Ethnomathematics And Its Place In The History And

2y ago
130 Views
5 Downloads
2.18 MB
5 Pages
Last View : Today
Last Download : 2m ago
Upload by : Luis Wallis
Transcription

Ethnomathematics and its Place in the Historyand Pedagogy of MathematicsUBIRATAN D'AMBROSIOI. Introductory remarksIn this paper we will discuss some basic issues which maylay the ground for an historical approach to the teaching ofmathematics in a novel way. Our project relies primarily ondeveloping the concept of ethnomathematics.·Our subject lies on the borderline between the history ofmathematics and cultural anthropology. We may conceptualize ethnoscience as the study of scientific and, by extension, technological phenomena in direct relation to theirsocial, economic and cultural backgrounds [I). There hasbeen much research already on ethnoastronomy, ethnobotany, ethnochemistry, and so on. Not much has been donein ethnomathematics, perhaps because people believe inthe universality of mathematics. This seems to be harder tosustain as recent research, mainly carried on by anthropologists, shows evidence of practices which are typicallymathematical, such as counting, ordering, sorting, measuring and weighing, done in radically different ways thanthose which are commonly taught in the school system.This has encouraged a few studies on the evolution of theconcepts of mathematics in a cultural and anthropologicalframework. But we consider this direction to have beenpursued only to a very limited and - we might say- timidextent. A basic book by R.L. Wilder which takes thisapproach and a recent comment on Wilder's approach byC. Smorinski [2] seem to be the most important attemptsby mathematicians. On the other hand, there is a reasonable amount of literature on this by anthropologists. Making a bridge between anthropologists and historians ofculture and mathematicians is an important step towardsrecognizing that different modes of thoughts may lead todifferent forms of mathematics; this is the field which wemay call ethnomathematics.Anton Dimitriu's extensive history of logic [3] brieflydescribes Indian and Chinese logics merely as backgroundfor his general historical study of the logics that originatedfrom Greek thought. We know from other sources that, forexample, the concept of "the number one" is a quite different concept in the Nyaya-Vaisesika epistemology: "thenumber one is eternal in eternal substances, whereas two,etc., are always non-eternal," and from this proceeds anarithmetic [4, p. 119]. Practically nothing is known aboutthe logic underlying the Inca treatment of numbers, thoughwhat is known through the study of the "quipus" suggeststhat they used a mixed qualitative-quantitative language[5).These remarks invite us to look at the history of mathe-44ma tics in a broader context so as to incorporate in it otherpossible forms of mathematics. But we will go further thanthese considerations in saying that this is not a mere academic exercise, since its implications for the pedagogy ofmathematics are clear. We refer to recent advances intheories of cognition which show how strongly culture andcognition are related. Although for a long time there havebeen indications of a close connection between cognitivemechanisms and cultural environment, a reductionist tendency, which goes back to Descartes and has to a certainextent grown in parallel with the development of mathematics, tended to dominate education until recently, implying a culture-free cognition. Recently a holistic recognitionof the interpenetration of biology and culture has openedup a fertile ground of research on culture and mathematical cognition (see, for example, [6]). This has clear implications for mathematics education, as has been amplydiscussed in [7] and [8].n. An historical overview of mathematics educationLet us look very briefly into some aspects of mathematicseducation throughout history. We need some sort of periodization for this overview which corresponds, to a certainextent, to major turns in the socio-cultural composition ofWestern history. (We disregard for this purpose other cultures and civilizations.)Up to the time of Plato, our reference is the beginningand growth of mathematics in two clearly distinctbranches: what we might call "scholarly" mathematics,which was incorporated in the ideal education of Greeks,and another, which we may call "practical" mathematics,reserved to manual workers mainly. In the Egyptian origins of mathematical practice there was the space reservedfor "practical" mathematics behind it, which was taught toworkers. This distinction was carried on into Greek timesand Plato clearly says that "all these studies [ciphering andarithmetic, mensurations, relations of planetary orbits]into their minute details is not for the masses but for aselected few" [9, Laws Vil, 818) and "we should inducethose who are to share the highest functions of State toenter upon that study of calculation and take hold of it, .not for the purpose of buying and selling, as if they werepreparing to be merchants or hucksters" [9, Republic Vil525b]. This distinction between scholarly and practicalmathematics, reserved for different social classes, is carriedon by the Romans with the "trivium" and "quadrivium"and a practical training for laborers. In the Middle Ages weFor the Learning of Mathematics 5, I (February 1985)FLM Publishing Association, Montreal, Quebec, Canada

begin to see a convergence of both in one direction: that is,practical mathematics begins to use some ideas from scholarly mathematics in the field of geometry. Practical geometry is a subject in its own right in the Middle Ages. Thisapproximation of practical to theoretical geometry followsthe translation from the Arabic of Euclid's Elements byAdelard of Bath, (early 12th century). Dominicus Gomdissalinus, in his classification of sciences, says that "it wouldbe disgraceful for someone to exercise any art and notknow what it is, and what subject matter it has, and theother things that are premised of it," as cited in (10, p. 8).With respect to ciphering and counting, changes start totake place with the introduction of Arabic numerals; thetreatise of Fibonnaci [ 11, p. 481] is probably the first tobegin this mixing of the practical and theoretical aspects ofarithmetic.The next step in our periodization is the Renaissancewhen a new labor structure emerges: changes take place inthe domain of architecture since drawing makes plansaccessible to bricklayers, and machinery can be drawn andreproduced by others than the inventors. In painting,schools are found to be more efficient and treatises becomeavailable. The approximation is felt by scholars who startto use the vernacular for their scholarly works, sometimeswriting in a non-technical language and in a style accessibleto non-scholars. The best known examples maybe Galileo,and Newton, with his "Optiks".The approximation of practical mathematics to scholarly mathematics increases in pace in the industrial era,not only for reasons of necessity in dealing with increasingly complex machinery and instruction manuals, butalso for social reasons. Exclusively scholarly trainingwould not suffice for the children of an aristocracy whichhad to be prepared to keep its social and economical predominance in a new order [ 11, p. 482). The approximationof scholarly matheinatics and practical mathematics begins to enter the school system, if we may so call education inthese ages.Finally, we reach a last step in this rough periodizationin attaining the 20th Century and the widespread conceptof mass education. More urgently than for Plato the question of what mathematics should be taught in mass educational systems is posed. The answer has been that it shouldbe a mathematics that maintains the economic and socialstructure, reminiscent of that given to the aristocracy whena good training in mathematics was essential for preparingthe elite (as advocated by Plato), and at the same timeallows this elite to assume effective management of theproductive sector. Mathematics is adapted and given aplace as "scholarly practial" mathematics which we willcall, from now on, "academic mathematics", i.e., themathematics which is taught and learned in the schools. Incontrast to this we will call ethnomathematics the mathematics which is practised among identifiable culturalgroups, such as national-tribal societies, labor groups,children of a certain age bracket, professional classes, andso on. Its identity depends largely on focuses of interest, onmotivation, and on certain codes and jargons which do notbelong to the realm of academic mathematics. We may goeven further in this concept of ethnomathematics toinclude much of the mathematics which is currently practised by engineers, mainly calculus, which does notrespond to the concept of rigor and formalism developed inacademic courses of calculus. As an example, the SylvanusThompson approach to calculus may fit better into thiscategory of ethnomathematics. And builders and welldiggers and shack-raisers in the slums also use examples ofethnomathematics.Of course this concept asks for a broader interpretationof what mathematics is. Now we include as mathematics,apart from the Platonic ciphering and arithmetic, mensuration and relations of planetary orbits, the capabilities ofclassifying, ordering, inferring and modelling. This is avery broad range of human acitivities which, throughouthistory, have been expropriated by the scholarly establishment, formalized and codified and incorporated into whatwe call academic mathematics. But which remain alive inculturally identified groups and constitute routines in theirpractices.III. Ethnomathematics in history and pedagogy and therelations between themWe would like to insist on the broad conceptualization ofmathematics which allows us to identify several practiceswhich are essentially mathematical in their nature. And wealso presuppose a broad concept of ethno-, to include allculturally identifiable groups with their jargons, codes,symbols, myths, and even specific ways of reasoning andinferring. Of course, this comes from a concept of cultureas the result of an hierarchization of behavior, from individual behavior through social behavior to culturalbehavior.The concept relies on a model of individual behaviorbased on the cycle. reality - individual - action reality . , schematically shown asINFORMATIONff-REIFICATION---STRATEGYIn this holistic model we will not enter into a discussion ofwhat is realtiy, or what is an individual, or what is action.We refer to [12). We simply assume reality in a broad sense,both natural, material, social and psycho-emotional. Now,we observe that links are possible through the mechanismof information (which includes both sensorial andmemory, genetic and acquired systems) which producesstimuli in the individual. Through a mechanism of reification these stimuli give rise to strategies (based on codes andmodels) which allow for action. Action impacts upon real-45

ity by introducing facti into this reality, both artifacts and"mentifacts''. (We have introduced this neologism to meanall the results of intellectual action which do not materialize, such as ideas, concepts, theories, reflections andthoughts.) These are added to reality, in the broad sensementioned above, and clearly modify it. The concept ofreification has been used by sociobiologists as "the mentalactivity in which hazily perceived and relatively intangiblephenomena, such as complex arrays of objects or activities,are given a factitiously concrete form, simplified andlabelled with words or other symbols" [13, p. 380]. Weassume this to be the basic mechanism through whichstrategies for action are defined. This action, be it throughartifacts or through mentifacts, modifies reality, which inturn produces additional information which, through thisreificative process, modifies or generates new strategies foraction, and so on. This ceaseless cycle is the basis for thetheoretical framework upon which we base our ethnomathematics concept.Individual behavior is homogenized in certain waysthrough mechanisms such as education to build up societalbehavior, which in turn generates what we call culture.Again a scheme such as,EVENTS".1\-coMMUNICATION'allows for the concept of culture as a strategy for societalaction. Now, the mechanism ofreification, which is characteristic of individual behavior, is replaced by communication, while information, which impacts upon an individual,is replaced by history, which has its effect on society as awhole. (We will not go deeper here into this theoreticalframework; this will appear somewhere else.)As we have mentioned above, culture manifests itselfthrough jargons, codes, myths, symbols, utopias, and waysof reasoning and inferring. Associated with these we havepractises such as ciphering and counting, measuring, classifying, ordering, inferring, modelling, and so on, whichconstitute ethnomathematics.The basic question we are then posed is the following:How "theoretical" can ethnomathematics be? It has longbeen recognized that mathematical practices, such as thosementioned in the end of the previous paragraph, areknown to several cuturally differentiated groups; and whenwe say "known" we mean in a way which is substantiallydifferent from the Western or academic way of knowing46them. This is often seen in the research of anthropologistsand, even before ethnography became recognized as ascience, in the reports of travellers all over the world.Interest in these accounts has been mainly curiosity or thesource of anthropological concern about learning hownatives think. We go a step further in trying to find anunderlying structure of inquiry in these ad hoc practices. Inother terms, we have to pose the following questions:l. How are ad hoc practices and solution of problems developed into methods?2. How are methods developed into theories?3. How are theories developed into scientificinvention?It seems, from a study of the history of science, that theseare the steps in the building-up of scientific theories. Inparticular, the history of mathematics gives quite goodillustrations of steps I, 2 and 3, and research programs inthe history of science are in essence based on these threequestions.The main issue is then a methodological one, and it liesin the concept of history itself, in particular of the history ofscience. We have to agree with the initial sentence in Bellone's excellent book on the second scientific revolution:"There is a temptation hidden in the pages of the history ofscience - the temptation to derive the birth and death oftheories, the formalization and growth of concepts, from ascheme (either logical or philosophical) always valid andeverywhere applicable . Instead of dealing with real problems, history would then become a learned review of edifying tales for the benefit of one philosophical school oranother" [14, p. I]. This tendency permeates the analysis ofpopular practices such as ethnoscience, and in particularethnomathematics, depriving it of any history. As a consequence, it deprives it of the status of knowledge.It is appropriate at this moment to make a few remarksabout the nature of science nowadays, regarded as a largescale professional activity. As we have already mentioned,it developed into this position only since early 19th centuiy.Although scientists communicated among themselves, andscientific periodicals, meetings and associations wereknown, the activity of scientists in earlier centuries did notreceive any reward as such. What reward there was camemore as the result of patronage. Universities were littleconcerned with preparing scientists or training individualsfor scientific work. Only in the 19th century did becoming ascientist start to be regarded as a professional activity. Andout of this change, the differentiation of science into scientific fields became almost unavoidable. The training of ascientist, now a professional with specific qualifications,was done in his subject, in universities or similar institutions, and mechanisms to qualify him for professionalactivity were developed. And standards of evaluation of hiscredentials were developed. Knowledge, particularly scientific knowledge, was granted a status which allowed it tobestow upon individuals the required credentials for theirprofessional activity. This same knowledge, practiced inmany strata of society at different levels of sophisticationand depth, was expropriated by those who had the responsibility and power to provide professional accreditation.

We may look for examples in mathematics of the parallel development of the scientific discipline outside the established and accepted model of the profession. One suchexample is Dirac's delta function which, about 20 yearsafter being in full use among physicists, was expropriatedand became a mathematical object, structured by the theory of distributions. This process is an aspect of the internaldynamics of knowledge vis-a-vis society.There is unquestionably a timelag between the appearance of new ideas in mathematics outside the circle of itspractitioners and the recognition of these ideas as "theorizable" into mathematics, endowed with the appropriatecodes of the discipline, until the expropriation of the ideaand its formalization as mathematics. During this periodof time the idea is put to use and practiced: it is an exampleof what we call ethnomathematics in its broad sense. Eventually it may become mathematics in the style or mode ofthought recognized as such. In many cases it never getsformalized, and the practice continues restricted to theculturally differentiated group which originated it. Themechanism of schooling replaces these practices by otherequivalent practices which have acquired the status ofmathematics, which have been expropriated in their original forms and returned in a codified version.We claim a status for these practices, ethnomathematics,which do not reach the level of mathematization in theusual, traditional sense. Paraphrasing the terminology ofT.S. Kuhn, we say they are not "normal mathematics" andit is very unlikely they will generate "revolutionary mathematics." Ethnomathematics keeps its own life, evolving asa result of societal change, but the new forms simplyreplace the former ones, which go into oblivion. Thecumulative character of this form of knowledge cannot berecognized, and its status as a scientific discipline becomesquestionable. The internal revolutions in ethnomathematics, which result from societal changes as a whole, are notsufficiently linked to "normal ethnomathematics". Thechain of historical development, which is the spine of abody of knowledge structured as a discipline, is not recognizable. Consequently ethnomathematics is not recognizedas a structured body of knowledge, but rather as a set of adhoc practices.It is the purpose of our research program to identifywithin ethnomathematics a structured body of knowledge.To achieve this it is essential to follow steps 1, 2, and 3above.As things stand now, we are collecting examples anddata on the practices of culturally differentiated groupswhich are identifiable as mathematical practices, henceethnomathematics, and trying to link these practices into apattern of reasoning, a mode of thought. Using both cognitive theory and cultural anthropology we hope to trace theorigin of these practices. In this way a systematic organization of these practices into a body of knowledge mayfollow.IV. ConclusionFor effective educational action not only an intense experience in curriculum development is required, but also inves-tigative and research methods that can absorb andunderstand ethnomathematics. And this clearly requiresthe development of quite difficult anthropological researchmethods relating to mathematics, a field of study as yetpoorly cultivated. Together with the social history ofmathematics, which aims at understanding the mutualinfluence of socio-cultural, economic and political factorsin the development of mathematics, anthropologicalmathematics, if we may coin a name for this speciality, is atopic which we believe constitutes an essential researchtheme in Third World countries, not as a mere academicexercise, as it now draws interest in the developed countries, but as the underlying ground upon which we candevelop curriculum in a relevant way.Curriculum development in Third World countriesrequires a more global, clearly holistic approach, not onlyby considering methods, objectives and contents in solidarity, but mainly by incorporating the results of anthropological findings into the 3-dimensional space which we haveused to characterize curriculum. This is quite different thanwhat has frequently and mistakenly been done, which is toincorporate these findings individually in each coordinateor component of the curriculum.This approach has many implications for research priorities in mathematics education for Third World countriesand has an obvious counterpart in the development ofmathematics as a science. Clearly the distinction betweenPure and Applied Mathematics has to be interpreted in adifferent way. What has been labelled Pure Mathematics,and continues to be called such, is the natural result of theevolution of the discipline within a social, economic andcultural atmosphere which cannot be disengaged from themain expectations of a certain historical moment. It cannotbe disregarded that L. Kronecker ("God created the integers - the rest is the work of men"), Karl Marx, andCharles Darwin were contemporaries. Pure Mathematics,as opposed to Mathematics, came into consideration atabout the same time, with obvious political and philosophical undertones. For Third World countries this distinctionis highly artificial and ideologically dangerous. Clearly, torevise curriculum and research priorities in such a way as toincorporate national development priorities into the scholarly practices which characterizes university research is amost difficult thing to do. But all the difficulties should notdisguise the increasing necessity of pooling human resources for the more urgent and immediate goals of ourcountries.This poses a practical problem for the development ofmathematics and science in Third World countries. Theproblem leads naturally to a close for the theme of thispaper: that is, the relation between science and ideology.Ideology, implicit in dress, housing, titles, so superblydenounced by Aime Cesaire in La Tragedie du Roi Christophe, takes a more subtle and damaging tum, with evenlonger and more disrupting effects, when built into theformation of the cadres and intellectual classes of formercolonies, which constitute the majority of so-called ThirdWorld countries. We should not forget that colonialismgrew together in a symbiotic relationship with modernscience, in particular with mathematics, and technology.47

References(!)(2)(3)(4)[5][6][7]Ubiratan D'Ambrosio, Science and technology in Latin Americaduring discovery. Impact of Science on Society, 27, 3 (1977) p.267-274R.L. Wilder, Mathematics as a cultural system. Pergamon, Oxford(1981). C. Smorynski, Mathematics as a cultural system. Mathematical Intelligence, 5, I ( 1983) p. 9-15Anton Dimitriu, History of logic (4 vols). Abacus Press, Kent,(1977)Karls H. Potter (ed.) Indian metaphysics and epistemology.Encyclopedia of Indian Philosophies. Princeton University Press,Princeton, N .J. ( 1977)Marcia Ascher and Robert Ascher, Code of the Quipu. The University of Michigan Press, Ann Arbor (1981)David F. Laney, Cross-culturalstudiesincognitionandmathematics. Academic Press, New York (1983)Ubiratan D'Ambrosio, Socio-cultural bases for mathematicseducation. Proceedings /CME 5, to appear(8](9][IO][ 11][12)(13](14]Ubiratan D' Ambrosio, Culture, cognition and science learning.To appearPlato, Dialogues, ed. E. Hamilton and H. Cairns. PantheonBooks, N. Y. (1963)Stephen K. Victor (ed.) Practical geometry in the high MiddleAges. The American Philosophical Society, Philadelphia (1979)Ubiratan D' Ambrosio, Mathematics and society: some historicalconsiderations and pedagogical implications. Int. J. Math. Educ.Sci. Technol. II, 4, pp. 479-488 (1980)Ubiratan D'Ambrosio, Uniting reality and action: a holisticapproach to mathematics education. In: Teaching Teachers,Teaching Students, L.A. Steen and D.J. Albers (eds.) Birkhauser,Boston (1980) p. 33-42Charles J. Lumsden and Edward D . Wilson, Genes, mind andculture. Harvard University Press, Cambridge (1981)Enrico Bellone, A world on paper. The MIT Press, Cambridge,( 1980; orig. ed. 1976)ContributorsA.J. BISHOPDepartment of EducationUniversity of Cambridge17 Trumpington StreetCambridge CB2 lPT, U.K.R. BORASIFaculty of Educational StudiesSUNY at Buffalo468 Baldy HallBuffalo, NY 14260, U.S.A.A. BOUVIERDepartement de mathematiquesUniversite Claude Bernard43, boul. du 11Novembre191869022 Vil/eurbanne Cedex, FranceU. D'AMBROSIOCoordinador Geral dos InstitutasUniversidade Estadual de CampinasC.P. 117013100 Campinas SP, BrasilP.GERDESFaculty of EducationEduardo Mondlane UniversityC.P. 257Maputo, Mozambique, AfricaR.CEPCL54, boul. Raspail75270 Paris Cedex, France48

and Pedagogy of Mathematics UBIRATAN D'AMBROSIO I. Introductory remarks In this paper we will discuss some basic issues which may lay the ground for an historical approach to the teaching of mathematics in a novel way. Our project relies

Related Documents:

01 The right place for the right housing 18 02 A place to start and a place to stay 20 03 A place which fosters a sense of belonging 22 04 A place to live in nature 24 05 A place to enjoy and be proud of 26 06 A place with a choice of homes 28 07 A place with unique and lasting appeal 30 08 A place where people feel at home 32

Towns and urban design are arranged focusing religious and political hierarchy and play a fundamental role in society. Since early dwelling through nowadays, the mutual influence of urbanization and mathematics is evident. Both polis and urbs reflect power structure and political organization.

Journal of Mathematics & Culture ICEM 4 Focus Issue ISSN-1558-5336 309 of research is the acknowledgement of the fact that people in several walks of life perform mathematical activities out of school, at home, and at work. Ubiratan D’Ambrosio, widely known as the father of ethnomathematics initiated the

The common calendar of the settlers included a seven-day week, the days being named after the Norse gods (Bjornsson, 1990, pp. 71–74; 1993, pp. 18–19, 665–660): Sunnudagur, Sunday, the day of the sun. Manadagur, Monday, the day of the moon. Tysdagur, Tuesday for Tyr, the god of war. Odinsdagur, Wednesday for Woden, the cunning god.

Compare Three-Digit Number Vocabulary Partner B Write ., ,, or 5 to compare the numbers. When I numbers, I always start with the place value. The greatest place value in 367 and 376 is the place. Both numbers have a 3 in the hundreds place. So I look at the tens place. The number 367 has in the tens place. The number 376 has in the tens place.

First place: Neal Huffman Second place: Blayne Chastain Third place: Chris Lee Fourth place: Richard Burnosky Fifth place: Jim Monaco Sixth place: Tom Siler Seventh place: Adam Quennoz Eight place: Randall Everly Senior Flyer: Dominick Lewis The team award went to Team One consisting of Jeff Carr, Jim Monaco, Blayne Chastain, and Neil Huffman.

animal. Say the good qualities of the 2nd place animal over the 1st place animal. List why the 2nd place animal does not win the class. (bad qualities) Say why 2nd place animal beats 3rd place animal by stating only the good qualities of the 2nd place animal. Say the good qualities of the 3rd place animal over the 2nd place animal.

Introduction to Literature, Criticism and Theory provides a completely fresh and original introduction to literary studies. Bennett and Royle approach their subject by way of literary works themselves (a poem by Emily Dickinson, a passage from Shakespeare, a novel by Salman Rushdie), rather than by way of abstract theoretical ideas and isms. In 32 short chapters they focus on a range of .