Children in our schools come from a multiplicity of ethnic, racial, and economic backgrounds. How can our teaching of mathematics respond to the diverse needs and learning styles of the school population?
Mathematics adapts readily to a multicultural context. Children learn how mathematics arose from the needs of the societies of the past and of today. Other cultures’ numeration and measurement systems, games of chance and games of skill, and patterns in art and architecture are all sources of learning experiences.
Similarly, we can profitably relate the study of mathematics to other disciplines. Activities and games can correlate with social studies, language arts, fine arts, and industrial arts, encouraging students to appreciate the many aspects of mathematics.
Investigations into sex-role stereotyping reveal that boys are more likely than girls to receive exposure to such informal sources of mathematics learning as the construction of toys, puzzles, and sports, while girls are socialized to care more about people. By integrating mathematics education with the study of culture and history, we may motivate more girls to become involved in mathematics.
Numbers in Other Languages
Ask your students whether any of them can count in languages other than English. These talented children can then take pride in their ability to teach the counting words to their classmates.
Comparison of the number words in a foreign language with the corresponding English words encourages analysis of the structure of the numeration systems.
English and most other European languages base their systems on grouping by 10s and powers of 10. Why is 10 commonly used as the base? Is it because we have 10 fingers (digits)? But the French word for 80 means “four 20s!” And in the Mende language of Sierra Leone, the word for 20 means “a whole person” – all 20 fingers and toes-and 40 is expressed as “two whole persons.”
Grouping by 20s is common among the peoples of West Africa and Middle America (Maya, Aztec, etc.) Perhaps they went barefoot in their warm climates, and therefore found it logical to count both fingers and toes. However, the Inuit Eskimos also have a numeration system based on 20.
In English we use a combination of multiplication and addition to form the higher numbers: 45 means “four times ten plus five.” In some languages the construction of large numbers involves subtraction. The Yoruba (Nigeria) word for 45 means “take five and ten from three times twenty.”
Ask your students to imagine how and why people first found it necessary to use number words. Even the experts cannot agree on the origins of counting. Some theories are: to count one’s private possessions; to keep track of the group’s livestock; for ritual and religious purposes; to trade with other individuals or groups. Your students’ theories may be just as valid as those of the experts.
The question of origin is sure to engender a lively discussion and to put across several useful ideas. Children will learn to appreciate that people invent new ways of doing things because of objective needs and that each group responds to these needs in its own unique manner.
Mathematical concepts are related to real life and develop as the society develops.
Having recognized the need to count, people invented number words, and then organized them into systems with well-defined structures. Written symbols and computational methods became necessary, as well as systems of finger and body gestures. Children should begin to recognize that mathematics is a growing body of knowledge to which even they can contribute.
Students can investigate the number symbols (numerals) of ancient societies and compare them with the Hindu-Arabic system in use today. They might consider such aspects as the number of different symbols, the order in which they are written, the use of zero, positional notation and place value, and convenience in keeping records and carrying out calculations. Figure 1 shows some ways of expressing 2369.
Although thousands of different languages exist today, we would have little difficulty in reading examples of computation in almost any arithmetic textbook in the world.
Most people now use the 10-digit place-value system originated by the Hindus in India and transmitted by the Arabs
to Asia, Africa, and Europe. By the time northern Europe adopted it, about the year 1500, the system was almost 1000 years old. This cultural borrowing is typical of the way in which ideas spread from one region to another.
Native American tribes speaking hundreds of different languages occupied the territory that later became the United States. The Indians of the Great Plains invented a sign language, including gestures for numbers, that enabled them to carry on complete conversations without having to say a word.
African peoples speak over l000 different languages, and also use gestures for communication. Although two different ethnic groups live in adjoining areas, they might use entirely different systems of gesture counting. In some cases the finger signs are related to the number words, in other cases they are quite different.
Happily, teachers no longer go along with the taboo on finger counting imposed on previous generations of students. We recognize the value of “hands-on” materials in helping children to learn arithmetic operations. What materials are “handier” than the fingers?
After investigating some systems of gesture counting, children might also invent their own gesture system and number words, and try them out on their classmates. They can learn to compute with their fingers, as in Chisanbop (also called “Fingermath”), a system based on the Korean and Japanese abacus.
Shops in Asian countries and the Soviet Union often have an abacus and a calculator sitting side by side next to the cash register. Although the Chinese suan-pan differs somewhat from the Japanese and Korean soroban, both are based on calculation by groups of 5’s and 10’s. The Russian scet has 10 beads on each strand, with the two middle beads colored differently for convenience in counting. Children can make their own abaci, and use them for computation.
Geometry and Measurement
An investigation of styles of housing in different cultures is a valuable source of experiences with shapes and sizes as well as with perimeter and area concepts. It can also develop skills in approximation and estimation. We are accustomed to right angles and straight sides in our homes and furniture. Can we imagine living with other shapes? Ask your students to draw floor plans of a simple home. What shapes do they envision?
Let students pretend that they are living in a society where they must produce almost everything by their own efforts and where a house must be constructed with as little material as possible. The task is to determine which shape will afford the greatest amount of floor space for a given investment of labor and materials.
They can test some of the possible shapes, to see which gives the largest area for a given perimeter, a given amount of material.
Distribute grid paper, and have them follow this procedure:
- Choose a perimeter — for example, 32 units. Tear off a strip of graph paper at least 32 units in length to serve as a tape measure, or use a piece of string of the required length.
- Form a circle with a circumference of 32 units, and sketch it on the graph paper.
- Sketch several rectangles with a perimeter of 32 units, having different dimensions.
- Draw several other shapes having a 32-unit perimeter.
- Compute the area of each figure by counting the number of small squares enclosed by each perimeter.
- Arrange the results in a table. Which figure has the greatest area? Which rectangle encloses the greatest area? What other conclusions can be drawn?
- Compare the areas by constructing a bar graph.
Children will raise many questions: How can we make the circle perfect? How can we trace around the string when it keeps moving? How shall we count the fractions of small squares? Counting all these little squares is boring; isn’t there a short cut?
As children carry out the activity and discuss their problems, they discover that the circle has the greatest area, and the square is the largest rectangle. To draw a circle, one need only mark a few points on the circumference while a classmate holds the string in place; then the circle can be drawn freehand. An adequate approximation of the area results from counting only those fractions of squares that exceed half the square, and ignoring the smaller fractions. Examining the symmetry of the shape allows one to avoid counting all the little squares. The number of squares in a rectangle is the product of the two dimensions, actually the area formula. Children learn the difference between area and perimeter, and when to use square units as distinct from units of length.
Students will conclude that people who build round houses achieve the greatest possible floor space for a given amount of building material. Why, then, is the rectangle shape most common in our society?
This project can expand in many directions.
Children can construct African-type compounds of round houses with conical roofs, decorated in geometric patterns of various colors—math, social studies, and art are all combined in one lesson.
Students can investigate the influence of the environment and of technology on styles of homes in different cultures. They can estimate areas and perimeters of objects in the classroom, and then measure to check their estimates. Students will suggest other applications.
Examining the geometric forms of symbols, such as the ankh, the cross, the star, and the peace sign is another worth-while activity. Children can look for symmetry of design and repeating motifs in decorated objects and cloth, and then produce their own versions. Geometric concepts have real meaning when they are presented in such a concrete and relevant form.
Games of Chance and Skill
From time immemorial human beings have tried to divine the future. Some divining practices led to games of chance, and eventually to the important field of mathematical probability and statistics. A toss of a coin is the simplest form of gambling. Some societies used cowrie shells, others used nuts, still others threw knucklebones, which then developed into dice.
Two excellent games of chance are the Jewish Dreidel game associated with the Hannukah festival, and the Mexican game called Toma-Todo. Both are played with tops or spinners, a four-sided dreidel or a six-sided topa. Any number of children can try to predict the outcome before each spin and keep a record of success or failure, thus adding to the excitement of the game. Once they have mastered the rules, they can make up variations.
Three-in-a-row games can be traced as far back as 1300 B.C., when diagrams for the games were chiseled into the roof slabs of the temple to Seti I in ancient Egypt. Tic-tac-toe is the most popular form in the United States, and one of the simplest to play. Investigations into the versions that children play in other lands—Morris in England, Mill in other European countries, Shisima in Kenya, Nerenchi in Sri Lanka— provides a good social studies lesson at the same time that children improve their decision-making skills.
The ancient stone game is a good way to introduce a unit on African heritage and Africa’s contributions to world culture.
Known by its generic name Mancala, an Arabic word meaning “transferring,” it is called Wari or Oware or Adi in Ghana, Ayo by the Yoruba people of Nigeria, and Giuthi by the Kikuyu of Kenya. Kalah is a commercial version. African captives brought the game to the New World, and social scientists have used the style of playing to trace the ancestry of black people now living in the Caribbean, the United States, and regions of Brazil and other countries of South America.
Playing the game depends entirely upon skill, not at all on luck, yet some versions are simple enough for first graders. The popular West African versions can be played with an egg carton as a gameboard and band beans for playing pieces. At the beginning it may be advisable to distribute sheets of paper on which the “board” has been drawn, so that children can see exactly what is going on.
(For the game rules see Zaslavsky, 1973 or Dolber, 1980.) On the simplest level, the game affords practice in counting and reinforces the concept of one-on-one correspondence. At a more advanced stage, a child uses all the operations of arithmetic.
Games of strategy help children to acquire skills in logical inference and decision making, important training for solving problems in the technological society of the future. They can vary the method of play by changing the rules, the shape of the board or the number of counters, each version requiring a different strategy. After all, that’s how new games are invented!
Through activities such as those described here, our students become aware of the role mathematics plays in all societies and of the need for mathematical problem solving and decision making in real life.
Furthermore, children learn to appreciate the contributions of people all over the world. Geography and history take on new meaning as children learn other number systems, analyze styles in housing, and trace the dispersion of games. They can take pride in their own heritage, as they become familiar with other cultures.
Dolber, Sam. From Computation to Recreation Around the World. San Carlos, CA: Math Aids, 1980.
*Feelings, Muriel. Moja Means One: Swahili Counting Book. New York: Dial, 1971.
*Lumpkin, Beatrice. A Young Genius in Old Egypt. Chicago: Dusable Press, 1979.
Zaslavsky, Claudia. African Counts: Number and Pattern in African Culture. Westport, CT: Lawrence Hill, 1979.
Zaslavsky, C. Preparing Young Children for Math. New York: Schocken, 1979.
*Zaslavsky, C. Count on Your Fingers African Style, New York: Crowell, 1980.
*Zaslavsky, C. Tic-Tac-Toe and Other Three-in-a-Row Games. From Ancient Egypt to the Modern Computer, New York: Crowell, 1982.