A comparison of American and Taiwanese students: their math perception.

The major purpose of this study was to attempt to understand some of the reasons for Mathematics perception of Taiwanese children compared to American children. The study was conducted with elementary schools in the Denver metropolitan area and Taipei, Taiwan in which fifth graders in each city (21 and 37 respectively) were selected as target subjects in the study. To determine if attitudes and beliefs have this profound of effect on American students' performance in mathematics, research believes that it may be helpful to compare American students to Chinese students. By providing comparative data, the researcher found marked differences in the beliefs of American and Taiwanese students in four areas under investigation: how to do well in mathematics, what math solutions should be, motivation. The present study makes a potentially important contribution to our understanding of child development and education in two cultures.

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Poor performance by American students on tests of mathematics and science has reached the level of a national crisis. Why is this? Study after study has reported on one or another facet of the low standing of Americans in international competition. For example, in a recent cross-national study of mathematics achievement, American students in the eighth and twelfth grades were below the international average in problem solving, geometry, algebra, calculus, and other areas of mathematics. In contrast, Japanese eighth graders received the highest average scores of children from 20 countries, and, at the twelfth-grade level, Japanese students were second only to Chinese students in Hong Kong (Garden, 1987). We must ask why this is the case. Why are Chinese students consistently among the top scorers in cross-national studies of achievement and American students consistently below the international average? The primary purpose of this research project was to attempt to provide some answers to this question. The researcher was interested in exploring cross-cultural differences in mathematics perception and attitude of younger children. Our major concern was to describe the context in which different levels of achievement occur in these two cultures. The researcher sought to identify not only contexts that appear to be important in explaining differences that we observed at the early years but also those that might be related to the cross-cultural differences in achievement that have been found among older children and youth. What effect does it have on our children's performance in mathematics? The researcher hope's these question s can be answered in further research.

Literature Review

Logically, children's academic achievement is related to three major factors: their intelligence, their experiences at school, and their experiences at home. With regard to the first factor, it seems unlikely that cross-national differences in academic achievement among Chinese, Japanese, and American children can be accounted for by differences in general intelligence. There is no evidence that Chinese and Japanese children are more intelligent than American children.

According to Schoenfeld (1989), the way people engage in mathematical activities is shaped by their conceptions of mathematics. There have been many studies that confirm that affective factors shape how students behave. For instance, perceived personal control (Lefcourt, 1982), and perceived usefulness of mathematics (Fennema & Sherman, 1978) are all positively correlated with achievement in mathematics (Schoenfeld, 1989). However, it is unclear if there is a cause and effect relationship between affective factors and achievement in mathematics. It is interesting to note that Schoenfeld (1989) found that the strongest correlation was between mathematical performance and perceived mathematical ability.

Children's academic achievement is given a more central role in some cultures than in others. In developing countries such as Taiwan, personal advancement is closely linked to academic achievement, and there is great emphasis on education. In Japan, where natural resources are limited, progress in technology and science is essential for the nation's economic health, and such progress is highly dependent on having a well-educated work force. Other cultures have different goals. Some value experiences that stimulate children to think and build up a broad fund of knowledge, regardless of whether such experiences result in higher school grades; others stress the importance of children developing a sense of self-worth. The goal of education in these societies is not only the acquisition of specific types of knowledge but also the development of children who feel good about themselves and their capabilities; self-confidence is believed to facilitate later learning. In other words, while some cultures value activities that help a child master prescribed skills, others, such as that in the United States, value experiences that will make a child more creative and confident (Stevenson, Lee, Chen, Stigler, Lee, Hsu, & Kitamura, 1990).

Chinese students must go through a series of rigorous entrance exams in order to get into a top university. Forty percent of Chinese students will make it to one of these universities. It is clear that these exams provide intense motivation for students in China to perform well. It would appear that the teachers in China put a great emphasis on teaching towards these national exams. They do in fact do this, but they emphasize conceptual understanding and applications of math to the real world instead of just computations.

Reynolds & Walberg (1992) found that motivation and home environment have a strong indirect effect on achievement. They also found that motivation appeared to be a stronger indicator of mathematics attitude than home environment. This would suggest that the student is responsible for changing their attitude towards mathematics and that their home environment is less influential. However, this could be very difficult for a student in the United States to do if they are consistently heating that it is okay to not be good in mathematics.

American teachers spent twice as much time on educating students in language arts as they did on mathematics (Stevenson, Lee, & Stigler, 1986). Chinese teachers, spent equal amounts of time on language arts and mathematics. Further teachers in the U.S. are allowed to organize their classrooms according to their own desires, not to a national standard (Stevenson, Lee, & Stigler 1986). Also Stevenson, Lee, and Stigler (1986, 1987) find that American teachers only spent about 22% of their time in the classroom imparting information, whereas Chinese teachers spent about 60% of classroom time imparting information. Chinese students spent 240 days a year in school whereas American students only spent 178 days in school (Stevenson, Lee, & Stigler 1986; 1987). Upon closer analysis of schools in China and the U.S., it has been found that students in Taiwan spend 5 and a half days in school a week compared to only 5 days a week for American students. Further, children in Taiwan spend more time in the day studying mathematics then their American counterparts (Stigler, Lee, Lucker, & Stevenson, 1982; Stigler, Lee, & Stevenson, 1987).

There appear to be issues such as how teachers', parents' and society's attitudes and beliefs affect our children. Research completed by Haladyna, Shaughnessy, and Shaughnessy (1983) found that there is a strong association between teacher quality measures and attitude toward mathematics. It is interesting to note that when teachers were asked what factors may influence students' performance in mathematics, 41% of American teachers believed that innate intelligence was more important than studying hard which was just the opposite of Chinese teachers (Stigler, Chen, & Lee, 1993). It was also found that American mothers feel that success in school is attributed to ability and Chinese mothers felt that success was to due to effort. Stigler, Chen& Lee (1993) found that even though Americans seem to be aware of the trouble that the American educational system is in, they still feel that the schools are doing a "good" or "excellent" job in educating their children. Parents even have a high regard for their children's academic performance even though they continue to be outperformed by their Chinese counterparts.

In Taiwan, the Ministry of Education specifies the curriculum for all schools in great detail. The ministry publishes all textbooks; further every school in Taiwan uses the same textbooks (Stigler, Lee, & Stevenson, 1986). However, in the United States there is a great deal of variation among textbook series (Stigler, Lee, Lucker, & Stevenson, 1982). This is due to the fact that there is not a national curriculum in the United States. All the textbooks in both countries use a common international system of mathematical notation and Arabic numerals (Stigler, Lee, Lucker, & Stevenson, 1982). Stigler, Lee, Lucker, and Stevenson, (1982) also found that the Taiwanese curriculum lagged behind the United States curriculum when concepts and skills were introduced. Therefore, it doesn't appear as if only the curriculum in the United States should be blamed for American students' low scores in comparison to Chinese students.

In studies of parental evaluations of children's capabilities, the closer the match between parental evaluations and the child's ability, the better the developmental outcome (e.g., Miller, 1988). Parents whose views are realistic are more likely to adapt interactions with their child to a level appropriate to the child's abilities than are parents who overestimate or underestimate what their child is capable of doing. The same effect would be expected at the societal level: members of some societies may generally be realistic in evaluating themselves and their children, and others may be biased and give excessively favorable or unfavorable ratings. Realistic evaluation should create the more positive environment for academic achievement (Stevenson, Lee, Chen, Stigler, Lee, Hsu, & Kitamura, 1990). Parents' perceptions of their child's capabilities are an important factor in their expectations for that child. For instance, the U.S. culture has a tendency to place a higher value on achievement in sports than in mathematics (Geary, 1996). The Asian culture, on the other hand, prioritizes mathematical learning. In fact, the elders of the culture believe that high achievement in mathematics is an important goal for the younger members of the culture (Geary, 1996). It is clear that education is highly valued in China. In fact, ongoing education is the norm.

Stevenson, Lee, Chen, Stigler, Lee, Hsu, & Kitamura, (1990) stated that the degree to which parents, family, and other members of society become involved in children's development and education is likely to differ, depending on the society's conception of the individual in relation to these entities. Chinese, and American children do have very different experiences at school. These differences in the cultural valuation of mathematics translate into differences in the investment of children, parents, and teachers in learning mathematics, and are likely to be the primary source of the mathematical ability differences comparing East Asian and U.S. children (Geary, 1996; Stevenson & Stigler, 1992) According to research by Hacker and Betz (1989), mathematics performance was significantly and positively correlated with attitudes toward mathematics. Stigler, Chen, and Lee concluded that the achievement gap is "unlikely to diminish until there are marked changes in the attitudes and beliefs of American parents and students about education" (Stigler, Chen,& Lee, 1993, p. 57).

Methodology

Participant

The study was conducted with fifth grade children from Taipei, Taiwan and Denver, Colorado. Taipei is a large modern city that is relatively comparable to Denver. The researcher selected one public elementary school from each city and surveyed one classroom from each school. The Taipei classroom had 37 students and the Denver classroom consisted of only 21 students as target subjects in the study.

The researcher chose elementary school children as the subject for two reasons. First, researcher wanted to know ff cross-cultural differences in achievement emerged during these early years of schooling. If this proved to be the case, it would be difficult to account for cross-cultural differences in achievement primarily in terms of the educational practices of the schools. A second reason for focusing on elementary school children was to gain some understanding of the early antecedents of the large differences that appear later in middle and senior high school.

The Instrument

A questionnaire containing 39 closed questions was developed by Alan Schoenfeld (1989) and was used with his permission in this study. All items were present in the form of a seven point rating scale, ranging from 1 = "strongly agree" to 7 = "strongly disagree". The questionnaire contained questions related to students' perception of what mathematics is and how to do well in it, what mathematics solutions should be, how math problems can be solved, how mathematics is learned, and student motivation. The students rated each of the first 33 questions on a seven point scale with one being strongly agree, four being neutral and seven being strongly disagree. The last six questions are concerned with gender, grades, and perception of their parent's attitude towards mathematics. The questionnaire was determined to be highly reliable with an alpha of 0.8468.

The Procedure

In this study, the fifth grade students were asked to answer the questionnaire. Due to the fact that the researcher was unable to go to Taipei to conduct the study, the teacher who is a friend of the researcher distributed the questionnaire at her convenience. However, the researcher was able to distribute the surveys in the Denver school. Typically, questionnaire took only 10-15 minutes to complete.

Data Analysis

The form of the data collected contained the attitude measure from a Likert type scale and personal information. The researcher compared the means of each question in the questionnaire and used a two-tailed t-test to analyze the data. Then researcher categorized some of the related questions to determine if there was a correlation between categories.

Result

Specific prior hypotheses were not developed prior because information from previous work was insufficient to allow confidence to be developed about the characteristic of each culture. Although it was not our purpose to evaluate the usefulness of the constructs that have emerged from this work explaining differences in achievement in the two cultures at issue, the researcher did use them to help organize some considerations. The researcher used two-tail t-test to compare the mathematical perceptions between Chinese and American's students. The means of each question in the questionnaire was also compared. Among the 33 items 24 items are significantly different between American and Chinese students' answers, these are items 1, 3,4,5,6,7,8, 12, 14, 15, 16, 18, 19,20,21, 22, 24, 25, 26, 28, 29, 31, 33.

The following items were grouped into six categories, with the thought that some questions asked might have the same concept. Table 1 displays the six categories and the mean score of the items that were significantly different for the two cultures.

Discussion of Results

The researcher found marked differences in the beliefs of American and Taiwanese students in many of the areas under investigation. The data shows significant differences in the means for the category of what mathematics is. It is interesting to note that the questions in this category suggest that mathematics is mostly number. The Taiwanes students tended to disagree or feel netutral about these statements. However the American children tended to agree with this perception of mathematics.

The sceond category of how to do well in mathematics also showed significant differences in the mean scores. The American students strongly agreed that memorization was the key to doing well in mathematics. If they could memorize all the formulas and how to do them, then they would do fine in the class, even if they didn't understand what they were doing. It is hardly the most creative or logical of acts, but it also is a creative disciple in math where one can discover, and learn to be logical. Their Taiwanese counterparts, on other hand, were again more apt to disagree with this belief.

The category of what math sloutions should be, encompassed whether there could be more then one right solution to a problem. Again a significant diffrerence in the mean score was determined. American students believe the solution to be right answers. Taiwanese students ' responses more are flexible. Taiwanese students stongly dis greeed with the idea that in mathematics they are either right or wrong. However the American students tended to strongly agree with this idea.

The category of how math problems can be solved and how mathematics is learned did not show significant differences for the two cultures. For the last category, the researcher broke up the categoy of motivation into two parts: negative motivation and positive motivation. There were significant differences between the two sub-categories for the two cultures. The negative motivation sub category encompassed the idea of learning math because it was required or because of fear of punishment. The Taiwanese answers showed that they are effected by this negative motivation. They tended to agree with these statements. However the American students strongly disagreed with these statements. Instead the American students were more influenced by postive motivation such as wanting to do well in class or impress the teacher.

There were a few individual questions that were of interest to us. Table 2 shows the mean score for each culture. There is a significant difference in the mean scores. American students believe that solving mathematics problem depends on knowing the rule. Mathematics is presumed to be more rules bound. Taiwanese students tended to agree that good teaching practice in mathematics consists of making sure students know how to use the rule. On the other hand, students also think that good teaching practice consists of showing students lots of different ways to look at the same question. The researcher also found the reason Taiwanese students believe school math is useful in real life, is because they may be more preoccupied with learning, an end itself rather than being concerned with self perceptions.

What is interesting to note is even though the Taiwanese students' perception of mathematics was overall more positive than their American counterparts, they seem to be learning mathematics mainly because of the fear punishment. This area should probaly receive more attention in future research studies. Even though the Taiwanese students hint that they learn mathematics for negative reasons, they do have a more positive perception of mathematics than their American counterparts. This is most likely due to the fact that their culture places such a high value on mathematics achievement and it in turn trickles down into the schools. The two cultures' perception of mathematics is clearly different, but the students did share some of the same beliefs which are the former that refer to mathematics that takes place inside class and the latter to mathematics that takes them outside.

From past international studies of achievement, we know that Taiwan continues to outperform the United States in mathematics achievement. The researcher also found, (Tsao, 2000), that there are marked differences in the attitudes and beliefs of the two cultures towards mathematics and in the students' attitudes and beliefs towards mathematics. This difference in mathematics achievement could be the effect of the Taiwanese's more positive perception of mathematics. Further it appears that the negative attitude of the American culture could be one factor causing the low international achievement scores in mathematics.

These differences in the cultural valuation of mathematics translate into differences in the investment of children, parents, and teachers in learning mathematics, and are likely to be the primary source of the mathematical ability differences comparing East Asian and U.S. children (Geary, 1996). According to research by Hackett and Betz (1989), mathematics performance was significantly and positively correlated with attitudes toward mathematics. Stigler, Chert, and Lee (1993) concluded that the achievement gap is "unlikely to diminish until there are marked changes in the attitudes and beliefs of American parents and students about education" (p. 57).

There is still a great need for further investigation into the differences between the two cultures' perceptions of mathematics. In further studies, researchers should do a series of questionnaires and interviews of students from elementary school through college, in order to pinpoint an age level or grade level at which students' attitudes and beliefs toward mathematics begin to change. Other studies should study the students' parents and teachers in order to get a better understanding of what their attitudes and beliefs are.

Limitation

The major limitation of this study is that they are convenient samples and small samples from each country. A larger number of samples would not only have resulted in more reliable and generalized conclusions about the effective of math perception between American and Taiwan students, but would also allow a more systematic study of the relation between math perception and performance outcome in mathematics.

Educational Implication

The goal of education in these societies is not only the acquisition of specific types of knowledge, but also the development of children who feel good about themselves and their capabilities. Self-confidence is believed to facilitate later learning. In other words, while some cultures value activities that helps a child master prescribed skills, others, such as that in the United States, value experiences that will make a child more creative and confident. The degree to which parents, family, and other members of society become involved in children's development and education is likely to differ, depending on the society's conception of the individual in relation to these entities. The firmness of boundaries separating individual, family, and group has important implications for children's development. In some cultures, such as that in the United States, the individual is deemed to be responsible for his or her accomplishments and difficulties; in others, such as the Chinese cultures, members of the family, teachers, or a larger group-such as pupils in the same classroom-are expected to assume some of the responsibility. As the interdependence among individuals increases, their mutual obligations to each other also increase. Individuals in such situations work hard not only to satisfy their own goals but also to meet the goals set by their families and success of the group is valued as highly as the-success of particular individuals within the group.

The greater the cultural emphasis on effort, the more likely it is that parents and teachers will believe that they can be instrumental in aiding children in their academic achievement. This belief is transmitted to children, and they, too, come to believe that diligence will lead to success, lf, however, adults believe that innate ability imposes critical limitations on children's progress in school, it seems unlikely that they would be motivated to make such strong efforts at assistance. Taiwan, like other countries influenced by the Confucian belief in human malleability, is among the cultures that place great weight on the possibility of advancement through effort.

It would be helpful if more studies that focus on affective issues would have stronger links to research on other topics related to the improvement of practice in mathematics education. Although little has been done to connect research on affective issues to these kinds of studies of cultural influences on mathematics learning, such connections should be able to link differences in achievement to beliefs that are connected to cultural influences.

Table 1. Six Categories: The Mean Score of the Items Shows Significant
Differences in the Two Cultures.

Number of    Concept                               Contained items
Category

1            What mathematics is                   1 *,2, 3 *,31 *
2            How to do well in it                  14 *,28 *
3            What mathematics solutions should     4 *,5 *,6 *,15 *
             be
4            How math problems can be solved       7 *,8 *, 11, 12 *,
                                                   13
5            How mathematics is learned            10, 27
6            Student motivation-negative           19 *,22 *
             Student motivation-positive           20 *,23,24 *

* p<.01

Table 2.

                        Question                             Mean Score
                                                             U.S.A

Math is mostly facts and procedures that have to be          2.3000
memorized
Math is just a way of thinking about space, numbers, and     2.2000
problems.
In mathematics something is either right or wrong            1.6500
The best way to do well in math is to memorize all of the    1.8500
formulas.
I'll get in trouble if I don't try to learn math             4.2000
Different math courses cover unrelated topics.               2.6500
Some people are good at math and some aren't                 1.3500
When you get the wrong answer to a math problem, it is       3.2105
absolutely wrong--there's no room for argument

                        Question                             Mean Score
                                                             Taiwan

Math is mostly facts and procedures that have to be          4.2432
memorized
Math is just a way of thinking about space, numbers, and     5.2222
problems.
In mathematics something is either right or wrong.           5.5833
The best my to do well in math is to memorize all of the     4.2162
formulas.
I'll get in trouble if I don't try to learn math             2.4324
Different math courses cover unrelated topics.               4.8649
Some people are good at math and some aren't                 2.9189
When you get the wrong answer to a math problem, it is       5.1351
absolutely wrong--there's no room for argument

                        Question                             p-value

Math is mostly facts and procedures that have to be          0.000 *
memorized
Math is just a way of thinking about space, numbers, and     0.000 *
problems.
In mathematics something is either right or wrong.           0.000 *
The best my to do well in math is to memorize all of the     0.000 *
formulas.
I'll get in trouble if I don't try to learn math             0.003 *
Different math courses cover unrelated topics.               0.000 *
Some people are good at math and some aren't                 0.000 *
When you get the wrong answer to a math problem, it is       0.000 *
absolutely wrong--there's no room for argument

* p<.01

References

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Dr. Yea-Ling Tsao, Math Computer Science Education Department, Taipei Municipal Teachers College.

Correspondence concerning this article should be addressed to Dr. Yea-Ling Tsao, Math Computer Science Education Department, Taipei Municipal Teachers College, No. 1, Ai-Guo West Road, Taipei, 100, Taiwan; Email: [e-mail]