within the history of mathematics education,been the focus of concern for different groups. Considering your aims foreducation is a further important part of clarifying the values that will informyour mathematics teaching.


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Table 1 The aims for education different types of development

Academic development

Education should help pupils to develop a thorough knowledge and understanding of the subject. At the same time, pupils are encouraged to form appropriate attitudes to-wards the subject. The desired outcome is sufficient people inducted into the academiccommunity, including an adequate supply of good teachers at all levels This is in orderthat the subjects place within our cultural heritage and its future development areguaranteed.

Vocational development

Education should provide pupils with the relevant knowledge and skills that they needin the world of work. The desired outcome is a suitably equipped work force ready toadapt to the needs of a growing economy.

Personal development

Education should provide opportunity for the all-round development of the individual.The desired outcome is fulfilled and autonomous people who have a well-developedself-awareness and who continue to grow and mature in adult life.

Social development

Education should provide the forum within which pupils can develop socially and findtheir roles within society. The desired outcome is individuals who will be confident intheir interpersonal relationships and in their role as critical citizens.

Further Issues

Your perspectives on the nature of mathematics and on the aims of education cometogether to form your aims for mathematics education. However, having establishedthe purpose of mathematics education a sense of why? there remain the furtherquestions, what? and how? The way in which you answer these questions will bestrongly connected to your aims. Table 2 lists questions that follow on from themore philosophical ones you have been considering. You will explore many of theseareas in more detail in later chapters of this book. However, you might immediatelybegin to see how different aims might lead you to answer these questions differently.You are faced with the challenge of developing your own considered responses tothese questions, consistent with your aims.


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Table 2 Fundamental questions

Area of interest Questions raised

Philosophy Why should pupils learn mathematics?

Curriculum What mathematics should be included in the school curriculum?Which pupils should learn which aspects of the subject?

Learning How do pupils learn mathematics?

Teaching What teaching methodologies will best support your aims?

Resources What resources are most appropriate to the tasks and thelearners?

Assessment How should/could/might pupils development in mathematicsbe measured?

Differentiation What accounts for the diversity of pupil response tomathematics?How should/could/might you respond to this diversity?

Task 1.3 The aims of the mathematics department

During your school experience, you will find out much about how your mathematicsdepartment works. It is also useful to find out what aims or vision the department hasfor mathematics education and how this comes through in its policies. Ask to read thedepartment handbook and in particular consider

what views about the nature of mathematics and education underpin any statementof aims,

how well policies and procedures relate to the departments aims the extent to which the official aims are shared by all members of the department whether classroom practice supports the achievement of the departments aims

COMPETING INFLUENCES ON THE MATHS CURRICULUM

This section includes various aims for education and some discussion of theimplications these could have on your work as a maths teacher in the classroom.Below are descriptions of the mathematical perspectives of four different socialgroups: the Mathematical Purists, the Industrial Pragmatists, the ProgressiveEducators and the Social Reformers. These groups are not real, organised associationsof people, but instead represent a categorisation offering a framework for exploringthe competing influences within mathematics education. No claim is made that allindividuals fit neatly into any particular group; indeed, as you read the descriptions,


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you may find you have certain sympathies with some or even all of the groups.However, the origins of much current practice in maths education have their rootswithin the opinions attributed to one of these four groups. Many of the ideasexpressed here are due to the work of Paul Ernest (for example, Ernest, 1991).

Mathematical Purists

This group is primarily concerned with the academic and some aspects of personaldevelopment of pupils. Its members strongly reject any utilitarian emphasis on workor the applications of mathematics for its justification as a school subject. They alsoassume that it is obvious that mathematics education has no role in the socialdevelopment of young people.

Members of this group have a long tradition within mathematics which can betraced back to Greek philosophy, where mathematics was seen to be educationallyvaluable in the development of thought rather than for learning about any

applications.

Now that we have mentioned the study of arithmetic, it occurs to me what a subtle andwidely used instrument it is for our purpose, if one studies it for the sake of knowledgeand not for commercial ends it draws the mind upwards.(Plato, trans. Lee, 1987, p 332)

The tradition of emphasising the importance of mathematics as a subject for themind continues to be maintained by some today.

The Mathematical Purists consider mathematics to be an objective form of knowledge, a complex hierarchical structure of ideas linked together through proof

and rational thought. They celebrate its significant contributions to our culturalheritage, identify it more as an art than a science and believe it to have aestheticqualities. The Cambridge mathematician G. H. Hardy wrote:

The mathematicians patterns, like the painters or the poets must be beautiful ; theideas, like the colours or the words, must fit together in a harmonious way. Beauty isthe first test: there is no permanent place in the world for ugly mathematics.(Hardy, 1940, p. 25)

They see, the role of the teacher as enabling the effective transmission of this bodyof knowledge and encouraging particular qualities in the pupils, such as concern forrigour, elegance and precision. This tends to involve a lecturing and explaining stylethat makes use of standard texts and traditional mathematical equipment butmakes little use of other resources. Teachers will have an enthusiasm for the subjectthat will be conveyed to the pupils.

This group supports major competitions for pupils such as Mathemtical Olympiads,partly because these help identify the next generations or mathematicians. Assessmenton the whole is not a major concern. However, it is important that qualitficationssuch as A-level mathematics preserve their high standards, and so members of the


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such as Cuisenaire rods and Dienes structural apparatus. Computers and calculatorsare considered important for offering new environments within whichmathematical exploration can happen. There is also an emphasis on developingcaring, supportive relationships in the classroom, with children being shielded fromsignificant social conflict.

Children are to be treated as individuals and allowed to learn at different rates. It isimportant to recognise and celebrate their success, so records of achievement andcriteria-based assessment are to be welcomed. External examinations are notconsidered helpful, as they have the potential for bringing discouragement anddisappointment to the child. They are also seen as skewing the curriculum towardsshort-term goals.

Social Reformers

This group is primarily concerned with the social development of pupils in the sense

that education should empower the individual to participate fully and critically in ademocratic society. Conseqently aspects of personal development are consideredimportant. Encouraging vocational and academic development is appropriate onlythrough negotiation with the pupil.

Members of this group have only relatively recently begun to influence mathematicseducation, but the origins of this group within education (also called PublicEducators) can be traced back to the nineteenth century. Then, the concern was foreducation for all. In more recent times Social Reformers have had more influence inthe emerging education structures of developing countries. For example, president

Julius Nyerere expressed the aims of a Tanzaman education programme as being:

to prepare people for their responsibilities as free workers and citizens in a free anddemocratic society, albeit a largely rural society. They have to be able to think forthemselves, to make judgements on all the issues affecting them; they have to be ableto interpret the decisions made through the democratic institutions of our society

(Ernest, 1991, p. 202)

The other stimulus to the work of the group of Social Reformers has been the needto work towards equality of opportunity for all within education. Withinmathematics education, significant work has been done in the field of gender issues,multicultural and anti-racist mathematics.

The view of mathematics held by some in this group is also relatively new.Mathematics is seen to be a social construction: tentative growing by means of human creation and decision-making and connected with other realms of knowledge, culture and social life (Ernest, 1991, pp. 207208). This offers a muchwider definition of mathematics than is the norm and challenges the MathematicalPurists exclusive ownership of real mathematics. Pupils should experiencemathematics as relevant to their own lives; as important in addressing wider social


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issues; as a vehicle for social emancipation, and as enabling a critical stance towardssociety.

They see the role of the teacher as faciliatiing pupils in both posing and solving theirown problems. This requires the teacher to set conditions in which pupils canparticipate in decisions about their learning and in which they feel able to questiontheir mathematics course and its associated teaching methods. Resources need to besocially relevant and include authentic materials such as newspapers and sources of real data. Discussion is central to the learning process and conflicting ideas arewelcome in promoting greater understanding.

Any form of assessment must be seen to be fair to all pupils and should notdisadvantage any social grouping. This requires a greater variety of modes of assessment, and so project work and the on-going assessment of coursework ishighly valued. The GCSE qualification initially had scope for a large percentage of assessment by coursework and so was welcomed by the Social Reformers. Because of the status that certain mathematical qualifications have within UK society, helping

pupils to pass external exams remains crucial, so teachers have to work within theexisting assessment system.

AGENCIES FOR CHANGE

Given that there is a range of views about the aims of mathematics education, youmay wonder how these competing influences actually bring about change in theschool curriculum and teaching approaches. In this final section, you will see howthe social groups discussed above have acted through different agencies to bringabout the reforms that they seek. You will also consider the influence of people whoexpress their expectations at ground level within your school context.

Mathematics Teaching Associations

The influence of different groups can be seen in the history of the two main mathsteaching organisations. The Mathematical Association (MA) was established in1871 as The Association for the Improvement of Geometrical Teaching withan overt agenda for change. The initial focus was on reforming the teaching of Euclidean geometry. The Association of Teachers of Mathematics (ATM), which wasset up in 1953 as The Association for Teaching Aids in Mathematics, was a splintergroup from the MA. Articles in the early issues of the journal of the newerassociation ( Mathematics Teaching ) show clearly the dominance of the Progressive

Educator group (Cooper, 1985, pp. 6990). These two organisations are discussedfurther in Chapter 12.

Curriculum Development Projects

Mathematics education has had its fair share of curriculum projects. Some succeedand have a large impact in schools across the country; others do not extend muchbeyond the initial project schools and are soon forgotten. In either case, projects will


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normally have the strong support of one of the four social groups listed above,which recognise that courses and materials are a significant way of influencing theway in which teachers work. The School Mathematics Project (SMP), established in1961, was perhaps particularly successful in establishing itself because it was theproduct of an alliance between members of both the Mathematical Purists and theIndustrial Pragmatists (Cooper, 1985 pp. 235266).

Official Reports

Every so often the climate within education reaches such level of concern that anofficial inquiry is commissioned. The reports then produced can become key pointsagainst which all futue proposals are tested. Consequently, many groups wish toinfluence the findings of these inquiries. Dr. W. H. Cockcroft chaired the mostinfluential inquiry into mathematics teaching in recent times. Its report was entitled

Mathematics Counts (DES, 1982). Its terms of reference included considering theneeds of employment, but perhaps its most famous paragraph (para. 243 see

Chapter 4) is about the need for a broad range of teaching methods. It can beinterpreted as being supportive of the view of both Industrial Pragmatistsand Progessive Educators (Ernest, 1991 pp. 220222).

Curriculum and Assessment Policy

All four groups described above have long recognised that qualifications and theirassociated exams have a significant impact on teaching and learning. Some of theirparticular views on assessment were outlined above. In comparison, the most far-reaching government policy in the area, the National Curriculum (NC) forEngland and Wales, and the corresponding for Scotland and Northern Ireland (see

Chapter 2), is still relatively new. Plainly, the NC will remain a crucial battlegroundfor competing groups and indeed the present NC can be seen to embody ideologicalaspects of some of the groups. Certain of the key debates surrounding the variousnational mathematics curricula are explored in the next chapter.

Local Expectations

Finally, it is important to recognise the influence of much more localised groups onyour teaching. Parents, colleagues in the mathematics department, and the schoolssenior management will all have expectations of you as a maths teacher. Theseexpectations may coincide with your own views or may be in conflict with them.

You will need to learn to negotiate with different individuals and groups, if you areto remain in contact with your own values while working to meet the legitimatedemands of others. In particular, pupils will certainly let you know their ownexpectations; this theme is addressed in Task 1.4.


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Task 1.4 What do the pupils want from their mathematics education?

Pupils are arguably the group that will most regularly express to you their views aboutmaths education and much else besides. It can be valuable for you to understandwhat they are wanting from their mathematics education. There is scope for exploring

this in every maths lesson, but there is not normally the time to consider the issue indepth with individuals.

Design an interview sheet that you could use with pupils to explore their views. Theprompts in Mathematical memories (Task 1 1) offer possible questions, but you mayalso want to add some of your own. If you ask questions about teachers, make sure thepupils understand that you do not want to know names; you just want to know whatteaching styles work well for them. If it is possible within your school, ask to interviewthree Year 7 pupils, three Year 10 pupils and three A-level pupils, and arrange with yourhead of department to speak with them for about fifteen to twenty minutes. Also checkwith your head of department that your interview questions are acceptable to the staff.

Record the results of your interview and write some notes in your journal, indicatingthe extent to which you believe you could accommodate the pupils perspectives withinyour teaching.

SUMMARY

The role of mathematics education in our society is complex. There is no simpleconsensus as to which mathematics is important or how it should be taught. Indeed,there is controversy about the nature of mathematics itself. Different social groupshave influenced and will continue to influence the shape of mathematics education.Maths teachers experience the influence of such groups through their teaching,associations, curriculum projects, recommendations of official reports andgovernment policy on curriculum and assessment.

The chapter title posed the question Mathematics education who decides? Nowyou must prepare to decide where you stand on the issues raised within it.


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FURTHER READING

Cooper, B. (1994) Secondary mathematics education in England: recent changesand their historical context in Selinger, M. (ed.), Teaching Mathematics London:Routledge 526.

This introductory chapter from an Open University reader for mathematicsstudents offers a helpful historical overview of the forces for change within Englighmathematics education. It also provides in extensive section of reference.

DES (1982) Mathematics Counts , London: HMSO.

This volume, often referred to as the Cockcroft Report, remains a key text formathematics education in Britain. It is much referred to, although often quiteselectively, and so it is worth being familiar with its contents.

Ernest, P. (1991) The Philosophy of Mathematics Eduation, Basingstoke: Falmer Press.This book is an ambitious work that seeks both to offer new philosophy of mathematics and to examine its impact on mathematics education. The influence of different ldeologies on mathematics education is explored in some detail.

MA (195a) Why, What How? Some Questions for Teaching Mathematics , Leicester: TheMathematical Association.

This booklet provides a good example of an outcome from the recent attempt byone of the mathematics teaching associations to produce a coherent rationale formathematics teaching. As well as offering answers to why, what and how in relationto mathematics teaching, it also includes a small number of classroom examples.


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Aims of Teaching Mathematics

Paul Ernst

The aims of teaching mathematics fall into groups with a common theme. The firstgroup concerns the giving of skills which will be useful to the student as a memberof society and help social development in general.

1. Utilitarian Aims

(a) To create numerate citizens. Students need mathematics for everyday life.They need to know about number and number operations, social arithmetic,measurement, approximation and estimation, basic geometry and graphsand relations. They will also need mental mathematical skills, and to be ableto apply all of this to practical situations.

(b) To give basic work skills. To be prepared for the modern world of work stu-dents may need more than (a) above. They may also need some algebra,trigonometry and ideas of computing and statistics. Mathematics teachingshould prepare them for employment.

(c) To help students who wish to study further beyond the end of secondaryschooling.They may need some mathematics as a basis for their studies. Forexample, computer engineers, electricians and geographers all use networktheory.

(d) To help students who will continue to study mathematics. They may becometeachers, actuaries or even mathematicians. These students will need a basisfor further study of mathematics.

(e) To foster physical skills such as using compasses or measuring instruments.

2. Cognitive aims

(a) To foster logical reasoning and critical thought. Students should be able tosupport their assertions and conclusions and be able to critically evaluate thearguments of others.

(b) To encourage a way of looking at the world. Students should acquire a rangeof spatial and other concepts with which they can order their experiencesand classify objects and events around them.

(c) To provide a concise language. Students should acquire a language withwhich to express concisely and communicate precisely relationships between

events, or ideas. They should be able to describe patterns and should gainimproved powers of abstract thought.

(d) The ability to formulate and solve problems. Students should acquire theability to express problems in mathematics. They should have a range of techniques for solving problems. They should be able to reapply theirsolutions to the real world.

Source: UWIDITE materials.


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In addition to the growth in their knowledge and skills, we would like students togrow and develop into whole, well-rounded people. The aim in this third groupincludes growth in personal areas like attitudes and mathematical self-concept.These areas are often grouped together under the title of the affective domain.

3. Affective Aims

(a) To give all our students experiences of pleasure, success and creativity inmathematics.

(b) To foster positive attitudes. Students should have their interest aroused andchallenged. They should have their open-mindedness, spirit of enquiry, self-reliance, persistence and feelings of worth encouraged.

(c) To foster social skills. Students should learn to cooperate, to listen, to discuss,to help, to value others suggestions and to put aside competition in favourof teamwork.

The final set of aims concerns the appreciation of mathematics in our culture.

4. Cultural Aims

(a) To appreciate the role of mathematics throughout our society.

(b) To appreciate the role of mathematics in providing a foundation for otherdisciplines and subjects.

(c) To appreciate the role of mathematics throughout history and its importancein the development of human thought.

(d) To appreciate the aesthetic side of mathematics: its beauty and its contribu-tion to music and art.


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Relational Understanding andInstrumental Understanding

Richard R. Skemp*

Faux Amis

Faux amis is a term used by the French to describe words which are the same, or veryalike, in two languages, but whose meanings are different. For example:

French word Meaning in Englishhistoire story not historylibraire bookshop, not librarychef head of any organization, not only chief cook

agrement pleasure or amusement, not agreementdocteur doctor (higher degree) not medical practitionerm decin medical practitioner, not medicineparent relations in general, including parents

One gets faux amis between English as spoken in different parts of the world. AnEnglishman asking in America for a biscuit would be given what we call a stone. Toget what we call a biscuit, he would have to ask for a cookie. And between English asused in mathematics and in everyday life there are such words as field, group, ring,ideal.

A person who is unaware that the word he is using is a faux amis can make incon-venient mistakes. We expect history to be true, but not a story. We take books with-out paying from a library, but not from a bookshop; and so on. But in the foregoingexamples there are cues which might put one on guard: difference of language, or of country, or of context.

If, however, the same word is used in the same language, country and context, withtwo meanings whose difference is non-trivial but as basic as the difference betweenthe meaning of (say) histoire and story , which is a difference between fact andfiction, one may expect serious confusion.

Two such words can be identified in the context of mathematics, and it is the

alternative meanings attached to these words, each by a large following, which inmy belief are at the root of many of the difficulties in mathematics education to-day.

*Richard Skemp is Professor of Education at the University of Warwick, Coventry, England.

Reprinted from: Mathematics Teaching , December 1976, 99. 9 15.


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One of these is understanding . It was brought to my attention some years ago byStieg Mellin-Olsen, of Bergen University, that there are in current use two meaningsof this word. These he distinguishes by calling them relational understanding andinstrumental understanding. By the former is meant what I have always meant byunderstanding, and probably most readers of this article: knowing both what to doand why. Instrumental understanding I would until recently not have regarded asunderstanding at all. It is what I have in the past described as rules withoutreasons without realizing that for many pupils and their teachers the possession of such a rule, and ability to use it, was what they meant by understanding .

Suppose that a teacher reminds a class that the area of a rectangle is given by A = L x B .A pupil who has been away says he does not understand, so the teacher gives him anexplanation along these lines. The formula tells that to get the area of a rectangle,you multiply the length by the breadth. Oh, I see, says the child, and gets onwith the exercise. If we were now to say to him (in effect) You may think youunderstand, but you don t really, he would not agree. Of course I do. Look: I vegot all these answers right. Nor would he be pleased at our de-valuing of his

achievement. And with his meaning of the word, he does understand.

We can all think of examples of this kind: borrowing in subtractions, turn it upsidedown and multiply for division by a fraction, take it over to the other side andchange the sign , are obvious ones; but once the concept has been formed, otherexamples of instrumental explanations can be identified in abundance in manywidely used texts. Here are two from a text used by a former direct-grant grammarschool, now independent, with a high academic standard.

Multiplication of fractions . To multiply a fraction by a fraction, multiply the twonumerators together to make the numerator of the product, and the two denominators

to make its denominator.

e.g. 234

5

2 4

3 5

8

15of = =

3

5

10

13

30

65

6

13 = =

The multiplication sign x is generally used instead of the word of .

Circles . The circumference of a circle (that is its perimeter, or the length of its bound-ary) is found by measurement to be a little more than three times the length of itsdiameter. In any circle the circumference is approximately 3.1416 times the diameter

which is roughly 31

2 times the diameter. Neither of these figures is exact, as the exact

number cannot be expressed either as a fraction o a decimal. The number is representedby the Greek letter (pi).

Circumference = d or 2 r Area = r 2


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The first of these will cause fewer problems short-term to the pupils, thought it willbe frustrating to the teacher. The pupils just won t want to know all the carefulground-work he gives in preparation for whatever is to be learnt next, nor his carefulexplanations. All they want is some kind of rule for getting the answer. As soon asthis reached, they latch on to it and ignore the rest.

If the teacher asks a question that does not quite fit the rule, of course they will getit wrong. For the following example I have to thank Mr. Peter Burney, at that time astudent at Coventry College of Education on teaching practice. While teaching areahe became suspicious that the children did not really understand what they weredoing. So he asked them: What is the area of a field 20cms by 15 yards? The replywas 300 square centimeters . He asked: Why not 300 square yards? Answer:Because area is always in square centimetres.

To prevent errors like the above the pupils need another rule (or, of course relationalunderstanding), that both dimensions must be in the same unit. This anticipatesone of the arguments which I shall use against instrumental understanding, that it

usually involves a multiplicity of rules rather than fewer principles of more generalapplication.

There is of course always the chance that a few of the pupils will catch on to whatthe researcher is trying to do. If only for the sake of these, I think he should go ontrying. By many, probably a majority, his attempts to convince them that being ableto use the rule is not enough will not be well received. Well is the enemy of better, and if pupils can get the right answers by the kind of thinking they are used to, theywill not take kindly to suggestions that they should try for something beyond this.

The other mis-match in which pupils are trying to understand relationally but the

teaching makes this impossible, can be a more damaging one. An instance whichstays in my memory is that of a neighbour s child, then seven years old. He was avery bright little boy, with and I.Q. of 140. At the age of five he could read TheTimes , but at seven he regularly cried over his mathematics homework. His misfor-tune was that he was trying to understand relationally teaching which could not beunderstood in this way. My evidence for this belief is that when I taught himrelationally myself, with the help of Unifix, he caught on quickly and with realpleasure.

A less obvious mis-match is that which may occur between teacher and text.Suppose that we have a teacher whose conception of understanding is instrumental,who for one reason or other is using a text which aim is relational understanding bythe pupil. It will take more than this to change his teaching style. I was in a schoolwhich was using my own text 1, and noticed (they were at Chapter 1 of Book 1) thatsome of the pupils were writing answers like

the set of {flowers} .

When I mentioned this to the teacher (he was head of mathematics) he asked theclass to pay attention to him and said: Some of you are not writing your answers


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difficult for most. They would give up the subject as soon as possible, and rememberit with dislike.

The other group is taught to associate certain sounds with these marks on paper. Forthe first few years these are audible sounds, which they make themselves on simpleinstruments. After a time they can still imagine the sound whenever they see orwrite the marks on paper. Associated with every sequence of marks is a melody, andwith every vertical set a harmony. The keys C major and A major have an audiblerelationship and a similar relationship can be found between certain other pairs of keys. And so on. Much less memory work in involved, and what has to beremembered is largely in the form of related wholes (such as melodies) which theirminds easily retain. Exercises such as were mentioned earlier ( Write a simpleaccompaniment ) would be within the ability of most. These children would alsofind their learning intrinsically pleasurable, and many would continue it voluntarily,even after O-level or C.S.E.

For the present purpose I have invented two non-existent kinds of music lesson ,

both pencil-and-paper exercises (in the second case, after the first year or two). Butthe differences between these imaginary activities is no greater than between twoactivities which actually go on under the name of mathematics. (We can make theanalogy closer, if we imagine that the first group of children were initially taughtsounds for the notes in a rather half-hearted way, but that the associations were tooill-formed and un-organized to last.)

The above analogy is, clearly, heavily biased in favour of relational mathematics.This reflects my own viewpoint. To call it a viewpoint, however, implies that I nolonger regard it as a self-evident truth which requires no justification: which it canhardly be if many experienced teachers continue to teach instrumental mathemat-

ics. The next step is to try to argue the merits of both points of view as clearly andfairly as possible: and especially of the point of view opposite to one s own. This iswhy the next section is called Devils Advocate . In one way this only describes thatpart which puts the case for instrumental understanding. But it also justifies theother part, since an imaginary opponent who thinks differently from oneself is agood device for making clearer to oneself why one does think that way.

Devils Advocate

Given that so many teachers teach instrumental mathematics, might this be becauseit does have certain advantages? I have been able to think of three advantages (as

distinct from situational reasons for teaching this which will be discussed).

1. Within its own context, instrumental mathematics is usually easier to understand :sometimes much easier. Some topics, such as multiplying two negative numberstogether, or dividing by a fractional number are difficult to understandrelationally. Minus times minus equals plus and to divide by a fraction youturn it upside down and multiply are easily remembered rules. If what is wanted


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is a page of right answers instrumental mathematics can provide this morequickly.

2. So the rewards are more immediate, and more apparent . It is nice to get a page of right answers, and we must not underrate the importance of the feeling of success which pupils get from this. Recently I visited a school where some of thechildren describe themselves as thickos . Their teachers use the term too. Thesechildren need success to restore their self-confidence, and it can be argued thatthey can achieve this more quickly and easily in instrumental mathematics thanin relational.

3. Just because less knowledge is involved, one can often get the right answer morequickly and reliably by instrumental thinking than relational. This difference isso marked that even relational mathematicians often use instrumental thinking.This is a point of much theoretical interest, which I hope to discuss more fullyon a future occasion.

The above may well not do full justice to instrumental mathematics. I shall be gladto know of any further advantages which it may have.

There are four advantages (at least) in relational mathematics.

1. It is more adaptable to new tasks . Recently I was trying to help a boy who hadlearnt to multiply two decimal fractions together by dropping the decimal point,multiplying as for whole numbers, and re-inserting the decimal point to give thesame total number of digits after the decimal point as there were before. This is ahandy method if you know why it works. Through no fault of his own, thischild did not; and not unreasonably, applied it also to division of decimals. By

this method 4.8 0.6 came to 0.08. The same pupil had also learnt that if youknow two angles of a triangle, you can find the third by adding the two givenangles together and subtracting from 180 . He got ten questions right this way(his teacher believed in plenty of practice), and went on to use the same methodfor finding the exterior angles. So he got the next five answers wrong.

I do not think he was being stupid in either of these cases. He was simply extrapolat-ing from what he already knew. But relational understanding, by knowing not onlywhat method worked but why, would have enabled him to relate the method to theproblem, and possibly to adapt the method to new problems. Instrumental under-standing necessitates memorising which problems a method works for and whichnot, and also learning a different method for each new class of problems. So the firstadvantage of relational mathematics leads to:

2. It is easier to remember. There is a seeming paradox here, in that it is certainlyharder to learn. It is certainly easier for pupils to learn that area of a triangle =1/2 base x height than to learn why this is so. But they then have to learnseparate rules for triangles, rectangles, parallelograms, trapeziums; whereasrelational understanding consists partly in seeing all of these in relation to the


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area of a rectangle. It is still desirable to know the separate rules; one does notwant to have to derive them afresh everytime. But knowing also how they areinter-related enables one to remember them as parts of a connected whole,which is easier. There is more to learn the connections as well as the separaterules but the result, once learnt, is more lasting. So there is less re-learning todo, and long term the time taken may well be less altogether.

Teaching for relational understanding may also involve more actual content. Earlier,an instrumental explanation was quoted leading to the statement Circumference =d. For relational understanding of this, the idea of a proportion would have to betaught first (among others), and this would make it a much longer job than simplyteaching the rules as given. But proportionality has such a wide range of otherapplications that it is worth teaching on these grounds also. In relational mathemat-ics this happens rather often. Ideas required for understanding a particular topicturn out to be basic for understanding many other topics too. Sets, mapping andequivalence are such ideas. Unfortunately the benefits which might come fromteaching them are often lost by teaching them as separate topics, rather than as

fundamental concepts by which whole areas of mathematics can be inter-related.

3. Relational knowledge can be effective as a goal in itself . This is an empiric fact, basedon evidence from controlled experiments using non-mathematical material. Theneed for external rewards and punishments is greatly reduced, making what isoften called the motivational side of a teacher s job much easier. This is relatedto:

4. Relational schemas are organic in quality. This is the best way I have been able toformulate a quality by which they seem to act as an agent of their own growth.The connection with 3 is that if people get satisfaction from relational under-

standing, they may not only try to understand relationally new material whichis put before them, but also actively seek out new material and explore newareas, very much like a tree extending its roots or an animal exploring a newterritory in search of nourishment. To develop this idea beyond the level of ananalogy is beyond the scope of the present paper, but it is too important to leaveout.

If the above is anything like a fair presentation of the cases for the two sides, itwould appear that while a case might exist for instrumental mathematics short-termand within a limited context, long-term and within a limited context and in thecontext of a child s whole education it does not. So why are so many children taughtonly instrumental mathematics throughout their school careers? Unless we cananswer this, there is little hope of improving the situation.

An individual teacher might make a reasoned choice to teach for instrumentalunderstanding on one or more of the following grounds.

1. That relational understanding would take too long to achieve, and to be able touse a particular technique is all that these pupils are likely to need.


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2. That relational understanding of a particular topic is too difficult, but the pupilsstill need it for examination reasons.

3. That a skill is needed for use in another subject (e.g., science) before it can beunderstood relationally with the schemas presently available to the pupils.

4. That he is a junior teacher in a school where all the other mathematics teachingis instrumental.

All of those imply, as does the phrase makes a reasoned choice , that he is able toconsider the alternative goals of instrumental and relational understanding on theirmerits and in relation to a particular situation. To make an informed choice of thiskind implies awareness of the distinction, and relational understanding of themathematics itself. So nothing else but relational understanding can ever beadequate for a teacher. One has to face the fact that this is absent in many who eachmathematics; perhaps even a majority.

Situational factors which contribute to the difficulty include:

1. The backwash effect of examinations . In view of the importance of examinationsfor future employment, one can hardly blame pupils if success in these is one of their major aims. The way pupils work cannot but be influenced by the goal forwhich they are working, which is to answer correctly a sufficient number of questions.

2. Over-burdened syllabi . Part of the trouble here is the high concentration of theinformation content of mathematics. A mathematical statement may condenseinto a single line as much as in another subject might take over one or two

paragraphs. By mathematicians accustomed to handling such concentratedideas, this is often overlooked (which may be why most mathematics lecturersgo too fast). Non-mathemticans do not realize it at all. Whatever the reason,almost all syllabi would be much better if much reduced in amount so that therewould be time to teach them better.

3. Difficulty of assessment of whether a person understands relationally orinstrumentally. From the marks he makes on the paper, it is very hard to make avalid reference about the mental processed by which a pupil has been led tomake them; hence the difficulty of sound examining in mathematics. In ateaching situation, talking with the pupil is almost certainly the best way to find

out; but in a class of over 30, it may be difficult to find the time.

4. The great psychological difficulty for teachers of accommodating (restructuring) their existing and longstanding schemas , even for the minority who know they need to,want to do so, and have time for study.

From a recent article 3 discussing the practical, intellectual and cultural value of mathematics education (and I have no doubt that he means relational mathematics!)


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between where I was staying and the office of the colleague with whom I was work-ing; between where I was staying and the university refectory where I ate;between my friend s office and the refectory; and two or three others. In brief, Ilearnt a limited number of fixed plans by which I could get from particular startinglocations to particular goal locations.

As soon as I had some free time, I began to explore the town. Now I was not wantingto get anywhere specific, but to learn my way around, and in the process to see whatI might come upon that was of interest. At this stage my goal was a different one, toconstruct in my mind a cognitive map of the town.

These two activities are quite different. Nevertheless they are, to an outside observer,difficult to distinguish. Anyone seeing me walk from A to B would have greatdifficulty in know (without asking me) which of the two I was engaged in. But themost important thing about an activity is its goal. In one case my goal was to get toB, which is a physical location. In the other it was to enlarge or consolidate mymental map of the two, which is a state of knowledge.

A person with a set of fixed plans can find his way from a certain set of startingpoints to a certain set of goals. The characteristic of a plan is that it tells him what todo at each choice point: turn right out of the door, go straight on past the church,and so on. But if at any stage he makes a mistake, he will be lost; and he will staylost if he is not able to retrace his steps and get back on the right path.

In contract, a person with a mental map of the town has something from which hecan produce, when needed, an almost infinite number of plans by which he canguide his steps from any starting point to any finishing point, provided only thatboth can be imagined on his mental map. And if he does take a wrong turn, he will

still know where he is, and thereby be able to correct his mistake without gettinglost, even perhaps to learn from it.

The analogy between the foregoing and the learning of mathematics is close.The kind of learning which leads to instrumental mathematics consists of thelearning of an increasing number of fixed plans, by which pupils can find theirway from particular starting points (the data) to required finishing points (theanswers to the questions). The plan tells them what to do at each choice point.And as in the concrete example, what has to be done next is determined purelyby the local situation. (When you see the post office, turn left. When you havecleared brackets, collect like terms.) There is no awareness of the overallrelationship between successive stages, and the final goal. And in both cases thelearner is dependent on outside guidance for learning each new way to getthere .

In contrast, learning relational mathematics consists of building up a conceptualstructure (schema) from which its possessor can (in principle) produce an unlimitednumber of plans for getting from any starting point within his schema to anyfinishing point. (I say in principle because of course some of these paths will bemuch harder to construct than others.)


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This kind of learning is different in several ways from instrumental learning.

1. The means become independent of particular ends to be reached thereby.

2. Building up a schema within a given area of knowledge becomes an intrinsicallysatisfying goal in itself.

3. The more complete a pupil s schema, the greater his feeling of confidence in hisown ability to find new ways of getting there without outside help.

4. But schema is never complete. As our schema enlarge so our awareness of possi-bilities is thereby enlarged. Thus the process often becomes self-continuing, and(by virtue of 3) self-rewarding.

Taking again for a moment the role of devil s advocate, it is fair to ask whether weare indeed talking about two subjects, relational mathematics and instrumentalmathematics or just two ways of thinking about the same subject matter. Using the

concrete analogy, the two processes described might be regarded as two differentways of knowing about the same town; in which case the distinction made betweenrelational and instrumental understanding would be valid, but not that betweeninstrumental and relational mathematics.

But what constitutes mathematics is not the subject matter, but a particular kind of knowledge about it. The subject mater of relational and instrumental mathematicsmay be the same: cars travelling at uniform speeds between two towns, towerswhose heights are to be found, bodies falling freely under gravity, etc. etc. But thetwo kinds of knowledge are so different that I think that there is a strong case forregarding them as different kinds of mathematics. If this distinction is accepted,

then the word mathematics is for many children indeed a false friend, as they findto their cost.

The State of Play

This is already a long article, yet it leaves many points awaiting further development.The applications of the theoretical formulation in the last section to the educationalproblems described in the first who have not been spelt out. One of these is therelationship between the goals of the teacher and those of the pupil. Another is theimplications for a mathematics curriculum.

In the course of discussion of these ideas with teachers and lecturers in mathematicaleducation, a number of other interesting points have been raised which also cannot beexplored further here. One of these is whether the term mathematics ought not tobe used for relational mathematics only. I have much sympathy with this view, butthe issue is not as simple as it may appear.

There is also research in progress. A pilot study aimed at developing a method (ormethods) for evaluating the quality of children s mathematical thinking has been


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finished, and has led to a more substantial study in collaboration with the N.F.E.R.as part of the TAMS continuation project. A higher degree thesis at Warwick Univer-sity is nearly finished; and a research group of the Department of Mathematics atthe University of Quebec in Montreal is investigating the problem with first andfourth grade children. All this will I hope to be reported in due course.

The aims of the present paper are twofold. First, to make explicit the problem at anempiric level of thinking and thereby to bring to the forefront of attention whatsome of us have known for a long time at the back of our minds. Second, to formu-late this in such a way that it can be related to existing theoretical knowledge aboutthe mathematical learning process, and further investigated at this level with thepower and generality which theory alone can provide.

References

1. R.R. Skemp: Understanding Meathmatics (U.L.P.)

2. For further discussion see R.R. Skemp: The Psychology of Learning Mathematics(Penguin 1972) pp. 43 46.

3. H. Bondi: The Dangers of Rejecting Mathematics (Times Higher EducationSupplement , 26. 3.76)


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Understanding

Michelle Selinger

Many teachers report that they suddenly have the feeling that they are the onlyperson in the classroom who cares whether the pupils can do mathematics or not.In front of them there is a sea of faces waiting for the next instruction. The pupilsseem to be saying:

You are in charge. Give me something to do and I will do it. When I ve finished this, you can give me something else to do.

These teachers are unsure whether it arises because their pupils do not see any pointin the mathematical topic or whether it is because they have no interest inmathematics at all. The teachers also believe that this cannot be a productive way tolearn mathematics, nor a way to discover the power of the subject. They want theirpupils to find out that working on mathematics can be challenging and exciting;that the connections and similarities between different aspects of mathematics canbe a constant source of amazement and new insights.

Before reading on, the reader is invited to construct a map of how pupils learnmathematics. This chapter describes some ways teachers have tried to encouragepupils to want to learn and to start asking questions about the mathematics theywere doing. Their methods are often indirect, relying more on stimulating pupils to

reflect on their work than on setting challenging mathematical problems.

TWO FORMS OF UNDERSTANDING

An article by Richard Skemp (1976) outlined two types of understanding,instrumental and relational . Instrumental understanding he described as ruleswithout reasons ; A = L X B, a 2 + b 2 = c 2, borrowing in subtraction, change the side,change the sign . On the other hand, pupils who had relational understandingwould be able to reconstruct forgotten facts and techniques: for example, theywould be able to demonstrate, perhaps by means of a diagram of a right-angledtriangle on which squares have been constructed on each side, the relationshipbetween the areas of the squares of the three sides. Being able to quote rules likePythgoras in parrot fashion does not constitute full understanding and a story of Skemp s highlights this strongly. He tells of a young child returning home fromschool and reporting that he now knew his four times table. Well done, said hismother, so tell me, if there are seven children at a party and they are each given

Reprinted from: Michelle Selinger, Teaching Mathematics . Open University, UK: Routledge, 1994.


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four balloons, how many balloons will there be? The child looked confused and hisface fell as he said to his mother But we haven t done balloons yet!

Skemp outlines three advantages for teaching instrumental understanding:

1. Within its own context, instrumental mathematics is usually easier to understand ;sometimes much easier. Some topics, such as multiplying two negative numberstogether or dividing by a fractional number, are difficult to understandrelationally. Minus times minus equals plus and to divide by a fraction youturn it upside down and multiply are easily remembered rules. If what is wantedis a page of right answers, instrumental mathematics can provide this morequickly and easily.

2. So the rewards are more immediate and more apparent . It is nice to get a page of right answers, and we must not under-rate the importance of the feelings of success which pupils get from this . . .

3. Just because less knowledge is involved, one can often get the right answer more

quickly and reliably by instrumental thinking than relational. This difference isso marked that even relational mathematicians often use instrumental thinking.(Skemp 1976: 23)

However, he argues that relational understanding can be

more adaptable to new tasks . Recently I was trying to help a boy who had learnt to multiplytwo decimal fractions together by dropping the decimal point, multiplying as for wholenumbers, and reinserting the decimal point to give the same total number of digitsafter the decimal point as there were before. This is a handy method if you know why itworks. Through no fault of his own, the child did not, and not unreasonably, appliedthis method to division of decimals. By this method 4.8 0.6 came to 0.08 . . . He was

simply extrapolating from what he already knew. But relational understanding, byknowing not only what method worked but why, would have enabled him to relate themethod to the new problem, and possibly adapt the method to new problems.Instrumental understanding necessitates memorising which problems a method worksfor and which not, and also learning a new method for each new class of problems. Sothe first advantage of relational mathematics leads to:

2. It is easier to remember . There is a seeming paradox here, in that it is certainly harder tolearn. It is certainly easier for pupils to learn that area of a triangle = 1/2 base x height than to learn why this is so. But they then have to learn rules for triangles, rectangles,parallellograms, trapeziums; whereas relational understanding consists partly in seeingall of these in relation to the area of a rectangle. It is still desirable to know the separate

rules, one does not want to have to derive them afresh everytime. But knowing alsohow they are inter-related enables one to remember them as parts of a connected whole,which is easier. (Skemp 1976: 23)

It might be thought that pupils would be attracted to relational understandingbecause it minimises their memory load and so might provide the key to a wayforward in motivating pupils to learn more effectively. But encouraging pupils tolearn more relationally can be problematic for both teachers and their pupils; the


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investment required to make connections is greater than in instrumental learningand the content of the curriculum needs to be considered so that connections canbe easily and readily established. The way a teacher works may affect the way theirpupils want to learn. A test can be set in such a way that instrumentalunderstanding will not be useful for all of the questions and relationalunderstanding is required in order to make sense of the questions.

MAKING CONNECTIONS

There are several ways in which relational understanding can be developed to helppupils make connections and which can be used to help understand concepts.Pupils can be set a range of tasks; some which will help them increase theirunderstanding of concepts within a topic and some which will encourage them tostand back from their work, view it as a whole and examine the way in which theymake sense of new concepts. It is such tasks which include the use of concept maps,matching statements, marking fictitious homework or using metaphors to describelearning which form the focus of this chapter. I have tried several of these and findthat although some tasks appear different from normal classroom practice they canbe successfully integrated.

Concept Maps

Concept maps present a method of visualising concepts and the relationshipsbetween them. Because concept maps are explicit, overt representations of theconcepts and relationships we hold, they allow teachers and pupils to exchangeviews on why a particular relationship is valid, and to recognise missing linkages

that suggest a need for further experience. Because they contain externalisedexpressions of relationships, they are effective tools for highlightingmisconceptions.

Concept maps are more than a mere overview, they can be used as tools fornegotiating meanings. They offer a method by which the relationships betweenconcepts can be shared, discussed and negotiated. One of the most important singlefactors influencing learning is what the learner already knows; consequently, many,teachers recognise that it is useful to have some idea of what their pupils alreadyknow (or misunderstand) before beginning a new topic. By using concept maps witha class, I can be provided with some of the necessary information which will helpme decide on a starting point that will involve all my pupils in building on fromwhat they know.

One approach I used with concept maps was to select a key idea from a new topicand to then invite pupils to construct a map showing all the concepts andrelationships they can link to this key concept (Figure 18. 1). At other times, inorder to assess the kind and extent of previous learning, I selected ten to fifteenconcepts from a new topic of study and asked pupils to construct a concept map


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using some or all of these concepts, and to add other concepts they think might berelevant. As a further extension, I sometimes ask pupils to construct three or foursentences linking two, three or more of the concepts.

Heather Scott (1991) introduced her pupils to concept maps through the termmathematical maps .

A mathematical map would consist of towns and various connecting routes. The townscould represent a variety of items . . . concepts knowledge, skills or processes. The routestoo may be symbolic of many things a connection, a need, or a particular method.(Scott 1991: 5)

Scott found that pupils maps highlighted understandings and misunderstandingsas well as vital gaps in areas of mathematics which would need to be closed forprogress to be made . Pupils were able to add towns and routes as they learnt andmade new connections or understood a concept more fully and the teacher s rolebecame that of provider of tickets to travel to more towns and upon different routes

on the map .

Once the notion of constructing concept maps has been established with pupils,concept maps can become a tool for them to use when they are faced with a conceptthey appear to have forgotten. They can be encouraged to try and recreate theirunderstanding by referring back to a concept map constructed earlier or to try toreconstruct their understanding through a new mental concept map. Pupils candelete towns or change their meaning as a result of new learning.

Matching statements

In the task of matching statements, I present pupils with a list of mathematicalstatements or number names and ask them to sort these in any way they like but tojustify the way they have been sorted. For example, this might be a list of fractionnames written as numbers and words, decimal names, ratios and percentages. Ininitial sorts many groups of pupils put the statements into separate sets of fractions,decimals, ratios etc., so I then invite them to sort them in another way. This timesome pupils start to group the equivalent fraction with its decimal, percentage andratio, e.g. three-quarters, 3/4, 0.75, 75% and 3:4 can be grouped together. I theninvite pupils to share their groupings with others and to search for similarities anddifferences and to point out where they think a grouping may be incorrect. Aconsensus as to what the correct groupings ought to be must then be reached. As a

result of these groupings and regroupings, pupils who had always seen conceptssuch as decimals, percentages and fractions as unconnected ways of representingtypes of numbers can now start to gain insights into the connections that will helpthem recognise percentages as another way of writing fractions or fractions asanother way of writing decimals. In other words their relational understanding willbe enhanced.


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Marking homework

Another way in which I help pupils to make connections or to see where amisconception has occurred is to give groups a fictitious homework to mark. Thismight have been compiled from errors made by a class in a previous homework thatI have collated and rewritten to ensure anonymity of the authors (and to avoidembarrassment). As a result, I hope that cognitive conflict might occur in whichthe pupils become aware of their own difficulties through the discussion andarguments that are generated. They see errors they might also have made themselvesand they are caused to rethink their understanding in order to make sense of theerrors and also to work on the correct solutions. Having to justify and explain willhelp pupils to make use of and enhance both their relational and their instrumentalunderstanding.

AWARENESS OF LEARNING

Asking pupils to consider how they learn or to think about one new thing theylearnt in a lesson can also help them consider the learning process. Below is atranscript of part of an interview I had with two 12-year-old girls about theirlearning.

MS: When you are doing a problem, do you ask yourself whether you areon the right lines?

LISA: You do, don t you? Because when you re in the middle of somethingand you re not sure, you do say, Am I doing this right?

MS: Do you ask yourself if you understand?

KATYA: Yes, I suppose so, cos if you don t, you go and ask the teacher andshe ll tell you. If you don t think about it, if you don t ask yourself if you understand you wouldn t go and tell the teacher, would you?

MS: Do you learn with a partner or a group of three or four?

LISA: Yes, I learn with a partner.

KATYA: Yes, that might be it, I don t learn when I m with a group of three orfour because everyone starts talking, so you don t get much done.

LISA: I think I learn best when I work with a partner because you have todiscuss. It s easier to discuss with two people because you don t haveas much argument.

KATYA: I think I like working with a partner best, I don t like working on myown at all.

MS: What do you like best about working with a partner? Say, comparinganswers?

LISA: Yes.

MS: Does that help you learn?


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KATYA: Yes because you go Ive got 94 and she s got 93 and then you go OKwell see who sright and do it again, and it helps you get thingsright and helps you learn.

MS: Do you learn when you listen to others?

KATYA: Yes, because If they got it right and they re explaining to you then

you re learning.MS: Is it better than a teacher explaining it to you?

LISA: Yes, because they re the same age as you and they use the words thatyou know. Basically they know all that you know . . . . So when theyexplain it to you . . . it ll be easier to understand.

MS: Does it help if you are shown how to do something?

LISA: Yes, then, because if they talk to you, you might not get it in yourhead, but if they show it to you, by writing it down or whatever (if you re allowed to do that in your maths book) then you understandit more clearly like dividing a chocolate bar into thirds.

MS: Does repetition help you learn?KATYA: No, that doesn t help you learn, cos when you re doing your maths

book and you ve got a whole page of timesing it doesn t help youlearn cos you think this is boring , you get about two done in a day.

LISA: It does help with your spellings though, doesn t it? If you dospelling over and over again you learn how to do it eventually.

MS: Does discussion help you learn?

KATYA: Yes, I think so, because you listen to other people s thoughts about itand then you put them together and you decide I think she s rightand you sort of mix them up and see what you get until you thinkeach other s right.

MS: If you are not understanding, does it help trying to work out foryourself?

LISA: Yes, because If you ve not understood something and then you cando it, you feel you ve really achieved something and you rememberit. You think it s the first time I ve achieved that, I ve really stuck myteeth into that and you I think, I ll remember that one.

MS: Does trying several ways to do a difficult problem help?

LISA: I suppose it does in a way.

KATYA: It means you ve learnt the first way, because you think about the firstthing you ve done and you . . .

LISA: If you ve worked out all the ways to do it then you ve got morechance of remembering what you had than if you had just learnt oneway, you d have more things in your brain toremember.

Finding out how pupils view mathematical learning can often open the door to theirfears and concerns as well as revealing what it is about mathematics that motivatesthem. Another way of exposing their views is to ask pupils to consider metaphors forlearning mathematics. For example, learning mathematics is like. . .


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wondering (sic) through a maze doing a jigsaw climbing a mountain a flower blooming filling a bucket

I recall a lesson in which I tried to find out about a new set of Year 10 pupils viewsof mathematics by introducing the idea of metaphors to describe their learning.Rather than give the pupils the bald statement, I had some cartoons illustratingthree or four of these metaphors mounted on card. I gave each group two of thecartoons and asked them to consider the following questions for each one and writea response.

1 In what ways do you agree with this statement?2 In what ways do you disagree with this statement?3 Try to think of a time when you felt like this about learning mathematics.

Describe the topic you were doing and what you were learning when you feltlike this.

The pupils were reluctant to start but once they started talking about each cartoon,the discussion seemed to be very valuable. One girl started to laugh and said, mybucket s got a great big hole in it .

The completed sheets certainly showed depth of thought I had not anticipated andhighlighted some of the difficulties they were experiencing particularly in the areasof algebra and loci. Here are some of the responses the pupils gave.

Learning mathematics is like . . . building a wall:

You add a new brick every lesson, as a brick represents knowledge and cement repre-sents understanding . . . Sometimes we have a lot of bricks but no ce ment to st ickthem together.

By the time you reach the top of the wall some of the knowledge is forgotten, andwhere we didn t understand, there may be bricks missing.

. . .being a sponge:

learning maths is not really like this because when reading a book aboutmaths it goes in one ear and out of the other, but when working somethingout practically, you thoroughly understand it.

I feel like this when revising for a test.

We all shared the comments and then I invited the class to invent their ownmetaphors and explain why the metaphor was appropriate. The following areexamples of some of the responses. Learning mathematics is like:


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a gold ring . . . because you never think you will get it, but when you do, you can bevery proud of it;

a rock . . . flipping hard;a tree . . . one minute you are upright and fine, and the next minute when

the wind blows, you are confused and sad;a foggy day . . . sometimes it is clear, sometimes it is notmaking a cake . . . when you mix the right things together, the result is

perfect.

This lesson had allowed the pupils to find a way to articulate their feelings aboutlearning mathematics, to express their difficulties and anxieties as well as theirenjoyment. I think it helped them to become more comfortable in mathematicslessons because they were more aware of how others felt about mathematics andrealised they were not alone. The metaphors had also provided the means by whichthey could describe their learning in the future. As a teacher I was more aware of how individuals viewed mathematics; I gained some insight as to whether theyviewed mathematics as a collection of facts and skills to be learned and practiced

(instrumental understanding) or whether they saw it as a collection of inter-relatedconcepts which could be drawn on when a new concept was introduced to helpmake sense of it and to incorporate it into their understanding (relationalunderstanding). I was able to adapt my teaching style to help individual pupilsovercome anxieties which had been highlighted as a result of the exercise and towork on the idea of learning mathematics more relationally. It also helped me todiscuss and describe individual ways of learning when the class was working onmathematical activities. I could say to one pupil, you are in a maze today, how doyou think you could get out of it? , or to another, You re trying to soak up too muchin one go, think about how you could wring out your sponge without losing whatyou have learnt so far . By using metaphors which both the pupils and I understood,

we could talk about their learning. Alternatively pupils could describe theirdifficulties to me with a metaphor so that we immediately had some sharedunderstanding of these difficulties. It offered me an insight into each pupil sthinking and enabled me to help them find a suitable strategy to resolve theproblem.

SUMMARY

Instead of summarising how I believe these strategies have offered pupils anopportunity to learn and understand mathematics relationally, I invite the reader toreflect on how their own concept map of pupil learning could be reformulated by

reading this chapter.

NOTE

Much of the work described in this chapter is based on research undertaken at theShell Centre for Mathematics Education for the ESRC funded project Pupils Awareness of Learning .


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What does it mean to understand something andhow do we know when it has happened?

Anne Watson

Words which appear to describe cognition, such as understanding and knowing are used throughout educational literature, and in teachers shared discourse, withflexibility and fluency. When we try to use them precisely they become problematic,as they can take slightly different meanings, but we communicate effectively aboutthem by elaborating what we mean. However, once they enter the statutorylanguage through official documents which describe what education should beachieving they can no longer be used casually. Teachers are accountable for the waysin which they fulfill the statutory requirements, and need to have a worked-out andjustifiable view of what understanding means. Phrases such as knowledge and

understanding and mathematical understanding are used in the Initial TeacherTraining National Curriculum (ITTNC) (TTA, 1999), and the Mathematics NationalCurriculum (NC) (QCA, 1999) refers frequently to pupils ability to use andunderstand concepts and to assessing such ability. These requirements suggest thatthere is a state called understanding and we can know when it exists and when itdoes not exist.

Understanding as a State

In this chapter I am going to argue that the idea that pupil progress in mathematicscan be seen by assessing recognizable states of understanding is an over-

simplification of how learning happens.

It is very common for new teachers to find themselves thinking, I never reallyunderstood addition of fractions (or calculus, or graph-plotting etc.) until I had toteach it! In other words, the thinking involved in planning to teach (such asworking out how to explain or exemplify and predicting what pupils will finddifficult) has enabled the teacher to re-examine existing knowledge and look at in anew way that is recognized as being deeper, more connected and more secure thanprevious experience. Possibly the teacher has easily remembered how to addfractions, but thinking about how to teach has led to considering why it is done thatway and brought new insights into the importance of equivalence, or has raised an

awareness of the numerical value of the fractions. And yet the teacher has been ableto add fractions, pass examinations involving this skill and be thought of as

Reprinted from: Linda Haggarty (ed.), Teaching Mathematics in Secondary Schools, A Reader . Routledge Falmer,2002, pp. 161 175.


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understanding adding fractions . What is being recognized here is that, even whenone is extremely competent in a mathematical technique, there are still ways inwhich understanding can grow in a new situation, when one looks at the topicdifferently. Understanding is not static.

Marton and Salj (1997) classify learning as surface (learning procedures anddescriptions) or deep (learning about connections and relationships with previousknowledge). This kind of distinction can be useful when planning how to teach, butfails to take account of the fact that mathematical procedures consist of strings of simpler procedures which could be described as previous knowledge. To continuethe example of adding fractions, one has to multiply and to add, to identifymultiples and factors, to find common multiples and common factors . . . alldependent on previously acquired knowledge and skills. In this sense, learningmathematical procedures inevitably involves connecting and employing previously-learnt procedures. What is missing from this observation, but is implied in Martonand Salj s distinction, is a sense of underlying meaning allowing us to explain whywe add fractions this way and justify the answers we get.

Nevertheless, most mathematicians do not explain their actions when addingfractions. It is usually enough to know how to do it and to understand that themethod works, but being able to reconstruct explanations, if needed, can contributeto future learning. So here we have two meanings for understanding: I understandthat in situation X I need to do Y and I understand why I need to do Y in situationX. Ryle (1949), in describing types of knowledge, referred to these as knowing-that (factual, definitional) and knowing-why . He also describes a third type, knowing-how ,which is the knowledge required to carry out the chosen action.

Examples of knowing-that can be found in the NC, for example Understand that

percentage means number of parts per 100 (p. 59). In this case understandingappears to mean knowing a definition of a word where the definition gives us someclues (but very few) about what we can do with it mathematically. Some studentsmay be able to construct everything they need to do with percentages from this fact,others may need much more help, but all can be tested on whether they can repeatdefinitions and correctly use procedures in particular circumstances. There iswidespread agreement that what is being tested is not understanding , which relatesto more complex forms of knowledge, but whether pupils can act in a certain way inthe very precise circumstances of the test a very localized knowing-that .

A state of understanding would include knowing facts and procedures, butmight also include a sense of underlying meaning, some connection to previousknowledge and, possibly, the ability to explain. However, as shown above in thedescription of previous knowledge links in adding fractions, making connections isnot dependent on a sense of meaning or knowing-why . It is possible to progressin mathematics to some extent by performing increasingly complex proceduresand hence displaying a kind of behavioural, fluent, automatised understandingof how to enact mathematical algorithms.


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ultimately everything is connected to everything else, hence growth of understanding relates to an increase in the number of links one makes. This [is] auseful metaphor in mathematics because ultimately the links themselves can benamed as mathematical objects (such as are expressed through abstract algebra,morphisms, networks, etc.,). Since we do not know how much there is to know,there is no end to the growth of understanding.

Pirie and Kieren (1994) have developed a theory of the growth of mathematicalunderstanding as a whole, dynamic, levelled but non-linear, transcendentallyrecursive process (p. 62). This hierarchical model has been used to relate differentlevels of understanding to what can be observed in pupils behaviour, i.e.,descriptions of observable actions of mathematical understanding that expressbackground processes. It provides a structure for considering questions such asWhat can be said about the understanding of a pupil who chooses to use symbolicforms, or manipulates familiar formats to adapt them to a new situation, or derives anew fact from some previous knowledge?

They describe stages of primitive knowing, image-making and -having, property-noticing, formalizing, observing, structuring and inventising. Primitive knowing iswhat is known so far, making distinctions in existing knowledge and using it in newways leads to formation of new images . Images can be manipulated and comparedand lead to new properties being noticed by the learner who then abstractssomething to be said about them, thus moving to a level of formalizing . Reflectingon, and expressing, such formal thinking is called observing , and developing theseobservations as theory is called structuring . After this the learner can create newquestions and new lines of enquiry, which they call inventising . These processes,although increasingly complex, do not necessarily follow each other. In practicethere is a lot of toing and froing between levels.

In secondary school mathematics it is rare for teachers to have the opportunity toobserve pupils closely enough to be precise about their understanding. The simplermodels of Bruner (1960), who sees learning as a process of developing iconic andthen symbolic representations of enacted experiences , with the help of interactionwith others or Floyd et al. (1981) who see learning mathematics as a process of manipulating , getting-a-sense-of and articulating , might be easier to use in theclassroom. Once learners can articulate or symbolise a mathematical idea, they areready to manipulate it further to gain more understanding, or to treat it as the rawmaterial for abstraction or more complex manipulations.

Understanding in Context

Some teachers may interpret relational understanding to be entirely aboutappropriateness in a context, which could be mathematical or real world , whileothers may look for genaralised arguments or descriptions of underlying structure.To interpret understand as able to use in a real context implies that allmathematics can be useful outside classrooms, which is dubious, and that pupils canapply what is learnt in one place to another, dissimilar situation. The implication is


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that relational understanding enables instrumental use of mathematics. But formalmathematics is rarely used outside classrooms (Nunes, 1993; Watson, 1998b), so therequirement to use it might be artificial and unrealistic. Further, Mason and Spence(1999) point out that none of the components of relational understanding (knowingthat, how and why) necessarily lead to doing the most appropriate, sophisticated orefficient action in a particular situation. For a variety of ad hoc reasons the featuresof the situation just may not trigger a particular pupil to use the hoped-formathematics. Cooper (in Chapter 13 of this volume), Christoforou (1999) andWatson (1999), among others, show that students responses may be as much due totheir social backgrounds and the way the math

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