In recent years, the outstanding perfor-mance in math by Asian students has been attracting a lot of notice. The ROC didn't join the world-famous International Math Olympiad until six years ago, but it has performed as well as such veteran participants as England and America.

For many people, the multiplication tables are among the earliest memori es involving a "fear of math.".

But there's another side to this coin: elementary and junior high students in Taiwan continue to have negative feelings about math. Research carried out by Yang Jung-yang, a professor at National Taiwan Normal University, shows that even students who perform exceptionally well in math competitions "totally hate math." Why is this so?

Cramming different numbers of chickens and rabbits into the same cage, plane geometry, the Pythagorean theorem, trigonometry. . . do you remember your junior and senior high school math classes? What have been your experiences with math? And what have been the experiences of your children, neighbors, brothers and sisters?

A tenth-grade teacher says she has a student who starts to shake as soon as she gets a math test paper. When the teacher asked her what was the matter, the girl explained that she had been a bad math student in junior high and had been beaten because of it, so that now when she confronts a math test, even just a quiz, she shakes with fright.

With after-school classes that train children to do calculations in their head, students become very quick with arithmetic. It's hard to say whether such classes also l ay the seeds of math anxiety.

This is an outward symptom of "math anxiety," but there are even more examples of people repressing this anxiety. For those who have grown up on the examination culture of Taiwan's schools, the following nightmare is nothing unusual: A barrage of xyz graph coordinates, square roots and right-angled triangles stare up at you from a test paper. If there are not simply far too many questions to answer, then the sweat running from your palms manages to make the answers you have written with a No. 2 pencil illegible. "Those who fail math will have to repeat this grade," is a phrase that stays in one's memory. Years after graduating from high school, many still suffer from subconscious fears about math. Math from the mathematician's angle

Do primary and secondary school students in Taiwan have especially serious cases of "math anxiety"?

In March of this year results for the Joint Entrance Exams for Taipei City Public High Schools showed that the sections for mathematics and natural science produced the lowest average scores. On both sections most students did not get a passing score. In the 1996 academic year, for instance, the average score for mathematics was 55.

Last year the Ministry of Education surveyed primary and secondary school students about their satisfaction with school curricula. The results showed that one-third of elementary school students and 46% of secondary school students were most scared of math class. Furthermore, the percentage of students who dislike math grows from grade to grade.

Chiu Shou-jung, an instructor in math education at National Changhua University of Education, points out that the term "math anxiety" comes from the West. It originated in the 1960s, when there was a shortage of math teachers in the wake of universal education, which led to the hiring of many unqualified teachers.

In truth, the primary and secondary mathematics curricula that everyone is familiar with, both in the West and here in Asia, has the same roots: the Industrial Revolution. Just like the modern curricula for languages, sciences, history, geography, art and physical education, the math curriculum has been designed to teach small numbers of the elite. Even in the 1960s, after the advent of universal education, the purpose of math education was to train scientists.

It's been no different in Taiwan. Down to the present, attempts to "reform and simplify" the mathematics curricula of primary and secondary schools have been aimed at getting away from "math education from the angle of mathematicians which overlooks the level of receptivity among the students," says Li Tzu-cheng, a graduate student in psychology at National Taiwan Normal University.

But simplifying teaching materials has not meant that Taiwan's students' interest in math has grown. "How the math teachers go about teaching is key," says Lin Kuei-jung, a teacher at Tatung Junior High. For instance, it's not easy to understand the formula for the volume of a sphere (4/3 r3), and some teachers will make the mathematical proof, but others make practical examples that "start from the perspective of the children themselves."

Take, for instance, the identical equation (a+b)2=a2+2ab+b2, which is studied in junior high. If students use graph paper to plot out examples, then it's easy to understand and won't become just a dry, meaningless formula.

Current texts still include rigorous proofs involving complex topics such as laws of mathematical induction, which are extremely difficult for students to understand.

Huang Min-huang, an associate professor of mathematics at National Taiwan University, believes that proofs of this sort may be still included because the writers of the texts think that the logic behind them is "beautiful." Huang notes, "Those who have studied math want to increase exposure to the legacy of great mathematicians."

If these were left in textbooks just "to increase appreciation for math" or to present great mathematicians as role models of excellence, then it wouldn't cause students so much frustration. The problem is that teachers in Taiwan often like to show off, and after explaining these proofs to students, they then test the students with similar kinds of questions. The students naturally find this troubling.

Scared of math

At an academic conference, Wu Po-lin, a professor of applied mathematics at National Chengchih University, spoke about the "arrogance" of math teachers and sarcastically mocked what he saw as "abnormal developments" in math education: "Teachers can't stand the idea that young people today are more and more intelligent. As a result they are doing all they can to teach students to be stupid."

He argues that as part of this all-out struggle for idiocy, math teachers are in love with using such awe-inspiring mathematical symbols as z that cause students to think of math as something mysterious and inaccessible. In class they make students constantly copy what's on the black board so that there is no time to ask questions. The speed at which students are expected to solve problems and the deep faith placed in the authority of standard answers allow students no room for doubt. Not explaining the concepts behind definitions and theories, teachers make students blindly memorize formulas. Instead of testing with the problems covered in class or in the textbooks, they seek out test questions that have proven track records of making kids fail. . . . All this insures that "teachers will succeed in getting kids not to like math," Wu Po-lin says.

Although Wu is joking, he is also making a description of the current state of math education in Taiwan. Lin Fu-lai, a professor of math at NTNU, has observed a "math anxiety syndrome" among Taiwan's students. They develop a resistance to math, feeling that no matter how much they study they will never learn it. Contact with math causes fear, helplessness, confusion and nervousness. The syndrome first afflicts many students in junior high, particularly seventh grade. Others are hit earlier, in fifth or sixth grade.

The increased difficulty of course work is one of the causes of this syndrome. Lin Fu-lai points out that from fourth grade to fifth grade abstraction in the current math curriculum jumps a level, and then it jumps another level when students begin junior high school in seventh grade. "A lot of students can't make the transition to using letters such as x, y and z to represent numbers in algebra."

"The trouble I have with math, is that I can't accept that x,y, and z are numbers, and that I have to solve what they are," wrote one student in his class-assigned journal about his feelings on math."

When you start to study geometry in ninth grade, the level of abstraction is even greater, and the teachers' explanations become even more important. The abstract proofs of geometry all refer to concrete geometric reality, such as the proof that shows that "the opposite sides of a parallelogram are equal." If the teachers put out a little extra effort, then by rights students in seventh and eighth grade should all have experience making measurements with specific examples. This will make it much easier for them to understand the abstract concepts.

But if this previous experience is insufficient, or if the students are having trouble with step-by-step logical reasoning-such as understanding that if two sides of triangle are equal, then their corresponding angles are as well-then when they get to analysis required in more advanced geometry, they're going to have trouble," says teacher Lin Chia-jung.

Lin Chia-jung points out that junior high math teachers are too hurried in the way they teach. Students have had insufficient previous experience with abstract reasoning, and because complex geometrical concepts are unconnected to daily life, geometry becomes a source of math anxiety.

Multiplying letters?

The use of abstract mathematical language in algebra and geometry requires some mental adjustment, and students who are first encountering such mathematical concepts will need time to understand them. Hung Wan-sheng, a professor of math at National Taiwan Normal University, points out that when the Qing emperor Kang Xi first learned algebra from a missionary in the 17th century, he also had trouble with idea of "multiplying a with a, b with b-they're not numbers and when you multiply them you don't know the value of their product either." His troubles are quite like those of modern day junior high students.

If you want students to get over their problems with abstraction, "Teachers should by no means expect immediate results," Hung says. First, don't call them stupid if they ask why x+2 can't be written as 2x or x2. Realize that when people are studying something new, they will always require a long period of transition. "Do your best to look at problems from the students' perspective," Hung says. Currently, NTNU is urging its fifth-year student teacher trainees to "step out of themselves, and look at math from the angle of the students," says Hung.

There is another way of resolving the problem: "Don't wait for the problem to happen and then say, 'How is it that the students still can't make the jump by now?' Rather, we should be cultivating students' abilities to think abstractly from a young age," says Shih Ying, a professor of math at National Taiwan University, who has edited some of the new textbooks used in Taiwan's elementary schools. In the lower grades, the curriculum should still include such concepts as "right and left," and "the sum of two sides of a triangle is always larger than the third side" so that younger children can have contact with "abstract thinking."

Students knead their own clay

In recent years various models of math education have all aimed at getting this academic discipline of math "off its high horse" so that everyone can experience the joys of studying math. These put students as the focus of education, and aim to foster understanding rather than to force-feed knowledge. In 1996, the curricula of Taiwan elementary schools were revised so that they were oriented toward the students' perspective. This "structuralist" revolution in curriculum revealed positive efforts at reform.

What is "structuralism"? It stresses that education shouldn't be a passive activity but rather one in which students actively build and revise a cognitive structure.

Educational theory tends to be rather dry and abstruse. One teacher explains it with a simple metaphor. In the past, the educational process was like kneading clay, and the teachers were like holders of a book that outlined a manufacturing process step by step. The students needed only to follow along and things would work out. Today, the teacher's role is to explain how the amount of water and clay used and the speed of the potter's wheel will affect the final result, so that the students can decide for themselves what methods to use. The teachers are only responsible for guiding and not for making decisions. All in all, students are given more respect as the "main subjects" of this process.

A structuralist mathematics curriculum has been in place in Taiwan's elementary schools for two years. It may or may not prove to be effective, but at the very least teachers at every school have reached a consensus: "In teaching math today you don't necessarily just want to teach students to give the correct answers; it's even more important to help students understand how to think about mathematical problems," says a math teacher at Taipei's Lungan Elementary School. And in the math curricula of many elementary schools hands-on activities have replaced the old way of "copying exactly what the teacher says."

Tsai Shu-ying, a math instructor at the Taipei Municipal Teachers' College, points out that when she teaches elementary and junior high school students who are confused about fractions, instead of overemphasizing the textbook, she uses cup measurements or a ruler and rope so that students can "play" with the concept of proportion. Then she uses jigsaw puzzles that have been colored one color so as to give students an understanding of the principles behind mathematical operations.

Teacher Tsai Shu-ying states that the purpose of "starting from the students' previous experience" is to grab hold of students' "feeling for math." Once students have the feeling for this kind of participation, discussion and dialogue, their interest in math greatly increases. And at that point it's easy to explain fractional operations and introduce word problems.

No answer is wrong?

Kids aren't just "actively playing with math" in school. They're doing the same in homes where parents care about their children's mathematical development. "Math problems don't necessarily have one standard answer, and if the answer is incorrect, it doesn't necessarily follow that everything is wrong," says Chiu Shou-jung. The long-held idea that math requires "precision and speed" is no longer universally held.

For instance, previously teachers would tell students to "borrow a one" from the next higher decimal place if the digit being subtracted is larger than what it is being subtracted from. But now teachers hope that students go through their own process of mental "play" to reach an answer.

Professor Chiu Shou-jung once saw her child working with classmates to solve the problem 56-38. Chiu's child's method was to first calculate 50-30=20, then 20+6=26 and finally 26-8=18. Another child was even more inventive coming to the right answer of 18 through the intermediate steps of 8-6=2 and 10-2=8. A third child used the teacher's method of "borrowing a one" and figured 10-8=2 to come up with the answer "12," forgetting about the left over "6."

Chiu says that when children use their own experience "to process and adjust" what they have been taught, then they will have a greater sense of achievement than if they simply memorize knowledge imparted by their teachers.

Many children have the ability to use their own explorations as a bridge to move from the concrete to the abstract, "but the problem is, how much time and space do we give children when they are first coming into contact with math?" asks Huang Min-huang, a professor at NTU.

Li Ya-ching, a teacher at the Seedlings School, which is at the cutting edge of open education in Taiwan, gives an example: Her school had an eight-year-old who came up with an unusual explanation of "why dividing a number by 1/4 was the same as dividing it by 3/4 x 3." "Taking 1/4 as the unit of measurement," Li recalls, "he figured that if you take a cake and give it to a one-quarter person, the amount of cake he gets would of course be larger than if you divide it among three one-quarter people, and moreover three times as large." Li says that when children come to such a conclusion themselves, they naturally gain confidence, and with confidence they aren't plagued by math anxiety.

Babies eating steak?

Li gives another example: Once a child at the Seedling School wouldn't accept that 3/7?/2=3/7x2/1=6/7. The teacher turned to the concrete world and carved up a big cake, doing it one step at a time to let the child understand. It was a month before the kid really understood that "dividing with fractions is the same as multiplying with them upside down."

Yet there aren't even 30 students in the entire elementary school of the Seedling School. A typical public school with 30 children in a class may not have the time to wait for a child's development. Shih Ying believes that every child develops at a different speed, and in the process of his or her studies will get stuck at certain places. Instead of pushing children along so they appear to understand without really grasping the essence of the subject matter, "educators ought to let them proceed along a slow path of gaining real understanding," Shih Ying says.

Even if education revolutionaries frequently make loud calls for a revolution in math education, "utility" in the eyes of teachers and parents is the key for determining whether this wave of educational reforms can take hold.

In a Taipei private school, second-graders are being tested on questions such as, "if 26 whole apples are divided evenly among four people, how many are left over?" Chiu Hsiu-jung believes that this is a suitable question if children are able to use actual objects and slowly divide them, but if second graders are being tested on their familiarity with division, than it's going too far. "It's like trying to feed steak to a small baby. The baby is sure to throw up," Chiu says.

Lin Fu-lai holds that children start to resist math in fifth and sixth grade because both teachers and parents believe that their children have already grown up, and that sooner or later they are going to have to face joint entrance exams to further their studies. They want their children to have more math practice, but the result is that it ends up "spoiling their mathematical appetite."

"As far as children are concerned, if learning something only means advancing to even greater pressures, then why would they want to learn it?" Chiu makes this analogy: Her friend's child is very thin, and has never been a good eater. Later Chiu discovered that "the reason why he didn't want to finish his first bowl of rice was because it would be followed immediately by second and third bowls."

"When a child enters higher grades, adults seem to stop thinking about whether the child's own needs are being met and whether they are happy in their studies," says Lin Chia-jung. There is instead a practical goal: the focus becomes jumping over the bar that determines admission to the next level of study, and the actual purpose of the education itself is overlooked.

Exposing latent math talent

The problem lies in what exactly we expect from our children's math education.

From ancient Greek investigations about the principles behind the order of the universe, to Newton pondering an apple falling to earth in the 17th century, down to the 20th century study of particles, mathematicians have built on the work of those who came before them. By finding order and "making the complex simple," math is very appealing.

"When teaching any kind of formula, theorem or operational rules, all I think about is how to get the children to come to an intuitive understanding of the subject matter," says Tsai Shu-ying, an instructor at National Taipei Teachers' College. She hopes that in learning math children can come to realize that everyone has abilities, and there's no need for these to come through authorities-whether teachers, text books, reference books or parents.

In carrying out numerous educational experiments, reformers have found it impossible to conceal their joy in welcoming new methods. But along the way, they have also encountered new problems, many in places where past methods were particularly strong.

In 1997, after the new curricula for elementary schools had been fully implemented, Chung Ching, an assistant professor of math education at National Taipei Teachers' College wrote in a report he made on "The Transformation of the Culture of Math Class in the Lower Grades" that those students who have been educated under the new curricula are much more "able to voice their opinions, ask questions, work with others and learn from others' way of thinking" than earlier students who had received traditional educations.

But there was another side to it: children's ability to do calculations, the speed at which they were able to submit answers, and the amount of studying they did had all clearly dropped. These results caused the proportion of parents who felt "very fortunate that their children could be taught with the new curricula" to drop from 70% in the lower grades to 40% in the higher grades. "More than half the parents were worried about their children needing more time to solve questions, using more basic strategies for solving problems and solving problems more slowly," Chuang Ching wrote in a 1998 report A Survey on Parental Opinions about Math Curricula in Lower Grades Matched to Children's Intellectual Development. "Many parents couldn't abide that their children weren't making vertical arithmetical calculations and couldn't use short cuts. Furthermore, from the parents' perspective, the processes the children used to solve problems were complicated and prone to error."

From the first responses to the new curricula, it seems as if children's math anxiety has indeed been effectively treated. But could it be that lowering math anxiety will end up lowering the level of math achievement?

The strict math education in Taiwan has cultivated high scorers on tests, prompting Western educators to come and try to uncover how we do it. Yet Taiwan math education seems to be bouncing between the extremes of serving "math geniuses" and serving "math failures," and when Western scholars come to Taiwan to learn our secrets, they see strengths in the places that our reformers are bent on changing. How should we explain that?