A Peer Reviewed Journal - ISSN 1499-819X

Volume 11, Number 8

Fall 2007

© 2007 Rebecca Dumoulin and EGallery

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What is the Story of Ethnomathematics?

Culturally Responsive Mathematics in Elementary Classrooms

Semester 2 Curriculum Case Inquiry

By Rebecca Dumoulin

For Sharon Friesen and Karen Dittrick

April 2007


Can Yup’ik Eskimo storytelling help students to learn geometry?

Can Madagascar solitaire games teach about patterns?

Does the ancient Inca Empire have anything to teach us about numbers and record keeping?

bone
solitaire
quipu1
Ishango Bone
Madagascar Solitaire
Quipu

The term ethnomathematics first appeared to me in a reading I did early in my second semester of the BEd program. Sleeter (1997) writes that ethnomathematics challenges people to examine the history of mathematics, thus seeing math in its entirety, not just as a discipline developed by western knowledge. Sleeter explains that ethnomathematics engages students in practical problem solving, which is used by all people throughout the world. Ethnomathematics is a concept I would like to know more about, hence this inquiry. I am very interested in using knowledge from different cultures in my own classroom. I believe using this type of knowledge fosters respect for people from different parts of the world, broadens our current understandings, and increases our opportunities to engage all students. The concept of ethnomathematics seems to fit with my values and how I would like to see classrooms operate. I begin this writing by highlighting some definitions of ethnomathematics and then writing about my experience with math when I was a child.

What is ethnomathematics?

“The term ethnomathematics is used to express the relationship between culture and math” (D’Ambrosio, 2001, p. 308). Bishop (2002) describes ethnomathematics as a theoretical concept which addresses the idea that “mathematical knowledge is a cultural product and that mathematics education is culturally shaped” (p.120). I would say the flaw in Bishop’s definition is that currently mathematics education is heavily focused on European concepts and is not “culturally shaped.” However, I realize that ethnomathematics, as both theory and practice, encourages educators to teach math in culturally responsive and inclusive ways. Casey (2007) defines ethnomathematics as “the study of mathematics which takes into consideration the culture in which mathematics arises” (p. 1). According to Borba (1992), ethnomathematics is “a field of [mathematical] knowledge intrinsically linked to a cultural group” (p. 134).

Turner (1990) explains that the concept of ethnomathematics was created in the context of developing countries that were looking for a meaningful way to help their children learn math. I think ethnomathematics can be used, not only for children in developing countries, but also for students studying here in North America. Canada is increasingly more diverse; therefore, as teachers we are seeing more cultures represented among our students. D’Ambrosio (1999) writes that “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 view of mankind” (p. 35).

Bishop (2002) highlights some of the influences that a multicultural approach has on math education. He writes that ethnomathematics draws attention to the activities in society that concern math and that take place outside of school. The concept of ethnomathematics makes us more aware that math involves values and beliefs. Ethnomathematics facilitates our understanding that mathematics, as a discipline, has a history of cultural beginnings, as do other forms of knowledge. Ethnomathematics challenges educators to view mathematics in a different way than it has previously been looked at and encourages us to see that there is more to math than the math we learned about in our school textbooks.

How does ethnomathematics differ from mathematics as I understood it in school?

I must admit that, before I began this inquiry, I had no idea that mathematics could be understood in a culturally responsive manner. I have always thought of mathematics as numbers and a set of rules that must be memorized. I never put any thought into where these rules came from or why they exist as they do. I have simply believed all mathematics to be true, and that there is nothing left to discover in the mathematical realm. Borba (1992) says that ethnomathematics “challenges the idea that ‘true’ mathematics is a uniform, objective, and monocultural phenomenon” (p. 134). Dahl (2007) writes that the mathematics presently used in schools has been largely developed in Europe over the last 300-400 years, and ignores the cultural beginnings of the discipline.

What are the cultural beginnings of mathematics?

Though this inquiry is primarily about using ethnomathematics in the elementary classroom, the paper would not be complete with at least a brief look at the history of mathematics.  Given the extensive history of the discipline and the fact that the history of mathematics is not the scope of this paper, I would like to caution the reader by stating that this section is simply some highlights about the cultural beginnings of mathematics. 

According to Joseph (1992), mathematics has been “perceived as an exclusive product of European civilization” (p. 5).  Joseph writes that there is "a considerable amount of research evidence pointing to the development of mathematics in Mesopotamia, Egypt, China, pre-Columbian America, India and the Arab" (p. 5).  In Joseph's book, The Crest of the Peacock, he discusses in depth the contributions each of these cultures have made to the development of mathematics.  He begins with a discussion indicative of the need for ancient civilizations to count and record numbers.  One of the earliest pieces of evidence regarding mathematical knowledge is the Ishango Bone, found in Africa (Joseph, 2002; Mankiewicz, 2000; & Zaslavsky, 1999).  From the readings I have done it appears as though archaeologists and mathematicians are not certain of the purpose of the Ishango Bone, but it clearly is a tally of some sort.  Joseph writes that the markings on the Ishango Bone may display knowledge of numerical patterns, may be an arithmetical game, may be a calendar of ceremonial events, or may be a lunar or seasonal calendar.  It amazes me that awareness of mathematical concepts may have been recorded 20, 000 years ago or earlier.  Mankiewicz (2000) tells his readers of an even earlier numeric recording.  Mankiewicz writes, “The earliest evidence of numerical recording was excavated in Swaziland, southern Africa, and consists of a baboon’s fibula with 29 clearly visible notches, dating from about 35,000 BC.  It resembles the ‘calendar sticks’ still used in Namibia to record the passage of time.”  I think many elementary students would be interested in learning how some cultures kept tallies on bones.  Joseph illustrates other examples of the early development of counting and recording data, such as quipus.  “Quipus are lengths of rope or cord [which can be different colours] that have different kinds of knots tied into them” (Bazin et al., 2002, p. 38).  Quipus were used by the Inca, from the area now known as Peru, for keeping records, even though no written language existed.  Bazin et al. explain that quipus were “organized in a way similar to what we call a database today” (p. 38).

This brief introduction into the history of mathematics is captivating and interesting, but it really only skims the surface.  During my inquiry into ethnomathematics I have realized there is much more to math than I ever thought possible.  I wonder how I could have gone through 13 years of school and 5 years of post-secondary education without hearing about this subject’s rich history.  I was talking with a child from grade 2 about his ideas on what constitutes math.  He simply said, “It’s something that everybody has to do in school.”  He had no idea that math is interesting, with a lot of stories to tell and that math is used in our lives, including his life at age 7, everyday.  The history and stories of math are definitely an area worthy of further inquiry.

Does ethnomathematics fit with current mathematics curricula?

The mathematics curriculum identified in the Alberta Program of Studies (2005) does encourage students to make connections between home and school, though in limited ways. One general outcome says that students will “investigate, establish and communicate rules for, and predictions from, numerical and non-numerical patterns, including those found in the community” (p.24). I believe this to be an attempt by Alberta Education to include aspects of real world knowledge, with the knowledge being learned in schools. However, in my opinion, the curriculum fails to address real cultural contributions to mathematics. Zaslavsky (1999) writes, “Students of many backgrounds can take pride in the achievements of their people, whereas the failure to include such contributions in the curriculum implies that they do not exist.” How can students take pride in these contributions if they are not aware of them? “Students are frequently faced with the challenge of learning in an environment that may undervalue or ignore their own cultural values” (Bazin & Tamez, 2002, p.xiii).

Bishop (2002) writes, “Mathematics is a subject which is taught in all schools in the world, and a remarkable aspect of this is that school mathematics curricula look almost identical from one country to another” (p. 119). He continues by questioning why this is so, stating that mathematics, like other forms of knowledge, has been influenced by culture. Like Bishop, I wonder why “the mathematics curricula which exist in the countries of the world do not appear to be culturally responsive” (p. 121). I also wonder why our students believe math is only something people do in schools. “Bringing the perspectives of many cultures into the mathematics curriculum reveals to students the relevance of mathematical ideas and applications to the lives of people all over the world” (Zaslavsky, 1999, p. 286).

Fitzsimons (2002) writes that “mathematics curricula generally appear to be managed through technical procedures which not only avoid ethical or qualitative issues, but also serve to render them the province of experts” (p. 112). She explains that mathematics is used by certain interest groups to control knowledge and “legitimize the unequal distribution of cultural capital” (p. 112). Apparently, there is a hidden curriculum in mathematics. Bishop explains that mathematics curricula are generally created to prepare a small minority of students who will study mathematics in university. This sort of curriculum leaves many students disliking mathematics and misunderstanding the discipline. At best, it gives most students a superficial understanding of math.

D’Ambrosio (2001) boldly states that “today’s curriculum is so disconnected from the child’s reality that it is impossible for the child to be a full participant in it. The mathematics [being taught] in many classrooms has practically nothing to do with the world that the children are experiencing” (p. 308). In my opinion, for children to really understand math they need to see the relevance of it to their lives. Like the history of math, children too are growing and learning in a cultural context. When considering mathematics curricula, I believe it’s not only important to understand the cultures in which mathematics evolved, but also to understand the culture of the students. Borba (1992) states, “Therefore, an important task would be to look for ways in which we can understand students’ cultural ways of producing and expressing their mathematics” (p. 135).

How can ethnomathematics be used in the elementary classroom?

I think it is important to include activities that are meaningful to students when planning the lessons that will be addressed in mathematics learning. This means that as teachers we should consider the culture in which students are living. While the next two examples don’t draw on the students’ cultural backgrounds, I do believe they draw on the cultural situations that students may be living in.

As an example, teachers may choose to teach about angles using the skateboarding culture if there are students in the class who like skateboarding. As a specific outcome listed in the Alberta Program of Studies (2005), students will “Recognize angles as being more than 90 degrees, equal to 90 degrees, less than 90 degrees, equal to 180 degrees, greater that 180 degrees” (p. 33). Most skateboarding students will already know what a “180” is, or a “360”. The teacher can use this experience to help students learn by bringing their culture into the classroom.

Another example may be to use the toys students bring to school to help teach categorizing and recording concepts. This past year, in my student teaching experience, I noticed a lot of students bringing beanie babies to school. The students were always asked to put them away. I wonder how easily the students would learn to “sort, concretely and pictorially, using two or more attributes” (Alberta Education, 2005), if we had them sort their beanie babies (or Pokemon cards, or...).

These two examples represent ideas of how the students’ cultural situation can aid their learning in the classroom. However, I think ethnomathematics encourages us to go deeper and to understand how different cultures do math differently and to see that math is evolving, and has evolved through the contributions of different cultures. Bazin and Tamez (2002) suggest a number of different hands-on, multicultural activities.

One of the activities described by Bazin and Tamez (2002) is Madagascar solitaire. The authors say, “Like many other games, Madagascar Solitaire demonstrates patterns and rules based on important mathematical concepts” (p. 26). Bazin and Tamez describe how to play the game and give strategies for teachers to use when teaching students how to record information mathematically.

Zaslavsky (1999) also offers a number of suggestions involving multicultural games. She writes, “Mankala games help students develop their number sense and offer practice in computation” (p. 285). Mancala games include a whole category of games that evolved from ancient African pastimes. All mancala games share a similar gameboard, consisting of a series of shallow holes, with game pieces called "seeds". Playing the game involves moving the seeds from hole to hole according to the rules of the particular game being played. Versions of mancala games are played all around the world, and according to Zaslavsky, are popular games in Africa, Asia, the Philippines, the West Indies and South America. This activity is easily explored in elementary classrooms, as the board can be made out of empty egg cartons.

As reported by Lipka et al. (2001), storyknifing is a way of telling stories and used by the Yup’ik Eskimos. “Storyknifing is a way of telling a story to an audience while using a storyknife—a tool not unlike a butter knife—to etch symbols in the mud to illustrate and enhance the presentation” (Lipka et al., p. 338). This activity encourages children to practice transforming three-dimensional objects into two-dimensional symbols. In the Yup’ik culture, storyknifing is a way of creating and designing the geometrical patterns that decorate clothing and boots (Lipka et al.). According to Lipka et al., “Storyknifing also includes symmetry and motion geometry…using transformations such as rotations, slides and flips…” (p. 339). The authors suggest some ways of adapting traditional storyknifing for use in elementary classrooms.

Another activity suggested by Bazin and Tamez (2002) is to have students make their own quipus. The writers illustrate how to make a quipu and make suggestion on how to incorporate this activity into the classroom. Students can use a quipu to record things like the ages of their friends, the amount of time spent in school, or the number of items in a collection. This activity could be used to address the grade 3 outcome, as listed in the Alberta Program of Studies (2005), which states that students will “collect first- and second-hand data, display the results in more than one way, and interpret the data to make predictions” (p. 43). This activity can also be used to introduce grade 3 students to an inquiry project about the history of Peru, which integrates with the social studies curriculum. An example of a quipu is attached to this paper, with hints for understanding it recorded in Appendix A.

How can the concepts of ethnomathematics be used to enhance integration of different subjects?

Casey (2007) suggests a number of different activities that require some form of mathematical knowledge, all of which can be integrated with other subjects. She starts with the suggestion of architecture, or studying the building of houses, bridges, et cetera, which can easily be integrated with science. She also states that mathematical knowledge is found in ornamentation, such as the tilings found in the Islamic world, or Native American Beadwork. Both of these activities can be used to integrate art with math. The beadwork can also be used to integrate social studies, knowledge of First Nations people, with the math curriculum.

Turner (1990) reports conducting research in Bhutan which used games, mask songs, play activities and art to teach math to Bhutanese children. These activities were designed to educate the “whole” child, integrating art, drama, music, physical education, language, and mathematics. These activities would be aimed at helping the children to learn the mathematical practices of counting, sorting and measuring which are used by Bhutanese people during their market day.

The ideas behind storyknifing, which were described earlier in this paper, can be used to integrate math with art and language arts. As mentioned above, making quipus can integrate the math and social studies curriculum. I am certain that other math activities can be used to explore the other countries that are found in the Program of Studies, such as Greece, India, Tunisia and China.

What are the pros and cons of using ethnomathematical concepts?

Like most things in life, ethnomathematics offers a number of benefits and also has some drawbacks.

Some teachers may attempt to be culturally sensitive while delivering the mathematics curriculum, but don’t really understand what is meant by culturally responsive mathematics. Sleeter (1997) writes that textbooks often trivialize cultural diversity by simply including photos of culturally diverse populations, but fail to understand or explain to students how mathematics has been historically and cross-culturally developed. Eglash (2007) puts it like this, “What goes under the name of multicultural mathematics is too often a cheap short-cut that merely replaces Dick and Jane counting marbles with Tatuk and Esteban counting coconuts” (p. 3). Using the concepts of ethnomathematics in elementary classrooms requires teachers to inquire into the history of the discipline and to bring their newly found knowledge about mathematics to their students. Or better yet, have students inquire into the cultural history of math, with some guidance.

Owens (2007) worries that some educators feel that teaching math from an ethnomathematical perspective will reduce math to a social studies subject. He writes a journalist’s comments on the topic that “unless you wish to balance your checkbook the ancient Navajo way, it’s probably safe to ignore the whole thing” (p. 1). It seems as though some people believe that students will learn little about “real” math if an ethnomathematical approach is used.

D’Ambrosio (2001) also explains that teachers often engage their students in cultural activities only because of curiosity and those activities are often remote from the cultures of the children in class. It is important for teachers to understand the cultures that are represented by their students and to find activities that are culturally appropriate for these students. Otherwise, as exemplified by Eglash (2007), “a child from Puerto Rico may find herself confronted with Incan llamas, as if she should automatically be familiar with any artifact from the universe of Latin America” (p. 3). I don’t believe that only cultures represented in the class should be used when teaching math, but I do believe that teachers need to be careful about the activities they choose and the assumptions they may make.

Ethnomathematics can help students to not only gain a deeper knowledge of mathematical concepts and how different cultures contributed to the development of these concepts, but also to gain a greater respect for cultures different than their own. D’Ambrosio (1999) writes, “Ethnomathematics is important in building up a civilization that rejects inequity, arrogance and bigotry” (p. 35). Using ethnomathematics, as a way to teach math, may help teachers address the vision of the social studies program - to emphasize “the importance of diversity and respect for differences” (Alberta Education, p. 1).

According to D’Ambrosio (2001), teachers who truly understand ethnomathematics are able to “extend their mathematical perceptions and more effectively instruct their students” (p. 308). Sleeter (1997) explains that ethnomathematics can be used to improve student achievement and motivation. This is done by offering students an opportunity to delve into inquiry projects that are culturally meaningful and personally engaging.

What are my personal thoughts about ethnomathematics?

In my opinion, through ethnomathematic inquiry, all students can begin to understand that mathematics has progressively developed throughout history, with contributions from many different cultures. This approach will help students to be excited about math and the possibilities that lay ahead. Ethnomathematics can encourage students to inquire into mathematical concepts and understand how they, too, can contribute to this progressive and dynamic discipline. “When cultural characteristics of the children’s invention, experience, and application of mathematics are realized and respected, these students more closely resemble the budding mathematicians we desire” (D’Ambrosio, 2001, p. 310).

I used to believe that were few or no connections between mathematics and culture. I believed mathematics to be “universal and culture-free…mostly numbers and formulae” (Fitzsimons, 2002, p. 110). D’Ambrosio (2001) would say that I “believed mathematics to be acultural, or without cultural significance” (p. 309). “Let us enjoy the diversity and beauty of mathematics as expressed in numerous ways by different cultures. Let this be a way for us to learn more about the nature of mathematics” (Dahl, p. 11). I now realize that a rich history provides a strong foundation for current mathematical inquiry and that each learner can make a personal connection to ethnomathematical inquiry.

References

Alberta Education. (2005). Program of Studies: Elementary Schools. Edmonton: the Crown of Right in Alberta.

Bazin, M., Tamez, M., & the Exploratorium Teacher Institute. (2002). Math and science across cultures: Activities and investigations from the Exploratorium. New York: The New Press.

Bishop, A. (2002). Critical challenges in researching cultural issues in mathematics Education. Journal of Mathematics Education, 23 (2), 119-131.

Borba, M. (1992). Teaching mathematics: ethnomathematics, the voice of sociocultural groups. The Clearing House, 65 (3), 134-135.

Casey, N. (January 11, 2007). Ethnomathematics [on-line]. Available (March 2007): http://www.cs.uidaho.edu/~casey931/seminar/ethno.html

Dahl, B. (April 1, 2007). Cultural diversity and international education: The case of ethnomathematics [on-line]. Available: http://www.multicultural.vt.edu/proceedings/2006/Dahl-BettinaSoendergaard-MACSD2006.pdf

D’Ambrosio, U. (2001). What is ethnomathematics and how can it help children in schools? Teaching Children Mathematics, Feb 2001, 7 (6), 308-310.

D’Ambrosio, U. (1999). In focus…mathematics, history, ethnomathematics and education: A comprehensive program. The Mathematics Educator, 9 (2), 34-36.

Eglash, R. (February 18, 2007). Multicultural mathematics: An ethnomathematics Critique [on-line]. Available: http://www.rpi.edu/~eglash/isgem.dir/texts.dir/multcrit.htm

Fitzsimons, G. (2002). Introduction: Cultural aspects of mathematics education. Journal of Mathematics Education, 23 (2), 109-118.

ISGEm : The international study group of ethnomathematics [on-line]. Available: http://www.rpi.edu/~eglash/isgem.dir/isgem.2.htm

Joseph, G.G., (2002). The crest of the peacock: Non-European roots of mathematics. Toronto: Penguin Books Canada Ltd.

Lipka, J., Wildfeuer, S., Wahlberg, N., George, M., & Ezran, D.R. (2001). Elastic geometry and storyknifing: A Yup’ik Eskimo example. Teaching Children Mathematics, 7 (6).

Mankiewicz, R. (2000). The story of mathematics. London: Cassell & Co. Owens, K. (ed.) (February 19, 2007). Ethnomathematics: A rich cultural diversity [on- line]. Published by the Australian Academy of Science. Available: http://www.Science.org.au/nova/073/073key.htm

Sleeter, C. (1997). Mathematics, multicultural education, and professional development. Journal for Research in Mathematics Education, 28 (6), 680-696.

Turner, J. (1990). Ethnomathematics, complementarity and Bhutan. Originally published in the International Study Group on Ethnomathematics (ISGEm) Newsletter, Volume 5, Number 2, May 1990. [on-line]. Available: http://www.ethnomath.org/resources/ISGEm/045.htm

Zaslavsky, C. (1999). Africa counts: Number and pattern in African cultures (3rd ed.). Chicago: Lawrence Hill Books.

Appendix A
Understanding the Quipu*

quipu2

  1. This quipu is a calendar.
  2. The main rope is purple.
  3. The white cords are the principal cords; they record the month number.
  4. The mutli-coloured cords are the subsidiary cords; they record the number of days in the month it is attached to.
  5. Different knots represent different numbers:
    - The top row of knots represents numbers in the hundreds, the middle row represents numbers in the tens, and the bottom row represents numbers in the ones.
    - Figure eight knots = 1
    - A single knot in the tens row = 10; a single knot in the hundreds row = 100
    - A long knot = the number of times the cord is wrapped around itself between the numbers 2 and 9, inclusively.
  6. The first subsidiary cord is a total of the other subsidiary cords, in the case of this calendar, it is the number of days in a year or all 12 months.
  7. Making a quipu can help to address the following strands in the mathematics program: number concepts, number operations, patterns and relations, and data analysis.

*adapted from Bazin, Tamez and the Exploratorium Teacher Institute (2002).

These pages are maintained by Michele Jacobsen

© 2007 Rebecca Dumoulin and EGallery