Nature of mathematical knowledge and peoples views on it

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The date is December, 1790; the place is Philadephia; the event is the obituary of Thomas Fuller, Died-NEGRO TOM, the famous African Calculator, aged 80 years. He was the property of Mrs Elizabeth Cox of Alexandria. Tom was a very black man. He was brought to this country at the age of 14, and was sold as a slave with many of his unfortunate Countrymen. This man was a prodigy. Though he could neither read nor write, he had perfectly acquired the art of enumeration. The power of recollection and the strength of memory were so complete in him, that he could multiply seven into itself, that product by 7, and the product, so produced, by seven, for seven times. He could give the number of months, days, weeks, hours, minutes, and seconds in any period of time that any person chose to mention, allowing in his calculation for all the leap years that happened in the time; and would give the number of poles, yards, feet, inches, and barley-corns in any given distance, say the diameter of the earth's orbit; and in every calculation he would produce the true answer in less time than ninety-nine men in a hundred would produce with their pens. And, what was, perhaps, more extraordinary, though interrupted in the progress of his calculation, and engaged in discourse upon any other subject, his operations were not thereby in the least deranged, so as to make it necessary for him to begin again, but he would go on from where he had left off, and could give any, or all, of the stages through which the calculation had passed. His first essay in numbers was counting the hairs in the tails of the cows and horses which he was set to keep. With little instruction he would have been able to cast up plats of land. He took great notice of the lines of land which he had seen surveyed. He drew just conclusions from facts; surprisingly so, for his opportunities. Thus died Negro Tom, this self-taught Arithmetician, this untutored Scholar!-Had his opportunity of improvement been equal to those of thousands of his fellow-men, neither the Royal Society of London, the Academy of Sciences at Paris, nor even a NEWTON himself, need have been ashamed to acknowledge him a Brother in Science. The date is July, 2007; the place is University of Dakar; the event is the French president Nicolas Sarkozy's compelling speech which was directed to the elite young African college students at the University. 'Le drame de l'Afrique, c'est que l'homme africain n'est pas assez entre´ dans l'histoire. Le paysan africain, qui depuis des mille´naires, vit avex les saisons, dont l'ide´al de vie est d' ˆ etre en harmonie avec la nature, ne connaˆıt que l'e´ternel recommencement du temps rythme´ par la re´pe´tition sans fin des mˆemes gestes et des mˆemes paroles.' 'The tragedy of Africa is that the African man has not fully entered into history. The African peasant, who for thousands of years has lived according to the seasons, whose life ideal was to be in harmony with nature, only knew the eternal renewal of time, in rhythm with the endless repetition of the same gestures and the same words.'1 With these words, Sarkozy positions Africa on the threshold of history, identifies it with rural life and a pure and natural state of humanity, and describes it as locked in the eternal return of the same. The feeling that one might have read these words before is confirmed by Achille Mbembe, who points out that Sarkozy - or rather his scriptwriter Henri Guaino - 'contented himself to lifting, almost word for word, passages from the chapter Hegel devotes to Africa'.2 Sarkozy's speech provides a good opportunity to reconsider the relationship between the colonial view of Africa that is compellingly captured in Hegel's writings and its current, post-colonial life. To show that racist representations of Africa still prevail, however, is only one side of the coin. The other, far more problematic, issue concerns the entanglement of Africa's exclusion with Europe's modernity. In this article, I argue that the uncanny return of Hegel's Africa ought not be characterized only as the problem of Europe's new right. Through a close reading of Hegel, I show that Africa's position on the limit of history is crucial for the self-understanding of modernity and thus deeply entangled with current notions of history, development and progress. The spirit of Eurocentric modernity that continues to recur in prominent political discourses brings back memories not only of intentional racist positioning but of a deliberate devaluation of other less "affluent" societies. The quest for liberation through transformative educational endeavors seems almost certainly to be a figment of pure imagination. The same discourse proliferates to one of the most globalised disciplines vis. Mathematics. D'Amborsio states that perhaps one of the most serious problems facing humanity is survival with dignity. This chapter is not dedicated to only brilliant African American children who have been largely disadvantaged by existing educational systems that do not recognize their full potentials. This chapter is a tribune to children in many marginalized societies in many countries all over the world that are caught in never ending political, social, and ideological struggles. What does it really mean to atlk about educating African Amererian Children? What talking about educating any race/ ethnioc group one will fall in the trap of essentailizing edcauiotn for e The most mentally charged debate in the history of mathematics revolves around the nature of mathematical knowledge. While some historians adopt a positivistic view of the nature of mathematical knowledge (Cajori, 1991), others believe that knowledge, in general, and mathematical knowledge in particular, is far from being an unproblematic, static concept (Furinghetti & Radford, 2002). As an important tool in globalization , mathematics has achieved a status of an international language independent of cultural affiliation and context of development (English, 2002). In many societies that have been recovering from the debris of colonization, mathematics is still considered as a major barrier for access to power resources. Additionally, exponential advances in technology is constantly contributing to the dominant knowledge-power nexus by improving the life of those who have access and control over it while marginalizing others. (Fasheh, 1997) As a reaction to traditional Eurocentric views, Joseph (1997) suggested an alternative unbiased trail to the history of mathematical development. This new trajectory shifted the attribution of mathematical knowledge from Western Europe to a recognized contribution of different cultures particularly emphasizing the role of Arabs in refinement and dissemination of mathematical knowledge. However, efforts fall short in acknowledging the vibrant body of mathematical knowledge implicit in traditional or small-scale cultures, lesser known to the mathematical community. Ascher (2002) indicates that there 6000 cultures that existed within the last 500 years and she capitalizes on the riches that can be cultivated from studying cultural practices and heritage. Cultures such as 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 Ascher(2002) explains, enrich and add nuances to mathematical knowledge. In this context, 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); Informal mathematics; Spontaneous mathematics ; Oral mathematics; Oppressed mathematics (of the Third World during the colonial occupation, that were not recognized as mathematics by the dominant ideology inspired from work of Freire, 1970) ; and Non-standard mathematics (Carraher, 1985; Jurdak & Shahin, 1999) in which other mathematical forms have developed in the 'streets', outside the school context . The role of Ethnomathematics D'Ambrosio(1999) coined the term ethnomathematics and defined it 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). Ethnomathematics draws on the belief that humans, from prehistoric ages, have been accumulating knowledges 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 way. (Borba, 1997) Perhaps one of the most important philosophical difference between a traditional and an ethnomathematical perspective is that ethnomathematics promotes the belief that all people are capable of doing mathematics in their own unique and personal perspective. As a result, ethnomathematics is seen as emerging 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 amount 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 closely tied to issues of access and equity. (Anderson, 1997) With a history fraught with conflicts and contradictions, and seen from a multiple lens perspective, the question of what mathematics do we teach in our diverse classrooms becomes of an unprecedented importance. D'Ambrosio(1985) also proposes a broader conceptualization of mathematics which encompasses mathematical knowledge of diverse cultural groups that sustain democratic access of knowledge by all students irrespective of their socio-cultural backgrounds. Additionally, Ascher (1991) explains that broadening the perspective to encompass other cultures extends the contribution of all classes of societies and not only the elite. Throughout history, math has been viewed as a catalyst fueling the debate on the knowledge power nexus and contributing to the production of a number of social and political conflicts (Vithal & Valero, 2003). Major conflicts that prevail until our present day including poverty and wealth, gender inequality and social injustice have been traced to the origin of modern Western world. A closer look at the history of mathematics provides a rationale for the role that mathematics has played in the process of globalization. Racism, sexism, and social and ideological injustices are nothing but the creation of colonial prejudices denying communities and cultures their history and intellectual heritage over the years. If we accept the claim that mathematics is a ticket for aspiring individuals and countries for technological and hence economic development then we should demand that all peoples should have the right and proper means to access it. We have witnessed, and are still witnessing, how our humanity is slipping away under the rhetoric of progress and under the "drums of triumphant technology at the expense of ecology, culture, and peoples". (Fasheh, 1997). Moreover, if we believe in what it means to be mathematically literate, then perhaps we need an inclusive vision of mathematics, a pedagogy built around a set of core values that enhance life and protect humanity. More importantly, we perhaps need a mathematics pedagogy that reclaims our cultural spaces by "… developing more than just mathematicians in the very strict sense of the word, [but] critical intellectuals who are scientists, who are not only apt in their discipline, but also see the work that they are doing as connected to the society they're in, and see their society as connected to other societies on the planet." (Powell & Frankenstein, 1997).

Research on the impact of using culturally relevant material

"A common misconception in the teaching of math has been, and still is, the belief that math can be taught effectively and meaningfully without relating it to culture or to the individual student". (Fasheh, 1997, p.281) The impact of using ethnomathematics techniques and practices in the mathematics classroom is far from being speculative. A handful of research has shown the importance of integrating cultural practices that resonate with students' ethnic and background experiences in everyday instruction. As a matter of fact, in a final report of The NCTM Achievement Gap Task Force (October 2004), the committee recognized that " … the achievement gap is not a result of membership in any group, but rather is the result of the systematic mistreatment of learners caused by racial and class bias-conscious and unconscious, blatant and subtle, personal and institutionalized. There is plentiful evidence of deep structural injustices in how the U.S. school system distributes opportunities to learn mathematics" (pp.2-3). One of the committee's recommendations is to expand efforts to examine research areas that integrate race, ethnicity, social class, and language issues pertinent to closing the mathematics achievement gap. The claim that has been forwarded is that children from diverse backgrounds have different modes of thinking, possess diverse perceptual abilities and spend differential efforts on tasks depending on personal criteria of perceived usefulness. (Lamon, 2003) In his book, the Geography of Thought , Nisbett(2003) expands the view that people coming from different ecologies and social structures hold different cognitive and affective systems. In striking a comparison between Easterners and Westerners Nisbett(2003) explains: " The collective or interdependent nature of Asian society is consistent with Asians' broad, contextual view of the world and their belief that events are highly complex and determined by many factors. The individualistic or independent nature of Western society seem consistent with the Western focus on particular objects in isolation from their context and with their Westerners' belief that they can know the rules governing objects and therefore can control the objects' behaviour" (p.17). With such diversity in modes of thinking and in abilities, all previous views seem to support the claim that there is no universal way to teach mathematics for all children. More interestingly, perhaps, is the belief that even mathematics has no generally agreed upon definition. The only truth is that "…the category 'mathematics' is Western and so is not to be found in traditional cultures". (Ascher, 1991, p.3) However, there have been many success stories in the literature that reports enhanced performance when cultural practices from ethnomathematics are integrated in the math instruction. In a study investigating the effect of using African culture on African students' achievement in geometry, Snipes (2005) implemented ethnomathemaitcs practices with 107 5th grade students in a public elementary school. Classes were randomly assigned to either the Mathematics With Culture (MWC) group or the Mathematics Without Culture (MWOC) group. The researcher served as the teacher for both groups in order to minimize teacher effects. The mathematics lessons lasted approximately 120 minutes for both groups. Both groups completed an entire geometry unit created by the researcher. The MWC group completed the geometry unit with some facet of African culture integrated into each activity. The MWOC group completed the same unit without the African culture component. Results of the study showed that students' achievement scores increased as they learned about African culture. Information from this study contributes to the growing role that culture can play in the mathematics classroom and in African American students' achievement. This study, as well as many others that advocate the use of ethnomathematics in the classroom, promotes NCTM's equity principle and calls for raising self esteem of students and thus empowering them by acknowledging their cultural backgrounds . In a similar study, Ensign (2003) describes the efforts of integrating personal, every day practices of students into classroom mathematics in an urban school setting. The author describes the culturally related technique of teaching used as "culturally connected" which aimed at finding creative and meaningful problems that help mediate difficulty that students face in understanding mathematical concepts. The study was carried out with second, third, and fifth graders in two urban schools in New Haven, Connecticut. Results reported gave evidence of a raised interest on the part of students in solving their personal math problems than those cited in their texts. In addition to social and emotional effects, results of students' tests indicated a trend toward higher test scores on text publishers' unit tests when personal experiences are included in lessons versus when only text problems are used. Implications of this argument for our multicultural classrooms are tremendous. Nesbitt (2003) views ethnic diversity as enriching, particularly in problem solving where different cognitive orientations complement and build on the strength of each others. He further explains that the key is providing the proper setting and the appropriate materials that capitalize on and support students' abilities.

Use of culturally-relevant materials

There is ample evidence in literature that supports the claim that culture can be implemented in the mathematics classroom in various ways and with a wide range of varying resources. Kim (2000) proposes two ways to integrate ethnomathematics in the process of teaching and learning in the math classroom: First through the use of inventive ideas inspired from one's own culture; Second by the exploration of new ideas in other cultures. Kim (2000) also emphasizes the role that ethnomathematics' materials play in the enculturation and acculturation processes within and across diverse cultures. Research on the ethnomathematics of different cultures has provided a wealth of creative and thought-provoking materials such as number systems, folk games and puzzles, kinship relations, divination systems, symmetric strip decorations and many others that can be actively used by students in the classroom to enhance their learning of math. Ascher (2002) argues that what makes culturally derived mathematical ideas so powerful is that they are entrenched in contexts in which they arise as part of the complex of ideas around them. Such contexts include divination, calendrics, religion, social relations, decoration and many others. One of the most important artifacts that is most powerful in unfolding the eccentricities and creativity as well as rigor of cultural heritage are exquisite games. Kim (2000) cites more than 200 games that are inspired from the history of mathematics across many cultures. The power of games in teaching lies in their ability to raise students' interest and provide a tool that brings into play many thought-provoking strategies that hone their mathematical skills. Temple & Powell (2001) reported that when playing games, children develop: "… intellectual frameworks that enable them further to construct and comprehend complex mathematical ideas, strategies, and theories" (p.369). The authors describe one such game, which is called oware, an African board game that offers rich opportunities for building and extending arithmetical ideas and strategic thinking. Another intriguing artifact that can be utilized in the classroom to represent number notations in decimal systems is the Inca Quipu. Gilsdorf (2000) provides a detailed description of the various mathematical underpinnings embodied in this record keeping device, he explains: " On quipus there are three types of knots: Simple knots representing powers of ten, long knots with several loops representing digits between 2 and 9, and figure eight knots representing the number one. The spacing of knots relative to other knots indicates value with respect to other knots, so in this way the Inca system is positional" (p.195). Building on the ingenuity of the Andean people, Ascher (2000) proposed lessons using quipus to enhance fifth graders' comprehension of positional decimal notation and place value. Two grade 5 teachers, a math teacher and a humanities teacher, and 31 grade 5 students participated in this project. After reading about the history of the Inca civilization, students were given activities on how to read the Inca quipu and how to do mental addition and subtraction while keeping record of the answers using knots on the quipu strings. At the end of the project, students wrote a summary of what they learned using the Inca quipu in their journals. Positive comments from the teachers indicated the success of using quipu to enhance student's understanding of place value and the merit of exposing them to novel ways that other cultures practice. Other forms of folk games have been documented in the literature that relate to the teaching and learning of mathematics in so many ways. Sizer (2000) cites several games of chance and of strategy as well as puzzles which were developed and used by the Pacific cultures and which can serve as tools to represent basic notions of probability, expectation, and fairness. These include the Hawaiian game of lu-lu, the Zambales dice game, the Hawaiian game of Konane , the Main Machan board game which was played by the Iban tribe of Borneo, and the Hawaiian Pu-waa-pa cord and block puzzle. Other sources of knowledge that can enrich and invigorate the learning of mathematics include mathematical ideas inspired by the logic underlying kinship relations. In examining the logical structure of the kinship system of Warlpiri native Australian tribe, Ascher (1991) describes a highly sophisticated dihedral mathematical group of order 8 with specific matricycles as well as patricycles that guide interpersonal relationships. When using such a logical system and in order to unbutton its peculiarities, the author draws heavily on ideas from graph theory as well as group theory. Additionally, Eglash(1997) investigated the most mathematically significant aspect of doubling in African religion that occurs in the divination ("fortune-telling") techniques of : Vodun , Ifa, and Bamana. He described the technique of Ifa as purely stochastic i.e. operates by pure chance and that of Bamana as systematic, highly compact oracles which follow laws of recursion. Asher (2002) also introduces a twin of the Bamana technique 5,000 miles to the east in Malagasy Sikidy. The rich available literature presents ample evidence that these major African divination systems were all transformations of the Arabic system of 'Sand Science' ('ilm al-raml) or 'Sand Calligraphy' (khatt al-raml) which spread from Abbasid Iraq all over the Islamic world, the Indian Ocean region, and Africa from the late first millennia CE onwards (Van Binsberg, 1999). Partly rooted in simple chance procedures (like hitting the earth, throwing tablets, beans, shells etc.), ilm al-raml is also a binary system of 16 figures (Al-Tokhi, 1991). Each figure is 4 rows high and each row consists of either one dot or two dots. The figures are determined through various methods both ancient and modern. The procedure is called darb al-raml or the forceful 'hitting of the sand' with a stick, in order to produce a chance number of dot traces or marks, which can then be scored as either odd or even. There is no doubt that various uses of materials represent invaluable clues to cultural connections and continuities through space and time. Besides the more obvious advantages concerning the possibility for providing a better understanding of tools and resources of civilizations, culturally relevant materials contain within it the keys to concepts which are believed to be at the heart of some sophisticated mathematical notions. This presents an open invitation to educators to bring unconventional mathematics into the classrooms and to build on the mathematical ideas that students bring from their experiences in their homes and in their communities Theories of Learning outside the school Researchers influenced by Vygotsky have basically emphasized the role of cultural practices in analyzing the relation between culture and cognition (Millroy,1992). Their investigations have generally focused on studying people's use of math outside the classroom ( Gay & Cole, 1967; Scribner, 1986; Lave, 1988; Saxe, 1991, Millroy,1992). The practice of math has been explored in the contexts of everyday activities. Two main groups of researchers have explored the use of math in settings outside school, 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. 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. A significant number of studies on cognitive development have shown that schooled people perform better than unschooled people in a variety of cognitive tasks (Greenfield, 1966). However, schooling effects are not always observed. Lave (1988), for example, did not observe schooling effects on every day math in her project with adults in California, although there was an effect on their mathematical abilities in a school-type test. In this sense, Nunes ….(1993) argue that schooling may influence general cognitive development even though it may not always give people an advantage in cognitive tasks. Different hypotheses have been stated about the origin of these effects. A first line of explanation is cognitive in nature. Some authors (Bruner, 1966, Vygotsky, 1962) propose that the effect of schooling on cognitive development is related to the means of representation used in school. Bruner (1966) , for example, argues that school learning is distanced from reality, it is carried out in the context of general representations of situations. Vygotsky(1962), also, argues that school learning leads to more general and abstract thinking because it is mediated by language rather than evolving from representation of reality. Furthermore, Scribner and Cole (1974) explored the influence of schooling and other cultural activities on perception and cognition through their description of school learning in contrast to informal learning. They describe school learning as based on verbal explanations and involving little direct demonstration. Informal learning is learning in context, by observation and imitation and rarely involves any verbal explanations. Studies on oral and written math (Nunes et al., 1993) are suggestive of yet another possibility. Learning in school and out-of-school is connected to specific problem-solving practices, a point made quite strongly by Lave (1988). The differences between these problem solving practices have an impact on what type of learning takes place and how knowledge is used. Not only issues of learning were the only primary concerns of theorists, but concerns for the transfer of this learning was also emerging along the way. The idea of students' transfer of basic mathematical knowledge to the real world has been the concern of researchers and educators, particularly in the late 1970's. Several research projects questioning the perception that school math can be easily transferred to the real world. Boaler (1993) adds: "observations that mathematical performance is markedly inconsistent particularly across what may be termed as school and everyday situations have suggested that it is the environment in which math takes place not the problem to which it is applied which determines the selection of mathematical procedures"(p. 341). The interaction of students with the contexts of tasks becomes interesting for, if students can learn math in such a way that enables them to see implicit similarities between problems in different situations, they'll be able to transfer their knowledge between different contexts namely from school to the real world. Boaler (1993) also asserts that, advocates of every day math generally believe that questions related to the real world provide learners with a link between the abstract role of math and their role as members of society. He talks of many research projects which suggested that a successful context can increase applicability. However, many teachers believe that whenever contexts are used in examples, students will be able to transfer the content learned to 'real world' situations. Bishop (1988), on the other hand, proposes that students will only be able to transfer math if they have developed meaning which is dependent on mathematical environment. This can only be done when the learner is involved with math through communication and negotiation of ideas and not through meaningless, decontextualized rules and procedures.