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In the last 20 years there has been a growing literature deal-ing with ethnomathematical issues, or about studiesinvolving culture and mathematics education. The growingcommunity of those interested in the field held the ThirdInternational Conference on Ethnomathematics (ICEm-3) inFebruary 2006, at the University of Auckland, New Zealand,convened by Bill Barton from the Department of Mathemat-ics. ICEm-1 was in Granada, Spain, in 1998, organized byMaria Luiza Oliveras and entitled Research, curriculumdevelopment, teacher education. ICEm-2 was in Ouro Preto,Brazil, in 2002, organized by Eduardo Sebastiani Ferreiraand entitled A methodology for ethnomathematics.

Here is an overview of ICEm-3, reflecting on the growthof the field and some key questions surrounding ethnomath-ematics and its significance for mathematics education. Didthe conference meet the participants’ expectations? Did any-thing new emerge? What are the challenges that now facethe ethnomathematical community? There are various direc-tions that can be taken for such an overview. We haveattempted to identify categories of interest, rather than par-ticular presentations – especially where these categorieshave shifted since the last conference. We have also lookedfor the questions that recurred during the conference, and thekey themes that emerged, both as a result of the way the con-ference was designed, and serendipitously.

We have used the ICEm-3 theme – Cultural connectionsand mathematical manipulations – to interpret the manyconference activities. We first present an analysis of partici-pation, and then discuss the cultural connections madeduring the conference. Next the mathematical themes thatemerged are presented, and finally the conference is viewedboth retrospectively and with an eye to the future.

The presentationsThere were 76 participants from 19 countries, with, newly,significant participation from Scandinavia, the Pacific andSouth Africa, as well as the usual large Brazilian contingent.During ICEm-3, 24 presentations were given. One third ofthese were from Asia or the Pacific, and the rest split evenlybetween North and South America, South Africa, and Europe.Six countries were represented in the plenary sessions:

1. Jerry Lipka, Dora Andrew, Evelyn Yanez (USA,Alaska): A two way process for developing cultur-ally based math: examples from math in a culturalcontext

2. Indigenous Knowledge Panel. Willy Alangui, Chair(The Philippines); Dora Andrew, Evelyn Yanez(USA, Alaska); Salinieta Bakalevu (Fiji); Colleen

McMurchy-Pilkington (New Zealand); Joel Mar-tim (Brazil); Mogege Mosimege (South Africa)

3. Mathematical Workshop. Hariata Adams, NgatiManiapoto (New Zealand): Raranga Harakeke(Flax Weaving)

4. Symposium in memory of Claudia Zaslavsky. Ubi-ratan D’Ambrosio, Chair (Brazil): The work ofClaudia Zaslavsky; Maria Do Carme Domite(Brazil): Indigenous intercultural program of edu-cation, elementary teacher undergraduatecertification; Kay Owens (Australia): Interna-tional contacts in ethnomathematics; LawrenceShirley (USA): Ethnomathematics in global edu-cation programs

5. Mathematical Workshop. Filipe Tohi (Tonga):Lalava (Rope Lashing)

6. Gelsa Knijnik (Brazil): Ethnomathematics and theBrazilian Landless Movement

7. Ubiratan D’Ambrosio (Brazil): The scenario 30years after

To try to get an overview of the presentations, we classifiedthem by the apparent preoccupation of the presenter, aswell as by the subject-matter concerned. Six preoccupa-tions emerged: mathematics learning; teacher education;politics; sociology/anthropology; philosophy/critique; andethnomathematical theory development. The subject-matterof presentations was divided into four categories: theknowledge of a specific cultural group; academic mathe-matical knowledge; classroom mathematical knowledge;and ethnomathematics as a field (see Figure 1).

CULTURAL CONNECTIONS AND MATHEMATICAL MANIPULATIONS

BILL BARTON, CAROLINE POISARD, MARIA DO CARMO DOMITE

For the Learning of Mathematics 26, 2 (July, 2006)FLM Publishing Association, Edmonton, Alberta, Canada

Figure 1: Table showing classification of presentations byauthor preoccupation and subject-matter.

Author preoccupation (%) Paper subject-matter (%)

ICEm- 2 3 ICEm- 2 3

Mathematics Learning

40 50 Specific GroupKnowledge

48 29

Teacher Education 12 8 Academic Mathe-matical Knowledge

7 4

Politics 10 8 Classroom Mathe-matical Knowledge

35 54

Sociology/Anthropology

24 13 Ethnomathematicsas a Field

10 13

Philosophy/Critique 7 8

EthnomathematicalTheory

7 13

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By far the majority of authors were preoccupied withmathematics education, although most were motivated bymore than one preoccupation. The subject-matter was mostoften classroom knowledge, but a significant proportiondealt with the knowledge of a specific group. The groupsrepresented in this work included: indigenous cultures fromthe Philippines, Mexico, Brazil, USA, and South Africa;national groups of Japan, China, Greece, and Israel; and thesocial groups of mathematicians and of nurses. Comparedwith ICEm-2, there has been a clear move towards educa-tional issues, and a small shift towards theoretical issues.There is a lower proportion of anthropological studies ofspecific group knowledge.

Cultural connectionsOne way to describe the conference is to talk about the var-ious kinds of cultural connections that were made during it.We can acknowledge at least four types: different indigenousgroups talking with each other; academics intersecting withindigenous groups; mathematicians connecting with artistsand mathematics educators; and the critiquers and advocatesof ethnomathematics engaging in debate. These connectionswere not always easy because of language, philosophicaltensions, or differing agendas. But they were always good-natured and, more importantly, extremely productive.

The conference opened with a Powhiri (formal Maoriwelcome) that was replied to by Joel Martim from theGuarani people of Brazil (see Figure 2).

With the first languages heard in the conference being twoindigenous languages, the scene was set for a critical role tobe played by indigenous representatives. This interactionwas highlighted in the plenary panel on indigenous know-ledge. Representatives from six groups talked aboutdifferent political, practical, and schooling issues of ethno-mathematics embedded in cultural knowledge as seen fromthe point of view of an indigenous person. This high degreeof interaction amongst people from different countries con-tinued in informal gatherings during the conference.

The effect was to draw indigenous perspectives into acade-mic ethnomathematics. Nowhere was this better illustratedthan in Willy Alangui’s presentation of his work on the rice ter-racing in the Cordillera region of the Philippines. Willy is bothan Igorot (indigenous Philippino), and Chair of the Departmentof Mathematics at the University of Philippines, Baguio. He

introduced us to his idea of “mutual interrogation”. Willy hadcollaborated with an applied mathematician to develop amodel of water distribution over a system of rice terraces. Healso examined the system in practice. He then set up a dialoguebetween the two systems, and found that each contained con-cepts that were not present in the other. This dialogue enhancedboth the applied mathematician’s understanding of modelling,as well as the rice farmers’ understanding of water systems.Such a deliberate interaction between world-wide conven-tional knowledge and indigenous practice presents a new wayto think about the task of ethnomathematics.

The conference allowed mathematicians, artists, teach-ers, and mathematics educators to explore the experiencesand thoughts of each other’s social-cultural groups. It wassignificant that the conference was hosted by NZ’s leadinguniversity Mathematics Department and included interestedand enthusiastic involvement on the part of some academicmathematicians. Interactions with craftspeople are des-cribed below, but, as with earlier conferences, the teacherspresent brought a practical focus. A wide range of teachingconcerns were evident in papers on multicultural class-rooms in Sweden, education of indigenous mathematicsteachers in Brazil, language of instruction in Japan, a Mayanautonomous school in Mexico, and an ethnomathematicalcurriculum project in Alaska.

ICEm-3 saw the appearance of debates between critiquersand advocates of ethnomathematics. Kai Horsthemke andMarc Schäfer critiqued the conceptualisations and results ofethnomathematics from social, political, and epistemologicalviewpoints. They were concerned about the problems of rel-ativism, whether all mathematical skills are culturallyembedded, and the threat of marginalisation in a culturallynamed mathematics. While these critiques have appearedbefore in the literature, it is a measure of the scholarship ofour field that these are welcomed at our conferences and gen-erated considerable informal debate. Thus several movementswere unleashed at ICEm-3 in terms of cultural connections –thereby meeting one of the aims of the conference.

Mathematical manipulationsThe second subtitle of the conference gives another means todiscuss its proceedings. There were two plenary workshopsduring the conference, and two presentation workshops: allfour captured people’s imagination and set participants reflect-ing on mathematical manipulations in the physical sense.

The plenary workshops were designed to give an experi-ence of Maori and Polynesian craft that could havemathematical interpretations. In each case, the presenterswere professional craftspeople and the mathematical partwas done by the participants. The Raranga Harakeke (flaxweaving) workshop took place in the University of Auck-land Wharenui (Maori Meeting House), and the Lalava(rope lashing) workshop in the Fale (Pasifika meetinghouse). These beautiful ceremonial buildings exhibitedmany examples of craft and design other than weaving andlashing, and were inspiring places to engage with the depthof knowledge and skill that these crafts require.

The Raranga Harakeke workshop was preceded by a talkby the Maori weaver Hariata Adams that discussed the con-ventions of weaving, and we learned, for example, about

Figure 2: Joel Martim in the Wharenui (Maori MeetingHouse)

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the contemporary environmental issue of the disappearanceof pingao, the grass used for the decorative panels in thewharenui. This reminder of the interconnection betweencraft with other aspects of life set the scene for a discussionabout the origins and social importance of the conventionssurrounding weaving. In the workshop we were able toundertake small weaving exercises of a much more creativenature than forming flat mats: flowers and fishes emergedfrom the long flax leaves.

Mathematical discussions varied from the practical ques-tions (how to form shapes and angles that were not thefamiliar rectilinear ones? how to create 3-dimensionalobjects? what are the limits on the mathematical shapes thatcould be formed?), to those about theory and context (howdid weaving conventions relate to the knowledge beingdemonstrated?; what would be necessary to bring these prac-tices into a mathematics classroom?; would these practicesbe useful in a mathematics classroom?).

The Lalava workshop was also introduced by a talk. ArtistFilipe Tohi from Tonga – a well-known sculptor – describedthe ways he uses the ideas of lashing to develop the huge 3-D sculptures for which he is famous.

His emphasis on the 3-dimensional nature of the apparently2-dimensional patterns formed during lashing, and his discus-sion of the construction of the shapes, took many of us bysurprise. Not only were the effects visually stunning but thegeometric complexity was much greater than expected, andthe analysis fascinating. Some participants made links to thirdyear university abstract algebra and to school geometry.

Discussions amongst participants concerned the mathe-matical aspects of these crafts, but often the talk becamedebates about the relationship between mathematics andcraft and the consequences of decontextualisation or math-ematisation when used in educational situations.

The other two workshops concerned origami and theChinese abacus respectively. Again, our hands-on work gen-erated conversations – this time much more firmly directed

at educational outcomes and relationships with school math-ematics curricula.

The abacus was handed out with no instructions, as it hadbeen to children in a study reported by the presenter. Wewere asked to make sense of it as much as possible, and tohypothesise on the techniques of adding, subtracting, multi-plying and dividing, reflecting on the way that the abacuscould be used (or not) to develop ideas of place value. Thegroup reproduced (so we were told) the stages that youngerchildren go through in their understanding. One of the manydiscussions was about the enhancing potential of culturalinstruments such as the abacus, but also reflected on the pos-sibility that such instruments may have a closing downeffect on other mathematical ideas.

The origami workshop involved models of strictly geo-metric shapes, with the intent of asking how and why thesemight be used in a classroom. The main presenter was anorigami specialist with a huge range of techniques, and a

Figure 3: A flax flower.

Figure 4: Lalava lashing.

Figure 5: Tohi sculpture based on Lalava structures.

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repertoire that included complex geometric models.A co-presenter had knowledge of the school mathematics

curriculum, and discussed the use of this art form to fulfilconventional learning objectives. The discussions concernedhelping each other when it was realised that what lookedsimple was not easy, and realising that the professionalismof all the craftspeople disguised a complexity that wasalmost impossible to reproduce just by watching.

Indigenous presentationsMathematical manipulations also occurred in another sense.The conference began with a plenary session of Yup’ik Inuitcultural practices presented in the context of the curricu-lum development work carried out by Jerry Lipka and histeam over a 25-year period. Evelyn Yanez and Dora Andrewdiscussed measurement systems for making clothing andfor building, and demonstrated design techniques used bytheir elders.

During the conference, many other presentationsdescribed indigenous (and other cultural) practices that con-tain mathematical interest. Through the conference theseexperiences developed into discussions about the relation-ship between cultural practices and mathematics: how couldone transform into the other?; can it be a two-way process?;what are the benefits and drawbacks of these transforma-tions, both for the cultural practice and for mathematics?

There was also a later debate about whether the point ofethnomathematics was to enhance conventional mathemati-cal understanding or not. Another presentation from theYup’ik curriculum group provided comprehensive statisticalevidence for the efficacy of ethnomathematical materialsand approaches for students measured against the criteria ofsuccess in standard mathematics tests. While it was agreedthat this evidence was a strong basis on which to pursue anethnomathematical educational agenda, the idea that con-ventional mathematics should be the ultimate outcome wasquestioned. The aims of an ethnomathematical educationwere restated in terms of humanist principles in D’Ambro-sio’s final plenary talk: that mathematical knowledge is notisolated, but forms part of the fabric of society and must betaught so this is inherent.

Looking backwards and forwardsThis was a smaller conference than ICEm-1 and ICEm-2. Thishad both advantages and disadvantages. On the one hand theintimacy provided more depth of discussion, and follow-updiscussions with the same people. It also allowed participantsto hear most presentations, with parallel sessions mostly hav-ing two presentations rather than the five at ICEm-2. On theother hand we missed the politically-oriented contributionsfrom North America, and fewer new researchers attended –as can be expected in a country of low population density.

Reflecting back on the three conferences, the coverage ofEurope, South America, and Oceania is an encouraging signof the broad base of the field. Looking forward, the willing-ness of Lawrence Shirley to organise ICEm-4 in the USA,and Mogege Mosimege to work towards South Africa beingthe following venue, continue this trend.

But what can be said about the direction of ethnomathe-matical study? Ubiratan D’Ambrosio’s final plenary was

titled as a retrospective look, however, as is his habit, hehad a lot to say about where we are heading. The predomi-nant theme, accepted by the conference in general, was oneof ethical values. It represents a turn away from studies ofcultural practices as mathematics, now that the case has beenmade for a variety of systems of knowledge. Rather we needto be looking at more fundamental characteristics of knowl-edge systems – namely, their relationship with society. Whatcan other knowledge systems teach us about that?

Ubiratan began by noting that history shows us that math-ematics is (and always has been) intrinsically involved withactions that deny human life (for example, war and envi-ronmental destruction). He argues that mathematics,mathematicians, and mathematics educators cannot act freefrom consideration of issues such as national and personalwelfare and security, governmental politics, economics, rela-tions between nations, relations between social classes, andthe preservation of natural and cultural resources. We areall deeply involved with these issues, and such issues affecthumans and their relations with nature.

Ubiratan reinforced the links between mathematics andsociety by reviewing the roots of mathematical activity. Indi-viduals create instruments (such as mathematics) to enhancethe possibility of survival and the transcendence of time andspace. In this search, people attempt to explain the phenom-ena they encounter. These models and explanations are anattempt to know (and control) the future. Such conceptsimplicitly include religious and value systems. These sys-tems of knowledge are the essence of ethnomathematics.

One group of concepts, including classifying, ordering,comparing, measuring, quantifying, inferring, and inventing,when organised in a specific way with a specific set of val-ues (for example, rationality) constitute what is called(Western) mathematics. It is widely agreed that, as a way oforganising this set of concepts, mathematics is a universalmode of thought – just as survival with dignity is agreed tobe a universal problem for humankind.

Ubiratan claims that we should establish a priority to lookat the relations between these two universals, that is, at therole of mathematicians and mathematics educators in creatinga civilisation with dignity for all, free of inequity, arrogance,and bigotry. He offers ethnomathematics as a proposal. Eth-nomathematics is thus a project in the history and philosophyof mathematics with pedagogical implications. The peda-gogical aims are to promote creativity in helping people reachtheir mathematical potential and citizenship by transmittinghuman values and responsibility within society.

Ubiratan acknowledged that we are currently in a statewhere critical views of mathematics education are meetingnostalgic and obsolete views of what mathematics is. Thiscauses confusion between ethnomathematics and ethnicmathematics. The latter is characterised by ethnographicstudies unsupported by theoretical foundations. He thereforeproposes that we give more attention to the theoretical strandof the programme ethnomathematics as a research pro-gramme with pedagogical implications.

At a post-conference event, Ubiratan D’Ambrosio, BillBarton, and Bengt Johansson recorded a two-hour discus-sion of what exactly this might involve. The trialogue will bethe subject of a follow-up paper.

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