LITERACY, MATHERACY, TECHNORACY: A TRIVIUM FOR TODAY*

by Ubiratan D'Ambrosio

Summary: This paper focuses on the relations between mathematics and mathematics education, and human behavior, societal models and power. Based on a critical analysis of school systems, and of mathematical thinking, its history and its socio-political implications, I suggest a new concept of curriculum, organized in three strands: literacy, matheracy and technoracy. I see education and scholarship as pursuing a major, comprehhensive, goal of building up a new civilization which rejects arrogance, inequity, and bigotry. Since the development of mahematics has been intertwined with all forms of human behavior in the history of mankind, it is relevant to discuss mathematics and mathematics with this major goal in mind.

INTRODUCTION

This paper deals basically with the global responsibility of mathematics educators. It raises issues of ethics and values. The guiding question is "What are our commitments to mankind, in particular to children, as mathematics educators?"

I see a double commitment to children: to convey a broad view of the world and of mankind in general and to promote values. It is practically impossible to exert our profession without a reflection about the perception of the state of the world and of its future. This is essential to help the children to acquire consciousness of their position in the world. And to be a mathematics educator, we must be convinced that mathematics can help to fulfil the commitment to children and to promote equity and democracy, dignity and peace for all of mankind. Otherwise, why teach mathematics?

* Mathematical Thinking and Learning,1(2),1999; pp.131-153U.D'Ambrosio 1

The paper is structured in three parts: reconceptualizing the curriculum, mathematics as human activity, and learning from the past in building up the future. In these parts I analyze school systems together with the evolution of society and of mathematics. The processes of generation of knowledge and of its societal implications guides much of my reflections.

Mathematics is seen in a broader way, in the framework of the Program Ethnomathematics.

PART 1: RECONCEPTUALIZING THE CURRICULUM

Historical review of the curriculum

Educational systems throughout history and in every civilization have focused on two purposes: to transmit values from the past and to prepare for the future. The strategy through which educational systems pursue these goals is the curriculum. The focus of this paper is a new conceptualization of the curriculum (for some of my early thoughts on this topic see D’Ambrosio,1981).

Let us examine briefly the notion of curriculum in the course of Western history. School systems in classical antiquity, particularly in Rome, aimed at preparation for citizenship, as is clearly recognized in the Roman trivium: Grammar, Rhetoric and Dialectics. To read and write well, to discourse and to argue were enough to exercise the political participation upon which Roman society was built. In the Middle Ages, the main goal of building up a Christian Philosophy needed much more than the trivium, hence the importance given to further studies, organized as the quadrivium. Under the influence of Hellenism, the Medieval quadrivium comprised Arithmetic, Music, Geometry and Astronomy. This higher education was essentially restricted to monasteries.

The encounter with Islam, the expansion of commerce, and the new needs of colonial administration, as well as the social consequences of modern science, required a larger portion of the population to have more education. New methods of teaching were needed, thus the birth of modern didactics, whose main representative figure was the Czech

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theologician Jan Amos Komensk (1592-1670), better known as Comenius, who wrote “The great didactic” (1628-1632).

In the early part of the 19th century, most of the colonies of the Americas achieved their independence. Particularly important for Education was the expansionist politics of the new United States of America. Both territory and population increased dramatically in the early decades of independence.1 The ideal of a same school for all was necessary for the consolidation of a new nationality among European immigrants in the new territories. The need for a trivium which could be a response to the needs of the new population was immediately felt. It would not be satisfactory to rely on home schooling neither on the then current European model. The new population had to read and write the same language and to engage in commerce and the practical needs with the same counting and measuring system. A curriculum based on Reading, Writing and Arithmetic would be the answer.

The three R’s were soon to spread to the newly independent countries in the Americas and eventually to the entire world. School systems all over the world basically pursue the three R’s. The basic question is: Do the three R’s prepare for citizenship in the present world? There is evidence that the answer is “no”. I propose another trivium, consisting of Literacy, Matheracy and Technoracy, which responds to the needs of the emerging age. The three together constitute what is essential for citizenship in a world moving fast into a planetary civilization.

Indeed there has been a lack of new ideas since then. Much of the advances in communications and in the overall model of society have had a very modest impact in education. Much of the effort of a so-called innovation in curricula have stressed preservation of old styles.

New curriculum.

Literacy

Clearly, reading has a new meaning today. We “read” a movie or a TV program, as reflected, for example, in the title of the influential book “How to read a movie: Movies, media, multimedia”(Monaco, 1998). It is common to listen to a concert with a new “reading” of Chopin! Also, socially, the concept of literacy goes through many changes (e.g.

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Hollingsworth & Gallego, 1996). Nowadays, “reading” includes also the competency of numeracy, the ability of understand graphs, tables, the condensed language of codes, and other ways of informing the individual. These competencies have much more to do with screens and button than with pencil and paper. Nowadays, “reading” and communicating through the new media precede the use of pencil and paper, and numeracy is dealt with using calculators. There is no way of reversing this trend, just as there was no successful censorship to prevent people having access to books in the last 500 years. Nor did the restrictive ordinance governing financial procedures issued by the City Council of Florence in 1299 impeded the widespread use of the newly introduced Arabic calculations in the 14th century (Menninger,1969,p.426). But, if dealing with numbers is part of modern literacy, where has mathematics gone?

Matheracy

The capabilities of drawing conclusions from data, making inferences, proposing hypotheses and drawing conclusions from the results of calculations are as important as simply “reading” data. This set of capabilities is the first step towards an intellectual and critical posture, which is almost completely absent in our school systems. Even conceding that problem solving, modeling and projects are currently seen in some mathematics classrooms, the main emphasis in the mathematics curricula continues to be numeracy, or the manipulation of numbers and operations. It impoverishes mathematics to restrict it to purely manipulative techniques and utilitarian purposes.

The broader concept of matheracy comes closer to the way mathematics was present in classical Greece. Its objective was not emphasize counting and measuring. This was part of common knowledge. Matheracy, in the sense I am giving to it, was appropriate for philosophers and concerned with higher objectives than utilitarianism. As we read in Plato: “Their [common men’s] language is most ludicrous, though they cannot help it, for they speak as if they were doing something and as if all their words were directed toward action. For all their talk is of squaring and applying and adding and the like, whereas in fact the real object of the entire study is pure knowledge.” Of course, Plato was concerned with the education of the elite and with values, as it is clear when he writes “What we have to consider is

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whether the greater and more advanced part of it [geometry and calculation] tends to facilitate the apprehension of the idea of good” (Hamilton & Cairns, 1961, p.759). But isn’t this a valid goal for our educational systems?

Although in Greek society this view of mathematics was restricted to philosophers, it is about time, two thousand years later, to aim higher than mere utilitarianism for a larger part of the population. Otherwise, we will remain very far from equity and democracy. It is true that there has been much effort to introduce philosophy in elementary schools and also to make philosophy available to a large part of the population. But it is undeniable that mathematics is the essence of modern philosophy. Not only the philosophers of classical antiquity were mathematicians, but most of the modern philosophers have discussed philosophy of mathematics in their works. A mathematical framework is an important tool to deal with many central issues of our society, which is increasingly mathematised. Hence my proposal of including matheracy, with its critical focus, as part of the elementary curriculum.

Technoracy

I understand technology as the organized ensemble of intermediacies which men created to extend their physical and sensorial capabilities. There is no need to emphasize that technology, in multifarious ways, is present in practically all of our daily activities, for everyone and everywhere in the world, although sometimes it goes unnoticed. The presence of technology is frequently taken for granted in human behavior.

Almost every cause of pleasure or dissatisfaction in the world can be traced, respectively, to the use or misuse of intermediacies, which I call technology, created by men. I discussed this is relation to the essence of being human in the UNESCO Forum on Science and Culture (D’Ambrosio 1996). Thus there is a need, from childhood, for critical familiarity with technology. Of course, the operative aspects of it are, in most cases, inaccessible to the lay individual. But the basic ideas behind the technological devices, their possibilities and dangers, and the moral issues underlying the use of technology are essential questions to be raised among children from a very early age. History shows us that ethics and values are intimately

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related to technological progress. For example, the development of swordplay in classical Japan -– which is a manifestation of technology -- imply specific codes of ethics and values.

Goals of an educational system.

Educators need to be seriously concerned about mankind as a whole, respecting the specificity of each culture. Hence the importance of regarding children as individuals with hopes and expectations of a future which reflect their own individual and cultural history. The following questions are thus essential for educators: What do we know about the children we are teaching and their cultural background? What do we know about the future? What is the state of the world? What does it mean for mankind? What is our role, as teachers, in influencing the future?

In one way or another, these question have partially guided the preparation of teachers. Psychology and Learning, Sociology and Curriculum Development became part of teacher training programs as a response to these questions. But all these disciplines, particularly when components of programs of mathematics education, have a propaedeutical character, giving more emphasis to contents and methodology and less attention to overall goals and objectives of mathematics education.2 Education can be interpreted as a strategy created by societies to promote creativity and citizenship. Education cares about the individual raising to the maximum of his/her capabilities and at the same time learning about their assimilation into societal life. In other words, education aims equally at the new (creativity) and at the old (societal values). But both aims must be pursued with special care, to avoid both irresponsible creativity (for example, because we do not want our students to become bright scientists creating new weaponry), and docile submissiveness (because we do not want our students to conform to rules and codes that violate human dignity). This double aim is the challenge.

As mathematics educators we are particularly interested in mathematics and in education. Mathematics has taken different styles and different forms in different natural and cultural environments, and also in different times, “there being...as many mathematics –- as many number-worlds –- as there are higher cultures” as the historian Oswald Spengler put it in his seminal book “The Decline of the

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West” (Spengler,1932,p.52). Hence my proposal of Ethnomathematics, which obviously does not exclude school mathematics as practiced today.3 For an introduction to Ethnomathematics, see D’Ambrosio (1990;1998a)4. We have to teach today’s mathematics to today’s children, always with a view to their future. Today’s mathematics is justified because it is the basis of modern science and technology. When we try to justify the presence of mathematics in the schools because it is part of our cultural traditions we indulge in the most flagrant cultural aggression. A cultural bias attributes to mathematics the highest intellectual achievement of mankind. I am referring to a form of knowledge which was originated in a small portion of the planet around the Mediterranean and which arrogates to be the standard of rational behavior. Indeed, modern science and technology are coming under a severe criticism, from which mathematics is not immune. It is naïve for us, mathematicians, to react to attacks on the misuses of modern science and technology by claiming the neutrality of mathematics.5

Questioning the school system.

When we question the current social, economical and political order in the face of a real threat to the continuation of Western civilization, we are essentially questioning its righteousness. How is it then possible to avoid questioning science and mathematics, which are the pillars of Western civilization? Yet, as I discuss later, we see an attempt to exclude non-scientists and non-mathematicians from the questioning of these pillars. The resource to arguments of authoritative competence lead to passionate and intimidating arguments. Questioning opens the way for the new. How can we reach the new by discouraging, refusing, rejecting, denying the new? Indeed, a subtle instrument of denial is discouragement through intimidation. Language plays an important role is this process, as every school teacher knows. Particularly in mathematics, the use of a formal language inherent to academic discourse has been a major cause and instrument of determent. School organization and curricula have been instruments for the denial of participation in the dynamics of society. The criticism inherently resulting from the needed re-establishment of the connections between the sciences, technology and human values is causing unavoidable conflicts. This is no less true about mathematics, in which the acknowledgement of human

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attributes has been conspicuously absent in its discourse. This has not been so throughout the course of history. Mathematics, the same as the other sciences, used to be impregnated with religious, as well as social and political considerations.

There have been few writings about values attached to mathematics and even less about the moral quality of our actions. 6 To search for a correlation between the current state of civilization and mathematics has been uncommon among mathematics educators (this point is touched in Skovsmose, 1998). It is interesting that the important pedagogical studies of Paulo Freire, Michael Apple, Henry Giroux and other educational reformers have been silent about mathematics. Recently, they have focused on some problems of mathematics education. Relevant to this paper is the affirmation of Paulo Freire: “In my generation of Brazilians from the North-East, when we referred to mathematicians, we were referring to something suited for gods or for geniuses. There was a concession for the genius individual who might do Mathematics without being a god. As a consequence, how many critical inteligences, how much curiosity, how many enquirers, how many capacities that were abstract in order to become concrete, have we lost?” (Freire, 1997).

A reason for the silence of the eminent educators mentioned above may be their feeling of exclusion. The same strategy now being used in the so-called Science War, deflagrated in its highest intensity by Alan Sokal’s hoax and his recent book on French current thought.7 Social and political scientist Marcus G. Raskin and physicist Herbert J. Bernstein, in their analysis of the linkage between knowledge generation and political directions, claim that "science seeks power, separating any specific explanation of natural and social phenomena from meaning without acknowledging human attributes (such as love, happiness, despair, or hatred), the scientific and technological enterprise will cause profound and debilitating human problems. It will mask more than it tells us about the universe and ourselves."(Raskin & Bernstein,1987,p.78).

It is strange how these criticisms have been practically ignored by mathematicians and mathematics educators. Even less has been done with respect to the relations between mathematics and employability, a serious concern of our society. Most of the programs aiming at equity see

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preparation for better jobs as a valid objective of mathematics education. True. But how can we hope to fulfill this objective teaching our students obsolete mathematics? An explicit reference to this important aspect of societal life was made by the science educator Michael R. Matthews: "There is a fiscal crisis of the State, and a structural unemployment crisis. Increasingly, there will be a small percentage of people whose labor time is virtually priceless and a vast percentage whose labor time is basically worthless."(Matthews,1980,p.197).

Discussing employability, Harvard Professor Robert B. Reich, former Secretary of Labor of the USA, proposes a broad conception of curriculum focused on what he calls symbolic analysis (Reich,1992), which has important implications for mathematics in a broad sense. The fact that mathematics educators, and mathematicians as well, have simply ignored these reflections reveals a sort of aloofness. Indeed, we may be cheating our youth when we say that mathematics, as taught in our schools, opens good perspectives of employment for them. Probably the best research in this field is due to Paul Willis, who shows how the practices in schools inculcate conformist ideology and the normality of prevailing social order(Willis,1977). Again, Willis does not refer specifically to mathematics. It seems that mathematics remains marginal to these critical considerations of society. Adriana C.M. Marafon (1996), in her dissertation, studied how working conditions in the family have a direct influence on the way students perceive mathematics and its relation to employability. The mathematics learnt at home helping in the family shop is not regarded as mathematics, although it is considered a more valuable knowledge than school mathematics. The same theme is studied by Cláudio José de Oliveira (1998) in his dissertation, working with students in a school of the periphery of Porto Alegre, a major industrial city in Southern Brazil. Students are there in the school, just awaiting for the legal age to get a job. Mathematics has nothing to do with the “world outside”, where they will get jobs. Certainly, the possibilities of jobs for these students will increase if they learn mathematics, but not the mathematics currently taught in schools. They will need a mathematics of keys and keyboards, of scanning machines and of taking decisions about optimal strategies. They need critical intelligence, curiosity, inquisitiveness, capacity to be abstract in order to become concrete, using the same categories that Paulo Freire used. This absolutely does not

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exist in the traditional mathematics curricula. But these are the foci of matheracy.

PART 2: MATHEMATICS AS HUMAN ACTIVITY

Mathematics, rationalism and ideology.

To a great extent, the polemics around the postmodern discourse of sociologists of science reflect the ideology intrinsic to words. Indeed, language has been in the course of history the main instrument in denying free inquiry. There is an implicit intimidating instrument in the language of academia and society in general. Remember that the Free Speech Movement was a major factor in the

1 The expansion is evidenced by the fact that population and occupied territory increased from 3,929,214 inhabitants in 888,811 sq. mil. in 1790 to 23,191,876 inhabitants in 2,992,747 sq. mi. in 1850.2 In ICME3, in Karlsruhe, 1976, I prepared a working document on "Overall Goals and Objectives of Mathematics Education." for the Working Group on "Why Teach Mathematics?". A summary of the working document was published as D’Ambrosio (1976) and the full version as D’Ambrosio (1979).3 I have been proposing Ethnomathematics as a research program in the history and philosophy of mathematics with pedagogical implications. The general idea is to look into the full cycle of knowledge, from its generation, intellectual and social organization, and diffusion. Obviously, school mathematics can be seen and interpreted according to this program. So, Ethnomathematics do not refer only, as some believe, to the mathematics of illiterate or indigenous cultures.4 D’Ambrosio (1998a), which is a translation of D’Ambrosio (1990) can be obtained through Patrick Scott, ISGEm, Box 30001 MSC 3CUR, Las Cruces, NM 88003, USA. Send a check of US$10.00 (domestic postage included; foreign add US$5.00).5 This is the motivation of a special section on “Mathematics, Peace and Ethics”, Zentralblatt für Didaktik der Mathematik,30,(3).6 Usually these appear in the context of mathematics education when goals and objectives are discussed. Revealingly, these practically ignore deeper issues relating to man and society and the political role of mathematics education. These matters have been discussed in D’Ambrosio (1979).7 Alan Sokal is professor of Physics of New York University. He published a paper entitled “A Physicist Experiment with Cultural Studies” in Lingua Franca 6(5), July-August 1996, pp.54-64, in which he reveals that he had written a deliberately absurd article entitled “Transgressing the Boundaries: Towards a Hermeneutics of Quantum Gravity”, which was accepted and published in the prestigious journal Social Text. Sokal’s view is that the publication of his article reveal a decline of standards of rigor in certain circles of the social sciences and the humanities. The controversy, which became known as “Science War”, was fueled by the publication of a book which is seen as an attack on post-modern French scholars, entitled Impostures Intellectuels [Paris: Odile Jacob, 1997].U.D'Ambrosio 10

confrontations of the sixties. This intimidatory use of language particularly affects schools, as embodied in the controversy over bilingual education, for example. However, although considering that bilingual education is of fundamental importance, particularly for mathematics education, I will not get into this theme in this paper.

There has been a resurgence of interest in the intuitive, sensorial (e.g. as reflected in hands-on projects) and emotional/affective aspects in mathematics education. While in the recent development of mathematics we see increased recognition of intuition, there is less to the senses and practically nothing concerning the emotional/affective.

Hassler Whitney has devoted much effort to an elaboration of affective components in mathematics education. After several years of research with children in elementary school, he said about problem solving: “First of all, replace ‘solve it’ by ‘play around with it.’ Half the difficulty is now over. Make it concrete: act out the story. Have courage to try the story in different ways, getting used to its various features. When things turn out wrong, be interested in how they are wrong, and try changes; act out the story again. Now if you ask what was wanted, you may be ready to see or quickly find the answer. Ii is really basically as simple as that. Courage to play with and try different things is the keynote” (Whitney,1974,p.41).

Through her experience in research and undergraduate and graduate teaching, Fam Chung Graham, of the University of Pennsylvania, says that students learn Mathematics “in courses that have been in existence for 30 or 40 years without much change in the curriculum” and “there is a non-trivial gap between classroom mathematics and the math used in current technology.” Later on she says “The Mathematics community needs to reach out”, otherwise the current decline will continue (Graham,1998).

Both Hassler Whitney and Fam Chung Graham, reflecting upon the extremes of elementary school mathematics and research mathematics, point to the need for a radically new approach to mathematics, relying on the conceptual, the creative, the critical, rather than the routine. This view is in line with the proposal of matheracy.

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We are unable to say clearly how can emotions play a role in mathematics. What had Gustave Flaubert in mind when he wrote "Mathematics: the one who dries up the heart"?8 And it is difficult to deny that the opinion of eminent art theorist Robert Vischer, when he wrote in 1874 that “even with regard to the so-called pure forms, where such [mathematical] analysis is much more at home than it is with the concretely vital forms, Mathematics can still only play a subordinated role in aesthetics precisely to the extent that we always perceive these forms, as we have said, with the participation of imagination, together with the movements of our sensible and ideal inner life” (Harrison, Wood, & Geiger,1998,p.693) still prevails. Where have mathematicians’ imagination, sensibility, and inner life gone?

The reaction I usually hear to these comments is: "But mathematics is the quintessence of rationalism." Indeed, much of the polemics going on relate to the prevailing acceptance of the superiority of rationality over other manifestations of human behavior. This was one of the main concerns of the mathematician-writer Robert Musil in his masterpiece “The Man Without Qualities”. Commenting on scientists and engineers, the main character Ulrich asks: “Why they do seldom talk of anything but their profession? Or if they ever do, why do they do it in a special, stiff, out-of-touch, extraneous manner of speaking that does not go any deeper down, inside, than the epiglots (sic)? This is far from being true of all of them, of course, but it is true of a great many...They revealed themselves to be men who were firmly attached to their drawing-boards, who loved their profession and were admirably efficient in it; but to the suggestion that they should apply the audacity of their ideas not to their machines but to themselves they would have reacted much as though they had been asked to use a hammer for the unnatural purpose of murder.” (Musil,1980 (originally 1930),p.38). Musil's oeuvre antecipates the intellectual framework of Nazi Germany, in which he identifies incapacity to tolerate pluralism. Indeed, many of the reactions against irrationalism are mixed with a latent emotional incapability of accepting the different. The denial of access to knowledge is a strategy for the exclusion of the different.(This paragraph is taken verbatim from D’Ambrosio,1998b).

8 Flaubert (1987). This was the motivating caption of D’Ambrosio (1993). U.D'Ambrosio 12

We are now living very interesting times of challenging systems of knowledge, which to a great extent have been generated to give to the prevailing social, economical and political order a character of normality. Inspirational reflections and research which are the bases of both the religions and the sciences are essentially a process of dismantling and reassembling systems of knowledge with the undeniable purpose of justifying the prevailing order. Religion and Science have focused in giving a sense of normality to prevailing individual and social human behavior.

Ethics and mathematics.

We are primarily individual human beings fully integrated in mankind and in nature in the broad sense. As individuals we have to relate with nature in order to survive and with the different other (woman/man), in order to give continuity to the species. This encounter with the very different is followed by encounters present in coping with survival, such as hunting. And the evolution to even more sophisticated encounters, such as laboring the land. These encounters with the very different other and the recognition of his/her essentiality are the first steps towards social relations. Thus, social behavior (relations with the different, with others) starts to take shape and gives rise to society. The survival of every species depends on the relations of groups (sharing a social behavior) with nature, which is regulated by ecological principles, and with other groups, giving rise to power. The possibility of human beings (substantive) being humans (verb) depends on the capability of managing these relations, which can not be dealt separately. To fundamental question about this capability resides in the relations between brain and mind. It is possible to know much about the human body, its anatomy and physiology, to know much about neurons, and yet have not the slightest idea of why we see something red (see the important oeuvre of Oliver Sacks, particularly, Sacks,1995). These considerations underlie modern theories of consciousness, which can be claimed to be the last frontier of scientific research, giving rise to much academic controversy (e.g. Papineau’s (1996) review of Chalmers(1995)).

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Through a sophisticated communication system and other organic specificity,9 our species probes beyond the span of one's existence, before birth and after death.10 This search outside the limits of lifetime is called transcendence. The symbiotic drives towards survival and transcendence constitute the essence of being human and regulate the relations

individual nature

other/society

The origins of Mathematical thinking have much to do with this symbiotic relation although current histories and epistemologies rarely touch this point. It is a mistake to claim, as many mathematicians do, that the reflections above belong to other disciplines and that mathematics has little to do with these developments. Individual and social behavior result from the system of relations represented above. For generations and generations these relations have been controlled by moral and material instruments, among them norms and codes, language and literacy, regulated by systems such as religions, sciences and technology. Paradoxically, the same instruments which were fragmentarily constructed to preserve the prevailing order have become so complex that they are no longer effective and became increasingly permeable.

The new technologies of communication allows information to flow through barriers. The main reason of the fall of the Berlin Wall was the fact that it was obsolete, both

9 How did we reach this specificity? Both evolutionism and creationism recognize specificity in human intelligence (see Deacon 1997).10 Here we find the origins of cults, traditions, religions, arts and sciences. These are undistinguishable in their first manifestations in mankind throughout history and in child development. Essentially, this is a search for explanations, for understanding, which go together with the search for predictions. One explains in order to anticipate, thus building up systems of explanations (beliefs) and of behavior (norms, precepts). These are the common grounds of religions and sciences, until nowadays. U.D'Ambrosio 14

technologically and intellectually. Boundaries and property are questioned because they cannot be maintained, and national and international finances are in disarray because money -- invented as a representation for real goods -- became virtual.

Analyzing these historical phenomena may offer a hint for dealing with increasing school avoidance, alienation and evasion. Interestingly, the more and more frequent analyses of evasion drawn from current events as reported by the media are giving a broader perspective on the problem than the academic analyses, which are practically immobilized by obsolete research methodologies.

It is an undeniable right of every human being to have access to all the natural and cultural goods needed to her/his material survival and intellectual enhancement. This is the essence of the Universal Declaration of Human Rights (1948), to which every nation is committed. The educational strand of this important profession on mankind is the World Declaration on Education for All (1990), to which 155 countries are committed. Of course, there are many difficulties in implementing United Nations resolutions and mechanisms. But as yet this is the best instrument available that may lead to a planetary civilization, with peace and dignity for all mankind. Aren't these the most fundamental principles to which we subscribe? Regrettably, these documents are generally not known to most mathematics educators (Haggis, Fordham, & Windham,1992).

Even more regretable is that global ethics, such as the ethics of diversity (respect for the other (the different); solidarity with the other; cooperation with the other), which may lead to quality of life and dignity for the entire mankind, have been absent from the reflections about mathematics and mathematics education. To be highly provocative, I invite people to consider the claim that most despicable human behavior in recent times have been located among people who have attained a high level of cultural development, particularly excellence in mathematics. Let me make it very clear that this is not an insinuation of an intrinsic malignity of mathematics. But it is clear that mathematics has been a companion in the historical events which we all criticize. I see mathematics playing an important role in achieving the high humanitarian ideals of a new civilization with equity,

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justice and dignity for the entire human species without distinction of race, gender, beliefs and creeds, nationalities and cultures. But achieving these goals depends on the way we understand how deeply related are mathematics and human behavior, a question normally untouched by mathematicians, historians of mathematics and mathematics educators. These considerations underlie the search for a mathematical ethics subordinated to the ethics of diversity, to be pursued by mathematicians and mathematics educators, opening up a new guiding principle in our practice.

Knowledge and power

The state of the world is disturbing. It is becoming common to express discontent, in fact disenchantment, with the course civilization is taking, by chastising science and technology, which are recognized as the embodiment of modern society. Science and technology are thus blamed for the malaise of humanity. Mathematics is obviously directly affected by these attacks.

Such criticism is faced with a mounting reaction. It is difficult to deny that modern science tries to organize facts under principles and laws and that Western thought has focused on experiences which, under specifiable conditions, are available to everyone. This has been interpreted, for centuries, as opening doors to heresy. Nowadays, “anti-science”, “irrationality” and “pseudosciences” are labels frequently used. This is clear when Carl Sagan cautions about the lure of new directions of inquiry in his recent book “The demon-haunted world: Science as a candle in the dark” (Sagan,1996). In his denouncement of the "new Dark Age of irrationality", Sagan claims that "Each field of science has its own complement of pseudoscience. Geophysicists have flat Earths, hollow Earths, Earths with wildly bobbing axes to contend with, rapidly rising and sinking continents, plus earthquake prophets. Botanists have....."(Sagan,1996,p.43). The tone of the criticism of Carl Sagan pervades a skeptical attitude about new approaches to the relations between science and society. I cannot see how this criticism fits sociologists and humanists, the target of this reaction.11

11 The Science War, mentioned in Note 7 above, clearly identifies these academics as the “enemies” of scientific rationalism.U.D'Ambrosio 16

A similar argument is sometimes used in referring to contextualized learning and to Ethnomathematics. In March 1994 I was invited to give a plenary talk on Ethnomathematics in the annual meeting HIMED 94, organized by the British Society for History of Mathematics. At that time London University had issued a course guide for a diploma which included: political roles for mathematics education; motivation, affect and inequalities; mathematics and gender; Ethnomathematics; and several other new subjects. Coincidentally, in the eve of my talk, The Observer (London,27/03/94) published an editorial entitled "Education's guerillas prepare for war". There we read the following comment: "Ethnomaths? This is the maths we pick up by chance in day-to-day life, said to be as valid, if not more so, than the maths we're taught in school. So it follows, as the guide helpfully explains, that classroom teaching merely confuses and demoralizes the pupil. Education is thus reduced to no more than the serendipity of random experience."

Indeed, this aggressive posture gives rise to a climate of war. It is interesting to notice that in the Research Pre-session of the 76th Annual Meeting of the NCTM, in Washington D.C., April 1998, mathematics educators Edward A. Silver and Judith T. Sowder and mathematician Hyman Bass joined in a symposium entitled “Reflecting on the Math Wars: Perspectives on the Role of Research and Researchers in the Public Discourse about mathematics education Reform.” The tone of the discussion was the distance between the work of mathematicians and mathematics educators, associated with a general lack of mutual respect.12 I have to draw attention also to the lack of engagement of students and the public -– who feel the effects -– in this conflict. For how long will the public be kept away from this war?

The crux is the challenge to the prevailing system of knowledge. To challenge scientific knowledge does not mean retrogress. The same is true with religious, artistic, social, political, historical and any form of knowledge. Challenge has always been a coherent response to the state of society and it can be understood, if we look into the full cycle of knowledge in a historical perspective. Of course, this can be mounted effectively only if we free ourselves of epistemological biases which are adapted to 12 The attitudes of mathematicians toward mathematics educators was discussed in D’Ambrosio and D’Ambrosio (1994).U.D'Ambrosio 17

justify the prevailing order. In the specific case of Science, the argument is that science is an object of knowledge of a different nature, in the realm of the ratioid. The appeal to its rational nature is frequent when we discuss mathematical knowledge, but rationality should not imply the denial of social and environmental concerns.

Knowledge is generated by individuals and by groups, and is intellectually and socially organized, and diffused. This full cycle of generation, organization and diffusion intertwines with needs, myths, metaphors, and interests. Like other animal species, the human species developed strategies of hierarchical power. In the human species the control of knowledge has been the main instrument of hierarchical power (see Restivo,1991).

The usual concept of power leads to the notion of "empowerment", in the sense of being establish in the dominant power structure. The trap of empowerment leads to efforts for ascending the ladder of hierarchical power, and this is the subtlest instrument of domination. Although sharing knowledge is essential for the survival of the group (family, tribe, and society), the ascension to power almost invariably leads to co-option through an elaborated system of filters based on a twining of grades, diplomas and certifications. The assessment industry in the society reflects the importance of this system of filters for the dominant power structure. The matter is essentially political and is central when we probe into the reasons for the denial of access of large portions of the human population to the full cycle of knowledge.

Mathematics is key in all this. Mathematics has grown parallel to the build-up of what we call modern civilization. But there has been reluctance among mathematicians, and among scientists in general, to recognize the symbiotic development of mathematical ideas and models of society (D’Ambrosio, 1992). This symbiosis is amply recognized by the eminent historian Mary Lefkowitz when she says that "the evolution of general mathematical theories from those basics [mathematics of Egyptians, Sumerians and others] is the real basis of Western thought"13. When we try to locate Mathematics in the complexity of the actual World, it is not so important that the Egyptians, the Sumerians and other civilizations were ahead of the Greeks, but that the contribution to build up 13 Interview given to Ken Ringle, The Washington Post, June 11, 1996.U.D'Ambrosio 18

general mathematical theories was indisputably Greek, as Lefkowitz says. Likewise, the claim that the medieval scholars received Euclid through the Arabs is not of central importance. The centrally important fact is that, whatever its roots, mathematics as it is recognized today in academia developed parallel to Western thought (philosophical, religious, political, economical, artistic, cultural). I do not need list evidence of the presence of mathematics in all aspects of Western civilization -- simply because they belong to each other.

PART 3: LEARNING FROM THE PAST IN BUILDING THE FUTURE

Origins of mathematics

It is practically impossible to have a complete and structured view of the role of mahematics in building up our civilization. The development of mahematics has been intertwined with all forms of human behavior in the history of mankind. History is a global perspective in time and space. It is misleading to see history only as a chronological narrative of events, focused in a short span of time in narrow geographic demarcations. The course of the cultural evolution of mankind, which cannot be separated from the natural history of the species and of the planet, reveals an increase interdependence of cultures and civilizations across time and space. Cultural specificity, which is now an important category for the study of human diversity, reveals how the cultural encounters have played a role in the evolution of the species (see Cavalli-Sforza,Menozzi,&Piazza,1996).

I have been using the word Ethnomathematics as modes, styles, techniques (tics) of explanation, of understanding, of coping with the natural and cultural environment (mathema) in distinct cultural systems (ethnos). I see this search of explanation, of understanding, of coping with the natural and cultural environment of explanation, of understanding, of coping with the natural and cultural environment as the imprint of our species. This might well suggest Alustapasivistykselitys. My use of Greek roots comes because I am more familiar with these words and because it reminded me of mathematics is a most familiar system of explanations and because the bases of the dominating culture nowadays comes from ancient Greece,

U.D'Ambrosio 19

unquestionably with the presence of much contribution from other cultures which have been exposed to each other in antiquity. The species homo sapiens sapiens appeared on the planet between 50,000 and 100,000 years ago. Its cultural evolution has been, everywhere, the result of the drive towards survival and transcendence, that is, on the one hand to cope with the surrounding environment to find the needed means to survive in the span of a lifetime and, on the other hand, to probe before and after the life span. The development of techniques for survival and the search of explanations of our origins and of the future, depend on understanding and describing the natural environment and the observable phenomena. They all focus on the individual, the relations of individuals with others and the relations between individuals and groups of individuals with nature, both the more immediate, like the elements, plants and animals, and the more remote, that is the skies, as well as the occurrence of birth and death.

These questions and consequent human behaviors have all grown together, in perfect symbiosis and in response to the proposed answers, which are obviously different according to the distinct natural environments of the planet. As a result knowledge is generated. The process of the intellectual organization of knowledge results from a cumulative process which depend both on communication among individuals who either are contemporaneous or belong to different generations of the same culture and on a dynamical process of encounters of different cultures, giving rise to corpora of knowledge.

The development of communication allowed for an increase in the possibilities of interchange of experiences among contemporaries (speech and discourse) and among generations (cryptography and writing). Travelling has been responsible for proximating cultures and for interchanges. Racial purity, hitherto restricted to kin, was soon replaced by mixed procreation, indeed intrinsically desired by the species, since this determines its creative vitality. Since prehistoric times the search for the different as essential for creativity has been a moving force in the history of mankind. Fertility concerns, both in the species (associated with the menstrual cycle) and in the production of food (agriculture), motivated earlier steps towards time measurement. The obvious correlation of women and land

U.D'Ambrosio 20

fertility was determinant in the early religions. Mathematics, more specifically numeracy and geometry, are, together with religion, part of the system of explanations and techniques to cope with the natural and cultural environment of each community, tribe, nation, civilization.

Development of mathematics around the Mediterranean

Just like every civilization in the entire planet, the civilizations around the Mediterranean developed techniques for survival and a set of structured explanations to understand and describe the natural environment and observable phenomena. We have registers of the techniques and explanations of the Greek and Roman civilizations from some 3,000 years before the Christian era. These were clearly associated with systems of explanation and of representation of nature and the natural phenomena, and with the consequent human behavior. These structured systems are, obviously, all interrelated and holistic. The disciplinary boundaries which we now call philosophy, mathematics, religion, arts, medicine, sciences, technology, and other classificatory schemes, were undistinguishable. Their modes of explaining and coping with their environment were built upon their own intracultural evolution, incorporating other forms of knowledge as a result of the cultural dynamics of encounters with other civilizations, mainly those of Egypt and of Babylonia. Equally influential were the encounters with pagan civilizations in the Northern part of Europe and in the Southern part of Africa and in the East. The achievements of the civilizations of Egypt and Babylonia are recorded since about 3,000 BC. The encounter of the Mediterranean civilizations is well documented since about 500 BC, when the early civilizations of Egypt, Babylonia and Judaea were in their apogee and the Greeks and Romans were settling. Less attention has been given to the encounters with civilizations of the North, of sub-tropical Africa, and of the Far East.14 Undeniably, all those encounters were determinant in building up the corpus

14 The following are important books discussing these other cultural environments: Jones & Pennick (1995); Joseph (1992). In discussing African heritage, the books Bernal (1987/1991) and Diop (1991) brought much controversy. The main reaction to Bernal and Diop has been by Lefkowitz (1996). There is a large number of articles on this topic and much of the arguments focus on the concept of historical knowledge. U.D'Ambrosio 21

of knowledge which we today identify as European medieval science. This corpus of knowledge relies on characteristic styles of observation, of identification and description, of experimentation and of theorization in the search for survival and transcendence, with the purpose of prediction and explanation of natural phenomena and of coping with the environment.

In the 15th and 16th centuries, the combination of the encounter of Europeans with completely different natural and cultural environments, after the discoveries and the consequent conquest and colonization, and the intracultural developments in medieval Europe was responsible for establishing the bases of what is now called modern science, or simply science. It is practically impossible to distinguish mathematics from other forms of knowledge in these early moments of modern civilization.

The appropriation of mathematical knowledge

The encounters revealed that, in their search for survival and transcendence, other cultures had developed different styles of observation, of identification and description, of experimentation and of theorization for the prediction and explanation of natural phenomena and for coping with the environment, and had built up different systems of knowledge for these purposes. Parts of the systems of knowledge of these cultures were expropriated by the alien conquerors and colonizers during the process of cultural dynamics during the encounter and were incorporated to the corpus of knowledge which was evolving in Europe to become modern science. As such, this knowledge is diffused among the populations, structured in a culturally aloof, mystified and mistifying way. In particular in the Americas, the systems of knowledge encountered by the conqueror and the colonizer were coherently organized by the indigenous cultures through centuries in response to their specific natural and cultural environments (see,e.g.,Urton,1997). Most of these systems of knowledge, which include what we now call religions, arts, sciences (obviously, we might as well say ethnoreligions, ethnoarts, ethnosciences and so on) have been ignored and even suppressed, and replaced by the systems of knowledge of the colonizer. They were, and in most cases still are, labeled superstitions and folklore. Many of these systems of knowledge still persist in the cultural memory of the people, a few preserving their original structure, but most

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of them modified by the same process of cultural dynamics. This is Ethnoscience and, in particular, Ethnomathematics. As a practice, Ethnomathematics, and the same goes for Ethnoscience in general, deals with components which were expropriated by the power establishment in shaping, through a complex system of cultural dynamics, mathematics as it is accepted today in the academia. It is easy to see that these components, recognized contributions of peoples -- hence of other cultures -- to the construction of mathematics, have been ignored. Historical sources have been "filtered" in a way that the contributions of the peoples, as well as of the dominated cultures, are ignored when not denied.

The effect -- indeed the intention -- is to convey the feeling that mathematics is a body of knowledge reserved for a few, in some way "superior" to others. Not only individuals doing well in mathematics (for obvious reason, whites and males) are judged superior, but mathematics itself, hence everything quantifiable, rigorous, unambiguous (which are values attached to the prevailing hierarchical and economical model), are judged superior. Innumerous cases of success are accepted as "scientific evidence" -- in the Popperian style -- that education is the road of access. If you work hard, as taught by me, you will get there! Throughout history this co-optive strategy has been used. But educators, particularly mathematics educators, are reluctant to recognize that this hinders the way to a dignifying school. Borrowing a neologism introduced by Jean-Yves Leloup, we may use the term “normosis” for the societal acceptance that it is normal the fact that only a few can make it. Leloup claims this that normotic behavior is the most serious cause of perverse societal behavior.15

The acceptance of the normality of this hierarchical model is intrinsic to the evolution of the colonial order. Trade was first dominated by transporting overseas goods needed to maintain this order: arms to insure their authority and objects to ensure their comfort and leisure. The objects included enslaved human beings. Return cargoes were much more profitable, ensuring their economic prosperity. Religious heralds were bringing new philosophies, religions and sciences, including mathematics. It is quite relevant the fact that the first non-religious book printed in the 15 Jean-Yves Leloup is a French theologician and psychologist, founder of the International Holistic University, Paris.U.D'Ambrosio 23

New World was an Arithmetic, by Juan Diez Freyle, the Summario Compendioso de las Cuentas..., printed in Mexico in 1557. This was an Aztec Arithmetic, useful for Spaniards to trade with the natives, who were responsible for the production of gold and silver. Less than a hundred years later this book practically disappeared and was substituted by an European Arithmetic. This clearly shows that the production was no longer in the hands of the natives. Soon, a new merchandise came into the trade, namely human beings. A new technology to design appropriate ships for this new merchandise was required. And a new commercial arithmetic was needed for this trade.16

It is impossible to understand the process of exclusion of large sectors of the population which now compose the peoples of the Americas and other regions of the globe, and the style of knowledge practiced in the World, without a much deeper reflection on the colonial period. It is not the case of putting the blame in one or another, nor is it an attempt to redo the past. Rather, the main goal is to choose a different path for the future. To persist is irrational and may lead to disaster. Maybe the real threat to humanity is not people looking for aliens coming in UFOs, but the nostalgic earthlings. A new world order is urgently needed. Our hopes for the future depend on learning -- critically! -- the lessons of the past. And mathematics permeates our past and our future.

CONCLUSION

Since mathematics is the imprint of the Western thought, as it was already emphasized above, our responsibility as mathematicians and mathematics educators is a major one.We should conduct our teaching in mathematics, and the same is true in other subjects, taking all these reflections into consideration. Instead, what prevails in designing curricula is a proaedeutical style, which justifies every step as a necessary preparation for the next. It is important that the teacher sees him/herself as a researcher of his subject and of the cultural history and understanding of mathematical concepts by the students, and not as a mere transmitter of frozen knowledge.17

16 I owe much for these comments to Charles Morazé and contributors (Morazé,1979).U.D'Ambrosio 24

In conceptualizing a curriculum for the new era we are now entering, it was clear to me that mathematics as conceived in the current school systems is both insufficient and discriminatory to the majority of the population. In proposing matheracy the intention is to give a much broader dimension to mathematical thinking, stressing its value as an instrument for explaining, understanding and coping with reality in its broad sense. Matheracy is the main intellectual instrument for the critical view of the world.

Ethnomathematics, as a program in history and epistemology with an intrinsic pedagogical action, is a proposal which fits the objectives of matheracy. Ethnomathematics responds to a broader conception of mathematics, taking into account the cultural differences which have determined the cultural evolution of mankind and political dimension of mathematics. With the growing trend towards multiculturalism, Ethnomathematics is recognized as a valid school practice, which enhances creativity, reinforces cultural self-respect and offers a broad vision of mankind. In everyday life, Ethnomathematics is increasingly recognized as systems of knowledge which offer the possibility of a more favorable and harmonious relation in human behavior and between humans and nature.

The historiographical proposal of Ethnomathematics looks into the various bodies of knowledge which are incorporated in the syncretic evolution of academic/school mathematics. It allows for a better understanding of the cultural dynamics which generate knowledge. Even more than the externalist approach, the proposed historiography can be seen as a transdisciplinarian and transcultural approach to the history of mathematics. Hence, history of mathematics can hardly be distinguished from the broad history of human behavior in definite regional contexts, recognizing the dynamics of population exchanges. This is fundamental for multicultural education.

This broader view may help in identifying the origin of exclusion of populations and of entire civilizations through denial of knowledge, and this analysis leads to proposals for corrective measures.

17 This is the main focus of the methodology of teacher preparation as proposed by Beatriz S.D'Ambrosio & Tania M.M.Campos (1992).

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Research methodology in Ethnomathematics borrows from anthropology and ethnography. The historiography relies on the identification of other historical sources and on a reinterpretation of texts usually not regarded as relevant to the history of science. The religious sources for Western science and mathematics are important, particularly the Bible, the Talmud and the Qu'ran. It is very difficult to appreciate the contribution of al-Kwarizmi to the founding of Algebra without an appreciation of the Qu'ran. When we try to penetrate other cultural traditions, mythology and common practices are quite important. For example, much can be understood from the Farmer's Almanachs and the likes. Particularly interesting is “The 1902 Edition of the Sears, Roebuck Catalogue”, published 1986 as a facsimile by Bounty Books, in New York. It helps to understand the ethos and pathos of modern society.

All these proposals may be voided without a serious reflection on the relation of the individual with others and with nature. Clearly, these relations are strongly influenced by the creation and use of technology. A critical analysis of technology and particularly of its consequences for mankind as a whole must be a priority of school systems. Intrinsic to this we have a discussion of values and ethics. Matheracy is a most essential instrument for this analysis. This must involve children since an early age. This is the essence of technoracy.

The denial of knowledge which affects entire populations is of the same nature as the denial of knowledge to individuals, particularly children. Hopes for survival with dignity of the human species depend on proposing directions to counteract ingrained practices. This is the major challenge for educators, particularly for mathematics educators.

NOTES

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