Ethnomathematics: a Dialogue
MARCIA ASCHER, UBIRAT AN D'AMBROSIO
Instead of each of us writing a paper for this special i
For the Learning of Mathematics 14,2 (June, 1994) 36 FLM Publishing Association, Vancouver, British Columbia, Canada Ubi: Through the work with the quipus and your fur find is counting. That is too small; it is much too ther work in other cultures you were able to gen small. Certainly mathematical ideas of number, erate a new conception of mathematics. logic, and spatial configuration organized into Marcia: Yes, I think that is very important for us to dis systems, include counting. But cotmting doesn't cuss .. One of the hardest problems I encountered, include all those wonderful things that are in particularly because I began by working with a mathematics and that are mathematical ideas. person who was not a mathematician, was that This is a worry to me because so many people he felt I should or could state for him a definition who are inspired by the concept of ethnomathe of mathematics. But there is no clear definition matics are involved in the elementary school of what mathematics is .. The problem then mathematics level and they are, perhaps, looking for too restricted a set of ideas becomes how to say that something is mathemat ical or a good illustrative case .. It meant resorting Ubi: They look at mathematics in a narrow way We to a belief that, from my training as a mathemati may express it differently, but there is much cian, I would know it when I saw it even if I coincidence in our views. When I look at the ety couldn't say what it was. So I worried a lot about mology of ''mathematics" I recognize, in mathe a working definition of mathematics and it kept ma or mathemata, what is usually recognized as changing a bit It wasn't until I was satisfied explaining, understanding, a broader sense of with my statement that I could sit down to write coping with the many aspects and challenges the Ethnomathematics [2] book that reality presents We know very little of I believe that what is used as a working defini Ancient Greek but the texts where something tion of mathematics is very important for the that looks like what we now call mathematics various aspects joined together under the appear support my interpretation of the umbrella of ethnomathematics. The word "math mathema The intellectual venture of explaining ematics" for many people is linked very closely it is the result of a challenge and generates ways to what they learned in school. Depending on of facing these challenges. When you use mathe their level of schooling, the word means quite matical ideas you are broadening the merely different things to different people Mathemati technical way of solving a problem. It is more cians, as such, rarely define mathematics .. When than the technique: here I prefer to use the Greek they do, it is either quite broad and includes word techne that stands for both the technique everything or it is utterly narrow But surely, to and the art This is why I see your use of the most of them, it started with the Greeks As a term "mathematical ideas" as very close to my result, the word ''mathematics" starts arguments interpretation of mathema Clearly all this is of ownership-that is, who owns mathematics? attached to particular cultures, which suggested Is it mathematics if people weren't doing to me the use of the prefix ethno In this way I proofs?, and so forth So, to avoid those tangen coined the word ethno mathema tics! Although tial arguments, I concern myself instead with the origin of the term is different from yours, the mathematical ideas 'That focuses on, and talks idea is the same and the link is your concept of to, what people have in common Those ideas mathematical ideas. have to do with number, logic, and spatial con Most people, among them mathematicians, are figuration and, very important, the combination subordinated to the same way of interpreting and or organization of those into systems and struc doing mathematics-to the same techne, in my tures. language-and they fail to recognize that others I am concerned that some of the other working are also doing mathematics definitions of mathematics that are used by the Marcia: Yes Let's take symmetry as an example. Sym group of people who talk about ethnomathemat metry is around people in the natural world. We ics are much too specific to the elementary observe symmetry, create symmetry, impose school level and, as a result, minimize the math symmetry. It has aesthetic components; it may ematical ideas that are really out there in the have practical components .. But the idea of sym non-school settings. metry, the concern for symmetry, the recognition Ubi: They start to work with something and out of of symmetry and, of course, in mathematical the this the discipline is identified Others doing ory building, the observation and seeking of things that are not exactly what they have identi symmetry or even, sometimes, the feeling that fied, although with the same objectives, the same some theory may be a bit wrong it if doesn't purposes, and even achieving the same results, exhibit certain symmetries-that doesn't have to they don't recognise as mathematics do with technique If you pin it down just to counting, measuring, and solving problems, you Marcia: And they don't seek it If you say, for example, lose the broad concept of symmetry. that mathematical ideas have to do with counting I also use the example of symmetry in order to and then you go out and look for culturally embedded mathematical ideas, what you will bring in aesthetics. I believe that many mathe-
37 matical ideas are totally unnecessary By unnec what separates the mathematician in Greece essary I mean that we do not need beautifully from the carpenter. fhe carpenter definitely is patterned mosaics on the floor in order to walk dealing with a mathematical idea; the mathe It doesn't make walking any easier .. But penple matician who set those strictures on the problem put in a great deal of effort to create these beauti was dealing with an idea .. They are both impor ful floor mosaics .. One, therefore, has to enlarge tant, but they are different.. And, they are linked the idea of problem solving All human beings, Ubi: Yes, that's important all cultures, order the space in which they live They need order on their space This is not prob Marcia: Our discussion, I think, underscores the kind of lem solving in the strict sense. broadening needed. Referring simply to problem I don't think that there is disagreement on this; solving can miss all these different ramifications I think, instead, it points to a need for broaden we are elaborating ing the way things are expressed. Ubi: The problems that were philosophical impasses Ubi: The idea of problem solving is too narrow. If to the Greeks are indicators that Greek mathe you focus on mathematics as the solution of matics was very close to the idea of a game. problems you will never justify the decoration of These intellectual games were related to divine a floor and the important branch of mathematics activities, activities which were typical of the called symmetry Thus mathematical ideas are gods .. I am very much influenced by the scenario much more than solving problems .. This is why I given by Herman Hesse in his "Magister Ludi" give so much importance, in developing my The most important intellectual activity in ideas of ethnomathematics, to history. I see the Castalia was the glass-beads game, which has challenges presented in the current histories of much of the same characteristics of the mathe mathematics as much broader than mere problem matical games the Greeks were playing.. This is solving-in particular, say, the Greeks dealing related to physical games, the Olympic games, with the classical problems: trisection of the which again followed strict rules, as was appro angle, quadrature of the circle, and duplication priate for the perfection attributed to the gods. of the cube. For example, there is the huge And the same with the sense of beauty which led importance given by them to trisecting an angle to these physical exercises. By playing these with ruler and compass. If you really want to games they were approaching the gods, both the solve the problem of the trisection of an angle, inte11ectua1 games-mathematics, in the sense of you can easily do it, as many people did But the the mathema-and the physical games The big moment you limit yourself to the use of ruler and objectives of athletics were not to win the compass you are at another level of reflection, medals as a reward for fitness but to get closer to trying to go beyond, to transcend, the immediate the gods. The athlete, as an athlete, and the needs. This is the same for several cultures that philosopher are following the same practice, have ritual geometries For example, in Amazo though with different drives. It's the same thing nian tribes color plays an important role This is that doing mathematics was not aimed at solving much more than a matter of their immediate daily, everyday, problems-this was done with needs. This is why aesthetics plays an important another kind of mathematics-but to be closer to part in raising and building up what you call the gods mathematical ideas and I refer to as ethnomathe Marcia: To me, the idea of intellectual games is a very matics important part of mathematics in any culture. For Let me elaborate on the importance given by example, I've written about logical puzzles-the the Greeks to the famous problems and to their challenge of a logical puzzle and the fact that solution strictly with ruler and compass: the only these can occur in different cultures with different explanation for this is that they were interested stories around them One can say that the story in something else. The same with the so-called puzzles are being used to educate, or as amuse philosophical impasses of infinity and the irra ment, but they are, in any case, logical play .. As tional. It is a big mistake to look at these out of another example: among the Malekula there are context We cannot separate Greek philosophy paths one must trace to get to the Land of the from Greek mathematics-or Greek medicine or Dead This, too, is intellectual play Not play in Greek drama. Indeed, in no culture, can it be that it isn't serious. 'These are how one-to use support<;
38 Ubi: Yes, we can use the word play in this broader Marcia: The important point that etbnomathematics adds, sense of transcending, which is pure existence. It however, is that, nonetheless, expressions of leads to the concepts of religion, which are clear mathematical ideas do have content and cultural ly of the same nature as those of the arts, sci context In the past, this has been a large source ences, and everything else. In every culture you of misunderstanding .. An excellent example is find some form of god. Every culture tries to the re-study by Sylvia Scribner [4] of Kpelle reach above the mere earthly needs of survival. understanding of logical implications and syllo i We have found in every culture these kinds of gisms. An earlier study concluded lack of under I intellectual exercises, these plays and these standing because of unsatisfactory responses. In games, as practices that allow people to one instance, the Kpelle respondent would not approach the considerations that go beyond pure answer when given the premises "All Kpelle I survival. This I see as the need to transcend men are rice farmers," "Mr. Smith is not a rice one's existence This is always associated with farmer," and then asked whether Mr. Smith was the search for explanations, for understanding a Kpelle One part of the respondent's explana and meeting the challenges, which I call the tion was that if you know a person and were mathema .. Mathematical ideas, in your sense, are asked about him, you could answer, but that was clearly associated with this Regrettably, the his not the case if you didn't know him. The logic of tory of mathematics, and history in general, has the response was fine, but, clearly, answering the put so much emphasis on the need of man to sur original question conflicted with Kpelle values vive, as if survival and transcendence were sepa That is, to the Kpelle, the form and the content rate states of human behavior. The originality of of the syllogism were both of importance. man among the other species is the association Ubi: This goes back to the beginning of our conversa of drives towards survival and towards transcen tion People from a different culture, particularly dence; man's behavior reveals both components those Westernized by such an important mode of For example, man eats ritually, man plays ritual thought as mathematics, do not recognize in ly. Other species have eating and playing habits, Kpelle reasoning any form of logic, mainly but none associated with transcendence, none rit because syllogisms are intrinsically both form ualized. Historical shortsightedness has led to and content for them Their inferences are loaded philosophical dichotomies such as "to know/to with ethics-or, we might even say, with pas do", "manual work/intellectual work", sion-while in the Western mode of thought ''mind{body", which are so prevalent in Western passion and even ethics and rationality are mutu civilization I call Western civilization that ally exclusive which developed, with so much cultural inter change, in the Mediterranean region This led to Marcia: The concern fOr context is also evident in the a sort of ethics and social values reflected in many cultures whose languages have numeral modes of property and production-that is, in classifiers-where different endings or different the division of labour-and the consequent eco number words are used depending on the catego nomic systems .. ry of the things that are being counted. With But this kind of discussion would lead us into classifiers, one somehow still relates the number another strand of ethnomathematics-very to the context so that the context is not forgotten ample-which I would rather leave to another or overlooked. Most mathematicians believe that discussion The important point is that the math what gives mathematics its power is the manipu ema was responsible for some body of knowl lation of symbols standing for anything and hav edge within a certain context, and this has not ing no context In a problem, x is x and nothing been clearly recognized in current historiography more However, I believe mathematics could be which is so impregnated with the dichotomic even more powerful by retaining some recogni style. Ethnomathematics relates to life in all its tion of what the symbols stand for and gearing aspects Indeed, it is a description of the evolu the approaches used to that If, for example, you tion of mankind through its diverse ramifica are dealing with x's referring to numbers of tions-that is, the civilizations, cultures, commu human beings, you should only be seeking inte nities, families, and individuals This calls for ger solutions and selecting solution methods deeper recognition than is found in most anthro accordingly pologies .. It is quite interesting that the publishers Ubi: something that has a past, and seems more of my book Ethliomathemdtica [3] had trouble natural The mind of a child seems more recep knowing whether to place it as a book about tive to the idea of number as attached to some mathematics or history or anthropology .. I felt thing else This is true too in many cultures .. This tempted to say it was a book about values. The has been characterized as a lack of capability for ultimate goal, I think, is to propose attitudes abstract thinking, which is an obvious historical deprived of arrogance and of bigotry perversion. Simply said, we use numbers accord-
39 ing to what we want to design or classify with at a few parameters, and apply the solving tech the number The fact that we deal with numbers niques according to those parameters .. But we by insisting on their abstract meaning may reveal know that the real situation is so complex, has a distortion in our civilization, the distortion of such a multiplicity of parameters, that in simpli decontextualization, essentially, which is a form fying it we inevitably limit the overall view of reductionism This criticism leads to what Marcia: I agree that this is a very important issue.. Partic some have called a holistic view. ularly in school settings, people try to make the Marcia: I also think that, especially in the education of ideas relevant to the students' world by creating children, values are being taught through our word problems or story problems I generally emphasis on contextless numbers. Very often find that students can more easily grasp the more school examples are phrased in terms of money. theoretical topics like number theory .. It is at the In the examples, numerical equivalents are stated interface between mathematics and unreal "real" for labor, for objects, for health care, for food, problems that the greatest confusion and over etc .. Then the students are told to focus on and simplification occurs It is exceptionally difficult manipulate only these numerical equivalents. As to take abstract mathematical statements and such, it is more than just teaching the mathemat apply them in a real world context It would, per ics for itself; it is teaching the students to view haps, help if more time was devoted to teaching the quantities and their manipulations as context about the creation of mathematical models. Then less and divorced from meaning I think this is a students could learn to be more critical of them very important issue for people in education to and could learn to distinguish between the valid consider. ity of the mathematics and the sufficiency of the Ubi: The absolute suppression of context, and the model When mathematical models were based primarily on physical theories, the omitted vari quantification of values for comparison~ in order to value something more than another, is what I ables represented smaller effects and so what consider a sign of the philosophical damage resulted was a first approximation As the appli done to modern thought; it leads to a world cations are moved into social, economic, or deprived of human values, even of human feel global settings, the problems become more diffi ings .. Everything valuable has to be quantified cult. There are far more variables, far less clarity on the hierarchy of their effects, and more value Marcia: One of my favorite examples is from linear pro judgements are involved in deciding which gramming in which you are minimizing or maxi effects to consider or ignore mizing some fUnction under linear constraints One early application was the creation of a diet Ubi: Putting all this in the context of ethnomathemat for pigs. The constraining equations were their ics: this is the reason I call ethnomathematics a daily nutrient requirements and then, using cur program in the history and philosophy of mathe rent prices, the most economical diet was found matics. It's a program with a holistic approach, But the pigs wouldn't eat it! There are a number much broader than current historiography and of similar examples where the mathematics was epistemology which have clearly selected only a absolutely correct but the solutions couldn't be few variables for analysis This program has used because the audiences and their tastes and implications for pedagogy I think we do not dis values were ignored during the mathematical agree on this. formulation Marcia: I very much agree .. This is why I believe there Ubi: This has also much to do with the problems we are two distinct aspects of ethnomathematics face with the environment The way we look at which are definitely related but sometimes have our behavior, in general, trying to quantify it, to be more clearly separated One aspect is seek without attaching any value-in the ethical ing understanding of the relationship between sense-to the quantifiers, is probably the main mathematical ideas and culture. For this under reason why we have been so unwise in dealing standing, more investigations, study, and with the environment research, and more thinking about these histori calfphilosophical issues are needed Once there Marcia: I also think that this is one of the causes of the is this deeper understanding, then can come the dislike of mathematics among young people educational aspect which addresses the question There is a feeling, often articulated by those of how to incorporate it; how should we or how alienated by mathematics, that it is emotionless do we modify education? The two aspects share and lacks feeling .. Even students who like mathe in seeking a broader view of mathematics in matics seem to associate it with a certain inhu which mathematical ideas are not restricted to manity. any one group, profession, culture, or historical Ubi: We do mathematics systematically out of con time. But they differ in that modification of edu text We put ourselves inside the discipline, look cation depends largely on the goals of the educa tor and the setting of the education This is not a
40 question of research but of clarifying one's goals talk about all of the time and devotion that was specifically enough to develop methods or prac put into constructing the squares by people in tices that move towards those goals diverse cultures and that, when they did this, it was to order the universe or with some other Ubi: The essential point on this is that there has been religious motivation and feeling Students are a big misunderstanding of the way mathematics very surprised that such feelings are involved has developed since the 16th century .. People fail because most things in school are presented as to realize that mathematics then was suitable for rational-and the emphasis is that rational is bet very limited purposes, the systems appropriate to ter .. fu class, we Often get into a discussion as to those purposes were very rough, very limited whether they believe in magic They immediate The mathematicians of those times were dealing ly say "No!" because in college you are not with things, with systems, that could be under allowed to admit this; you are supposed to be stood with a very rough degree of approxima scientific. But then there emerge examples of tion. Their perception of the world was very nar lucky pencils for exams, special objects or words row .. It was our big mistake to build up a cunicu associated with certain sports events, and so on lum based on this kind of mathematics This All of this is a big problem for mathematics mathematics was important in the 16th and 17th because so many people point to it as the ulti centuries when it was essential fOr that world, mate in rationality people in other fields, for with many limitations in practices and in knowl example, show surprise when mathematicians edge To focus, to reduce down, and to center the put less credence than they do in judgments educational system on this narrow perception of based on numerical evaluative schemes the world was a mistake. Ubi: Yes, we know the limitations. Marcia: But it is also a narrow perception of mathematics because the scope of mathematics has broadened Marcia: Instead they think we are the advocates and mathematics changes In many ways, much Ubi: I think the richness of ethnomathematics, and the of the educational system is left with 17th/18th reason fOr the acceptance of it, is that we are century mathematics. For example, there have looking at mathematics in this much broader been many changes incorporating probabilistic way. thinking I am shocked when I talk with college students who have never heard of quantum theo Marcia: And also because we are adding the idea of cul ry. They still have a Newtonian view of the tural variety. There is no question that no culture world has remained pure or unchanged. The students are more diverse and many students are part this Ubi: There has been an evolution in mathematical and part that-a wonderful, beautiful mixture thinking, an evolution that has incorporated all The more we ignore the variety of cultures, the that resulted from the technology that was devel less the students will understand what we are oped from those same theories-mainly the talking about And, here I use the word "culture" capability of making observations We have to in a very broad sense take into consideration the fact that much of Newtonian science, and the mathematics devel Ubi: We should address another area in which I think oped to deal with it, was the result of a sophisti we have agreement. Ethnomathematics in cated scheme of observation, experimentation, schools is not an attempt to replace mathematics and mathematization with rudimentary tools. Of Our observations apply to other forms of knowl course these tools improved over the following edge too .. This misunderstanding is a danger we centuries, but the school system insisted on face. almost ignoring that fact We are now able to see Marcia: I, too, am quite concerned when I realize that things that were impossible to perceive before, some people think ethnomathematics is trying to and we can create experimental conditions that replace mathematics I have heard comments were unthinkable before And the same goes for saying, for example, that students coming from the treatment of the information obtained .. There traditional cultures could excel in modem mathe is much obsolescence in schools. There is a gulf matics but that ethnomathematics was trying to between the modem world and the school world, bring them down to a lower level. This is part of particularly in mathematics .. And probably this is why, in my work, I stress broadening the histo the basic factor causing the disaster in current ry-the global and ongoing history. On several educational systems. occasions I have used the example of graph theo Marcia: And there are the other aspects we were talking ry. When most students learn about graph theo about earlier that have been greatly downplayed ry, they are told about the Konigsberg bridge because of the extreme rationalistic approach. problem. That is an excellent opportunity to tell For example, in the non· Western math course them instead, or in addition, about the Malekula that I teach, one of the topics is magic squares .. I who posed essentially the same problem in try-
41 ing to get to the Land of the Dead .. In part, it is different approaches to it. Not only are their like Claudia Zaslavsky's goal of bringing the gods different, their everyday lives are differ world into the classroom, but, more particularly, ent-and these two major differences are related. it is to tie into humanity This is a question that The way they live their everyday lives is related human beings have been interested in-not just to the way they try to transcend them. We con some mathematician or some people in Prussia cern ourselves with arts, morals, religions, sci in 1730 .. In my view, it makes the question more ences, production, and so on, often without pay wondrous because it is the same question. Or, as ing attention to the fact that all this comes in my earlier example about magic squares; it is together Breaking down these different forms of that people do these things for intellectual rea thought into separate disciplines may be the sons, emotional reasons, aesthetic reasons, reli cause of most of our troubles in dealing with civ gious reasons It broadens the realm of the possi ilization as a whole It naturally leads to mistak ble for students As a byproduct, it also provides enly recognizing the prevalence of one form of a different way to introduce students to mathe thought over another, and even to the valuation matical ideas even if they were put off by the of one form over the other. We see manual work algebraic/symbol manipulation aspects of school less valued than intellectual work, leading to an mathematics The point you raise is true and unjust social ethics; we see material assets more very serious. How do you think this danger can valued than cultural assets; we see a dichotomy or should be dealt with? between material and spiritual concerns: at the one extreme we see people regarding material Ubi: We shouldn't try to define ethnomathematics values as absolute while at the other we see peo or mathematics. Your use of the concept of ple treating mind as disembodied, as if the mind mathematical ideas and my use of mathema, and the body were separate entities. which have such an affinity, recall man's efforts to transcend his existence through explaining Marcia: Within the mind/body distinction, I think of it and understanding what is going on, looking for more specifically as drawing too rigid a line new ways of coping with what reality is impos between creations of the hands and creations of ing on him, but at the same time going a step the mind They are both creations and they are beyond the mere solution of the problem. This related through visualization and planning As can restore to the mathematics in every cultural with the quipu-they embody a logical-numeri system-that is, to ethnomathematics-its cal system but creating them and "'reading" them breadth: it goes beyond the solution of the prob involves the tactile lem, it restores the higher dimension of an intel Ubi: Ibis leads to some reflections about cognition, lectual exercise which has been studied in such a narrow way Marcia: But at the same time as we value the aspect of The process of creation, typical of human mathematics that transcends daily problem solv beings, is the result of a conjugation of the sen ing, we should recognize and value the daily sorial with the intuitive and with the emotional aspects There are many groups of people who and with the rational dimensions of our behavior engage in mathematical thinking but who We do not create, hence we do not build knowl express it in their specific ways. I include them edge, and we cannot act, without the conjugation when I refer to the need for more studies in eth of all these. But we tend to place each manifesta nomathematics As you frequently have men tion of culture in a single dimension Religion is tioned, there are, for example, people in particu purely emotional, science-and particularly lar fields of endeavor who share mathematical mathematics-is purely rational. This is not so, ideas and who transmit them to each other. 'That and has not been so in history.. Yet historiogra is part of their apprenticeship as in architecture, phy and epistemology both disregard this boat building, carpentry, or sailing. There has to pluridimensional aspect of building up knowl be more understanding of their ideas, as they are edge. Look into Greek geometry, for example. A culturally embedded, so that their mathematical triangle comes from three sticks and a string aspects can be recognized .. Take weaving for joining them Greek geometry is always regard example. That surely involves geometric visual ed as having ignored the sticks and the strings. ization. It not only requires the creation and con And, surely, these sticks and strings are made of ception of a pattern, but also requires knowing some materials, have colors in them. These are what mov_es to execute or colors to use to cause suppressed in the abstract version of geometry the pattern to emerge. In effect, the weaver is Amazonian geometry is full of colors. The geo digitalizing the pattern. The weaver expresses metric properties are there in both the Greek pos the visu~lization through actions and materials ture and the Amazonian posture, but they are Ubi: It is not a lower level of thinking, it is a different regarded as different expressions of human way of thinking. If an Eskimo and an Amawni mind: one is science, the other is at best att, and an Indian face a similar problem they will follow is more commonly called folklore. And if you
42 move to the Hippocratic writings, this is early school, an institution which addresses not indi scientific medicine, while the use of rituals and vidual action but cooperative action .. Because roots by the Amazonians is called sorcery. I see ethnomathematics is not passive it is loaded with ethnomathematics dealing with all these consid critical components .. But most importantly, the erations .. This is why I like to classify it as a pro gains and advancements will be collective and gram in history and philosophy. History and phi not individual. While keeping capabilities very losophy of mathematics? No, not only of mathe individualized (each individual is different from matics, but of everything And since mathemati the other) we have to generate, through this cal dimensions are just about everywhere, in the socializing school, respect for the other with all cosmos, in nature, in our actions, the name for his/her differences, solidarity with the other in such a program is not inappropriate. her/his pursuit of satisfying the needs of sur vival, and transcendence of their material and Marcia: I would like to raise one final point about an spiritual needs, learning how to act in coopera educational aspect of ethnomathematics. It is tion with others, putting together the physical generally assumed that grade-levels in school go with particular ages. However, there are a grow and intellectual resources to reach common goals .. These three components: respect, solidari ing number of people seeking education or reed ty, and cooperation, constitute an ethics for a ucation who are not in those age grades. You global civilization and serve as the basis for my find, for example, more students in college who model of the school of the future are 35 or 45 or 65.. Whether on the college or Of course this is intrinsic to my view of ethno elementary level, in the U S. or in other coun mathematics. I entirely agree with you that the tries, I believe that the teaching of the adult revitalization of mathematics through ethno learner is of growing importance and needs spe mathematics will be the result because ethno cial attention .. At all levels of schooling, the mathematical pedagogy is an active one.. One is teaching of subject matter is intertwined with the in each moment putting all one's efforts, intel teaching of culture But these people are already lectual and manual, to the perfOrmance of a com fully enculturated .. As a result, ethnomathematics mon goal. The place of adults in this scheme is may have even more important ramifications fOr essentiaL But we have to treat them, these adults, their education If part of the pedagogical pur as individuals with lifetimes of experiences pose of ethnomathematics is a revitalization of We have covered many points in this conver mathematics education, that aspect would be sation. Now, even more than befOre, I think we particularly important for this older group. And, are very convergent even though from different since they have had many life experiences, some roots And I am very happy to see this of which may have involved utilizing mathemat ical ideas, they could make significant contribu Marcia: Yes, I agree And we are particularly convergent tions to our understanding .. Have you thought in our effOrt to see ethnomathematics as a revi about this? talization of mathematics in school. We do not want to see mathematics disappear as a school Ubi: I have been thinking a lot about this. My view of the future of education points to a different con subject cept of the levels of schooling. Labor will be something very different from what it is now, offering learning opportunities which are nowa References days not even considered I see every moment of [1} Ascher, Marcia and Robert Ascher. Code of the quipu a study in life as a learning experience, and this is different media, mathematics, and culture, University of Michigan Press, Ann Arbor, 1981 from individual to individual So I see school as [2} Ascher, Marcia, Ethnomathematics: a multicultural view of mathe a kind of meeting place where people with dif matical ideas, Brook,./Cole, Belmont, California, 1991 ferent experiences come together to socialize [3] D'Ambrosio, Ubiratan, Etnomatemdtica: Arte ou ticnica de their experiences Thus they begin another expe explicar e conhecer Editora Atica, Sao Paulo, 1990 [4] Scribner, Sylvia, "Modes of thinking and ways of speaking: culture rience, which is to put their capabilities together and logic reconsidered,"' in Thinking: readings in cognitive science, to function at a common task. Ethnomathematics ed. by P.N Johnson-Laird and P.C. Wason, Cambridge University is a most suitable pedagogy for this kind of Press, New York, 1977, pp 483-500
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