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Leonardo

Symmetry and Dissymmetry in Mathematics Education: One View from EnglandAuthor(s): Mary HarrisSource: Leonardo, Vol. 23, No. 2/3, New Foundations: Classroom Lessons inArt/Science/Technology for the 1990s (1990), pp. 215-223Published by: The MIT Press

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PROGRAMS FOR THE COMING ERA

Symmetry a n d Dissymmetry

n Mathematics Education

n e i e w f r o m ngland

Mary Hams

work in the field of mathematics education.

It is a complex field in which we try to make sense of

mathematics, its place in society and the factors that affect

its learning. Thus we have to look at social and political as

well as intellectual factors that determine the mathematics

that should be taught in school and why, and at social and

psychological factors that influence the theory and practiceof the cognitive growth of individualslearning mathematics.

Since mathematics is aligned with science and education

is aligned with the humanities, the whole profession, inthe words of Stephen Brown [1], is almost bound to be

schizophrenic.

My particular work is concerned with realism, one of

today's overworked words. I am concerned with mathemat-

ics as a mode of thinking, and I try to make closer the

relationship between the mathematics studied in schools

and the mathematics practiced outside them. Most of themathematics curricula in schools consist of dilutions of the

academic mathematics the majority of students will never

study, together with sets of isolated 'basic skills' demanded

by industry, often without any evidence for their use in

industry. This unsatisfactory mixture ensures that manystudents leave school at age 16 without either the skills or

the interest in mathematics that the academic aspects of thecourse are supposed to inspire. So I am concerned with

meaning, with making explicit connections between school

syllabus items and the real world, which of course includes

mathematicians doing mathematics. In England, mathe-

matics education still suffers from class distinctions laiddown in the nineteenth century when state education be-

gan: pure mathematics as intellectual discipline for future

leaders, practical mathematics for the middle class occupa-tions of industry and commerce, and basic skills for theworkers who are not expected to rise above their station.The curriculum content of those at the bottom is defined

by those at the top, who of course know what is best for

everybody.In

my view,mathematics is

somethingthat

goeson in our heads; we all have one of those, and no outsider

has the right to tryto limit the education of the individual.In England and in many other countries today there are

two main approaches in mathematics education: construc-tivist models, which see mathematical development as the

understanding that grows in individuals, and 'empty jug'models, in which the teacher pours in knowledge and skills.The former depends on understanding and the latter on

memory, which iswhy those of us in the trade sometimes use

HilaryShuard'sapt phrase 'leakyjug' [2] instead. The latteris much espoused by the present British government, and

1990 ISASTPergamonPress plc. Printed nGreat Britain.0024-094X/90 $3.00+0.00

there is much talk of skills and

applications; we are expectedto teach children specific skills

and then teach them how

to apply them. One of the ef-

fects of this system is to gen-erate learning materials that

offer many false applications.These tend to be contrived to

fit the mathematical skills be-ing taught and tend to be writ-

ten by people who believe that

they know whatgoes on in worksituations but have not checked

lately. The skills are then pre-sented to the children as real,but the children, who know

that they are not, thus become

ABSTRACT

Mathematicseducations afield hat ries o reconcilehe con-

flicting emands fdifferentec-tors ofeducation ndsocietywhilemaintaininghe best climate or heintellectualrowth f allchildren.An nvestigationy heMathsnWork roject ftheDepartmentfMathematics,tatistics ndCom-

puting ftheUniversityf LondonInstitutefEducationnto hemath-ematical ature nduses of clothconfirmeddeepbias nmathemat-icseducationgainstraditionalwomen'soncerns.Theproject ro-duced earningesources hatusetextiles oteachmathematics,utitsmajormpact rose romanexhi-bitionalledCommon hreads,whichouredEnglandrom1987-1990 andwill ubsequentlyourtheworld nder heauspicesoftheBritish ouncil.

even less impressed than before by their mathematics les-sons. My approach is to look at what people do at work,

working alongside them when I can, so that I can analysethe mathematical thinking in what they are doing. Manyofthese people deny that they are doing anything mathemati-

cal, either because they do it so routinely that they have

forgotten how they worked it out originally or because theyhave such unhappy memories of school mathematics that

they are convinced that they cannot do it. From the ex-

perience of work, I produce learning materials that involvechildren in practical tasks from which teachers drawout themathematics at the level that is appropriate for the children.

A surprisingobstacle to fresh developments in myfield is

the masculine nature of mathematics-surprising becausein this abstract and highly rational subject there is norational basis for it.There are twoaspects of this masculinity,one within the practice of the school subject and one withinthe organisation of the profession itself. Within the subject,textbook writers

and many teachers assume that they areaddressing boys only, and many of the examples are takenfrom sport, war and science. This is now well documented,and there is considerable activityaround the world to makemathematics more 'user-friendly'to girls.The second aspect

MaryHarris (mathematics educator), Mathsin Work Project, Department ofMathematics, Statistics and Computing, Universityof London Institute of Education,28 Woburn Square, London WC1H OAA, United Kingdom.

Received 9June 1989.

Based on a paper titled Artand Skill of Symmetryin Old and Modern TextileTechniques read at the interdisciplinary symposium Symmetryof Structure held in

Budapest, Hungary, August 1989. The symposium marked the inauguration of theInternational Society for the InterdisciplinaryStudyof Symmetry (ISISS).

LEONARDO,Vol. 23, No. 2/3, pp. 215-223, 1990 215

This content downloaded from 158.109.199.19 on Mon, 18 Nov 2013 07:47:31 AMAll use subject to JSTOR Terms and Conditions




8/13/2019 1578608


is just as intractable. Like all pro-fessions, mathematics and mathematics

education are ruled bya clearlydefinedand somewhat defensive hierarchywiththe usual power groups and peckingorders. Both of these features have neg-ative effects on the work of womenand on women themselves. Because thefield is dominated by men, women

generally do not reach the level of

achievement of their male colleagues,and when women do perform well,their work is seen as having no intellec-tual content or as being trivial, unim-

portant and easy.

THE MATHEMATICS AND

TEXTILES PROJECT

My initial reasons for looking at the

stereotypically female topic of textilesas a resource for learning mathematicswere not feminist. I had already pub-lished learning materials on mathemat-ics and packaging and on the mathe-matics of getting around an inner city,

Fig. 1.

Turkish Kelim

(Anatolia),wool,approximately90 x 140 cm,modern. Averi-table symphony of

symmetry. Eachsmall motif hasits own internal

symmetry.These

symmetriesare

talkingEowoven into the

total symmetriesof the borders andcentre panel ofthe rug. Such rugs

the ............n are made by girlsand women withlittle or no formal

education, workingfrom memory. Theincomplete designin one corneroffers a clue as toits construction.

and I wanted to develop materials ontextiles because I knew this field to be

richly mathematical.The first aim of my textiles project

was to produce a pack of materials.Since I have always made my own andmy family's clothes, I had had somepersonal experience with textiles; bytalking to other women who do thesame thing, and byvisiting textile exhi-bitions, ethnographic museums andfactories where people make ties, socksand shirts, I was able to add an in-dustrialdimension. Myoffice filled withinteresting pieces of cloth as I began todemonstrate the geometry of neckties,

the algebra of knitting and the sym-metries in African textiles to the mem-bers of my patient steering committee.They suggested that I set up an exhibi-tion as well as publish the learningmaterials.

Preparing a learning resource tohang on the wall is vastlydifferent frompreparing single sheets designed to be

reproduced on a school photocopier[3]. The exhibition was to be aimed

both at schools and at the general pub-lic. For the former my goal was to showthe mathematical nature of cloth, so

ordinary and widespread a part of theenvironment that it is taken for grantedand certainly normally ignored as amathematics resource. For the generalpublic my goal was to show that there ismore to mathematics than arithmetic,and that many people are doing some-

thing mathematical much of the time,whether or not

theychoose to see it so.

From the mathematics education pointof view, I needed to handle at least partof the exhibition in enough depth forthe mathematicians in the education

hierarchy to take it seriously, and I alsoneeded to handle it in a developmentalway that would enable teachers to in-

corporate textile work into their teach-

ing at a number of levels. And finally Iaimed to demonstrate and demand re-

spect for the large number of women

throughout the world who clothe theirfamilies and thereby introduce theirchildren

dailyto

geometry.Thus I

needed a theme that ran horizontallythrough mathematics and the wholeof life but that would not be dominated

by the mathematics. It also had to bea theme that could be handled verti-

cally to demonstrate, and keep acces-sible and inviting, the special nature ofmathematical thinking at all levels. Icannot remember when the theme of

symmetryemerged, because it did notcome in a blinding flash. Itjust slid into

place as the obvious unifying force inwhat I was trying to do, together witha title for the exhibition-CommonThreads: Mathematicsand Textiles. Tobalance the theme of symmetryand toensure that the school syllabuscould becovered adequately, I chose four other

underlying themes of mathematics:

Number, Creativity,Information Han-

dling, and Problem Solving; of courseall five themes overlap.

Basicallythere are two ways of mak-

ing cloth: interlacing several threads or

looping a single one. Both of these aredetermined by the form in which natu-ralfibres appear: hair of animals or ribsof leaves. Both

weavingon a loom and

knitting on two or more needles tendto produce rectangles unless one does

something to alter the shape. While

rectangles have their own symmetries,the very act of structuring cloth has

symmetrybuilt into it: for the maker ofcloth rectangles, the physical act of

manipulating the threads with twohands causes the bilaterally symmetri-cal human body to interact with thefibres.

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Bydeliberately manipulating colour,texture and grouping of threads, clothworkersplaywith the symmetries inher-ent in the structure to produce designs.As Gombrich [4] reminds us, decora-tion in craft is subservient to utility.But

decoration, like the sonata form,thrives under constraints. One can dec-orate a cloth rectangle by modifyingtexture or colour or both (see for ex-

ample the weaving analysed by Wash-burn [5]) or one can print on it. Most

printing techniques demand that a de-

sign, even if it has no symmetryin itself,be repeated systematically along the

length of the cloth, ensuring that even-

tually there is symmetry in the whole

pattern.

THE COMMON THREADS

EXHIBITION

The Common Threads exhibition be-

gan with a rectangle of cloth, a pieceof cross-stitch

embroidery done by aYugoslavwoman on a piece of factory-woven cloth. There are several possibleways to construct the symmetrical de-

sign the embroiderer used on the

piece. As part of the exhibition, I

merged some of these ways as I re-worked the design in stages to demon-

strate how embroiderers build their de-

signs. In the accompanying captions, I

deliberately mixed the discourses of

mathematics and textiles, and at one

stage I did so on the cloth itself byembroidering x and y axes on it.

Incounted-thread work, this is oftenhow people proceed, though they do

not normally embroider their axes ofreflection into the cloth itself. They usethe grain of the cloth formed by the

warp and weft threads as 'mirror lines'for their symmetricaldesigns. Formanypeople who saw the exhibition, the em-broidered axes were an astonishing rev-

elation. Some were quite shocked be-

cause I had abused their mathematics

by taking it from its private academicbox and treating it so lightly. Others

were amused. Many teachers, students

and members of the public told me thatthis was the first time they had lookedat a design analytically.Yet the world isfull of cross-stitchembroidery made byboth amateur and professional mathe-

maticians, although people are sur-

prised to hear that anyone should thinkof embroiderers as mathematicians at

all. Hungary is particularlyrich in cross-stitch embroidery; indeed, Hargittaiand Lengyel have published all seven

one-dimensional and all 17 two-dimen-

sional plane symmetries found in

Hungarian needlework [6].Other such mathematics-based art

forms are also used by people tradition-

ally assumed by colonialists to be not

verybright. Examples of such art formswould be raffia cloths from Zaire [7]and Maori rafters [8].

Designs on cloth may be embroi-dered on or they may be woven orknitted in. A rug from Turkey (Fig. 1)is a veritable symphony of symmetry.Unlike the mathematician working on

paper, the illiterate weaver has to workwithin a frame defined by a specificnumber of warp threads, the cost of

which, in human and economic terms,she well knows. Within this frame sheworks symmetrical designs ('crystallo-

graphic patterns' [9]) within a sym-metrical centre panel, surrounded bya symmetrical border containing sym-metrical designs. Unlike mathema-ticians working on paper, the weaverscannot rub their designs out and start

again-they cannot cut bits off one end

and stick them on the side. Neitherdo they unpick and start again; thiswould be uneconomical in time andmaterials. Too much unpicking weak-

ens the woolen threads that wereso laboriously sheared, cleaned, spun,dyed and woven. The full conception isin the weaver's head before she starts.

People who have studied these rugsandtheir construction liken the work of theweavers to that of conducting a sym-phony orchestra without a score.

The weaving from Bangladeshshown in Fig. 2 is quite different. Herethe designs of the frieze patterns aresocial comments; fish from the Ganges,the flower that gets into the rice crop,footprints of birds and animals, the

pattern on an imported biscuit. Theweaver SaratMala Cakma'seye is tuned

humourously and affectionately to thefine detail of her world. She works herthreads in pairs until she gets boredwith the groups of even numbers andthen plays with groups of five or seveninstead.

Knitters also play with numbers in

their construction of symmetries, but

usually only for the first few rows of a

garment, while the designs are beingset. After all, a sweater has to fit a per-son, and that means working within a

prescribed number of stitches. Withinthat number the knitter works her de-

signs so that in a Fair Isle sweater (Fig.

Fig. 2. SaratMalaCakma,weaving(Bangladesh), cotton, approximately50 x 190 cm, 1986. The weaving is a blendof traditional and modern designs inwhich the weaver weaves pictures from her

village life into the strip symmetries. Sheuses a simple backstraploom, fixing one

end to a tree and maintainingtensionat the other end by leaning into a beltattached to the loom. She raises groupsof warp threads by means of short sticks,thereby allowing the shuttle to passunderneath the threads.

3) the horizontal lines of pattern areunbroken (because it is knitted as a

cylinder, or rather a helix) and in anAran sweater the vertical lines are ar-

Hams, Symmetryand Dissymetry n Mathematics Education 217





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ranged symmetricallyon each side of asymmetrical central panel. There are a

number of mathematical themes beingworked on here. I leave the readerto ponder the reasons that so manyknitted designs are worked on an odd

number of stitches. Modern domestic

knitting machines are making these

patterns now, as the brains of skilled

knitters are 'picked' and programmingof the machines becomes more so-

phisticated. The woman who made the

designs based on the Y shape (Fig. 4)told me that her machines can cope

with reflections but not rotations. Shealso told me that she was no good at

mathematics.

Once one has woven and decorated

a rectangle of cloth, one can make it

into a garment. There are two ways of

doing this, both adapted to the sym-metries of the human body. One can

take whole rectangles and drape them

in some way, or one can cut them into

pieces and sew the pieces together to

make a fitted garment. The ingenuityof garment makers working against

practical constraints is remarkable. A

sari drapes asymmetrically, but do weever stop to read its border patterns?How many ways can we find of ma-

nipulating one, two, three or fourwhole rectangles to make a poncho?How much extra cloth is needed when

pleating a kilt, so that the design on the

cloth is maintained on the pleated gar-ment?

Cutting up cloth to make garmentsinvolves a range of other problems

Fig. 3. Hand-

blocked Fair Islesweater in shadesof purple and

grey, donated to

the CommonThreadsexhibition by theShetland FairisleKnitwearAssocia-tion. In a tradition-

al hand-knitsweater made inone cylindrical(helical) piece, thedesigns workedinto the strip sym-metries form con-tinuous bands thatare worked on thecorrect number ofstitches for the gar-ment to fit thewearer.

concerned with maintaining the two-di-mensional symmetry of the pattern on

the cloth while clothing the three-di-

mensional symmetry of the body. Take

a shirt, for example, or a woman's

blouse. Designing the shape of the tem-

plate pieces (called pattern pieces by

garment makers) is a science in itself.Our arms maybe symmetricallyplaced,but they are not simple cylinders; for

example, the shoulder is rounder at theback than at the front of the body.Placing the template pieces on the

cloth presents a further set of problems

more sophisticated than the well-known mathematical problem of plac-ing templates most economically on

sheet metal. Sheet metal does not have

the grain of warp and weft threads,neither does it have pile like velvet andother textured cloth, nor does it tend

to be printed with daisies with theirstalksrunning one way.Most people do

not want the stripes on their shirts run-ning verticallyup one side of the front,horizontally across the other side and

spiralling around the sleeves.

I am often amazed by the number

of mathematics teachers who havedonned a shirt or T-shirt every day for

the past 30 yearsyet who think that the

head emerges symmetrically from theshoulders. They seem to assume that

the backbone goes up the middle of the

body andjoins the head centrally at itssouth pole. As its name suggests, it is abackbone, and most of the body andhead lie forward of it. The neck, in fact,starts ower in the front of the body than

in the back, so that the hole left for itin the garment is not circular butcurved asymmetrically.The design and

construction of neckties caters to this ina quite sophisticated way.

Socks are cylindrical and interestingmathematically. In industry there are

cylindrical knitting machines that canbe taught to put in heels. A toe is simplya heel sewn the other way. Socks

squashed flat provide a splendidly fa-miliar

wayof

demonstratingfrieze

pat-terns to children (Fig. 5). Feet and legsare symmetricalabout one verticalaxis;the art of putting a right-angled bendinto a cylinder of knitting-so that the

shaping is symmetrical about the heeland the insulating single layer of sockfits neatly-has exercised the ingenuityof hand knitters throughout the worldfor centuries. Paul Cochrane, a sculp-tor turned knitter, has researched 30

solutions, of which 12 were knitted forand graphed in the exhibition. Readerswho do not think that this is a mathe-

maticalproblem

areinvited to take asheet of A4 2-mmsquared graph paper,

roll it into a tube, form a right-angledbend in the tube and devise a methodfor completing the surface.

Industry has been looking at new

ways of making garments that can ex-

ploit the capabilities of programmedmachines and can eliminate the labourof sewing pieces together. If a cylindri-cal knitting machine can make a heel,then it can make a shoulder. Can it alsobe taught to make a whole, sleevedsweater-three cylindersjoined at the

shoulders by 'sock heels'? With a cylin-drical machine one can in theoryspecify a diameter and (using fusible

thread) knit an artificial artery, fuellines for an aircraft, perhaps even achannel tunnel. What is aglove afterall,but a five-junction manifold byanothername?

Real skill is required also for design-ing three-dimensional garments onflat-bed (two-dimensional) machines.

Byholding stitches (rather than fasten-

ing them off) and picking them upagain later, one can manipulate the

geometryin

almost any way. Byknittingsets of right-angled triangles so that the

hypotenuse of one triangle isjoined tothe long side of the next, one can knitbrake pads, skirts or whatever is

desired, using flexible discs or cones

(this cannot be done in metal, of

course, but in knitting one can take upthe slack if the difference in the lengthof the sides is not too great).

In other words we can print out inthree dimensions once we admit that

218 Harris,Symmetry and Dissymetry n Mathematics Education





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the knitting machine is in fact a printer.Schools generally have the software but

not yet the hardware for this. We can

study and generate transformations on

the microcomputer and print them out

on paper or on a knitting machine;

given enough memory on the ma-

chines, we can shape the garment on

which to knit our own design at the

same time. I look forward to the time

when we can teach our Logo software

to 'triangle', teach our triangle to 'tetra-

hedron' and then print out tetrahedra,

fully fashioned.

It is difficult to gauge the effect

of Common Threads by traditional

measures. It was seen by about 10,000children and probably about 1,000teachers and other adults. It has been

booked to capacity wherever it has

gone, and currently it is being re-

designed for a world tour to include

countries in Europe, east and southern

Africa, India, Pakistan, Malaysia,Thai-

land, Singapore, Australia and New

Zealand.The enthusiasm it generates is

remarkable, particularlywhen we note

that the exhibition is about mathemat-

ics and even has the dread word in its

subtitle. Teachers go to it for a number

of reasons: because they alwaysknew of

the mathematical nature of textiles and

a public exhibition supports and rein-

forces that knowledge; because it offers

inviting, fresh and stimulating yet famil-

iar materials for revitalising the school

curriculum; because it is good to be

able to go on a mathematics uting for

once. When the exhibition came back

to my office for repair between visits, I

used to clean the small finger marks off

the mirrors and mend and refresh the

cloth, so I know that it has been

handled in the way that was intended.

Cloth is very friendly and invites the

sort of manipulation of plane surfaces

that many would like to see in mathe-

matics lessons.

matical, and the children, when de-

monstrating it, talk very capably about

its construction and about the need for

accuracy. In this sort of practical work,children's mathematical ideas grow as

they argue and discuss and learn to see

how things work and how they do not.

Teachers talkof the fears of mathemat-

ics being removed, of increases in the

children's confidence and motivation,of their pride in their work and their

willingness to talk about it-thingsoften missing from a mathematics class.Above all, such activities confirm child-

ren's own experiences in their homes

and in their livesoutside school, validat-

ing ordinary, everyday activities with

the respect due to them [11]. Mathe-

Fig. 4. KathrynHobbs, machine-knitted demonstration samples for the Common Threads

exhibition, brown on green wool-blend yarn. Hobbs's machine can reflect motifs about

horizontal and vertical lines, the mathematician's x and yaxes, but cannot rotate them.Nonetheless, complex designs, in which it is difficult to identify the original motif, can be

built up.

TEXTILES AND

MATHEMATICS IN SCHOOL

The effects of the exhibition in schoolshave also been remarkable. For ex-

ample, teachers have used patchworkas a way of reestablishing the value of

women's work and teaching the geom-

etry of polygons as required by the

school mathematics syllabus. Children

studying the symmetryand tessellation

of polygons thus have something beau-

tiful to show for their work; this is un-

usual in a mathematics course [10]. A

class patchwork is undeniably mathe-

Harris,Symmetryand Dissymetry n Mathematics Education 19~~~~~219





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matics is theirs. It is not the private

property of institutions.

The basis of the mathematics cur-

riculum is being challenged in a waythat is well illuminated by Brown's

[12] discussion of the work of Carol

Gilligan [ 13] on moral decision makingand the fresh perspective it can

throw on the problem-solving processin mathematics. The problem-solving

styles of mathematics tend to ignoremost 'non-mathematics education ter-

rain' that could inform them. Very

briefly, Gilligan's work challenges the

accepted scales of moral developmentestablished by Kohlberg [14]. The lev-

els on Kohlberg's scales were estab-

lished by analysisof reasons offered by

young people of different ages for mak-

ing decisions in various moral dilem-

maspresented to them. Inworkingwith

the same problems, Gilligan noticed

consistencies of response amongst her

female subjects, who on the Kohlberg

hierarchy seem to exhibit arrested

developmentand

logical inferiority.Referring to the original research, Gil-

ligan found that Kohlberg's sample was

solely male. What Gilligan suggests is

not only that the choices of men and

women represent differences in psy-

chological dynamics but that the Kohl-

berg scales are deficient in the moral

categories that women's responses de-

scribe and that these scales are urgentlyin need of them. The female responseis as systematic as Kohlberg's but or-

thogonal to it. Kohlberg's hierarchy is

based on the concepts of justice and

rights, Gilligan's on caring and re-sponsibility. Ingeneral, and oversimpli-

fying Gilligan'sviewfor the sakeof brev-

ity, the women tend to recontextualise

the problem, to seek outside it for

further information and to ask, for ex-

ample, how and why the moral

dilemma in question could have arisen

in the first place, and finally to seek

discussion and reconciliation in resolv-

ing it. The male response is to acceptthe problem and to look for rules, some

legalistic formulae, some decree from

authority, to apply to it. For the males,

the problems are mathematics prob-lems with humans attached. For the

women the problems are human prob-lems with mathematics as relevant.

In the standard leaky-jug model

of school mathematics and in the 'un-

folding' of the curriculum,children are

supposed to see similarity rather than

difference with past experience.'Noise' is ruled out, numbers are sim-

plified and invented, value-free ex-

amples masquerade as daily problems.

School mathematics is hierarchical,

science oriented, male dominated and

limited in vision. In the words of Brown,it offers little encouragement for stu-

dents to move beyond merely accep-

ting the non-purposeful tasks [15].The mathematics curriculum follows

Kohlberg's perspective. Not only does

it not support Gilligan's perspective, it

ignores half the world'spopulation and

its attendant mathematical concerns.

Mathematics is not just a search for

what is common among apparentlydifferent structures; it also reveals

differences among those that appear to

be similar. The school mathematics

curriculum presents a perfect exampleof asymmetryin its relations with itself,with the children who study it and with

the outside world. The girls making the

patchwork mentioned previously were

behaving in a Gilliganian way. Their

teacher saw to it that the Kohlbergian

syllabus was covered, and the girls

emerged with an increased under-

standing,not

onlyof the abstract math-

ematical concepts involved but also of

the significance of those concepts to

life outside the classroom and in ordi-

narypartsof their lives. In the words of

one of the teachers, the children had

made the mathematics their own [16].It is interesting to take a school

mathematics topic and follow its career.

Take the humble rectangle for ex-

ample. First we learn that it has squarecorners, that it has four of them and

four sides. Then we learn that its op-

posite sides are equal and parallel and

we learn how to spell parallel. Then welearn that its diagonals are equal in

length and bisect each other. We tessel-

late them and later transform them,

and that really is the end of the rec-

tangle, because we then move on to

more sophisticated polygons, general

polygons and so on. Most of this is done

in the abstract because mathematics

is an abstractsubject. One daywe hap-

pen to see a carpenter cross two equal

lengths of wood to test whether a door

frame is true, and we are astonished,

partly because we somehow never ex-

pected school mathematics to be prac-tical and partlybecause afterour school

experience we do not really expectto find a humble artisan behaving so

mathematically.As soon as children enter school, the

process of detaching them from their

roots begins. All too soon skills are

taught in the abstract and applied to

invented 'mathematical' situations that

often bear little relationship to the chil-

dren's experience. In primary school,

teachers are often taught to teach un-

derlying ideas of symmetrythrough the

use of activities prepared for the pur-

pose; the ideas are imposed throughspecific school activities with folded pa-

per and paints rather than drawnout of

familiarhome activitiesthathave mean-

ing for the children. In spite of ourcommon sense, we teach symmetryby

special apparatus, as if assuming that

our pupils have never made patternswith footprints in the sand or shadows

with their hands, or played with reflec-tions in a puddle; that they have never

noticed what they are wearing or what

their birthday present comes wrapped

up in; that they have never lain awake

reading the patterns on the bedroom

curtains.

By secondary school, the banishingof home and outside world iscomplete,so that the myth that mathematics is

something that is applied to the world,rather than derived from it, can be

firmly instilled. Symmetry becomes

partof lessons in which we take trian-

gles and put them around squared

paper, learning how to locate and label

the new positions of their vertices and

how they got there with various arcane

symbols that have to be stored in the

leaky jug, for they will rarely be used

again or for anything else. Mathematics

becomes subservient to the tyrannyof

the page.

My colleague Dietmar Kfichemann,as a member of the Concepts in Second-

ary Mathematics and Science Projectconducted in England from 1974

through 1979, asked children a seriesof questions about reflection and rota-

tion. Of the sample of 14-year-olds,

nearly all showed some understandingof reflection, but their performance de-

pended heavilyon features such as the

presence or absence of a grid, the slopeof the mirror line or the object and the

complexity of the object. A common

error was to ignore the slope of the

mirror and simply reflect the imagedown the page. The research showed

too that many children's difficulties

with such single transformations pre-

empt them from even considering theircombinations. By this stage of their

careers, of course, children knowwhen

they are doing mathematics tests, those

tests that so often measure only them-

selves. Mathematics tests often require

only that one push symbols around

and remember the approved way of

doing so.

As Kfichemann points out, the intro-

duction of transformation geometryinto the curriculum provides a per-

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Fig. 6. Dwennimen. A ram'shead. Designfor Adinkra cloth of the Ashanti people of

Ghana. The designs are carved on piecesof dried gourd, painted with dye and used

to print on the cloth. Each of the Adinkra

designs has its own symbolic meaning, and

many are symmetrical. Dwennimen is the

symbol of humilityand strength, wisdom

and learning, and forms part of the armsof the University of Ghana.

Fig.7. Duafe. A wooden comb. Adinkra

design. The comb, with its outstretched

arms, is the symbol of the wisdom,

patience and concern of women.

fect illustration of the hopes and dis-

appointments of those who have at-

tempted to revitalise the content and

teaching methods of secondary school

maths over the previous 20 years [17].Transformation geometry was in-

tended to replace the teaching of Eu-

clid's geometry, a tight deductive

system that most children could not

master except by rote, in the hope that

children would discover general rulesabout transformations that would lead

them eventually to the structure of

groups. Since this end is not realised on

most syllabi until after the age when

most children give up mathematics,

teachers and children were left workingtowards goals that were confused and

unclear. Thus one area of high-status

knowledge was superseded by another,

equally unsuitable.

This state of affairs reflectsthe pro-

fessional interests of mathematicians,

who particularly through the examina-

tion system still exert a strong control

over curriculum reform. Unfortunatelythese interests do not necessarily coin-

cide with those of the children who

study the subject [18]. The children

return eventually to their pattern-filledworld not only uninterested but unable

any more to notice the pattern on their

jumpers or the bedroom curtains, let

alone to ask how and why it got there.

Manyleave school in the situation Kap-praff [19] describes, capable of under-

standing complex ideas and applyingthem to practical problems but unable

to manipulate mathematical symbolsor

follow the narrow paths of mathemati-

cal argument. I cannot but agree with

Kuchemann that it is pointless to teach

transformations in a didactic expo-

sitory manner particularly when it is

the ideal subject for teaching in a prac-

tical investigative manner [20]. The

most accessible regularity in things and

phenomena is their symmetry [21] as

Mamedov states. We must develop this

fundamental idea in school, in its own

right, yet we successfully manage to

blind our children to it. It is, after all,

fundamental to human functioning. I

have a copy of a photograph taken

by war photographer Don McCullin in

1982 after the first bombings in Beirut.

It shows a small boy among the ruins of

a mental hospital. The dust had settled,

and there was nothing left except a few

rags and bits of broken crockery. The

boy took some cloth and some bits

of china and set about reestablishingorder in his insane world. The photo-

graph shows him contemplating the

ordered arrayhe has made in the form

of a sine wave.

Meanwhile in England we now have

a new national curriculum, forced

through with haste by a government

whose view of education is determined

by costs and not values. Symmetry is

included in the curriculum, but in the

fragmented way that a document con-

cerned with hierarchies of particularskills demands, indeed ensures. The

fact that children will be tested regu-larly in these skills from the age of seven

will also ensure the fragmentation of a

topic whose concern is with wholeness.

The public fear and ignorance of math-

ematics and the lack of people to teach

it is now a national disaster. As a wayof trying to correct the situation, an

international professional conference

on the popularising of mathematics

took place in England in September1989. True to the nature of the profes-

sion, though, the event was by invita-

tion only. Thus, out of so much poten-

tial for change one foot has taken a step

backwards and the other remains stuck

in its concrete boot.

SCHOOL MATHEMATICS,

SYMMETRY AND SOCIETY

The same Middle Eastern inferno that

trapped the small boy in Beirut caused

one mathematics educator of highqualification and commitment and rich

experience to start asking what mathe-

matics education should really accom-

plish. In a paper read to the Sixth Inter-

national Congress on Mathematics

Education held in Budapest (1988)

Fasheh [22] noted that it took the war

to reveal how little we, the formally

educated, knew. The New Mathemat-

ics that he had been introducing to

West Bank schools was alien, dry and

abstract. He had revitalised it by bring-

ing in cultural concepts and a wide

range of enlivening activities. Whathe had not done, however, was to ques-

tion the hegemonic assumptions be-

hind the mathematics itself. To quote

Fasheh:

While I was using math to help em-

power other people, it was not em-

powering for me. It was, however, for

my mother, whose theoretical aware-ness of math was completely un-

developed. Mathematics wasnecessaryfor her in a much more profound andreal sense than it was for me. Myilliterate mother routinely took rectan-

gles of fabric and with few measure-

ments and no patterns cut them andturned them into beautiful, perfectlyfitted clothing for people. In 1976 itstruck me that the math she wasusingwasbeyond mycomprehension; more-over, while math for me was a subjectmatter I studied and taught, for herit was basic to the operation of her

understanding. In addition mistakesin her work entailed practical con-

sequences completely different frommistakes in my math.

Moreover the values of her math andits relationship to the world aroundher was drastically different frommine. Mymath had no power connec-

tion with anything in the communityand no power connection with theWesternhegemonic culture which had

engendered it. It was connected solelyto symbolicpower.Without the official

ideological support system, no onewould have 'needed' mymath; itsvaluewas derived from a set of symbols cre-ated by hegemony and the world ofeducation. In contrast, my mother'smath was so deeply embedded in theculture that it was invisible througheyes trained by formal education....

What kept her craft from being fully a

222 Hamrs,Symmetry and Dissymetry n Mathematics Education





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praxis and limited her empowermentwas a social context which discreditedher as a woman and uneducated and

paid her extremely poorly for herwork. Like most of us she never under-stood that social context and wasvulnerable to itshegemonic assertions.She never wanted any of her childrento learn her profession; instead, sheand my father worked veryhard to seethat we were educated and did notwork with our hands. In the face of this,it was a shock to me to realize the

complexity and richness of mymother's relationship to mathematics.Mathematics was integrated into herworld as it never was into mine.

When I first started looking at tex-

tiles as a resource for mathematics

there were two important social issues

that, like any worker in mathematics

education, I took into account. These

were the issues of low attainment

among girls and low attainment among

some ethnic minority groups. In the

course of the year I met a large number

of people from many cultures verylike Fasheh's mother. By about halfway

through the year I knew I was wrongabout the two issues. There was in fact

one issue: why mathematics is so nar-

row-minded. I have hinted at some so-

cial and political reasons for this and

will not dwell on them more. I will how-

ever continue to bewail the results.

What my study of symmetry revealed to

me is a gross asymmetry within the very

subject that claims to own it. Not only is

there a lack of balance within mathe-

matics and mathematics education but

there is a lack of balance in their rela-

tions with the outside world.

Mathematics is but one set of strandsin a seamless robe and the seamless

robe itself is no mean metaphor. A robe

warms, embraces, nurtures and en-

hances. When we enter the world we are

wrapped in a shawl. When we leave it we

are wrapped in a shroud and one verysmall wheel turns. The hands that knit

the shawl and weave the shroud are the

hands that rear the children, make the

home and provide the protective care

without which independent thinking

and the experience of knowing cannot

grow.

In Ghana, people who make Adinkra

cloth carve designs on small pieces of

calabash, which they then use for print-

ing patterns on their cloth. Each design

has a name, a symbolism and a sym-

metry. The design shown in Fig. 6, a

ram's head, has two axes and rotational

symmetry, and it combines four ideas

in a refreshingly non-Western but

nevertheless closed form: strength

and wisdom, learning and humility.Another design (Fig. 7), a wooden

comb, is the symbol of the wisdom,

patience and concern of women. It has

bilateral symmetry-and is infinitely ex-

tensible.

APPENDIX

Note on SymmetryTermsIn mathematical terms, patterns are

produced when a motif is repeated

systematically. If a shape is moved to

another position along a straight line

with no change in its shape or size, then

that shape is said to undergo isometric

transformation. The only possible iso-

metric transformations are translation

(along a line), reflection (about an axis)or rotation (about a point). Transforma-

tions can be combined; for example,the combination of reflection and

translation along the axis of reflection

is called a glide. The acts of moving

shapes in these ways are known as

symmetry operations. There are only 15

ways of combining transformations

if these operations take place in one

dimension. These are usually com-

bined into what are widely known as the

seven frieze or strip patterns, demon-

strated by the baby socks in the Com-

mon Threads exhibition. Transforma-

tions in two directions are called the

plane symmetries because they cover a

plane surface.

References and Notes

1. Stephen I. Brown, The Logic of ProblemGeneration: From Morality and Solving to De-

posing and Rebellion , For heLearningofMathemat-ics4, No. 1 (February1984).

2. HilaryShuardmade the remarkabout leakyjugsin a BBC television programme in the Horizonseries called TwiceFive Plus the Wings of a Bird .The programme, a review of mathematics educa-tion, was firstbroadcast on 28 April 1986.

3. Mary Harris, Common Threads , Mathematics

Teaching 23 (June 1988).

4. E. H. Gombrich, The Sense of Order(Oxford:Phaidon, 1984).

5. Dorothy K. Washburn, PatternSymmetryandColoured Repetition in Cultural Contexts , Com-

putersand Mathematicswith Applications123, No.

3/4,767-781 (1986).6. I. Hargittai and G. Lengyel, The Seven OneDimension Space Group Symmetries Illtustrated yHungarian Folk Needlework , Journal of ChemicalEducation61 (1984) pp. 1033-1034; The 17 2-DSpace Group SymmetriesIllustratedby HungarianFolk Needlework ,Journal of Chemical ducation62

(1985) pp. 35-36.

7. Claudia Zaslavsky,AfricaCounts(Westport, CT:LawrenceHill, 1973).

8. F. Allen Hanson, When the Map is the Terri-

tory: Art in Maori Culture , in Dorothy K. Wash-burn, Structure nd Cognition n Art (Cambridge:Cambridge Univ. Press, 1983).

9. Kh. S. Mamedov, Crystallographic Patterns

ComputersndMathematics ithApplications2B,No.3/4,511-529 (1986).

10. MaryHarris, ed., Textiles n MathematicsTeach-

ing. Maths in Work London: Maths in Work, Uni-

versityof London Inst. of Education, 1989).

11. Harris [10].

12. Brown [1] pp. 11-14.

13. Carol Gilligan, In a DifferentVoice:PsychologicalTheory nd Women'sDevelopmentCambridge, MA:Harvard Univ. Press, 1982).

14. Lawrence Kohlberg, MoralStages in Motiva-tion: The Cognitive Developmental Approach , inThomas Lickona, ed., MoralDevelopment nd Be-havior:Theory,Research nd Social ssues(New York:Holt, Rinehart and Winston, 1976).

15. Brown [1] pp. 12-13.

16. Harris [10] p. 19.

17. Dietmar Kuichemann, Reflections and Rota-tions ,in K. M.Hart, ed., ChildrensUnderstanding fMathematics 1-16 (London:John Mturray, 981),p. 137.

18. Kfichemann [17] p. 138.

19. Jay Kappraff, ACourse in Mathematics of De-

sign , Computersnd Mathematicswith Applications12B, No. 3/4, 913-948 (1986).

20. Kiichemann [17] p. 157.

21. Mamedov [9] p. 514.

22. Mtinir Fasheh, Mathematics n a Social Con-text: Mathwith Education as Praxis vs. within Ediu-cation as Hegemony , paper presented at the SixthInternational Congress on Mathematical Educa-tion, Btudapest,1988.

Harris,Symmetryand Dissymetry n Mathematics Education 223

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