Women as the First Mathematicians
By Claudia Zaslavsky
Women were the first mathematicians ever! So claims Dena Taylor in an article entitled “The Power of Menstruation” (Mothering, Winter 1991).
- The cyclical nature of menstruation has played a major role in the development of counting, mathematics, and the measuring of time… Lunar markings found on prehistoric bone fragments show how early women marked their cycles and thus began to mark time. Women were possibly “the first observers of the basic periodicity of nature, the periodicity upon which all later scientific observations were made” (quote is from William Irwin Thompson: The Time Falling Bodies Take to Light, St. Martin’s Press, 1981, page 97).
Let’s review some of the evidence. In my book Africa
Counts: Number and Pattern in African Culture (L. Hill, 1979), I wrote about the Ishango bone, an artifact that has since found its way into books on the history of mathematics by Howard Eves, George G. Joseph, and others. This incised bone was discovered in the 1960s on the shore of a lake in northeastern Zaire. Originally described as a record of prime numbers and doubling (perhaps a forerunner of the ancient Egyptian system of multiplication by doubling), Alexander Marshack later concluded, on the basis of his microscopic examination, that it represented a six-month lunar calendar. The dating of the Ishango bone has been reevaluated, from about 8000 B.C.to perhaps 20,000 B.C. or earlier. Similar calendar bones, dating back as much as 30,000 years, have been found in Europe. Thus far the oldest such incised bone, discovered in southern Africa and having 29 incisions, goes back about 37,000 years.
Now, who but a woman keeping track of her cycles would need a lunar calendar? When I raised this question with a colleague having similar mathematical interests, he suggested that early agriculturalists might have kept such records. However, he was quick to add that women were probably the first agriculturalists. They discovered cultivation while the men were out hunting, So, whichever way you look at it, women were undoubtedly the first mathematicians!
(The above appeared in the Fall 1991 issue of the Women in Mathematics Education Newsletter.)
Author’s subsequent note: In his revised (1991) book, Marshack has an extensive note on the redating of the Ishango incised bone: “The date obtained suggest that the Ishango tool with its sets of marks, its inset points, and its association with bone harpoons, was 20000 to 25000 years old.”
“Mathematics” & Ethnomathematics: Zimbabwean Students’ View
By David Kufakwami J. Mtetwa
ethnic group, for example, the Shona people of Zimbabwe. Needless to say, interpreting Ethnomathematics that way is proscribing only a narrow view of it, incompatible with the now widely accepted definition presented above. Indeed, one might even consider any kind of mathematics, including “school” mathematics, “college” mathematics, or “professional” mathematics (mathematics as conceived and practiced by the professional mathematics community) as forms of Ethnomathematics. In other words, rather than seeing the situation as one which pits “mathematics” against Ethnomathematics, particularly in prestige (Fig. lA); it is probably more appropriate to see just Ethnomathematics, with what most people call mathematics (unprefixed but essentially referring to professional mathematics) as one of the many forms of Ethnomathematics (Fig. lB).
Figure 1: Relationship between
“Mathematics” and Ethnomathematics
Views on Ethnomathematics expressed by Zimbabwean secondary school (11th grade) students who participated in a recent study appear to be in line with the “appropriate” view described above (Mtetwa, 1991). Using in-depth individual interviews, the study’s objective was to explore the students’ beliefs and perceptions about mathematics. One aspect of the study included an exploration of the students’ perceptions of “out of-school” mathematics, in particular, the mathematics in Zimbabwean traditional life, modern and non-modern, for example during their forefathers’ time.
Discussion with the student participants for this aspect of the study was centered around three questions
concerning (1) whether mathematics can be done, practiced, or learned in the students’ own first languages, apart from English which is the medium of instruction in school and is also a second language for them); (2) whether there was any mathematics at all practiced in the lives of their forefathers, that is, in pre-colonial Zimbabwe; and (3) whether out-of-school mathematics (if, according to the students, there is any), in particular their forefathers’ (traditional) mathematics, is legitimate or “real” and “proper” mathematics.
Most of the 10 students (6 female, 4 males) inter- viewed readily concluded that mathematics can be done in other languages (including their own indigenous languages such as Shona) apart from English. “You would be saying the same things but using different words,” one student said. “Math is universal ….” said another. However, three of the students expressed the same conclusion only after deliberate reflection and after professing, “I don’t know,” or, “I am not sure.” One student suggested that although doing or learning mathematics could indeed be done in Shona, it would be exceedingly difficult to do so in Shona because Shona does not have a lexicon for mathematics. Overall, the student’s realization that mathematical processes are not expressible only in language medium, which in their case would be English, amounts to an implicit acknowledgement that mathematics is not just a feature of schools and classrooms. Mathematics exists in life out of classrooms.
This was confirmed and more explicitly when discussing the second of the questions stated above. The students unanimously agreed that inhabitants of precolonial Zimbabwe, that is, their forefathers, practiced mathematics. “Math didn’t start with schools, it existed long before that… children could make ten things….” one student said. A number of the students pointed to the existence of The Great Zimbabwe National Monument (remains of a 13th century castle in central Zimbabwe) as evidence of their claim.
However, the students granted the existence of mathematical practice in pre-colonial Zimbabwean life. About half of them suggested that although those tradi tional people practiced mathematics, they [traditional- ists) were not aware that they were doing mathematics. This is how student R put it: “They [traditionalists] didn’t know it [mathematicsj existed, they didn’t call it math, they thought it was something else…” Such remarks may lead one to think that to those students’ notion of mathematics was restricted to “school” mathematics, which the traditionalists would not have known or been aware of; though they practiced some mathematics.
The students’ responses to the third discussion question on the legitimacy of the mathematics practiced by the students’ forefathers (traditional mathematics) help to clarify their thinking. All ten students granted the traditional mathematics as legitimate mathematics. In addition, the students regarded the traditional mathematics as the “foundation” on which “school” mathematics developed and expanded into its current forms. All the students expressed that everyday life out-of-school mathematics in general and traditional Ethnomathematics in particular, is too trivial and elementary. An “infant” form of school mathematics; almost entirely “just addition and subtraction;” too ‘basic” while school math is more “advanced, developed, and complicated”; just “beginning math”; were some of the characterizations of traditional Ethnomathematics given by the students.
Because the students considered traditional mathematics as too trivial and elementary, though legitimate, some of the students felt hesitant to regard it as “proper” and “real” mathematics. For example, student T who had earlier defined mathematics as “reasoning” actually dismissed traditional mathematics as not quite” real” mathematics because “it is too simple and routine that it does not involve any reasoning at all.”
Three important observations can be made from these expressions of student perceptions of non-school Ethnomathematics discussed above. First, the students’ acknowledgement of the existence of both modern and traditional Ethnomathematics, and their belief that Ethnomathematics is the foundation upon which school nathematics evolved and developed is in line with what I earlier called the “appropriate” view the Ethnomathenatics – mathematics relationship. In this view, school mathematics is an outgrowth and subset of Ethnomathematics.
Second, by considering the difference between school mathematics and modern and traditional Ethnomathematics as that of level of complexity rather than kind, the students are seeing both school and out-of-school mathematics as different ends of the same thing, mathematics. This has an important instructional implication. It may make such students ready, even enthusiastic, to icorporate and learn non-school Ethnomathematics and school mathematics without treating the non-school ethnomathematics as perhaps irrelevant. The challenge is now for Zimbabwean researchers to find interesting and non-trivial non-school Ethnomathematics and make it available to the students, and for teachers to present it in constructive and challenging ways.
Finally, that the students did not denigrate traditional Ethnomathematics as perhaps substandard, unprestigious, unsophisticated, and worthless vis-a-vis school mathematics is a welcome observation. In a sociy that is still reeling from horrendous physical and psychological effects of colonialism which denigrated everything indigenous and glorified everything Western, this kind of pride in the students’ own out-of-school
mathematical realities, past and present, can be a useful source of a sense of self-concept and confidence, indispensable ingredients of a successful schooling career.
D’Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5(1), pp 44-48.
Mtetwa, D. K J. (1991). An investigation of Zimbabwean secondary school students’ beliefs about mathematics and classroom contexts. Unpublished doctoral dissertation, University of Virginia, Charlottesville, Va 22903.
The VIIIth Interamerican Conference on Mathematics Education – VIII IACME was held at the University of Miami, USA, from August 3-7, 1991. The following are examples of presentations related to Ethnomathematics that were presented at the VIII IACME (the proceedings should be available from UNESCO, 7 place de Fontenoy, 75700 Paris, FRANCE, in March 1992):
Panel A on Integration of the Sociocultural Context in Mathematics Teaching, Martha Villavicencio, Peru; Ubiratan D’Ambrosio, Brazil; and Elisa Bonilla, Mexico.
Introducao ao Sistema de Numeracao Hindo-Arabico Segundo o Processo Historico, Eduardo Sebastiani, Brazil.
Una Proposta Pedagogica en Etnomatematica, Ademir Donizeti Caldeira, Brazil.
Uma Proposta de Ensino de Matematica entre os Guarani, Jackeline Mendes, Brazil.
Ensenanza de las Matematicas y Conflicto Cultural: El Caso de la Educacion de Adultos, Alicia Avila, Mexico. Multicultural History Can Help Teachers to Implement the NCTM Curriculum Standards, Beatrice Lumpkin, USA
The 3rd Pan-African Congress of Mathematicians was held August 20-28, 1991, in Nairobi, Kenya. Paulus Gerdes made presentations on “Ethnomathematical Research: Prpeparing a Response to a Major Challenge to Mathematics Education in Africa” as part of a special Symposium on “Mathematics Education in Africa for the 21st Century”. Professor Gerdes also presented a paper “On the History of Mathematics in Subsaharan Africa”.
ICME-7 Working Group on Multicultural/Multilingual Classrooms
Plans are proceeding for Working Group 10 “Multicultural and Multilingual Classrooms” that will be part of ICME-7 in Quebec in August 1992. The Working Group will meet for four sessions of 90 minutes each. The first session will be for posters describing projects and displays of materials. The next two sessions will be in subgroups to facilitate discussion and the development of action plans. The fourth and final session will be a total group meeting with reports from the subgroups and discussion.
If you would like to participate in the Working Group on “Multicultural and Multilingual Classrooms” please contact the Organizer of the subgroup in which you have the most interest:
Curriculum, Resources & Materials for Multicultural\Multilingual Classrooms
1209 Doonesbury Dr.
Austin, TX USA
Teacher education for Math in Multicultural\Multilingual Classrooms
1809 Sagebrush Rd.
Billings, MT 59105 USA
Multicultural\Multilingual Classrooms & the Curriculum for the XXIst Century
Victoria University of Tech
Dept of Teacher Education
P.O Box 54
Footscray 3011 AUSTRALIA
Vernacular Language and Culture in the Math Education of Indigenous Groups
College of Education
University of New Mexico
Albuquerque, NM 87131 USA
If you need an official invitation to facilitate your participation please contact the Chief Organizer (Rick Scott, College of Education, University of New Mexico, Albuquerque, NM 87131 USA) who will communicate your request to the Program Committee.
ISGEm at NCTM Annual Meeting
ISGEm is sponsoring a Panel Discussion on Ethnomathematics at the NCTM 1992 Annual meeting in Nashville, Tennessee. The Discussion will be held from 10:30 to noon on April 2, 1992. Ubiratan D’Ambrosio will present the Cultural Dimension, Marilyn Frankenstein the Socio-Political Dimension, Arthur Powell the Historical and Epistemological Dimension, Gloria Gilmer the Classroom Dimension.
There will also be an ISGEm Business Meeting at the Nashville Doubletree Inn the same day from 4:30 – 6:00. All interested are encouraged to attend.
ISGEm and the Criticalmathematics Educators Group to Cosponsor Post ICME-7 Conference
ISGEm and the Criticalmathematics Educators Group will sponsor a one day Conference on August 24, 1992, immediately following ICME-7 in Quebec. For further information please write:
College of Public and Community Service
University of Massachusetts – Downtown
Boston, MA 02125 USA
Department of Academic Foundations
Newark, NJ 07102 USA
Have You Seen
“Have You Seen” is a regular feature of the ISGEm Newsletter in which works related to Ethnomathematics can be reviewed. We encourage all those interested to contribute to this column.
D’Ambrosio, Ubiratan (1990). Etnomatematica: Arte ou Tecnica de Explicar e Conhecer, Editora Atica, Caixa Postal 8656, Sao Paulo, Brazil.
In this volume Professor D’Ambrosio presents in Portuguese his vision of Ethnomathematics and supports each chapter with an extensive bibliography. Among the topics he treats are “Values in the Teaching of Mathematics”, “An Alternative Proposal”, “On Creativity and a Conceptual Transition in Modern Science”, “Some Reflections on the Future”, “An Anthropological Focus on Mathematics and its Teaching’, “Scientific Knowledge and the Search for Alternative Methodologies”, “Critical Vocabulary” and “Annotated Bibliography”.
Saavedra, Holger and Villavicencio, Martha. Hacia una estandarizacion de vocablos quechuas en Matematica, Lima, Enero, 1990, 85 pages.
This document contains a proposal of Quechua words to standardize in the Mathematic class in the Bilingual Primary School. This proposal has been formulated after research into the words that are used in the Quechua communities of Peru.
The document has been distributed free of charge by OEI in the “III Symposium so Iberoamericano de Educacion Matematica” developed in Sevilla from September 20 to 22, 1990.
Villavicencio, Martha (1990). La Matematica en La Educacion Biliugue: El caso de Puno. Lima, Peru. Edited with the support Gesellschaft fur Technische Zusammenarbeit (GTZ). 193 pages.
This document has been structured in three parts. In the first part an overview of the Mathematic Education of the Peruvian Indians from the Pre-Inca period to the Republic, and a summary of the history of the impact of Western Mathematics and the trends of the Mathematic Education since 1950.
The second part contains the systematization of the experiences in the implementation and development of a methodological alternative and educational materials for the Mathematics Education in the Experimental Project of Bilingual Education (Quechua-Spanish and Aimara-Spanish in the highlands of Puno in southern Peru, from 1980 to 1990. In this Project the Mathematic Education is based on the Quechua or Aimara culture.
In the last part a set of reflections are presented on the many needs and the tasks needed urgently to realize the improvement of the Intercultural/Bilingual Mathematics Education in the Indian population of Peru.
Mensh, Elaine & Larry Mensh (1991). The IQ Mythology. Class, Race, Gender and Inequality. Southern Illinois University Press, (800)848-4270, Ext. 950. 200 pages; ISBN 0-8093-1668, $19.95.
In their expose of IQ testing as a pseudoscience, the Menshes offer a wealth of evidence to show that IQ tests and tests of “aptitude,” such as the SAT, have as their prime purpose to justify the placement of children in either superior or inferior educational settings on the basis of their race, gender, and class. They are equally scathing in their denunciation of the hereditarian and the environmentalist schools of thought about the supposed inability of certain groups in the population to profit from the best educational resources. Whether the judgment that “this child can’t learn” is based on a theory that girls’ brains are not suited to learning math or that the “culture of poverty” prevents African-American children from profiting from an enriched educational climate, the outcomes are the same – some children are deprived.
The Menshes trace the history of IQ testing from the original Binet tests right up to recent court cases challenging the use of such tests to place children, as in California, or to award scholarships, as in New York State. Although the names and the forms of the tests may change in response to attacks on their use, the purposes and results are the same.
One chapter is devoted to the use of various tests in African countries, particularly South Africa, and should be of special interest to members of ISGEm.
One disturbing feature of the book is the tendency of the authors to use out-of-context quotations from other books in order to justify their arguments, thereby possibly distorting the views of other writers. In general, however, they make a good case for branding the whole field of IQ testing a pseudoscience.
Zaslavsky, Claudia (February 1991). “Multicultural Mathematics Education in the Middle Grades:” Arithmetic Teacher: pp 8-13.
The author introduced several activities based on African cultures to students in grades six and nine. The sixth graders carried out a project involving area and perimeter of homes constructed by some African peoples, and learned through their own hands–on experiences that the circular home would afford the largest amount of floor space for a given amount of materials for the walls. Yet teachers engaged in the same project in an inservice course took a short cut and merely applied formulas, entirely negating the goals of the project. Both sixth and ninth grade students analyzed sand drawings in the form of traceable networks made by children in Zaire in imitation of adult fishing nets and embroidery patterns, and by adults in Angola to accompany storytelling and learning experiences for the young. Students’ evaluations indicated that these experiences broadened their understanding of other cultures and engaged their interest mathematically.
Educators Against Racism and Apartheid (EARA) (1990). Apartheid Is Wrong: A Curriculum for Young
People. 308 pages in three-ring binder. Order from EARA, 164-04 Goethals Avenue, Jamaica, NY 11432
USA. $17.00 + $5.00 p&h.
Multidisciplinary, hands-on curriculum dealing with apartheid in South Africa and racism in the U.S., for teachers, librarians, community groups, and parents of young people. Includes lessons and projects in all subject areas, K- 12. Most of the mathematics lessons were written by ISGEm member Claudia Zaslavsky.
George Gheverghese Joseph (1991). Crest of the Peacock: Non-European Roots of Mathematics. I.B. Taurus, London, 367 pages.
At last, the book many of us have been waiting for is here. Crest of the Peacock is a survey, in some depth, of the non-European mathematics that built the foundation of the modern mathematics we enjoy today. The book is readable, and the mathematics clearly stated in a form accessible to anyone who remembers her high school mathematics. The poetic title comes from the Indian Vedanga Jyotisa, which proclaims, “As are the crests on peacocks, as are the gems on the heads of snakes, so is mathematics at the head of all knowledge.”
The quotation suggests a reverent, almost elitist concept of mathematics in early India. However, Joseph is far from elitist in his approach to the history of mathematics. On the contrary, he finds many opportunities to show that the development of mathematics is a response to the needs of people and is made possible by the accumulated productive skills of people. Above all, the weight of the evidence that he presents is a powerful
argument against what he calls the “Eurocentric model” of the history of mathematics, “with Greece as the source and Europe the inheritor and guardian of the Greek heritage.” Motivation for the “Eurocentric model” is not a mystery to Joseph who offers this explanation: “The contributions of the colonized people were ignored or devalued as part of the rationale for subjugation and domination.”
Main topics discussed in Crest of the Peacock include tally and string records, Maya numerals, the mathematics of ancient Egypt, Babylonia, India, China, and the “Arab” contribution. Given the author’s birthplace in Kerala, India, and his Syrian ancestry, it is understandable that he claims “By the second half of the first millennium, the most important contacts for the future of the development of mathematics were those between India and the Arab world. Here he is speaking of more than the development and transmission of Hindu-Arabic numerals whose importance, he says, cannot be overestimated.
Joseph outlines in some detail the early Indian trigonometry, and indeterminate analysis, and the later, little-known (1400-1600) infinite series expansions for trigonometric functions. The rich Chinese contribution is treated with equal respect and similar length (81 pages):
matrix methods for solution of systems of equations, indeterminate analysis, the Chinese anticipation by four centuries of the Horner-Rusini Method and the Pascal triangle for higher-order equations, the use of negative numbers, and double false position solution of equations were also part of Chinese mathematics.
The final chapter is “Prelude to Modern Mathematics – The Arab Contribution.” In just 47 pages, Joseph manages to compress the extensive contributions of Islamic math: decimal fractions, inheritance problems, number theory, figurate numbers, algebra, number theory, real numbers, conic sections, the introduction of six basic trigonometric flinctions and identities, and exploration of non-Euclidean geometry.
I’ll confess I would have liked to have found more on Abu Hamil, the “Egyptian calculator” whose work extended the algebra of al-Khowarizmi to use several variables, powers up to eight, and irrational solutions. I also missed references to the Cairo “House of Wisdom,” a Science Academy where in Yunus, al-Haytham, and other scholars worked. I wondered about the use of the term “Arab” instead of the more general “Islamic”. As the author points out, the mathematicians of this tradition came from Persia, Central Asia, Egypt, North Africa and Spain, as well as Arabic countries.
As is the case with some “standard” histories of mathematics, there is a chapter on ancient Egypt and one on Babylonia early in the book. Unlike the standard histories, the chapter on Egypt emphasizes the African origins of the Egyptian civilization. In general, the chapter follows the analysis made by Gillings’ in Mathematics in the Time of the Pharaohs. Babylonian mathematics is also discussed at greater length than in the usual text.
Joseph asks, “Is the overly critical attitude to Egyp- tian mathematics found in many textbooks an attempt to counteract the Greeks’ (and others’) generous acknow- ledgements of the great debt they owed these earlier civilizations. Were this debt acknowledged today, “This would undermine one of the central planks of the Euro- centric view of history and progress.” The chapter on Egypt ends with two sentences pointing out that Alexan – dria, Egypt, became the center of Hellenistic mathematics, combining the traditions of Egypt, Babylonia and classical Greece. He says no more about this period because it has been “extensively explored” in other books.
Unfortunately, this also leaves out the work of Hypatia, an opportunity to discuss a woman algebraist. In addition, this understandable omission may create the appearance of an abrupt end to Egyptian mathematics c 300, in contrast to other mathematical traditions whose development is shown as a continuum in India, China and Mesopotamia. The presentation of the African contribution could also have been strengthened by including the Egyptian and North African contributions to medieval Islamic mathematics. These omissions do not take away from the usefulness of this work but indicate possible areas of expansion in the future editions of Crest which I am sure will appear.
ISGEm Advisory Board
Gloria Gilmer, President
Math Tech, Inc.
9155 North 70 Street
Milwaukee, WI 53223 USA
Ubiratan D’Ambrosio, First Vice President
Universidade Estadual de Campinas
Caixa Postal 6063
13081 Campinas, SP BRAZIL
David Davison, Second Vice President
Department of Curriculum & Instruction
Eastern Montana University
1500 N. 30th Street
Billings, MT 59101-0298 USA
Luis Ortiz-Franco, Third Vice President
Dpeartment of Mathematics
Orange, CA 92666 USA
Claudia Zaslavsky, Secretary
45 Fairview Avenue #13-1
New York, NY 10040 USA
Anna Glosgalvis, Treasurer
Milwaukee Public Schools
3830 N. Humboldt Blvd.
Milwaukee, WI 53212 USA
Patrick (Rick) Scott, Editor
College of Education
University of New Mexico
Albuquerque, NM 87131 USA
Henry A. Gore, Program Assistant
Department of Mathematics
Atlanta, GA 30314 USA
Sau-Lim Tsang, Assistant Editor
ARC – Southwest Center for Ed. Equity
310 Eighth Street, #305A
Oakland, CA 94607 USA
Jerome Turner, NCTM Representative
St. Francis Xavier University
Antigonish, NS CANADA B2G 1CO
David K Mtetwa, Member-at-Large
14 Gotley Close
Marlborough, Harare, ZIMBABWE
Lawrence Shirley, Member-at-Large
Department of Mathematics
Towson State University
Towson, MD 21204-7079 USA