eglash - culturally situated design tools

8/6/2019 Eglash - Culturally Situated Design Tools

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Anthropology and EducationR O N E G L A S H

A U D RE Y BENNETTC A S E Y O D O N N E LL

SYBI LLYN J ENNINGSM A R G A R E T C I N TO R I N O

Culturally Situated Design Tools:Ethnocomputing from Field Site to Classroom

ABSTRACT Ethnomathematics is the study of mathematical ideas and practices situated in their cultural context. Culturally Situated

Design Tools (CSDTs) are web-based software applications that allow students to create simulations of cultural artsNative Americanbeadwork, African American cornrow hairstyles, urban grafti, and so forthusing these underlying mathematical principles. This articleis a review of the anthropological issues raised in the CSDT project: negotiating the representations of cultural knowledge duringthe design process with community members, negotiating pedagogical features with math teachers and their students, and reectingon the software development itself as a cultural construction. The move from ethnomathematics to ethnocomputing results in anexpressive computational medium that affords new opportunities to explore the relationships between youth identity and culture, thecultural construction of mathematics and computing, and the formation of cultural and technological hybridity. [Keywords: mathematics,computing, youth subculture, indigenous knowledge, identity]

C ULTURAL ANTHROPOLOGY has always dependedon acts of translation between emic and etic per-spectives. Some of these translations have become formalsubdisciplinesethnobotoany, ethnomedicine, archaeoas-tronomy, and so forth. The subdiscipline of ethnomathe-matics is more recent and much more controversial. First,some ethnomathematics research provides an unusuallystrong challenge to the primitivist view that indigenous(i.e., band or tribal) societies had only simplistic technolo-gies: It is one thing to claim that the natives have manyherbal cures, another to claim they are well versed in graphtheory or topology. The second controversy stems fromthe applications of ethnomathematics to contemporary

K12 mathematics education. Here, ethnomathematics en-ters the culture wars debates over classic curricula versusmulticulturalist revision. Finally, ethnomathematics alsoparticipates in thescience wars debate over thesocial con-struction of science and technology: Is math universal ordoes it varyfrom culture to culture?This articledescribes therole that these and other social science issues have playedin the development and evaluation of a suite of computersimulations of indigenous and vernacular artifacts and

A MERICAN A NTHROPOLOGIST , Vol. 108, Issue 2, pp. 347362, ISSN 0002-7294, electronic ISSN 1548-1433. C 2006 by the American AnthropologicalAssociation. All rights reserved. Please direct all requests for permission to photocopy or reproduce article content through the University of CaliforniaPresss Rights and Permissions website, at http://www.ucpress.edu/journals/rights.htm.

practices, termed Culturally Situated Design Tools (CSDTs).These design tools are not only promising in their initialevaluations of impact on minority academic achievementbut they also open new possibilities for anthropologicalresearch.

ETHNOMATHEMATICS VERSUS MULTICULTURALMATHEMATICS

Anthropologists and other researchers have revealed sophis-ticated mathematical concepts and practices in the activi-ties and artifacts of many indigenous and vernacular cul-tures (e.g., see Ascher 1990, 2002; Closs 1986; Crump 1990;DAmbrosio 1990; Eglash 1999; Gerdes 1991; Urton 1997;

and Zaslavsky 1973; for a theoretical overview, see Eglash1997b). These practices includegeometric principles in craftwork, architecture, and the arts; numeric relations found inmeasuring, calculation, games, divination, navigation, andastronomy; and a wide variety of other artifacts and proce-dures. In some cases, the translation to Western mathe-matics is direct and simple: as with counting systems andcalendars, for example. In other cases, the math is embed-ded in a processiteration in bead work, Eulerian paths in


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348 American Anthropologist Vol. 108, No. 2 June 2006

sand drawings, and so forthand the act of translation ismore like mathematical modeling.

What is the difference between ethnomathematics andthe general practice of creating a mathematical model of acultural phenomenon (e.g., the mathematical anthropol-ogy of Paul Kay [1971] and others)? The essential issue is

the relationbetween intentionality and epistemological sta-tus. A single drop of water issuing from a watering can, forexample, can be modeled mathematically, but we wouldnot attribute knowledge of that mathematics to the aver-age gardener. Estimating the increase in seeds required foran increased garden plot, on the other hand, would qual-ify. Some cases of indigenous practice are, however, not asclear-cut.

For example, in 19992000, a debate in Critique of Anthropology briey ourished between Stefan Helmreich(1999) and Steven Lansing (2000)over Lansings com-puter models of Balinese rice irrigation. Helmreich heldthat Lansings description of an evolutionary optimizationin irrigation schedules naturalized the Balinese in waysthat eliminated their intentionality. But Lansings own bookprovided detailed description of the indigenous knowledgesystems involvedin particular, the Tika calendar, whichis clearly articial and intentional. Should we then con-clude that the Balinese have an intentional, explicit mathe-matics of computational optimization? That would clearlybe going too far in the other direction. The epistemolog-ical status of Balinese irrigation mathematics lies some-where between unconscious social process and deliberate,explicit knowledge. 1 Ethnomathematics research often usesthe term translation to describe the process of modeling in-digenous systems with a Western (i.e., mainstream, aca-demic) mathematical representation; however, as with alltranslation, the success is always partial, and intentionalityis one of the areas in whichthe process is particularly tricky.

Such subtleties can be easily lost, however, when mov-ing from research to application in education. Given thestrict regulation of standards in the United States created bythe No Child Left Behind Act (Public Law 107110), math-ematics teachers are under a great deal of pressure to stickto a standardized curriculum. At the same time, the impor-tance of making real world connections in math instruc-tion, especially those making use of the heritage cultureof students, has been increasingly recognized. As a result,U.S. teachers have been attracted to the area of multicul-tural mathematics, 2 which often substitutes the anthro-pological specicity of ethnomath for a variety of dubiousshortcuts.

First, some examples from multicultural mathematicstake standard First World word problems and give thema supercial Third World gloss: Rather than Dick and Jane counting marbles, Rain Forest Mathematics gives usTaktuk andEsteban counting coconuts (Jennings1996).Sec-ond, it is easier to take a more literal-minded approach anduse strictly numeric systems (e.g., counting) than to lookat embedded mathematics (architecture, crafts, etc.), whichrequires modeling. Thus, there are texts such as Multicul-

tural Mathematics (Nelson et al. 1993), which emphasizeonly Chinese, Hindu, and Muslim examples, because theyhave far more numeric calculations than the indigenousband and tribal societies of sub-Saharan Africa, North Amer-ica, South America, and the South Pacic. Those particularAsian and Arabic empire civilizations produced mathe-

matics that easily translates into the standard curriculumbecause they were large state societies with the labor spe-cialization and associated tasks that tend to require exten-sive numeric calculations, similar to the society that themath teachers now inhabit. There is nothing wrong withincluding such examples in the cross-cultural mathematicsrepertoire; we would be remiss if we did not. But restrictingcurricula only to examples fromthese non-Western state so-cieties merely exacerbates the orientalist myth of abstractAsian and Arabic minds. By the same token, there is noth-ing wrong with including ancient Egypt as one example of math from the African continent, but if it is implied to bethe only math from Africa then it can inadvertently primi-tivize the rest of the continent. 3 Finally, in the few cases inwhich actual indigenous mathematical practices are used,most examples are restricted to lower primary school levelAfrican houses are shaped like cylindersagain reinforc-ing primitivism, rather than opposing it.

In short, although there is much good done under therubric of multicultural mathematics, it also runs the dangerof acting as a sortof safety valve, satisfyingdiversity require-ments without challenging the most deleterious miscon-ceptions. There is, however, a second, more subtle area inwhich the multicultural math approach has run into trou-ble, and that is in its implicit models for identity. Whenasked about the reasons why they might want to includeculture in mathematics class, teachers often respond bysaying something like students might relate better to ex-amples that come from their own culture. 4 There are im-plicit assumptions at work in such statementsalmost afolk anthropologythat maintain that we each have a spe-cic culture, and that it is a given, static entity to whichour personal identity is fused in a kind of mimetic rela-tionship. This stands in strong contrast to the portraits of race, class, and gender hybridity offered by research in con-temporary youth subculture (Davidson 1996; Gilroy 1993;Hebdige 1979; Pollock 2004). It is our contention that mul-ticultural education that abides by this folk-anthropologyand assumes a singular, static, pregiven identity will beless effective. Our goal in emphasizing the design aspectof CSDTsthe ways in which they allow students to uti-lize a synthesis of math, computing, and culture in creativeexpressionis to provide better support for students to takeadvantage of their self-making abilities in ways that canenhance academic performance.

In summary, we can contrast ethnomathematicsto multicultural mathematics using the following fourprinciples.

1. Deep design themes. When examined in their so-cial context, indigenous mathematical practices are not


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Eglash et al. Culturally Situated Design Tools 349

trivial or haphazard; they often reect deep designthemes providing a cohesive structure to many of the im-portant knowledgesystems (cosmological, spiritual,med-ical, etc.) for that society. Examples include the pervasiveuse of fractal geometry in African design (Eglash 1999)andthe prevalence of fourfold symmetry in Native Amer-ican design (cf. Daz 1995; Klein 1982; Witherspoon and

Peterson 1995). 52. Anti-primitivist representation. By showing sophis-ticated mathematical practices, not just trivial examples(e.g., African houses are shaped like cylinders), eth-nomathematics directly challenges the epistemologicalstereotypes most damaging to minority ethnic groups.

3. Translation, not just modeling. Often indigenousdesigns are merely analyzed from a Western view; for ex-ample, applying the symmetry classications from crys-tallography to indigenous textile patterns. Ethnomathe-matics also makes use of modeling, but here it attemptsto use modeling to establishrelationsbetween the indige-nousconceptual frameworkand the mathematics embed-ded in related indigenous designs, such that the mathe-

matics can be seen as arising from emic rather than eticorigins. This is critical in contesting biological and cul-tural determinism.

4. Dynamic rather than static views of culture.Although evidence for independent indigenous mathe-matics is crucial in opposing primitivism, it is also im-portant to avoid the stereotype of indigenous peoples ashistorically isolated, alive only in a static past of museumdisplays. For this reason, ethnomathematics includes thevernacular practices of their contemporary descendents;for example, the street mathematics of Latino pushcartvenders, grafti art, and so forth (Nunes et al. 1993).

FROM ETHNOMATHEMATICS TO ETHNOCOMPUTING

The computational component of this research began withthe research by one of the coauthors (Ron Eglash) on model-ing traditional African architecture using fractal geometry.Fractals are patterns that repeat themselves at many scales;they are usually used to model natural phenomena suchas trees (branches of branches), mountains (peaks withinpeaks), and so forth. Both computer simulations and mea-surement of fractal dimension of these traditional villagearchitectures showed several repetitions (iterations) thesame pattern at different scales: circular houses arranged incircles of circles, rectangular houses in rectangles of rectan-gles, and so on. Eglashs year of eldwork in West and Cen-tral Africa (sponsored by the Fulbright program) showedthat these architectural fractals result from intentional de-signs, not simply unconscious social dynamics, and thatsuch iterative scaling structure can be found in other ar-eas of African material cultureart, adornment, religion,construction, games, and so forthoften as a result of ge-ometric algorithms known (implicitly or explicitly) by theartisans (Eglash 1997a, 1999).

Although there are many potential implications andapplications of the African Fractals thesis, the strongestappeared to be the possibility for using this African mathe-matics in the classroom. With the help of a small seed grant(the AAA 2001 Integrating Anthropology into Schools

award), and collaborators in Utah and Idaho (professor James Barta and tribal member Ed Galindo), we also began anew effort toward Native American design tools. Expandingtheproject to Latino ethnomathematicsas well, andplacingthe proposed tools for all three ethnic groups under the col-lective title of CSDT, we applied for and received three fed-

eral grants: a Housing andUrban Development CommunityOutreach Partnership Centers (COPC) grant, a Departmentof Education Fund for the Improvement of Post-SecondaryEducation grant, and a National Science Foundation (NSF)Information Technology Work Force (ITWF) grant. The lat-ter also expanded evaluation of the impact from mathe-matics achievement to the science and technology careerpipeline: Would minority students exposed to these designtools increase their interest in information technology ca-reers? Although ethnomathematics is an established eld,the computational aspects of this research seemed to ask fora slightly different rubric: Matti Tedre of the University of Joensuu in Finland has suggested the term ethnocomputing ,which seems a better t (cf. Tedre et al. 2002).

The funding allowed us to hire ethnomathematics con-sultants at specic locales in which there was a signicantpopulation of one of our three target groups (African Amer-ican, Native American, and Latino). These included facultymembers at universities who were working with teachers,the teachers themselves, and community members such aselders and artisans. Together with the students, each localeconstituted a eld site in which we solicited ideas, gatheredfeedback on prototypes, evaluated student learning, andcarried out professional development. This also enabled thedevelopment of a cultural background sectionfor each de-sign tool, so that students could understand the social con-text of the practice as well as its underlying mathematics.

ANTHROPOLOGICAL RESEARCH AND THEORY ISSUESIN ETHNOCOMPUTING

At the same time that we carried out the application of CS-DTs to the problem of minority student achievement, wealso attempted to use this opportunity to explore the asso-ciated research and theory issues. In some ways these re-sembled any other cultural study: Representation is alwaysa critical issue in anthropology, and like any ethnographerswe grapple with the power relations and politics of sharedvoices. Some CSDTs, for example, concern Native Ameri-can tribes, for which those considerations are heightenedbecause of both political history and popular misrepresen-tations. A tool that represents a practice carried out by mul-tiple tribes, such as the Virtual Bead Loom, must also resolveintertribal conict over representations. Other CSDTs con-cern youth subculture, in that we are caught between theK12 school demands to avoid advocacy of inappropriateactivity (Cousins 1999) and the need to faithfully representsubculture practices such as grafti.

What made this project somewhat different from otherrepresentations of culture was the computational aspects.On the one hand, many anthropological projects have


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made use of electronic media for detailed cultural portraits,but even interactive media are typically a matter of userspressing a button to see an image or video clip (cf. Banks1994 for skepticism on its impact in cultural anthropol-ogy; Clarke 2004 on the impact of electronic media in ar-chaeology representations). On the other hand, the eld

of computational anthropology has taken advantage of computer simulations ranging from population dynamicsof the !Kung (Howell and Lehotay 1978) to agent-based sim-ulations such as Articial Anasazi (Axtell et al. 2002). Butin such computational anthropology, there is little role forethnographic portraits or representations of visual culture;its simulations are primarily composed of numeric variablesof kinship, reproductive rates, and other demographic data.Because the CSDT project combines both the representa-tional aspects of electronic media and the computationalaspects of these simulation systems, we might classify ourproject as something more along the lines of computa-tional ethnography, combining quantitative formal mod-eling with the kinds of collaboratively developed culturalportraits of the classic ethnographic approach.

Computational ethnography, as we envision it here,might span the range of endeavors from social studies of technology to technological studies of societies. At the so-cial end, ethnocomputing can act as a kind of culturalprobe: What happens when you ask traditional Afro-Cuban drummers to create a mathematical simulation of their rhythms or grafti artists to draw on a computerscreen? At the technological end, it is useful for probingour own convictions: How do we reconcile our sense thatzero degrees lies along the horizontal, with the Yupik un-derstanding that the horizontal is at 90 degrees? How do wereconcile our understanding of mathematics as a human in-vention with the Shoshone position that it existed beforehumans? Computational ethnography helps us make ourown assumptions visible; we begin to see that some self-evident aspects of math or technology are actually choicesthat could have been otherwise.

The following three sections describe some of the bestexamples from the design tools, highlighting the ways inwhich the tools emerged through the trialectic of com-puting, cultural anthropology, and math education.

AFRICAN AND AFRICAN AMERICAN DESIGN TOOLS

As noted previously, we began with the African Fractalsmaterial. 6 Presentations at meetings of the National Coun-cil of Teachers of Mathematics provided the opportunity todiscuss the material with middle school and high schoolteachers from inner-city schools and other areas with largeAfrican American student populations. They were excitedabout the new perspective on African math but skepticalabout its practical use in the classroom. Teachers cited thelack of t to the K12 standard curriculum(fractal geometryis typically a college-level course) and the lack of t to thestudents own cultural knowledge: They did not think their(predominantly) African American students knew much

about Africa, and some doubted they would care about itmuch either.

The cultural challenge of making the connection withAfrican American students who had little knowledge of Africa was solved by three innovations. First, followingteachers suggestions, we focused on the example of fractal

patterns in cornrow hairstyles (see Figure 1). Mathemati-cally, cornrows work as fractals because most styles alloweach crisscross (plait) of the hair to diminish progres-sively in size, creating many iterations of scale in a singlebraid. Culturally, cornrows work for these students becausethey are both deeply embedded in African culture (Boone1986) and an ongoing innovative practice in contemporaryAfrican American culture. We found that very few studentsknew that cornrows originated in Africa, so on the designtool website we developed a historical background page.This included images and information covering the indige-nous styles and their meaning in Africa, their rst appear-ance among early African Americans in the preCivil Warera, their survival and eventual rebirth in the civil rightsmovement and hip-hop culture. Finally, we also createda series of goal images: photos of styles for students tosimulate, including both professional styles and photos thestudents took of themselves and their friends. This helpedstudents to make the connection from contemporary ver-nacular identity to heritage identity, which then opened upa wide selection of indigenous African fractal patterns foradditional simulation. The most successful has been onebased on Mangbetu design (see Figure 2).

We solved the problem of curricular t by changingthe emphasis from fractal geometry to transformationalgeometryeach plait is given some particular rotation,translation (spacing), scaling, and reection. Because thesetransforms are part of the standard curriculum, it satisedthe math teachers. Starting with the braid designs, studentsquickly caught on to the art of creating simulations usingthese controls. One of the interesting conversations amongthe students (primarily African American in our workshops)was speculation on whether or not hair stylists could besaid to know these parameters. Some excitedly reported thatthey now realized they have been doing math all along andjust did not know it (that you had to think about how youwere rotating or scaling the plaits as you braided); otherssaid braiding was pure unthinking intuition (although theytoo reported that they enjoyed using the design tool). Wealso found that we couldalso use the simulation to pose eth-nomathematics research questions: As you move through aseries of braids, which parameters were altered and whichremained unchanged? How do hairstylists think about thesame issue?

After students create a successful hairstyle simulation,we then ask them to develop their own designs usingCornrow Curves. Figures 2 and 3 show two samples of thisstudent work. Figure 3A is from an African American stu-dent whose father came from Ethiopia; she titled this de-sign Tisissat and added the comment I named this af-ter the largest waterfall in Ethiopia. It shows strength and


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FIGURE 1. Cornrow Curves: at right an original hair style selected by the student; at left the students simulation, generated by theparameters in the center control panel.

FIGURE 2. Mangbetu Design pattern created by a student for artistic purposes rather than simulation of an artifact.
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FIGURE 3. A. Tisissat. B. Snowake.

holding people together. The second is from a student whoself-identied as Puerto Rican. He was a problem student,disrupting the class and not getting much done. He nallyhit on the idea of making simulations of snowakes. He re-searched real snowakes on the web, and gured out howto get the cornrow software to make the proper angles forsix-fold symmetry and the proper scaling ratios for its arms(see Figure 3B). It was an enormous triumph for him: Theother students gave him high ves when they saw what hehad done, his mother heard about it and came in to visitthe computer lab, and it greatly improved his overall atti-tude. The fact that he had violated our own focus on thesimulation of cultural artifacts was completely irrelevant.The fundamental goal for design tools is to empower thestudents sense of ownership over math and computing; onthe basis of that objective, it was a strong success.

After students have completed both the cornrow andMangbetu software experiences, we ask them why theythink they were able to use iterative scaling for both sim-ulations. They are quick to answer that it is because bothoriginate from the same African origins: 7 an indication thatfor these students math and computing has now becomea potential bridge to cultural heritage, rather than a bar-rier against it. Reecting on teachers warnings that AfricanAmerican students might not care about African culture, itseems less a question of ethnic pride than one of contextandmotivation: Given a chanceto incorporatesome agencyinto their encounterto creatively improvise with thesecultural materialsthese students often nd it fascinating.It is not simply a matter of using a static, preformed identityto lure students into doing math or computing. Identityis always in a process of self-construction (Hermans 2001).That self-construction is, of course, going on regardless of our presence, but as Foucault suggested in his phrase tech-

nologies of the self it is constrained and enabled by thevarious resources at hand. Our goal is in creating a culture-enriched computational medium that offers students newopportunities in identity self-constructionopportunitiesthat we hope will provide them with critical tools and per-spectives for both social and technical domains, as well asthe interrelationships between the two.

NATIVE AMERICAN DESIGN TOOLS

Our current Native American design tools includeSimShoBan (Eglash 2001), Yupik Navigation, Yupik ParkaPatterns, Alaskan Basket Weaver, Navajo Rug Sim, and theVirtual Bead Loom (VBL). Because of space constraints, wefocus on the VBL here. This portion of the project be-gan in the spring of 2000 at the Shoshone-Bannock sec-ondary school. We decided that the geometric patterns inShoshone-Bannock beadwork would be a good choice for anew design tool for the following reasons:

1. It is a vibrant contemporary art form on the reservationbut has deep (precolonial) historical roots.

2. The rows and columns of the loom are analogous to thedeep designthemeof fourfold symmetry in NativeAmer-ican cultures.

3. The two axes of the loom offer an analog to the CartesianCoordinate system and thus provide a good match forstandard school curricula.

In retrospect, we can see that these criteria were the sameas those used to select cornrows: It connects contempo-rary vernacular culture, traditional heritage culture, andthe standard curriculum. Back at Rensselaer, we assembleda software prototype for the VBL. The webpage begins byshowing the prevalence of fourfold symmetry in Native
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FIGURE 4. A comparison of uneven steps in a Virtual Bead Loom triangle (left) and even steps in Shoshone-Bannock beadwork (right).

American design in general and for the bead loom in partic-ular. The web-based software allows the user to enter x, y co-ordinates for bead positions;together withcolor choice, thisallows the creation of patterns similar to those on the tradi-tional loom. We also put together a cultural backgroundsection showing how the concept of a Cartesian layout canbe seen in a wide variety of native designs: Navajo sandpaintings, Yupik parka decoration, Pawnee drum design,andother manifestations of the Four Winds concept. Onenative student who had at rst been skeptical about com-bining technology with native design suddenly got itrealized the ethnomath claim that Native Americans haddeveloped an analog to the Cartesian coordinate systemand said They will never let you get away with this. Weasked, Who wont let us get away this? and she repliedWhite people (conversation with authors, November 42000).

However, the prototype only allowed creation of a pat-tern with single beads. This was clearly too tedious, and soafter discussion with potential users, we introduced shapetoolsenter two coordinate pairs for a line, three coordi-

nates pairs to get a triangle, and so forth. But these virtualbead triangles often had uneven edges, whereas the origi-nal Shoshone-Bannock beadwork always had perfectly reg-ular edges (see Figure 4). It turned out that our program-mer, STS graduate student Lane DeNicola, had looked up astandard scanning algorithm for the triangle generation;however, somehow the traditional beadworkers had algo-rithms in their heads that produced a different result. Aftera few conversations with beadworkers, it became clear thatthey were using iterative rulesfor example, subtract threebeads from the left each time you move up one row. We de-veloped a second tool for creating trianglesthis one usingiterationbut kept the rst in the VBL as well for compari-son. This has provided a powerful learning opportunity: If you tell someone that there is such a thing as a Shoshonealgorithm, they may balk at the suggestion; but let themcompare the standard scanning algorithm with that usedby the Shoshone beadworkers, and the ethnomathematicsimplications are difcult to ignore.

How should we regard this contrast between the twoalgorithms in terms of theories of knowledge? At one


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FIGURE 5. Ready Made Science versus Science in the Making.

extreme, we could simply dismiss the difference as a failureto properly specify software design goalsafter all, no onetold the programmer to make sure the software only pro-duces triangles with even stepping. At the other extreme,we could celebrate this as evidence for a mathematical ver-sion of cultural relativism, claiming that there is not oneuniversal math but only many locally produced maths.We reject both of these extreme positions, as neither re-ally provides an adequate social account of technologicaldesign.

The problem with the rst extreme position is nicelyillustrated with the Janus gure from Bruno Latours Sci-ence in Action (1987; see Figure 5). At the left is the face of Ready Made Science, at the right is Science in the Mak-ing. Latour produced this gure to describe an event fromTracy Kidders Soul of a New Machine (1981), in which hard-ware engineers battle marketing, management, software,and other corporate divisions to champion their new (yetto actually work) computer design. Our rst propositionthat we did not give the programmer adequate functionspecicationsis the view from the older Janus face at left.After the fact, it is easy to give purely technical specica-tions for what was required to make it work and thus toforget that the younger face was looking out over an uncer-tain terrain that was as much social as it was technical.

What of the second extreme position: that these twoalgorithms should be regarded as evidence for a relativistposition on mathematics? Consider the mangle, AndrewPickerings (1995) term for the ways in which nature, cul-ture, and technology combine in the creation of science.Pickering shows that scientists and mathematicians are of-ten proceeding along a line of inquiry when they run intosome kind of resistancea physical property is not quitewhat they thought it would be, a machine does not quiteact the way they thought it would, and so forth. They re-spond with an accommodation, a creative solution that

allows the inquiry to proceed through some alternativearrangement (which rearranges social as well as technicaland natural relations). These are contingent; some other ar-rangement might have also provided the accommodation. 8

In Pickerings framework, scientists are neither giving us atransparent window on the world nor are they merely ex-pressing a relative truth. Rather, the products of science area mangle created through contingent accommodationsbetween (nonuniversal) people and the (universal) worldthey inhabit. The Shoshone bead artisans used a differentalgorithm because they were in a different mangle.

Universallocal contrasts also arose in some teachingsituations. A math teacher at the Shoshone-Bannock schoolwas using the bead loom in early December, so she decidedto surprise the students by assigning a geometric taska re-peating series of triangleswhich would lead to their gen-erating an image of a Christmas tree. In my little whitemind, she said, there was just one obvious way to thinkabout that. She was amazed to see that the students reinter-preted herdirections, overlaying the triangles to createthatfeather pattern you see on much of the beadwork here(conversation with authors, October 4, 2002). Here was acase in which students had used the design tool to appro-priate the mathematics instruction, adapting it to suit theirown cultural priorities.

One of the schools serving the Uintah-Ouray Reserva-tion (Northern Ute) in Utah has also made signicant useof the bead loom; our attempts to accommodate their viewled to an interesting conict. Following our request for userfeedback, the Ute group replied that black beads were badluck and recommended that they be eliminated from theVBL. The Shoshone-Bannock group had earlier told us thatit was important to include the four original colors of red,black, white, and yellow, which they maintained were rep-resentative of the four races of people on earth. Thus, wehad one group rejecting black beads and another requiring


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FIGURE 6. A Puerto Rican ag (minus its star) on the Virtual Bead Loom.

them. The solution we devised was a color mixer, which al-lowed users to custom make their own bead colors. This en-hanced the utility for everyone and opened a new possibil-ity for adding the mathematics of proportion (because thecolor mix was controlled by ratios of red, green, and blue).

The most recent tribal collaboration has been with theOnondaga Nation school in upstate New York. An excitingcontribution from their group was the important histor-ical connections with the development of the U.S. Con-stitution, including a photo of the 1794 Iroquois treatywampum. They also suggested creating an option to workwith virtual wampum beads rather than spherical beads. Joyce Lewis, a tribal member and math teacher at a nearbyhigh school serving the reservation population, investi-gated traditional wampum and found that their height towidth ratio was about 2:1thus disrupting the one-to-onemapping of beads to integer coordinates that had made thecurrent VBL work so well in math classrooms. We decided tooffer two wampum choices: one with traditional dimensionand one with a modied bead with a height to width ratioof 1:1a sort of hybrid bead whose reconstructed identityechoed the cultural hybridity inhabited by many of the na-tive students.

The VBL has also been used in classes with Latino,African American, and white (or of European descent,see Preliminary Quantitative Evaluations below) majorities.It has had a high popularity with students of every eth-nic background. Again, appropriation is a strong theme:

We have seen several African American students from low-income urban areas use the VBL to write their initials, likethe grafti tagsa design not seen with any Native Amer-ican, white, or Latino students to date. Several students of Puerto Rican descent have used the VBL to create an imageof the Puerto Rican ag. One of the ags turned out to bea strong inquiry learning project (Brown and Campione1994; Lewis et al. 2004). The student rst looked up back-ground information on the web and found that the trian-gle in the ag had to be an equilateral triangle, because itrepresented the equal balance of powers between judicial,legislative, and executive branches. This turned out to be aconsiderable mathematical challenge, because you cannotsimply count the number of beads to get three equal sides:The beads along the diagonal are spaced farther apart thanthebeadsalong thevertical or horizontal, because each beadis at an intersection on a square grid. The students inno-vative solution (see Figure 6) combined geometric insightswith the use of a high-resolution option on the VBL.

In summary, the developmental history of the VBLis particularly helpful in illuminating how the universaland the local can be brought together into a productivetension. Through Pickerings mangle, we can see how thealgorithms of both bead workers and software programmersreect multiple local attempts to accommodate singularuniversal laws. Conversely, both students and teachersaccommodate the (one) software application by their(multiple) local appropriations. And appropriation itself


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FIGURE 7. Rhythm Wheels.

can be more clearly seen as a two-way street, creating newhybrids in both machines and people.

LATINO DESIGN TOOLS

We currently have two Latino design tools: (1) Pre-columbian Pyramids, in which students create three-dimensional simulations of architecture from the ancientcultures of Central America and (2) Rhythm Wheels (RW),which we discuss here. RW were the result of working withelementaryschool children of Puerto Rican heritage in Troy,NewYork. Unlike African or NativeAmerican heritage,theredid not seem to be any artifacts,other than the Puerto Ricanag, that the students could identify as specically PuertoRican. Not only was Puerto Rican society culturally hetero-geneous to begin with, but it was also distant in the sensethat many of these children are Nuyorican. We nally fo-cused on music, because that has both distinctively PuertoRican forms and allows for the hybrid blending that is char-acteristic of these childrens lives. The software makes use of the ratios between beats in percussion. The bembe ostinato,for example, has six drum beats for every eight clave beats.

Thus, the two instruments go out of phase, but come backinto phase after 48 beats. It is this impression of separatingand reuniting the rhythms that gives the music its hook;thus,the existence of a Least Common Multiple (LCM) is animportant part of any drummers understanding. Becauseratios and LCM is part of the standard math curriculum,this gave us an opportunity to link this ethnomathematicsto the classroom.

The RW software (see Figure 7) allows a student tochoose from a variety of percussion sounds (hitting thedrum with the heel of the hand versus the open hand, clave,tambourine, etc.) and to then drag each sound into a posi-tion on a rotating wheel. The students (who were in fourthand fth grade) select the number of beats per wheel (up to16), the number of simultaneous wheels total (up to three),the number of times each wheel loops, and the speed of thewheels (all wheels must rotate at the same speed). They canalso separately vary the volume of each sound for accents.There are a wide variety of rhythms that can be reproduced.

But we found that the music that is distinctively PuertoRican Bomba y Plena was not something the children


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identied with as their own; that was music their parentslistened to. The favorite songs list they generated was pri-marily hip-hop, with an emphasis on Spanish lyrics. Thenext generation of our software included both traditionaland hip-hop sounds (along with the ability to mix the twotogether).

Meanwhile, we found that by challenging the studentsto make both wheels stop simultaneously (e.g., by having athree-beat wheel loop four times and a four-beat wheel loopthree times), the tool allowed students to discover the con-cept of LCM on their own. Having students discover theconcept themselves is a much richer learning experiencethan simply memorizing a formula. They also discoveredthe cultural connection: During the lesson one of the chil-dren raised her hand and said excitedly, We could do therhythm we learned in drumming class on this! (observa-tion by authors, October 15, 2003).

The drumming class was also an outcome of the soft-ware. We were fortunate in working with the Ark Commu-nity Charter School in Troy, New York, which takes a posi-tive outlook on multicultural teaching strategies. They wereinterested in our report that the children did not identifywith traditional percussion and were able to obtain a lo-cal arts grant to hire a percussion teacher (our COPC grantfunds were used to purchase the percussion instruments).This placed the project in a more dialectical relationship:While we were working to adapt our software to reecta more hybrid version of cultural identity, the school re-sponded to our initial work by questioning the suppressionof the more traditional version of this heritage identity andworking to create an educational environment that offeredroom for its exploration and celebration. The project cul-minated in a drumming and dance performance for theparents of the children at the Troy arts center (includinga brief rhythm wheels software demo by two children andthe math teacher), an emotionally inspiring afrmation of the community value of this technological, academic, andcultural synthesis.

Two other classrooms with majorities of Latino stu-dents used the design tools, but they (or at least theirteachers, both of whom were Latina) opted for tools fromother categories. One teacher focused on the bead loom.She had both Native American and Mexican ancestry anddid a wonderful job of conveying the ethnomathematicsconcept through the software, to the point that one stu-dent, indignant that the coordinate system wasnamed afterDescartes and not Native Americans, shouted Weve beenripped off! 9 (conversation with authors August 5, 2003).The other teacher decided to utilize a tool based on graftiart (Grafti Grapher). We originally planned to place thistool under the culture category of African American, butafter interviews with various grafti artists who insisted onits multiethnic origins and community, we decided to cre-ate a new category called Youth Subculture. This teacherdecided to use it for her primarily Latino classroom andfound that students readily adapted it to express their owncultural sensibilities (see Figure 8).

PRELIMINARY QUANTITATIVE EVALUATIONSQuantitative Evaluation of Interest in IT Careers

Our current NSFITWF grant evaluates the effect of CS-DTs on student attitudes toward computers and IT careers.Our baseline data was generated by 175 randomly selected

eighth-grade students from low-income families complet-ing two surveys. The workforce survey questioned themabout their plans for taking math courses in high school,ideas about future career, their genders, ages, ethnicities,and parents levels of education. The Bath County Com-puter Attitudes Scale surveyed computer use and attitudes,including their work in mathematics on computers. In2002, 2003, and 2004, we ran workshops with grade-812minority students (primarily African American and Latino)from low-income families. They used the design tools twohours per day over a two-week period; afterward they com-pleted the same surveys. The collective average for the 24students surveyed in our 200204 Ark summer workshops

(83percent underrepresented minority students) did show astatistically signicant ( p < . 05) increase from the baselinemeasure. It is possible that thisdifference reects an increasein positive attitudes toward information technology careersfor the minority students because of their experiences usingthe design tools.

Another study, using a similar variety of the designtools with similar exposure, consisted of 25 students, al-most all of European descent. The same survey didnot showa statistically signicant difference in attitudes after takingthe course. Although the studies are too few to draw anydenitive conclusions, the contrast between minority andwhite students is worthy of further investigation. Although

white students used all the design tools, we did not haveany design tools based on European heritage. To compen-sate, we hadstudents conduct Internet explorations for can-didate artifacts (described in more detail in the followingsection, Why Do CSDTs Work?). It may be that the inclu-sion of design tools reecting white heritage (however de-ned) might improve those scores (again see Why Do CS-DTs Work? below for preliminary design tool candidates).Less formal quantitative evaluations also indicated similarincrease in career aspirations among high school studentsat the Shoshone-Bannock reservation in Idaho and at a lo-cal elementary school for low-income minority students inTroy, New York, after exposure to culturally situated peda-

gogy from our team and others.

Qualitative Evaluation of Mathematics Achievement

Middle school teacher Adriana Magallanes in Californiaran a quasi-experimental study of the VBL for her mas-ters thesis. Usingpretestposttest comparisons (CSDT n.d.),she compared the math performance of Latino studentsin two of her prealgebra classes, one using the bead loomand the other using conventional teaching materials. Shefound a statistically signicant improvement ( p < . 05) inthe performance scores of students using the bead loom.High school teacher Linda Rodrigues, also in California,


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FIGURE 8. Grafti Grapher design from Latino student.

compared grade levels for two classes, one using the VBLand Grafti Grapher, and one from the previous year with-out any design tools. She found statistically signicant im-provement in grades ( p < . 001) for the class using the de-sign tools (pretestposttest test comparisons also availablefrom CSDT n.d.). Two other teachers have conducted eval-uations of the VBL using a pretest to establish a baselineand a posttest to determine if materials made a signicantimpact on the students learning. Both found statisticallysignicant improvement ( p < . 001) in students scores.

WHY DO CSDTS WORK?

The question is perhaps premature. First, the teachers wehave recruited for this work are extraordinary, and it is notclear that the results could be replicated with the average

teacher in the average classroom. Second, we only havesome suggestive preliminary results, not wide-scale test-ing. But assuming that using CSDTs in the classroom actu-ally raise minority student math achievement and improvetheir technological career aspirations, it would be helpfulto understand why. One explanation is simply that we areusing a exible computational medium, which allows stu-dents to pursue inquiry and discovery learning (Brown andCampione 1994; Lewis et al. 2004), and that the culturalcomponent is irrelevant. We do believe that the exibilityand discovery learning aspects are critical to CSDT success,but we also see exibility and creativity as integral to cul-tural identity.

Although many problems in minority studentperformance can be directly attributed to economic


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circumstanceslack of school infrastructure, classroomovercrowding, difculties in the home environment, andso forththere are a cluster of problems that can bedescribed in terms of cultural barriers. Signithia Fordham(1991) and John Ogbu (1998) document the ways inwhich African American students perceive a forced choice

between black identity and high scholastic achievement.For example, high-achieving African American students areoften accused of acting white by their peers (Fryer andTorelli 2005). Rather than suffering from low self-esteem,many minority students maintain high self-esteem byasserting that their authentic cultural identity is in op-position to math and science. Although some researchers(cf. Ainsworth-Darnell and Downey 1998) have critiquedthis framework as conicting with the positive view of education reported on minority attitude surveys, RoslynMickelson (2003) has shown that there is a differencebetween what she terms abstract conceptions of education,to which all racial groups respond positively, and concreteconceptions of education, the responses to which differacross racial groups and correlate with disparity in aca-demic achievement. Danny Martin (2000) reports a similarnding in African American conceptions of the culturalownership of mathematics. Lois Powell (1990) foundthat pervasive mainstream stereotypes of scientists andmathematicians conict with African American culturalorientation. Eglash (2002) describes conicts in the identityof the black nerd in both popular imagination and reality.Additional conicts between African American identityand mathematics education in terms of self-perception,course selection, and career guidance have been noted (cf.Boyer 1983; Hall and Post-Kammer 1987).

Similar assessment of cultural identity conict in ed-ucation has been reported for Native American students(Moore 1994), Latino students (Lockwood and Secada1999), and Pacic Islander students (Kawakami 1995). AnNSF-sponsored study of how minority students are lost inthe science and technology career pipeline (Downey andLucena 1997) is also consistent with these results: Many mi-nority students with good math aptitude reported that theydropped out because their science and technology coursesdid not convey how this material would contribute to theirconcerns for social justice.

In addition to these conicts between cultural identityand mathematics education, a second component for thepoor mathematics performance in these minority groups isexamined in the work of David Geary (1994). His reviewof cross-cultural studies indicates that although children,teachers, and parents in China and Japan tend to view dif-culty with mathematics as a problem of time and effort,their U.S. counterparts attribute differences in mathematicsperformance to innate ability. This myth of genetic deter-minism then becomes a self-fullling prophecy, loweringexpectations and excusing poor performance. In contrast,ethnomathematics directly counters this myth by showingsophisticated mathematical practices in the students her-itage cultures.

In summary, these four cultural featuresthe actingwhite accusation, identity conict, social irrelevance, andmyths of genetic determinismcreate cultural barriers tohigh academic performance by minority students in sub-jects associated with science and technology careers. Wesuspect that CSDTs ameliorate these barriers. Although fur-

ther study will be needed to determine which ones or towhat degree, we see the following possibilities:

1. The acting white phenomenon. It is difcult to ac-cuse someone of acting whiteif they areusing materialsbasedon blackculturedifcult but not impossible.Suchpedagogies always risk the accusation of being patroniz-ing or illegitimately appropriating minority culture, andin some cases those accusations are correct.

2. Identity conict. Ethnomath examples can decreasethe perceived cultural distance between math and cul-tural identity (whether that is an experienced home cul-ture, an imagined heritage culture, or some hybrid). Thedistance can be diminished from either endthat is, stu-dents might change their perception of the minority cul-ture as more mathematical, or change their perception of mathematics as being more cultural.

3. Social irrelevance. Ethnomathematics is particularlyeffective in the context of class discussions of colonial-ism, primitivism, racism, and other histories of stereo-types. Its relevance is thus in its ability to provide alter-native portraits. There are also practical applications of ethnomathematics to design (e.g., architecture). Here thechallenges are commensurate with any such discussion(that some students will argue in favor of primitivism, orargue that such problems are in the past and thereforeirrelevant, etc.).

4. Myths of genetic determinism. Ethnomathematicsoffers strong counterevidence to primitivist and ethno-

centric portraits of simple cultures.

CONCLUSION

CSDTs provide a potent space for students to reconguretheir relations between culture, mathematics, and technol-ogy, and for anthropologists to carry out research in thesesame domains. Of particular interest to us as researchers isthe transition from indigenous and vernacular knowledgein situ to its public space as both education and electronicmedia: What is lost and what is gained in this translation?What are its politics?

One danger of this work is its potential to reify somecultural identities and make others conspicuously absent.The presence of white students in many of these class-rooms has motivated us to investigate how they think of culture, including their own ideas for design tools. One ob-vious contrast has emerged between students who thinkof their whiteness in terms of a specic European ethnicheritage and those who think of it as a more generic fea-ture, a bland homogeneous mix, or even an absence (Waters1990). Our presentation for the Appalachian CollaborativeCenter for Learning and Instruction in Mathematics Edu-cation in November 2003 promoted an ongoing discussionon the possibility of using design tools for Appalachian cul-ture, which is predominantly white but strongly marked by


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class as well (Stewart 1996; see Hartigan 1999 for relatedvariants in an urban setting). Another danger is that ourfocus on ethnicity has set aside important aspects of genderanalysis. Informal observations, for example, suggest thatgirls tend to prefer the cornrows and bead-loom softwareand that boys tend to prefer the grafti software. The lower

rate of participation of girls in the science and technologypipeline, particularly for math-oriented professions such ascomputer science and engineering (Commission on the Ad-vancement of Women and Minorities in Science Engineer-ing andTechnology2000), makes this an important frontierfor our future analysis.

On the positive side, wecan think about CSDTs in termsof the framework provided in the recent anthology Appro- priating Technology (Eglash et al. 2004), which examines awide variety of case studies in which groups at the mar-gins of social power appropriate science and technology forpurposes of resistance and even revolt. Some technologiesare built to prevent user appropriation: cameras, for exam-ple, which will only work with special lm from the man-ufacturer; some software only runs on the manufacturesplatform; and so forth. There is obviously a nancial ad-vantage to creating such lock-in for your customers. Butit is also possible to design technologies for appropriation,as we have attempted to do in the case of CSDTs.

Appropriation, however, is a much broader issue thantechnology design features. It is also an accusation hurledaround concepts of ownership and authenticity in cul-ture (Cuthbert 1998). Reconstituting Bourdieus culturalcapital as computational capital, we then need to askhow we can make sure that CSDTs are making that capitalfungible for its owners, rather than just extracting that cap-ital from them. Should the native beadwork algorithms andcornrow equations be protected by indigenous intellectualproperty rights, as proposed for native biological knowl-edge (Brush 1993)? For the moment, we can only say thatwe maintain it is possible to make respectful, ethical useof those materials, given a dialogue with community rep-resentatives in which we do the following: (1) Ask permis-sion for such use (although no one person could be said tohave sufcient ownership to be the exclusive grantor); (2)contextualize presentation of the materials to show theirassociated histories and meanings; and (3) prioritize use bythe community members themselves (in this case for ed-ucation, but there are other potential applications such asarchitecture, graphic design, etc.).

Our preliminary data appears promising. The trialec-tic of computer media, math pedagogy, and culture pro-vides a meeting place in which the praxis of social changeand the theory of cultural critique can generate new formsof hybridity and synthesis.

RON EGLASH Department of Science and Technology Stud-ies, Sage Labs, Rensselaer Polytechnic Institute, Troy, NY12180-3590

AUDREY BENNETT Department of Language, Literature andCommunication, Sage Labs, Rensselaer Polytechnic Insti-tute, Troy, NY 12180-3590CASEY OD ONNELL Department of Science and TechnologyStudies, Sage Labs, Rensselaer Polytechnic Institute, Troy,NY 12180-3590

SYBILLYN JENNINGS Department of Psychology, Russell SageCollege, Troy, NY 12180M ARGARET CINTORINO Independent Scholar

NOTES Acknowledgments. This material is based on work supportedby theNSF under Grant No. 0119880.1. To further complicate matters, intentionality itself is culturallyconstructed, and the Western focus on individualism may clashwith indigenous concepts of collective intentionality, particu-larly forstructures created over several generations.See Eglash1999for further discussion of this issue.2. Here we are drawing a strict contrast for comparative purposes;in the literature this distinction is often blurred, and in some casesauthors include ethnomathematics as a subset of multiculturalmathematics.3. Not to mention the problems in historical accuracy: but seefor comparison critique of the Portland Baseline Essays in Martel1994 and Oritz de Montellano 1993.4. Andrea Kelly, a graduate student at University of Colorado, col-lected a variety of such statements for her dissertation on CSDTs.On the one hand, she found that follow-up questions could un-pack the statements and reveal more nuanced thinking; on theother hand, she found that there was still a conspicuous absenceof certain frameworks. For example, few of the teachers mentionedany problems akin to cultural or biological determinism, and nonementioned anything like Fordham (1991) and Ogbus (1998) peer-proong thesis. See section Why Do CSDTs Work? below for fur-ther discussion.5. Outside of anthropology, it is often assumed that these math-ematical design themes only exist in traditional cultures, and thatcontemporary structures merely reect natural laws, rationality, orefciency. One interesting counterexample to this invisibility isBuckminster Fullers (1963) discovery that despite their increasedstrength to weight and cost to volume ratios, geodesic domes weretoo geometricallydifferent from the U.S. design theme of Cartesiangrids in materials, construction methods, architectural aesthetics,and land plots to have popular usage.6. Inadditionto several years of software development andculturalresearch, the following discussion makes use of eldwork carriedout from 200105 in summer, in-school, and after-school work-shops in Albany and Troy, New York, and Chicago, Illinois, withstudents ranging from middle school to high school, in coursesthat included mathematics, art, English, and design. All studentswere from low-income families; the majority were African Ameri-can, others were primarily Latino, although there were a few white

students as well.7. Of course iterative scaling can be used to model patterns inseveral other cultures as wellSouthern India, for one examplebut at this level of education, it is sufcient for students to havethis limited answer.8. Pickerings best example of contingency is probably the historyof Leonhard Eulers formula for polyhedra. In 1752, Euler proposeda relation of vertices (V), edges (E), and faces (F) for all polyhedra:V E + F = 2. In 1813, Simon Lhuilier found that the formuladid not hold for polyhedra with holes going through them, butit was generally agreed to restrict the formula to polyhedra with-out holes. In 1815, Johan Hessel noted that a cube with a cubichollow inside does not satisfy Eulers theorem. This produced acontroversy in mathematics: Should we give up Eulers theoremor redene polyhedra? The monster-barrers won out: Polyhedrawas redened as surface[s] made up of polygonal faces. Then in


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1865, August Mobius noted that two pyramids joined at the vertexalso dees Eulers theorem. Again arose a controversy in mathe-matics: Shouldwe give up Eulers theorem or re-redene polyhedra? Polyhedra are then redened as a system of polygons such that twopolygons meet at every edge and where it is possible to get fromone face to the other without passing through a vertex. At eachdecision point, there were two viable solutions to the resistanceencountered (to paraphrase Pickering, a resistance against the at-tempt to capture or frame mathematical agency): one of which is apath not traveled, a virtual mathematics that we could have todaybut do not.9. Of course, this triumphal chauvinism-in-reverse (as one of the AA reviewers of this article termed it) is not a desirable conclusion.It may be, however, a necessary stage in some students intellectualgrowth, but we hope teachers will be able to move them beyondthat.

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