A
Comparative Study of Ethnomathematics
SPENCER OREY
IB #
D-0518-186
Extended Essay
Mathematics
Ms. Balhorn
Abstract
According to
the founder of the field of ethnomathematics, Professor Emeritus Ubiratan
D’Ambrosio,[1]
the term “ethnomathematics” is used to express the relationship between culture
and mathematics. Ethnomathematics has
been broken down into five sub-fields: non-western mathematics, mathematical
anthropology, sociology of mathematics, vernacular mathematics, and indigenous
mathematics.[2]
While a large quantity of data exists that links mathematical understanding to
the cultural context in which the student learns, there is a large debate as to
whether an ethnomathematically designed curriculum would be feasible or
practical for modern mathematics education.
This paper analyzes what the possible role of ethnomathematics is in
modern education by focusing on the importance of culture in education with the
research question: Should mathematics in schools be taught using an
ethnomathematical approach and if so, is such an approach feasible? By comparing and contrasting arguments both
for and against ethnomathematics, this paper analyzes whether culture has an
effect upon mathematical understanding, and, what significance ethnomathematics
should have in school curriculum. Based on the results of the comparison,
ethnomathematics is shown to be an applicable and possible part of modern
mathematics education.
Background
Leigh Wood writes,
“All cultures are mathematised, in that people within any culture use ideas of
mathematics in their everyday life”.[3]
According to research of ethnomathematicians such as Ubiratan D’Ambrosio, the cultural
applications of mathematics are present in everyday life. He explains that when
the word ethnomathematics is broken down, the prefix “ethno” refers to natural
and socio-cultural environment. Ethnomathematicians strive to link the
culturally diverse background of the student to the curriculum taught in the
classroom.
Ferreira: "You have
one pen, and you obtain a second. How many pens do you now have?"
Indigenous woman: "I have
one".
Ferreira: "How did you obtain this answer?"
Indigenous woman: "I have
one pen, and I do not need the other". [4]
Based on an
experience like the one above, numerous researchers of ethnomathematics cite
how the cultural influence of the indigenous woman changed her outlook on what
was meant to be an applied mathematics problem. The expected answer was “two”
for the quantity of pens obtained. However, as can be ascertained from
Ferreira’s work, the
However,
others such as Dr. Ron Eglash at Rensselaer Polytechnic Institute cite the more
advanced mathematics that indigenous cultures have developed. In particular,
Dr. Eglash expresses how ethnomathematics provides a new picture of the
sophisticated mathematics of indigenous cultures that is often mistaken for
primitivism. In African Fractals, Modern Computing and Indigenous Design,
Dr. Eglash gives specific evidence to prove the existence of fractals in African
culture within hairstyles, architecture, sculpting, metalworking, and other
crafts.[5]
By focusing on the advanced mathematics that already exists in cultures,
ethnomathematicians can redefine what is often seen as pure primitivism.
Instead, evidence exists that many non-western cultures have developed sophisticated
mathematics that they require for the specific context of their living
situation.
In order to
expand the knowledge of students and teachers, much of the work currently being
done in ethnomathematics is connected to ideas related to multicultural
educational theory. While
ethnomathematicans propose the full integration of multicultural practices,
many social realities often make such integration extremely difficult.[6]
The
late philosopher Paulo Freire described teaching and learning as a relationship
in which both sides gain new knowledge.[7]
The use of various algorithms for such basic mathematical concepts as division
and multiplication draw attention to the need for more specialized education;
with so many methods of working, many think it is important to ensure that the
student is able to understand concepts from his or her own point of view. Because
of ethnomathematics, a new mathematical definition has arisen: “Mathematics is
a social product that originates from practical motivations” [8]
Investigation 1: Arguments Supporting Ethnomathematics in
Education
1.1 A Cultural Perspective
Researchers in
ethnomathematics express the need to teach mathematics in a culturally
sensitive manner, basing methodologies on culture and surroundings rather than
on traditional western mathematics. While most ethnomathematicians agree that
it is impossible to build a curriculum based on ethnomathematics because of the
simple fact that the cultural surroundings are what determine how material
should be taught, it is agreed that a curriculum can be taught from a multicultural
perspective.
1.2 Connections Between Culture and
Mathematics
Ubiratan
D’Ambrosio argues that much of the mathematical curriculum in modern schools
worldwide is so disconnected from a child’s reality that it is impossible for
the child to be a full participant in it: “The mathematics in many classrooms
has practically nothing to do with the world that the children are
experiencing.”[9]
Also, the content of many mathematics programs does little to help in preparing
children for the modern world and leaves out important connections to
technology. Instead, education serves as a filter, allowing or not allowing
students to pass from one grade to another or entrance into advanced
educational opportunities.
Modern
academic curriculum focuses on the use of mathematics without a focus on
culture. It is a discipline without cultural significance or connections, designed
to teach children. In most classrooms, children are expected to learn
prescribed arithmetic procedures without necessarily gaining a deeper and
conceptually significant understanding of what they are studying and without
forming linkages between what they learn and the world around them.
Ethnomathematics
bridges culture and math together, forming a connection between the
surroundings of the child and the world of mathematics. By allowing the student
to develop a personal understanding of why mathematics works, thereby forming
connections, the student is then able to explain and comprehend his or her work
on his or her own terms. In essence, the student personalizes the given
mathematical information in order to develop a stronger understanding of the
material, making it easier to remember and utilize by the way of connections to
reality.
1.3 Algorithms Show Cultural
Differences
Fundamental
differences in algorithms, precise, systematic methods for solving a class of
problems,[10] show
the effect that culture has on how students do math problems. Professor Daniel
C. Orey is the principal investigator and coordinator of the Algorithm
Collection Project at
For example, in
the operation of division (see figures below), students from various parts of
the world produced greatly differentiated algorithms for the same task. [12]
Diverse algorithms still yielded the same mathematical answer, proving the
point of many ethnomathematicians: there is more than one way to solve a
problem. Different teaching methods are needed to cater to the individual thought
process of the student.
Two Examples
of Algorithms
Both students from
Professor Orey’s sample attended Encina High School in Sacramento, California,
but because they both originally immigrated to the United States (originating
from Yugoslavia and Vietnam), they were taught to use alternative algorithms
that reflected the pedagogy of their native countries. These algorithms
differed from one another, but both came to the same answers at the end of the
problem. The difference in the mathematics that each pupil showed was a direct
consequence of the culture in which each student was brought up: using the
principles of ethnomathematics, it is possible to realistically hypothesize
that a single curriculum would be detrimental to both students; rather than
expanding upon the knowledge that each student had already obtained, a set
standard would force both students to regress backwards in their methods in
order to learn new algorithms. The importance of curriculum designed to reflect
culture is clear in situations involving algorithms, leading researchers to
propose that students need a more personal, connected explanation of mathematical
material in order to further successful learning of concepts.
1.4 Evidence in Favor of Ethnomathematics
Much
research has been done to show a correlation between culture and the ability of
the student to perform mathematics. In a study, Dr. Eduardo Jesús
Arismendi-Pardi of
However,
in studies involving education, there is generally an unknown variable in that
the group of students may have different learning abilities. In his study, Dr.
Arismendi-Pardi stated the assumption that the students in both classes were
comparable in mathematical abilities even though the first class (taught
without ethnomathematics) was taught in Fall 1996 while the second (taught with
ethnomathematics) was taught in Fall 2000. From an opposing standpoint, the results
reached by the survey can be brought under question by the examining the
fundamental differences between students four years apart and possible changes
in the teaching style of Dr. Arismendi-Pardi. It is also possible that the
quality of students at the college simply improved, bringing about higher final
grades in the Intermediate Algebra course. More studies of this sort are needed
to prove to skeptics that ethnomathematics can play a key role in improving the
process of educating the student.
A
similar example involves another study of Adrianna Magallanes’ at Torch Middle
School, in which students trained with an ethnomathematics software were
evaluated against a similar group of students taught only with traditional
methods. The students using the software obtained higher scores, showing
further evidence in support of ethnomathematics.[14]
Investigation 2: Arguments Against Ethnomathematics in
Education
2.1 Fun and Games in the Classroom
Arguments
against an integration of ethnomathematics into modern curriculum center around
concerns that through specialized curriculum adapted to culture, using
methodologies such as games and activities will result in the loss of academic
achievement. John Leo views ethnomathematics as a rant about “eurocentrism,” claiming
that with the focus on culture instead of pure curriculum, the knowledge of the
student is impaired by the inclusion of too much cultural information.[15]
Many
of those opposed to ethnomathematics state a similar argument. Marianne
Jennings cited her daughter’s textbook as being full of poetry, pictures, and
lectures on the environment. She also cites that the book, designed for
multicultural education, is 812 pages long as opposed to the average of 200
pages in an algebra textbook in
2.2 Cultural Misrepresentation
A
fatal flaw with ethnomathematics is that it relies upon “cultural diversity” as
a teaching tool. The more multicultural the material becomes and the more
diverse the audience that the teacher tries to appeal to, the greater the
possibility that the mathematical lessons will be both more shallow and more
superficial. While such exposure to diversity is meant to build connections
between the material and the everyday life of the student, the curriculum is at
risk of being lost in the process, leaving the student with at best, an
unsatisfactory understanding of what is being taught. Ethnomathematics also
precludes the single most important requirement for a successful education:
coherent means to a discernible end.[17]
In the era of standards and testing that have overcome most schools, these two
points are a large concern for educators.
2.3 The Difficulty of Designing
Curricula
There
are few if any textbooks published in English that teach mathematics from an
ethnomathematical standpoint. Similarly, there are very few classes taught at
universities designed towards multicultural mathematics for preparing teachers.[18]
With this lack of direction, it is difficult for teachers interested in
ethnomathematics to create a common curriculum. Also, due to the importance of
appealing to the students’ needs and ethnic backgrounds, it is almost
impossible to create a single curriculum for multiple classes. The sheer
difficulty of engineering material designed specifically for individual students
and assessment standards from multiple cultures places a large amount of stress
on the teachers.
Many
educators are unfamiliar with the links between math and the surrounding world.[19]
It appears from discussion related to ethnomath in the literature that there is
a concern that many teachers simply do not have the time to research and employ
strategies aimed at aiding all ethnicities of students in their classrooms.
Ethnomathematics
also preaches a de-emphasis on the authority of teachers.[20]
Rather than “teaching to” the students, the learning process becomes more of a “teaching
with” the interaction and help of the students. Ethnomathematics requires
teachers to learn more about their students before they can properly teach the
curriculum, meaning that teachers would be forced to throw away the established
shield of formality in order to help their students to understand mathematics
on a new, more familiar level.
Many
students have been taught to stay quiet and do not formulate questions in
class.[21]
The inclusive curriculum that is demanded by ethnomathematics only works when
both the teachers and students work together to further education. If students are
passive or fail to cooperate, it becomes seemingly impossible to teach them
using this system. And often, cultural differences can lead students to refuse
to work together, resulting in an unfriendly situation from what was meant to
be an inclusive and good-willed system. Students may also have difficulty in
comprehending curriculum designed for multiple groups. Being assaulted by
multiple subjects at once instead of pure mathematics may confuse some
students, resulting in a greatly stunted rate of learning.[22]
Much
of the activities published in English used in multicultural mathematics and
ethnomathematics have been designed for younger children, showing them as
simple and folkloristic introductions to “real” mathematics.[23] This assumption has the possibility of making
students uneasy towards such a curriculum. In order for a system based on
ethnomathematics to work in the classroom, there needs to be a unifying and
more expansive component that works beyond what currently exists.[24]
2.4 The Multicultural Mathematics
Dilemma
Ethnomathematics is meant to succeed
where previous multicultural mathematics education has failed. The idea of
connecting mathematics to cultural heritage invites examples which replace
class characters such as Dick and Jane counting marbles with Tatuk and Esteban
counting coconuts.[25]
Changing curriculum to reflect culture in such a superficial manner only serves
to reinforce views on primitivism. For ethnomathematics to work, it needs to
focus more on the math that is suitable to the particular context of the
classroom. Building on mathematics that has already been taught to children in
the context of everyday life may be beneficial to students, while forcing them
to learn ancient Egyptian math may have a detrimental effect. Picking and
choosing what and how students should be taught is a difficult process that,
while well-meaning, runs the risk of leaving many students behind.
In
individual cases, it is possible that some students may not do well with a
curriculum focused around ethnomathematics. Conversely, some may not do well
without it. In order to improve mathematics education, the solution cannot be
to simply choose which group should lose out on increased understanding.
Conclusion
When examining the pros and cons of
mathematics education, it becomes clear that ethnomathematics, while well
meaning, is not the panacea that can remedy any and all faltering that present
day mathematics education is experiencing. Multicultural education has failed
in the past by unintentionally emphasizing the false notion that mathematics
from indigenous cultures is at most primitive, especially when examples focus
only on lower primary school levels.[26]
However, ethnomathematics seeks to redeem multicultural education in the
classroom by focusing on the appropriate material for the context of the math
class rather than simplifying culture.
There
was also a greater amount of data supporting ethnomathematics than against it, with
studies such as that of Dr. Arismendi-Pardi demonstrating how the majority of
students tested, with exceptions appearing amongst individuals, appear to
consistently learn better with curriculum focused on bridging culture and
mathematics. Many of the arguments against ethnomathematical education seem
short-sighted, if not premature, focusing on the present difficulty of
converting curriculum rather than focusing on the benefits that could be
achieved in the long run. An approach towards teaching through ethnomathematics
has the possibility of revolutionizing mathematics education in schools, even
if such an approach may be difficult.
References
Arismendi-Pardi,
Eduardo Jesús,. “Comparison of the Final Grades of Students in Intermediate
Algebra Taught With and Without an Ethnomathematical Pedagogy” http://www.rip.edu/~eglash/isgem.dir/texts.dir/ejap.htm
Ascher,
Marcia., Mathematics Elsewhere: An Exploration of Ideas Across Cultures.
Ascher,
Marcia., Ethnomathematics: A Multicultural View of Mathematical Ideas
Burak, D.,
Basic criteria for the adoption of mathematical modeling in elementary, middle
and high schools. Zetetiké, 2(2): 47-60
D’Ambrosio,
Ubiratan., “What is Ethnomathematics, and How Can It Help Children in Schools?”
Eglash, R.
"When math worlds collide: intention and invention in
ethnomathematics." Science, Technology and Human Values , vol 22, no 1,
pp. 79-97, Winter 1997.
Eglash, R. African
Fractals, Modern Computing and Indigenous Design.
Famularo, T.
J. Should multiculturalism permeate the curriculum? In Noll, J.W (Ed). Taking
sides: clashing views on controversial educational issues (pp.106-122).
Freire, P., Pedagogia do Oprimido [Pedagogy of
the oppressed]. Rio de Janeiro: Paz e Terra, 1970
Gerdes,
Paulus., Awakening of Geometrical Thought in Early Culture. Marxist
Educational Press, 2003
Kitchen, R.S.
and Becker, J.R., Ethnomathematics. Journal for Research in Mathematics,
29(3): 357-363, 1998
Leo, John., “That
So-Called Pythagoras” in U.S. News & World Report, Inc., May, 1997
Magallanes,
Adriana, “Comparison of Student Test Scores in a Coordinate Plane Unit Using
Traditional Classroom Techniques Versus Traditiation Techniques Couplies with
Ethnomathematics Software at Torch Middle” http://www.ccd.rpi.edu/Eglash/csdt/na/loom/classrm/amm_abs.htm
Maurer, S. B.,
“What is an Algorithm? What is an Answer? In the Teaching and Learning of
Algorithms in School Mathematic.” 1998 NCTM Yearbook (Morrow & Kenney,
Eds.).
Orey, Daniel.,
The Algorithm Collection Project http://www.csus.edu/indiv/o/oreyd/ACP.htm_files/algprojexpla.html
Pedroso, S.R. Mathematical
modeling as a teaching and learning method: a report of a classroom experience.
ETECAP, 1(1): 1-10., 1998
Powell, Arthur
B., Ethnomathematics: Challenging Eurocentrism In Mathematics Education
Rosa, Milton.,
“Ethnomathematics: Teaching and Learning Mathematics from a Cultural
Perspective
Selin,
Helaine., Mathematics Across Cultures: The History of Non-Western
Mathematics
Zaslavsky, C.
Ethnomathematics and multicultural mathematics education. Teaching Children
Mathematics. 4(9): 502-504, 1998