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 PressStable URL: http://www.jstor.org/stable/1578608 .

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PROGRAMS FOR THE COMING ERA

Symmetry and Dissymmetry in Mathematics Education:

One View from England

Mary Hams

I 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 practice of the cognitive growth of individuals learning mathematics. Since mathematics is aligned with science and education is aligned with the humanities, the whole profession, in the 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 the mathematics 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 many students leave school at age 16 without either the skills or the interest in mathematics that the academic aspects of the course 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 laid down 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 the workers 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 something that goes on in our heads; we all have one of those, and no outsider has the right to try to 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 is why those of us in the trade sometimes use Hilary Shuard's apt phrase 'leakyjug' [2] instead. The latter is much espoused by the present British government, and

1990 ISAST Pergamon Press plc. Printed in Great Britain. 0024-094X/90 $3.00+0.00

there is much talk of skills and applications; we are expected to 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 what goes on in work situations 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

Mathematics education is a field that tries to reconcile the con- flicting demands of different sec- tors of education and society while maintaining the best climate for the intellectual growth of all children. An investigation by the Maths in Work Project of the Department of Mathematics, Statistics and Com- puting of the University of London Institute of Education into the math- ematical nature and uses of cloth confirmed a deep bias in mathemat- ics education against traditional women's concerns. The project pro- duced learning resources that use textiles to teach mathematics, but its major impact arose from an exhi- bition called Common Threads, which toured England from 1987- 1990 and will subsequently tour the world under the auspices of the British Council.

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 analyse the mathematical thinking in what they are doing. Many of these 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 they have 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 involve children in practical tasks from which teachers draw out the mathematics at the level that is appropriate for the children.

A surprising obstacle to fresh developments in my field is the masculine nature of mathematics-surprising because in this abstract and highly rational subject there is no rational basis for it. There are two aspects of this masculinity, one within the practice of the school subject and one within the organisation of the profession itself. Within the subject, textbook writers and many teachers assume that they are addressing boys only, and many of the examples are taken from sport, war and science. This is now well documented, and there is considerable activity around the world to make mathematics more 'user-friendly' to girls. The second aspect

Mary Harris (mathematics educator), Maths in Work Project, Department of Mathematics, Statistics and Computing, University of London Institute of Education, 28 Woburn Square, London WC1H OAA, United Kingdom.

Received 9June 1989.

Based on a paper titled "Art and Skill of Symmetry in Old and Modern Textile Techniques" read at the interdisciplinary symposium "Symmetry of Structure" held in Budapest, Hungary, August 1989. The symposium marked the inauguration of the International Society for the Interdisciplinary Study of Symmetry (ISISS).

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

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is just as intractable. Like all pro- fessions, mathematics and mathematics education are ruled by a clearly defined and somewhat defensive hierarchy with the usual power groups and pecking orders. Both of these features have neg- ative effects on the work of women and on women themselves. Because the field 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 textiles as a resource for learning mathematics were 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, approximately 90 x 140 cm, modern. "A veri- table symphony of symmetry." Each small motif has its own internal symmetry. These symmetries are

talkingE towoven into the total symmetries of the borders and centre panel of the rug. Such rugs

the .............n are made by girls and women with little or no formal education, working from memory. The incomplete design in one corner offers a clue as to its construction.

and I wanted to develop materials on textiles 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 and my family's clothes, I had had some personal experience with textiles; by talking to other women who do the same thing, and by visiting textile exhi- bitions, ethnographic museums and factories where people make ties, socks and shirts, I was able to add an in- dustrial dimension. My office filled with interesting pieces of cloth as I began to demonstrate 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 learning materials.

Preparing a learning resource to hang on the wall is vastly different from preparing 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 show the mathematical nature of cloth, so ordinary and widespread a part of the environment that it is taken for granted and certainly normally ignored as a mathematics resource. For the general public my goal was to show that there is more to mathematics than arithmetic, and that many people are doing some- thing mathematical much of the time, whether or not they choose to see it so. From the mathematics education point of view, I needed to handle at least part of the exhibition in enough depth for the mathematicians in the education hierarchy to take it seriously, and I also needed to handle it in a developmental way that would enable teachers to in- corporate textile work into their teach- ing at a number of levels. And finally I aimed to demonstrate and demand re- spect for the large number of women throughout the world who clothe their families and thereby introduce their children daily to geometry. Thus I needed a theme that ran horizontally through mathematics and the whole of life but that would not be dominated by the mathematics. It also had to be a theme that could be handled verti- cally to demonstrate, and keep acces- sible and inviting, the special nature of mathematical thinking at all levels. I cannot remember when the theme of symmetry emerged, because it did not come in a blinding flash. Itjust slid into place as the obvious unifying force in what I was trying to do, together with a title for the exhibition-Common Threads: Mathematics and Textiles. To balance the theme of symmetry and to ensure that the school syllabus could be covered adequately, I chose four other underlying themes of mathematics: Number, Creativity, Information Han- dling, and Problem Solving; of course all five themes overlap.

Basically there are two ways of mak- ing cloth: interlacing several threads or looping a single one. Both of these are determined by the form in which natu- ral fibres appear: hair of animals or ribs of leaves. Both weaving on a loom and knitting on two or more needles tend to produce rectangles unless one does something to alter the shape. While rectangles have their own symmetries, the very act of structuring cloth has symmetry built into it: for the maker of cloth rectangles, the physical act of manipulating the threads with two hands causes the bilaterally symmetri- cal human body to interact with the fibres.

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By deliberately manipulating colour, texture and grouping of threads, cloth workers play with 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 modifying texture 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 symmetry in 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 piece of cross-stitch embroidery done by a Yugoslav woman on a piece of factory- woven cloth. There are several possible ways 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 by embroidering x and y axes on it.

In counted-thread work, this is often how people proceed, though they do not normally embroider their axes of reflection into the cloth itself. They use the grain of the cloth formed by the warp and weft threads as 'mirror lines' for their symmetrical designs. For many people 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 academic box and treating it so lightly. Others were amused. Many teachers, students and members of the public told me that this was the first time they had looked at a design analytically. Yet the world is full of cross-stitch embroidery made by both amateur and professional mathe- maticians, although people are sur- prised to hear that anyone should think of embroiderers as mathematicians at all. Hungary is particularly rich in cross- stitch embroidery; indeed, Hargittai and 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 very bright. Examples of such art forms would be raffia cloths from Zaire [7] and Maori rafters [8].

Designs on cloth may be embroi- dered on or they may be woven or knitted in. A rug from Turkey (Fig. 1) is a veritable symphony of symmetry. Unlike the mathematician working on paper, the illiterate weaver has to work within a frame defined by a specific number of warp threads, the cost of which, in human and economic terms, she well knows. Within this frame she works symmetrical designs ('crystallo- graphic patterns' [9]) within a sym- metrical centre panel, surrounded by a symmetrical border containing sym- metrical designs. Unlike mathema- ticians working on paper, the weavers cannot rub their designs out and start again-they cannot cut bits off one end and stick them on the side. Neither do they unpick and start again; this would be uneconomical in time and materials. Too much unpicking weak- ens the woolen threads that were so laboriously sheared, cleaned, spun, dyed and woven. The full conception is in the weaver's head before she starts. People who have studied these rugs and their construction liken the work of the weavers to that of conducting a sym- phony orchestra without a score.

The weaving from Bangladesh shown in Fig. 2 is quite different. Here the designs of the frieze patterns are social comments; fish from the Ganges, the flower that gets into the rice crop, footprints of birds and animals, the pattern on an imported biscuit. The weaver Sarat Mala Cakma's eye is tuned humourously and affectionately to the fine detail of her world. She works her threads in pairs until she gets bored with the groups of even numbers and then plays with groups of five or seven instead.

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 being set. After all, a sweater has to fit a per- son, and that means working within a prescribed number of stitches. Within that number the knitter works her de- signs so that in a Fair Isle sweater (Fig.

Fig. 2. Sarat Mala Cakma, weaving (Bangladesh), cotton, approximately 50 x 190 cm, 1986. The weaving is a blend of traditional and modern designs in which the weaver weaves pictures from her village life into the strip symmetries. She uses a simple backstrap loom, fixing one end to a tree and maintaining tension at the other end by leaning into a belt attached to the loom. She raises groups of warp threads by means of short sticks, thereby allowing the shuttle to pass underneath the threads.

3) the horizontal lines of pattern are unbroken (because it is knitted as a cylinder, or rather a helix) and in an Aran sweater the vertical lines are ar-

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ranged symmetrically on each side of a symmetrical central panel. There are a number of mathematical themes being worked on here. I leave the reader to ponder the reasons that so many knitted 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 programming of 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. She also 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 ingenuity of garment makers working against practical constraints is remarkable. A sari drapes asymmetrically, but do we ever stop to read its border patterns? How many ways can we find of ma- nipulating one, two, three or four whole 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 garments involves a range of other problems

Fig. 3. Hand- blocked Fair Isle sweater in shades of purple and grey, donated to the Common Threads exhibition by the Shetland Fairisle Knitwear Associa- tion. In a tradition- al hand-knit sweater made in one cylindrical (helical) piece, the designs worked into the strip sym- metries form con- tinuous bands that are worked on the correct number of stitches for the gar- ment to fit the wearer.

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 may be symmetrically placed, but they are not simple cylinders; for example, the shoulder is rounder at the back 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 and other textured cloth, nor does it tend to be printed with daisies with their stalks running one way. Most people do not want the stripes on their shirts run- ning vertically up 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 have donned a shirt or T-shirt every day for the past 30 years yet who think that the head emerges symmetrically from the shoulders. They seem to assume that the backbone goes up the middle of the body and joins the head centrally at its south pole. As its name suggests, it is a back bone, and most of the body and head lie forward of it. The neck, in fact, starts lower in the front of the body than

in the back, so that the hole left for it in the garment is not circular but curved asymmetrically. The design and construction of neckties caters to this in a quite sophisticated way.

Socks are cylindrical and interesting mathematically. In industry there are cylindrical knitting machines that can be taught to put in heels. A toe is simply a heel sewn the other way. Socks squashed flat provide a splendidly fa- miliar way of demonstrating frieze pat- terns to children (Fig. 5). Feet and legs are symmetrical about one vertical axis; the art of putting a right-angled bend into a cylinder of knitting-so that the shaping is symmetrical about the heel and the insulating single layer of sock fits neatly-has exercised the ingenuity of hand knitters throughout the world for centuries. Paul Cochrane, a sculp- tor turned knitter, has researched 30 solutions, of which 12 were knitted for and graphed in the exhibition. Readers who do not think that this is a mathe- matical problem are invited to take a sheet of A4 2-mm squared graph paper, roll it into a tube, form a right-angled bend in the tube and devise a method for completing the surface.

Industry has been looking at new ways of making garments that can ex- ploit the capabilities of programmed machines and can eliminate the labour of sewing pieces together. If a cylindri- cal knitting machine can make a heel, then it can make a shoulder. Can it also be taught to make a whole, sleeved sweater-three cylinders joined at the shoulders by 'sock heels'? With a cylin- drical machine one can in theory specify a diameter and (using fusible thread) knit an artificial artery, fuel lines for an aircraft, perhaps even a channel tunnel. What is a glove after all, but a five-junction manifold by another name?

Real skill is required also for design- ing three-dimensional garments on flat-bed (two-dimensional) machines. By holding stitches (rather than fasten- ing them off) and picking them up again later, one can manipulate the geometry in almost any way. By knitting sets of right-angled triangles so that the hypotenuse of one triangle is joined to the long side of the next, one can knit brake 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 up the slack if the difference in the length of the sides is not too great).

In other words we can print out in three dimensions once we admit that

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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 10,000 children and probably about 1,000 teachers 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, particularly when 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 always knew 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 outing 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 talk of 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-things often missing from a mathematics class. Above all, such activities confirm child- ren's own experiences in their homes and in their lives outside school, validat- ing ordinary, everyday activities with the respect due to them [11]. Mathe-

Fig. 4. Kathryn Hobbs, 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 y axes, 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 schools have also been remarkable. For ex- ample, teachers have used patchwork as 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 symmetry and 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-

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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 way that is well illuminated by Brown's [12] discussion of the work of Carol Gilligan [ 13] on moral decision making and the fresh perspective it can throw on the problem-solving process in mathematics. The problem-solving styles of mathematics tend to ignore most 'non-mathematics education ter- rain' that could inform them. Very briefly, Gilligan's work challenges the accepted scales of moral development established by Kohlberg [14]. The lev- els on Kohlberg's scales were estab- lished by analysis of reasons offered by young people of different ages for mak- ing decisions in various moral dilem- mas presented to them. In working with the same problems, Gilligan noticed consistencies of response amongst her female subjects, who on the Kohlberg hierarchy seem to exhibit arrested development and 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 urgently in need of them. The female response is 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. In general, and oversimpli- fying Gilligan's view for the sake of 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 accept the 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's population and its attendant mathematical concerns. Mathematics is not just a search for what is common among apparently different structures; it also reveals differences among those that appear to be similar. The school mathematics curriculum presents a perfect example of asymmetry in 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 only of the abstract math- ematical concepts involved but also of the significance of those concepts to life outside the classroom and in ordi- nary parts of 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 square corners, 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 we learn 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 abstract subject. One day we 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 partly because after our school experience we do not really expect to 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 symmetry through the use of activities prepared for the pur- pose; the ideas are imposed through specific school activities with folded pa- per and paints rather than drawn out of familiar home activities that have mean-

ing for the children. In spite of our common sense, we teach symmetry by special apparatus, as if assuming that our pupils have never made patterns with 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 banishing of home and outside world is complete, 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 part of 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 tyranny of the page.

My colleague Dietmar Kfichemann, as a member of the Concepts in Second- ary Mathematics and Science Project conducted in England from 1974 through 1979, asked children a series of questions about reflection and rota- tion. Of the sample of 14-year-olds, nearly all showed some understanding of reflection, but their performance de- pended heavily on features such as the presence or absence of a grid, the slope of 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 image down the page. The research showed too that many children's difficulties with such single transformations pre- empt them from even considering their combinations. By this stage of their careers, of course, children know when 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 geometry into the curriculum provides "a per-

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Fig. 5. The seven frieze or strip patterns were demonstrated in the Common Threads exhi- bition by baby socks. These transformations are usually represented in mathematics texts by scalene triangles or other abstract symbols. Many visitors to Common Threads remarked on how powerful and help- ful to students such a homely representation is.

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Fig. 6. Dwennimen. A ram's head. Design for Adinkra cloth of the Ashanti people of Ghana. The designs are carved on pieces of 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 humility and strength, wisdom and learning, and forms part of the arms of 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 rules about 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 working towards goals that were confused and unclear. Thus one area of high-status knowledge was superseded by another,

equally unsuitable. This state of affairs "reflects the pro-

fessional interests of mathematicians, who particularly through the examina- tion system still exert a strong control over curriculum reform. Unfortunately these interests do not necessarily coin- cide with those of the children who study the subject" [18]. The children return eventually to their pattern-filled world 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. Many leave school in the situation Kap- praff [19] describes, capable of under- standing complex ideas and applying them to practical problems but unable to manipulate mathematical symbols or 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 reestablishing order in his insane world. The photo- graph shows him contemplating the ordered array he 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 particular skills 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 way of trying to correct the situation, an international professional conference on the popularising of mathematics took place in England in September 1989. 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 high qualification 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. What he 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 was necessary for her in a much more profound and real sense than it was for me. My illiterate mother routinely took rectan- gles of fabric and with few measure- ments and no patterns cut them and turned them into beautiful, perfectly fitted clothing for people. In 1976 it struck me that the math she was using was beyond my comprehension; more- over, while math for me was a subject matter I studied and taught, for her it was basic to the operation of her understanding. In addition mistakes in her work entailed practical con- sequences completely different from mistakes in my math.

Moreover the values of her math and its relationship to the world around her was drastically different from mine. My math had no power connec- tion with anything in the community and no power connection with the Western hegemonic culture which had engendered it. It was connected solely to symbolic power. Without the official ideological support system, no one would have 'needed' my math; its value was derived from a set of symbols cre- ated by hegemony and the world of education. In contrast, my mother's math was so deeply embedded in the culture that it was invisible through eyes trained by formal education....

What kept her craft from being fully a

222 Hamrs, Symmetry and Dissymetry in Mathematics Education

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praxis and limited her empowerment was a social context which discredited her as a woman and uneducated and paid her extremely poorly for her work. Like most of us she never under- stood that social context and was vulnerable to its hegemonic assertions. She never wanted any of her children to learn her profession; instead, she and my father worked very hard to see that we were educated and did not work with our hands. In the face of this, it was a shock to me to realize the complexity and richness of my mother's relationship to mathematics. Mathematics was integrated into her world 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 very like Fasheh's mother. By about halfway through the year I knew I was wrong about 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 strands in 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 very small 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 Symmetry Terms In 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 Problem Generation: From Morality and Solving to De-

posing and Rebellion", For the LearningofMathemat- ics 4, No. 1 (February 1984).

2. Hilary Shuard made the remark about leakyjugs in a BBC television programme in the "Horizon" series called "Twice Five Plus the Wings of a Bird". The programme, a review of mathematics educa- tion, was first broadcast on 28 April 1986.

3. Mary Harris, "Common Threads", Mathematics Teaching 123 (June 1988).

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

5. Dorothy K. Washburn, "Pattern Symmetry and Coloured Repetition in Cultural Contexts", Com- puters and Mathematics with Applications 123, No. 3/4,767-781 (1986).

6. I. Hargittai and G. Lengyel, "The Seven One Dimension Space Group Symmetries Illtustrated by Hungarian Folk Needlework", Journal of Chemical Education 61 (1984) pp. 1033-1034; "The 17 2-D Space Group Symmetries Illustrated by Hungarian Folk Needlework", Journal of Chemical Education 62 (1985) pp. 35-36.

7. Claudia Zaslavsky, Africa Counts (Westport, CT: Lawrence Hill, 1973).

8. F. Allen Hanson, "When the Map is the Terri- tory: Art in Maori Culture", in Dorothy K. Wash- burn, Structure and Cognition in Art (Cambridge: Cambridge Univ. Press, 1983).

9. Kh. S. Mamedov, "Crystallographic Patterns" Computers and Mathematics with Applications 12B, No. 3/4,511-529 (1986).

10. Mary Harris, ed., Textiles in Mathematics Teach- ing. Maths in Work (London: Maths in Work, Uni- versity of London Inst. of Education, 1989).

11. Harris [10].

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

13. Carol Gilligan, In a Different Voice: Psychological Theory and Women's Development (Cambridge, MA: Harvard Univ. Press, 1982).

14. Lawrence Kohlberg, "Moral Stages in Motiva- tion: The Cognitive Developmental Approach", in Thomas Lickona, ed., Moral Development and Be- havior: Theory, Research and Social Issues (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., Children 's Understanding of Mathematics 11-16 (London:John Mturray, 1981), p. 137.

18. Kfichemann [17] p. 138.

19. Jay Kappraff, "A Course in Mathematics of De- sign", Computers and Mathematics with Applications 12B, No. 3/4, 913-948 (1986).

20. Kiichemann [17] p. 157.

21. Mamedov [9] p. 514.

22. Mtinir Fasheh, "Mathematics in a Social Con- text: Math with Education as Praxis vs. within Ediu- cation as Hegemony", paper presented at the Sixth International Congress on Mathematical Educa- tion, Btudapest, 1988.

Harris, Symmetry and Dissymetry in Mathematics Education 223

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