Compared to typical Han Chinese el-ementary school students, aboriginal students in Taiwan do have a different "feel for math," and this difference reflects basic cultural differences.

"Our students are poor achievers in math here-and they have particular trouble with word problems, which they virtually can't do at all. Only about a third of our fourth graders can do arithmetic in their head. Another third can do it if they use their fingers to help. As for the remaining third, well, they simply can't do it," says a principal of an elementary school located in an Atayal tribal community in Hualien.

Six bananas traded for five yams

Indigenous children do indeed have more trouble with math than Han Chinese children. Research carried out from the 1960s to the 1990s by anthropologist Li Yi-yuan and education researchers Chen Chih-lieh and Lin Hsuan-cheng in different aboriginal villages revealed that average achievements in math did lag behind those of typical elementary schools in Han Chinese communities.

Apart from stereotypical notions that parents of aboriginal children are less inclined to supervise their children's homework and that indigenous culture is less mathematically stimulating, there is evidence that the nine tribes traditionally understood math somewhat differently from Han Chinese.

Chi Hui-ying, a doctoral candidate at National Hualien Teachers College now conducting research into the "Mathematical Conceptions of Aboriginal Children," explains that the tribes' traditional conceptions about math varied-but all were rather limited.

"They used quite simple methods to memorize numbers, and their mathematical vocabulary lacked specificity. In their traditional lives, they rarely used numbers over 1000, and very few people could count over 1000. Most people didn't know how to count over 100, and for larger numbers they would use such descriptions as 'as many as the leaves of a tree' or 'as many as the stones of a riverbed' or 'as many as ants,'" wrote Chi Hui-ying in a study she made at an aboriginal elementary school in Nantou County in 1995.

Fang Chi-hsiung, a Rukai who teaches at a junior high school in Hengchun, says much the same thing, noting that traditional Rukai life was simple, and conceptions about numbers had only developed to an ability to count whole numbers. Numbers from one to ten and on up to 1000 were all based on a "repeated counting method." For instance, the Rukai word for five was "lima" and for 50 was "malima." Between the first and second digit, they would place a "le," so that 57 is spoken "ma-lima-le-bido," or "50 plus seven," which is different from the direct way that Han Chinese memorize specific numbers.

"In regard to the fair exchange of goods, aborigines are concerned about their mutual needs and love, and they often overlook numbers or prices," says Chi Hui-ying. She quotes an Atayal teacher: "When bartering six bunches of bananas for five yams, as long as both parties are willing, it's a fair deal."

A watch: jewelry or timepiece?

But have conceptions about math that are rooted in traditional aboriginal life styles been passed down to aboriginal children now living sinicized lives in modern Taiwan? The effects of this legacy on the way aboriginal students study math is a focus of research.

Chi Hui-ying says that different researchers have come to vastly different conclusions about the reasons for the lag in aboriginal mathematical achievement, which vary according the relative experience of the researchers and the depth of their understanding about tribal people's lives.

For instance, at the aboriginal elementary school where she teaches, students describing the distance between two landmarks would say something like, "If you start off just as the sun is rising, you'll get there at noon." Or they might use the distance to another landmark as a standard of reference: "It's five times as far as going to Fushih Elementary School."

"When Ami describe the size of a fish they've caught, they'll say, "as thick as Bani's thigh [Bani being a fellow tribesman]," says Li Wen-cheng, principal of the Huajen Junior High School in Hualien.

Hung Yi-chang, assistant director of the aboriginal health center at the Tzu Chi College of Medicine, says that when Ami are asked how many fish they have caught, they rarely respond with the exact figure and instead just say "dulu." This literally means three, but when they reply that way it doesn't necessarily mean that they've caught exactly that number, but rather that they've "caught some."

"Their descriptions won't be based on units of measure (such as kilometers), but rather on comparisons to things as they exist concretely in the world," Chi Hong-ying explains. The life experience of aborigines thus affects their mathematical conceptions, and maybe even the way they study math.

Chi once played a coin game with some aboriginal first-graders. She wanted students to combine coins so that the total equaled NT$25. Some students were very clear that coins had different values of NT$1, NT$5 and NT$10 and quickly made correct combinations. But most of them didn't care about the face values of the coins, and merely counted 25 coins of different values. Chi argues that because aboriginal children rarely have pocket money, they have "limited experience with using coins."

Similarly, aboriginal children have a vague understanding of time as compared to Han Chinese children. In the classroom first-graders understand a model clock, but they don't understand time from their own watches. "The kids think of a watch as a piece of jewelry rather than as something that tells time," Chi says.

She holds that life experiences do indeed shape aboriginal children's understanding of math. Shih Ying, a professor of math at National Taiwan University, holds that although children in aboriginal villages may in fact have different life experiences from Han Chinese children living in the city, math teachers just need to make a little extra effort to put them on the right track. If teachers instead come to the conclusion that aboriginal children do not benefit from studying math, the teachers will be turning the students into victims of ethnic discrimination.

Looking carefully at cultural diversity

Shih Ying believes that at its basic level mathematics requires an understanding of the shape of numbers and the idea of quantity. The traditional lives of the aborigines were simpler than those of Han Chinese, and as a result there are, of course, differences between how they looked at distance and numbers. "The problem is in adjusting the curriculum, teaching materials and even teaching methods in accordance with our understanding of what these differences are," he says.

The lower scores of aborigines in mathematics involves "a web of interrelated factors" including traditional culture, life experiences, and family and social support networks," says Chi Hui-ying. "It's very difficult to distinguish between them." But what is certain is that the current curriculum, teaching materials and teaching methods are "not the most appropriate for aboriginal children."

Just as there are major gaps in outsiders' understanding of aboriginal culture in general, more detailed research needs to be carried out about the mathematical conceptions of aborigines. This research should be broken down according to age, tribe, area, social situation, and so forth. It's wrong to make a sweeping generalization and say that "aborigines' poor performance in math is or is not related to culture," argues Wang Wei-chang, a photographer who often visits aboriginal communities.

But many educators who work in aboriginal areas are worried about language difficulties in the many communities where the native languages are still the mother tongue.

This is revealed in their problems counting aloud: "A number quickly recited in Mandarin may take a long time to say in Atayal, which may mean that Atayal children have to spend more time to learn how to count," says Chi. Many aboriginal languages have a different word order than Mandarin, with the verb coming before its subject, and this provides quite an obstacle to the study of Mandarin. By the third or fourth grade, when there are more and more word problems, aboriginal students may not understand the meaning of the questions, which in turn can make them dislike math.

Chi notes that aboriginal children may perform better than their Han Chinese counterparts in astronomy and nature, physical education, and music classes. But this leaves aboriginal children with little room to make the most of their talents today, when the mainstream curriculum is still based on "pen and paper" examinations. The aborigines' non-mainstream mathematical conceptions are disappearing in the wake of exposure to modern curricula. In a society that is constantly stressing the importance of creating educational diversity, can this be viewed as something positive?

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Trade two bunches of bananas for five yams? The mathematical conceptions of aborigines are clearly closer to their life experiences.

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Is a watch a timepiece or a piece of jewelry? It's a relevant question both for city folk and the residents of aboriginal villages in the hills.