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Purpose - The purpose of this paper is to examine indigenous ethnomathematical ideas that thrive in existing traditional technologies and which are still used by South African indigenous and local people as means of survival and adaptation in a variety of environments.

Design/methodology/approach - Indigenous Knowledge Systems (IKS) has been recently established by the international world organizations as a top global priority for empowering traditional and local communities in their quest for survival and development. However, despite its highly proclaimed importance and sound pedigree of recognition for strengthening indigenous communities' preservation of social and traditional capitals towards more independence, no clear effort has been cited that magnifies and brings to light the contributions of indigenous cultures to the mainstream knowledge and epistemologies. This paper highlights some of these knowledge systems as prevalent in the beadwork and basket ware practices of the Zulu culture in South Africa. Initial data will be collected via ethnographic techniques where researchers will conduct focused participant-observations consisting of making field notes, interviews and informal conversations with different cultural actors and key knowledge holders as well as artifact collection.

Findings - The present study capitalizes on the role of indigenous knowledge (IK) structures and associated technologies in enriching and diversifying the teaching and learning of mathematics. Inspired from problem-based authentic situations traditional technologies present information media that preserve as well as epitomize the role that IK plays in cognitive development and learning in non conventional contexts. Such technologies that include beadwork and basket ware acquaint students with a broad perspective on knowledge systems and structures that cover a plethora of contents and contexts that incorporate mathematical artifices.

Research limitations/implications - The present study focuses on the riches of two traditional technologies as a medium for transmitting specialized mathematical knowledge produced and disseminated by indigenous people. Prospective work that investigates other indigenous technologies with underlying ethnomathematical ideas can be highly enlightening and reflect major breakthroughs in the field on indigenous mathematical knowledges.

Practical implications - The article addresses the role that indigenous technologies play in unfolding a specialized body of tacit knowledges continuously transformed and adapted in synchronization with internal and external circumstances shared within a community.

Originality/value - This work provides a novel stand that challenges the traditional epistemology of mathematical knowledge. By accentuating the use of traditional technologies, the paper justifies the humanistic perspective of what it means to do mathematics in the cause of survival and amidst limited resources and affordances.

Keywords Traditional technologies, Culture, Education, Ethnomathematics, Zulu Culture , South Africa

Indigenous Mathematical Knowledge Systems.

Ethnomathematics

As a new conceptual field and an emerging research program, ethnomathematics evolved from a rhetoric networking the interplay among mathematics, cultural anthropology, education, and politics.

While too many definitions and associated perspectives prevail in the literature attempting to delineate ethnomathematics, two dominant conceptualizations are in use. D'Ambrosio (1985), who coined the term, insists on ethnomathematics as a rather broad conceptualization of mathematics and defines it as embracing "…all culturally identifiable groups with their jargons, codes, symbols, myths, and even specific ways of reasoning and inferring" (p. 17). Furthermore, D'Ambrosio(2001) explains: " Ethnomathematics is the mathematics practiced by cultural groups, such as urban and rural communities, groups of worker, professional classes, children in a given age group, indigenous societies, and so many other groups that are identified by the objectives and traditions common to these groups" (p. 1). In this perspective, ethnomathematics exists at the connection between mathematics history and cultural anthropology. Additionally, D'Ambrosio(1999) described ethnomathematics as: " …a program in history and epistemology with an intrinsic pedagogical action….[it] responds to a broader conception of mathematics, taking into account the cultural differences that have determined the cultural evolution of human mankind and political dimensions of mathematics." (p. 150). In this respect, ethnomathematics draws on the belief that humans, from prehistoric ages, have been accumulating knowledge to respond to their drives and needs. Such knowledge varies from region to region and from culture to culture and is transmitted through generations in a more casual and less formal way.

Ascher( 1990), on the other hand, focuses more on the mathematics of indigenous societies and defines ethnomathematics as " the study of the mathematical ideas of non-literate people" (p.125) . Moreover, Ascher (2002) indicates that there are 6000 cultures that existed within the last 500 years and capitalizes on the riches that can be cultivated from studying cultural practices and heritage. Ascher (2002) proposes those diverse cultures including the Inuit, Iroquois, and Navajo of North America; the Incas of South America; and the Bushoong, Kpelle and Tshokwe of Africa; in addition to the Caroline Islanders, Malekula, Maori, Warlpiri, and Trobriand Islanders of Oceania and many others enrich and add nuances to a semi-structured core of mathematical knowledge.

Inspired by both definitions of ethnomathematics, various concepts have been proposed in discrepancy with the 'academic mathematics' or 'school mathematics' (Western-based mathematics). These have been given different names such as Indigenous mathematics ; Sociomathematics (of Africa) ( Zaslavsky, 1973); Informal mathematics; Spontaneous mathematics ; Oral mathematics; Oppressed mathematics (as inspired from works of Freire, 1970); Folk mathematics; and Non-standard mathematics (Gerdes, 1994; Carraher, 1985; Jurdak & Shahin, 1999; Jurdak & Shahin, 2001) in which different mathematical forms have developed on the streets, outside the school context .

One critical paradigm promoted by proponents of ethnomathmeatics is the belief that all peoples are capable of doing mathematics in their own unique and personal perspective. Concomitant with such a framework is the view that ethnomathematics emerges from within individuals while interacting with their cultural and physical environments. Nunes, Schliemann & Carraher (1993) argue that ethnomathematics develops when there is a discrepancy between people's need for problem solving and the level of mathematics they have learned in school i.e., when people become involved in tasks requiring problem solving skills that are not learned in school. In this argument, ethnomathematics is believed to be also closely tied to issues of access and equity (Anderson, 1997).

While it is recognized that definitions of ethnomathematics are contested, in some cases a figment and tool of political and social ideologies, however, one common theme essentializes the field namely that ethnomathematics employs the observation of mathematics that emerges across cultures as a creative expression of human thought.

Research on Ethnomathematics

Considering the fact that the controversial view of mathematics as "universal" has been largely dominant, research on ethnomathematics emerged later than other ethnosciences. Rist & Dahdouh-Guebas, 2006) defined ethnosciences as " a scientific realm which aims to understand how humans--in spite of their fragmented and limited interactions with the world--are developing different forms of knowledge and beliefs" (p. 472). Likewise, ethnomathematics calls for systematically taking into account the flux of mathematical ideas that emerge as humans interact and make sense of their environment.

However, studies on ethnomathematics have been condensing for more than three decades. Forerunners of ethnomathematics include mathematicians, ethnographers, psychologists, anthropologists, and educationalists. Such a repertoire of research, which has been launched in the cause of justifying ethnomathematics as a new, legitimate field of research on the international academic panorama and not a mere fleeting fad, has its roots in the writings of Lévy-Brühl. In his publication, How Natives Think, Lévy-Brühl (1966) suggests a divided world of thought between literates and "non-literates" where he described non-literate people as "pre-logical", primitive and mystical rather than logical and thus incapable of analytic reasoning. He further explains that the minds of primitives do not function as those of the literate people. Within few years of his writings, the ideas of Lévy-Brühl entered the mathematical literature in an extensive, uncritical manner.

Other counters to Lévy-Brühl's ideas challenged the classical evolutionist assumption that "non-literate" peoples are the earliest living humans along a straight, hierarchical, evolutionary path which ranged from savagery to civilization. Such countering views advocate the claim that different cultures exhibit a universal "natural rationality" that enables learning by making inferences from daily experiences. In this respect, Ascher & Ascher (1997) disputed the views of Tylor (1874) in his famous book Primitive cultures and that of Conant (1896), in his book The number concepts: Its origin and development which claim that the mathematical thought of non-literate peoples centers on number as number words and that their number comprehension is compared to a number sense found in other animals. Ascher & Ascher (1997) emphasize the role of language in providing insight into understanding the diversity and richness of different cultures and guard against erroneously inferring that the use of different number words for different categories of objects is an evidence of a lack of an abstract concept of number.

Extensive researches that emphasized the role of cultural practices in analyzing the relation between culture and cognition have been inspired from ethnographic literature. In his book, Mathematics as a Cultural System, Wilder (1981) asserts that each culture has its own mathematics which evolves and perishes with that culture. Such studies, and many others, reinforced the belief that the category "mathematics" is rather specific and that perhaps we should not expect to find anything so labeled by other peoples in other cultures. Mathematical ideas will have to exist implicit in other areas and activities and can be expressed in multiple and broadly diverse ways.

Orality:

At its inception, the evolution of ethnomathematics as an epistemology can be perceived as a testimony to a situation reflecting the competition between literacy and orality. While literacy in the information age requires the competency to assimilate information presented in words, sentences, numbers, tables, graphs, and diagrams, orality or orally-based information systems, on the other hand, signify collective expressions of specialized knoweldges that belong to a particular community of people. The vehicles by which such body of knowledge is preserved and transmitted are inextricably determined by formats fashioned and enacted by the community itself (Tisani, 1994). In the Zulu culture of South Africa, orally-transmitted and tactile-based traditional technologies play a major role in the production, dissemination and preservation of a rich body of information that is concealed and estranged from a foreign viewer and inextricably linked to everyday life of the people. The only way to decode the syntactic of this information system and retrieve meaning is to penetrate the Zulu culture's systems of social organization, symbolism, belief, and ideology. In this article, we will focus on beadwork and basket ware as two illustrations of existing traditional technologies that thrive in the Zululand and are popular among the women and which represent an intimate disclosure of ethnomathematical practices that signal a highly specialized knowledge that is owned only by the practitioners. The basic motivation…

literacy had greatly affected many aspects of Western society.

The gradual replacement of the oral

manner of transmission, of thinking and expression, and the speed at which the

transition from oral to literate occurred became the concern of yet other classicists

and philologists who attempted to trace the linguistic signature of orality in the

Jousse advances the

claim that among the primitives (whom he prefers to call "spontaneous") the oral

rendition of wisdom was, indeed, spontaneous and rhythmic, a pure expression of

the esprit du peuple, lost once writing was introduced.

[Jousse's] anthropological work on human expression led him deeper

into the "play" of mimicry and to distinguish between "cinemimism" or

mimicry of gestures, and "phonomimism" or mimicry of sounds, which

became language in the etymological sense of significative laryngobuccal

gesticulation. By echoing the ear, themouth becomes the resonator

of the sound of things.54

By reducing all the data about oral societies, recitation, and memory to one common "verbo-motor" function of the human psyche, Jousse is able to explain how humans without writing process and

transmit information.

the new research shows that oral performers are able to write down their own texts in

the manner in which they expect to perform: this ability explains the "oral" features of written poetry. Thirdly, it shows that literate singers can become masters of a complex Kunstsprache, the artificial language that manages to defy the linguistic laws of analogy and regular change over time.60

Zulu culture

I visited Durban South Africa to learn about the culture and to examine firsthand the intrinsic cultural artifacts including beadwork and basket ware that thrive on the streets of the coast of Port Natal in Durban. While walking on the beach front in Durban, I had the opportunity to converse with some Zulu women, who exhibited their beadwork and basket ware on the pavement under the shaded huts. The women who were so proud of their artistic work were keen on explaining the variety of formats and techniques that they use in the bead-woven bracelets, necklaces, picture frames, belts, bookmarkers, and head bands. The myriad techniques such as netting, wrapping fringing, braiding, and weaving introduced with many vivid designs and bright colors embody artistic expressions of social narratives and cultural discourses. As such the geometric structures and concepts such as symmetries and transformations prevalent in the beadwork designs represent cultural codes that convey expressions of thought from the Zulu ancient traditions and hold the key to intrinsic knowledge systems.

Beadwork technologies: Semantics and syntax

In the realm of oral traditions, the technology of bead working represents a form of an intercultural visual language that textualizes geometric patterns through ideas are communicated and exchanged vividly and meaningfully. As such this orally-based, naturalistic system of visual representations acts a mode of information dissemination but also as an embodiment of thought and belief systems drawn from lived experiences of ancestors across many generations. The role of these traditional technology systems imbibed with metaphoric connotations is significant in the meaningful portrayal, preservation and transmittal of complex insightful solutions created by the Zulu culture in response to emerging problems. The problems can range from everyday endeavors such as making embroidered and beaded covers for dishes to keep the flies away to more complex issues that addresses fertility (e.g. fertility dolls), individual identity as members of the community, and the need for survival (Sienaert, Cowper-Lewis, & Bell, 1994).

***

A structural analysis of one single motif in a beadwork pattern reveals a complex hybrid of intertwined multi-dimensional layering of information which, if appropriately decoded, helps interpret and make explicit the embedded meaning in a single motif. The first layer embodies the physical and material texture of the pattern as different materials might be incorporated in the design. In additional to glass beads which are most commonly used in South Africa today, other kinds of beads are locally made from wood, shells, animal teeth, seeds, metal rings and clay. The choice of the material is purposive and signifies the tone, depth, and hierarchy in power relations and social statuses. The second layer consists of an assortment of discursive visual patterns or structural units such as dots, lines, triangles, symmetries, dimensions, scales, angles, and proportions. The third layer comprises the colors which act as signifiers of visual metaphors indicating cultural position and personal and social accomplishments as well as provide a deeper, contextualized meaning. It's in the synergy of these layers that information on the identity of the individual who wore them viz. the gender, marital status, social status, occupation, and age (Morris & Preston-Whyte, 1994) is visually synchronized and transmitted to the viewer. The recurrence and nesting of motifs constitute a fluid visual narrative that represent momentous disclosures communicating a story, a social or religious event, or that reveals…

In terms of functionality, this visually-based information system is inherently diverse and serves many purposes. Be it as an ornament, a jewelry piece, a textual diction (for example, a Zulu letter), or a commemorative object, these functionalities provide various venues for exploring different narratives by virtue of its association with the context.

A common geometric motif that thrives in most of the Zulu Beadwork is the 'stepped diamond'. Constructed from the triangle as a "basic Zulu design unit", the diamond represents a signature image template as beadworkers begin experimenting with symmetry and learn to play with it" (Papini, 1994). Being a recurrent geometric structure across many

Basket ware technologies:

Common to all of them are the richness and intricacy of their patterns, produced with a limited range of colors which have symbolic meanings. Brilliant visual effects are created in geometric designs of diamonds, chevrons and zigzags. A knowledge of the local color code used in the beadwork is necessary before one can read the message in the tabs and strings. Regina Twala did field work in 1948 on the cipher and colors used in beadwork by the Emangwaneni tribes of the Bergville district of Natal, but her interpretation of the color codes often contradicts that of Rev. Mayr who also wrote from personal observation in 1907. Mayr, however, did not record from which groups of Zulu he drew his infor-mation; he seems to have assumed that the color symbolism was standard throughout the Zulu world and stated that "... the actual pattern does not appear to have any defined sig-nificance; it is rather the succession of the color and the rela-tive amounts of the colors, that express the tenor of the message." I Twala, however, felt that the interpretation of the colors varied with the pattern. Also, according to Mayr the border was merely decorative and the beaded string the most important message bearer, while Twala believed that the main message was in the tab. Regional variations and the difference in the dates of investigation are very likely re-sponsible for these discrepancies. However, certain colors seem to have retained general meanings which were shared by all Zulu. For instance, opaque white beads, Ihambo or "bone," stood for purity of love; pink symbolized poverty; vaseline-yellow signified wealth; and blue symbolized the dove. Mayr interpreted a string of beads in the following manner: "My heart is pure and white in the long weary days (white beads); I have be-come quite lean and sickly (green beads); If I were a dove I would fly to your home and pick up food at your door (blue beads); Darkness prevents my coming to you (black beads).'2 The entire message is repeated a number of times. The fol-lowing is Twala's interpretation of a design according to the physical arrangement of the beads: "(a) WHITE ... I say this with an open white heart. (b) BLUE ... I say, Oh for the dove that picks food (c) WHITE ... In the yard at your kraal. (d) RED ... I envy also the one who enjoys your fire-place. (e) WHITE ... Although my heart may be pure. (f) PINK... You are poor.

Despite apparent general similarities in meanings of col-ors, accurate interpretations can be made only by one who knows the exact local origin of the love letter and the color code peculiar to that place. Unfortunately, lacking a more specific provenance than the vast Transvaal, the color code to the pieces in the Fleming Museum must be considered lost. In many pieces throughout the collection as a whole, and especially in the love letters, some odd beads appeared to create a certain tension or imbalance in an otherwide regular pattern. Usually these stray beads were red, although occas-ionally blue or pink, and they occurred either singly or by twos. Since they are found most often in the love letters, it may be possible that they formed part of the message. How-ever, the beads may also have been deliberately placed to break the repetitive rhythm of a design on either aesthetic or magical grounds, or both

Basket weaving:

In the last decade, there has been a growing worldwide interest in South African beadwork. In Zulu Inspired Beadwork, the first book devoted to the beading techniques used by Zulu women, bead master Diane Fitzgerald shares her expertise on Zulu beadwork with 25 stunning projects, celebrating the culture of this indigenous population.

Begin with an introduction to Zulus and their beadwork, an inside look at the importance of beads in this South African culture, and the many beading techniques-some of which have never been published until now-used by Zulu women to create adornment. The author shares several dozen unique beading techniques garnered from years of visiting South Africa and countless hours spent examining-and dissecting-Zulu beadwork.

Next move into projects inspired by the author's visits to Africa, including netted diamond earrings, a netted triangle and swag bracelet, zigzag chain, and a Zulu wedding necklace. Techniques include netting, wrapping, fringing, braiding, and weaving. Projects are illustrated with easy-to-follow diagrams and supplemented with helpful hints. Readers will find gorgeous photographs of original Zulu beadwork juxtaposed with the author's interpretation of the design and techniques. Part how-to, part history, part travelogue, Zulu Inspired Beadwork is a beading journey in a book.

Reviews: "It's a stunner: her best yet. This book is a marvelous illustration of the inventiveness of beaders and what can be done with those beautiful little seed beads."-Bead (UK) magazine

Table of Contents

Foreword (Richard Green)- Ethnologist and Collector, Birmingham, England

Introduction- Zulu beadwork is multifaceted and has been studied, written about, and exhibited for cultural, aesthetic, and historic reason. However, I believe this book is the first to be devoted to the bead weaving techniques used by Zulu women. The skill and intricacy exhibited in Zulu beadwork is unlike that of any other culture I have see. -Diane Fitzgerald

Zulus and Their Beadwork- Beadwork is deeply embedded in the culture of the Zulu people of South Africa. It is unique and distinctive in its colors and patterns, and particularly in its structure.

Zulu Beading Techniques- On the following pages, you'll find instructions for the many techniques I discovered on my journey into the world of Zulu beadwork. I've included a photograph of the original Zulu pieces from which I deciphered the weaving methods along with instructions for contemporary projects.

Projects

Flowerette Chain- The Zulu Flowerette Chain is often the class favorite when I teach Zulu techniques.

Spearhead Chain- While this pattern looks symmetrical, each side is worked differently.

Lace Leaf Chain- The Lace Leaf Chain is worked in two rows.

Zigzag Chain- Diane found this Zigzag chain on a necklace supporting a bead-covered horn tip.

Ladder Chain- This is another typical "looped" Zulu technique, one that may be mistaken for tubular netting.

Tri-Leg Chain- Look at this chain from the end and you'll see that it has three legs that stand out from the center like a Y.

Square Tube- This chain looks great worked in one color or with two highly contrasting colors. It is a highly popular chain among Zulu beaders.

Triangle Tube- This delightful chain yielded a surprise when I took it apart.

Slinky Chain- This chain is a variation of the Square Tube and the Triangle Chain.

Double Weave- This chain is reproduced from a Zulu armband made of white, blue, red, and black beads.

Wrapped Rope- Wrapping a rope made of tightly rolled cloth with beads is a simple and quick way to make a beaded bracelet or necklace.

Netted Triangles and Swags- This graceful design, made of triangles and swags, can be made into a necklace or bracelet by varying the spacing.

Switchback Chain- The author's husband named this piece "switchback" because it reminded him of the steep mountain road they took to Machu Pichu, the famed ruins in Peru.

Bow Tie Chain- This piece is a variation of netting.

Trefoil Chain- This is a deceptively simple piece to create.

Hexagon Netting- Hexagon netting is found in many types of Zulu beadwork.

Netted Diamonds- This design was inspired by an old Zulu belt made with netted diamonds of black, white, and red seed beads dangling from a netted band.

Popcorn Stitch- The Popcorn Stitch is made with all size 6 seed beads so it goes fast and is fun to make.

Fingo Chain- The Fingo or Mfengu people are a subgroup of the Zulus who live in the Eastern Cape of South Africa. This bracelet pattern, which is derived from a Fingo Girl's Belt, is worked in two layers.

Zulu Love Letter Pins- The Zulu Love Letter pin is probably the most widely known item of Zulu beadwork.

African Netting Stitches- The following two projects are worked with adaptations of an unusual stitch of Zulu origin.

Ngwane Fringes- The Ngwane, a subgroup of the Zulus, make densely beaded capes and aprons with row upon row of a variation of brick-stitch fringe on fabric.

Everyday mathematics

Investigations that have generally focused on studying people's use of math outside the classroom (Gay & Cole, 1967; Scribner, 1986; Lave, 1988; Saxe, 1991; Millroy, 1992) are divided into two main groups namely, those interested in "everyday cognition" or "cognition in practice", where Lave is a prominent figure, and those interested in "ethnomathematics", where D'ambrosio is a key figure. Both groups of researchers call for a new conceptualization of math that is rooted in nonacademic practices. For instance, some research investigated the mathematical problem solving of Kpelle farmers of Central Liberia (Gay & Cole, 1967), while others examined the Southern California house wives' strategies for finding best buys in the supermarket (Lave, 1988). A second study explored the dairy workers' strategies for assembling products, pricing delivery tickets, and taking inventory(Scribner, 1986),while another inspected the arithmetic practices of Brazilian candy sellers (Saxe, 1991), and another investigated the mathematical ideas of carpenters in South Africa (Millroy, 1992). The work of these groups focuses on three main issues: Analysis of school practice, investigation of the transfer of school knowledge to out-of-school situations, and using the social theory of practice to challenge conventional cognitive theory.

Ethnosciences--A step towards the integration

of scientific and indigenous forms of knowledge

in the management of natural resources for the future

Stephan Rist Æ Farid Dahdouh-Guebas

Received: 18 November 2005 / Accepted: 30 January 2006 /

Published online: 3 June 2006

_ Springer Science+Business Media B.V. 2006

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