Focus east asian teachers and teachjing by Heurística Educativa

Factors Contributing to East Asian Students’ High Achievement: Focusing on East Asian Teachers and Their Teaching Paper presented at the APEC Educational Reform Summit, 12, January 2004

Kyungmee Park Department of Mathematics Education Hongik University, Seoul, Korea

1. Introduction East Asian students have consistently outperformed their counterparts in the West in international comparisons of mathematics and science achievement. The merits of such international studies do not lie in constructing a league table of countries, but in identifying factors that contribute to high achievement, and in understanding the practices in other countries for the sake of improving the education in one’s own country. Thus, an obvious question to ask of such international studies is what accounts for the high achievements of East Asian students. In this paper, the factors contributing to East Asian students’ high achievements in mathematics and science will be identified with the review of previous literatures. Since students learn most of their knowledge in the classroom, it is reasonable to expect that the instruction they receive should be a major factor in influencing their achievement. Therefore, among many factors, this paper focuses on the factor of East Asian teachers and their classroom teaching, using data from the recently published TIMSS 1999 Video Study and a small-scale comparative study in Hong Kong and Korea as examples.

2. East Asian Students’ Achievement in Mathematics and Science Two important and recent international studies in mathematics and science are the Third International Mathematics and Science Study (TIMSS) and its follow up study, and the OECD Program for International Student Assessment (PISA). 2.1. TIMSS East Asian students performed well in mathematics and science in TIMSS (Beaton et al, 1996; Mullis et al, 1997) and its follow up study (TIMSS Repeat, or TIMSS-R) (Mullis et al, 2000). As can be seen from Figures 1 and 2 in Appendix, East Asian countries top the list of APEC countries in mathematics and science (with the possible exception of Hong Kong in the area of science). However, the high achievements of East Asian students do not seem to be accompanied by correspondingly positive attitudes towards mathematics and science. According to the results of the TIMSS-R questionnaire, East Asian students (with the exception of the Singaporean students) ranked very low in the indexes of ‘students’ report on whether it is important to do well in mathematics and science’, ‘students’

positive attitudes towards mathematics and science’, and ‘students’ self-concept in mathematics and science’.1 2.2. PISA The performance of East Asian students (Korea and Japan) in PISA is similar, with high achievement in mathematics and science, and low interest in mathematics. The PISA results in mathematics and science are shown in Figures 3 to 4 in Appendix(OECD, 2001). As can be seen from Figures 1 to 4, East Asian students performed very in these international comparative studies. In the next section, the factors contributing to East Asian students’ success in international mathematics and science comparative studies will be reviewed and discussed.

3. Factors Contributing to East Asian Students’ High Achievement 3.1. The Number System Two characteristics of the Chinese, Korean and Japanese number system may have contributed favorably to the high level of performance in mathematics and probably in science as well: the simple pronunciation of numbers and the regularity of the number system. First, the pronunciations of numbers in East Asian languages are relatively simple. For example, Korean numbers up to 10 are all pronounced in one syllable, making it rather efficient for students when they are handling with numbers. This is not so with Western languages such as English, where the pronunciations of some numbers may be more complicated (e.g. the number “seven” has two syllables). The second characteristic is the regular nature of the number system. In Chinese, Korean and Japanese, a consistent rule is applied to all numbers. For example, for numbers between eleven and nineteen, they are all pronounced as “ten followed by a single digit number”. There are no irregular numbers such as ‘eleven’ or ‘twelve’ in Korean. Similarly, numbers between twenty-one and twenty-nine are expressed as “twenty followed by a number”, and so on. In contrast, for numbers between thirteen and nineteen in the English language, they are pronounced as “a single digit number followed by ten”. For example, ‘16’ is pronounced as ‘sixteen’, with “six” followed by “teen” (which means ten). However, when it comes to numbers between twenty-one and twenty-nine, the pattern is changed to “twenty followed by a number”. This seemingly trivial inconsistency may cause problem for children when they learn and handle 1

It may perhaps be wrong to accept the above student questionnaire results at face value and conclude that East Asian students have a more negative attitude than their counter-parts in Western countries. In fact, the East Asian culture stresses the virtue of humility or modesty and hence there may be tendency for their students to underestimate their ability. Traditional teaching in East Asia requires teachers to teach students not to be conceited while imbuing a proper level of confidence and modesty into them, and sparse praise for students.

numbers, especially for young children at the lower grades of primary school. In sum, the number system in East Asian countries is a simpler and more logical system, easier for students to work with (Park & Leung, 2003). 3.2. Examination and Selection One of the major factors influencing students’ high achievement in mathematics and science in East Asian countries appears to be the national enthusiasm for education, the eagerness for study, and the ethics of hard work (Education Review Office, 2000). It is a well-known phenomenon that the focus of primary and secondary education in East Asian countries is on subjects required in the national college entrance examination. Since students are more discriminated by mathematics and science scores than by scores in other subjects, mathematics and science becomes most effective when it comes to a selection-oriented education environment, and schools tend to place a relatively high importance on the subjects of mathematics and science. In addition, there are many private institutions or tutoring courses dedicated to preparing students for mathematics and science in college entrance examination. East Asian students and parents take education very seriously, and these private institutions or tutoring courses, in parallel with regular schooling, become major elements of education in East Asia. For example, according to a survey administered by the Korean Institute of Educational Development (KEDI, 2000), 81.2% of the Korean primary and secondary school students are receiving at least one private lesson beyond school work. While private education courses cover a wide range of subjects, mathematics and science are the most common, and most secondary school students attend additional mathematics and probably science private institutions or receive tutoring outside school hours. This reflects East Asian students’ emphasis on mathematics and science, and results in East Asian students having more exposure to mathematics and science instruction and practice. 3.3. Attitudes of Students towards the Test Students’ performance in a test is obviously affected by their attitudes towards the test, and this applies to testing in TIMSS-R and PISA as well. In many western countries, in order for students to take the international comparative tests, the national study centers need to get prior approval from the parents concerned. This “voluntary” nature of taking the tests may have sent a signal to students that this is an activity that “does not count”, and hence students may tend not to take the tests seriously. In contrast, East Asian students, who are raised in the Confucian culture, are educated to take testing very seriously. Their attitude towards the TIMSS and PISA tests may have been influenced by this general serious attitude towards testing and may have contributed positively to their performance. Supervising the TIMSS-R tests in Korea, the author observed a school principal encouraging his students who were taking the test, saying ‘You should take pride in taking the test since you are representing your country. Do your best.’ This anecdotal incident shows that East Asian students may be taking the TIMSS-R and PISA tests more seriously than their counter-parts in the west.

3.4. Selection of Pre-service Teachers How selective a country is for candidates into the teaching profession also contributes to the quality of the teachers, and hence play an important role in students’ achievements. In Korea, for example, secondary teacher education is mostly provided at the colleges of education at comprehensive universities.2 Most students who enter the department of mathematics education or the department of science education at the colleges of education are from the upper group in College Entrance Examination, and this trend is becoming more evident since the financial crisis in the late nineties. Moreover, some students enter the department of mathematics education and science education after completing their bachelor’s degree in the relevant fields at a comprehensive university, making entry into these teacher training courses, and hence into the profession, very competitive. Furthermore, completing the four-year education at a college of education does not in itself qualify the graduates for teaching in public schools. The graduates are only awarded a teacher’s certificate which enables them to be eligible for teaching in private schools, but to qualify to teach in public schools, certificate holders are required to pass a very demanding national examination, the Teachers Employment Test (TET). Most certificate holders choose to take the TET. The TET consists of three parts, encompassing general educational theory or pedagogical knowledge, subject matter knowledge, and pedagogical content knowledge (see Table 1 below). It ensures that teachers (in the public sector at least) have a firm grasp of the pre-requisite knowledge before they enter the profession. As pointed out above, the TET is very demanding. For instance, the TET for secondary school mathematics teachers that took place in December 2003 had a passing rate of only one in seven. The low success rate of TET has earned it a nickname of the ‘bar exam’ of the College of Education. Table 1. Characteristics of items in TET Percentage Item type Relevant knowledge Content of items Multiple General pedagogical Education in general 30% choice items knowledge Mathematics, Open-ended Subject matter History of mathematics, 49% items knowledge Secondary school mathematics Open-ended Pedagogical content Mathematics Education3 21% items knowledge In summary, the keen competition for teacher education and the demanding entry test ensure that teachers are selected from a pool of candidates with high scholastic 2

Elementary teacher education is offered by normal universities in Korea, and some East Asian countries provide both elementary and secondary teacher education by normal universities. 3 An examples of a “Mathematics Education” item is as follows: “What follows is the proof of a statement using a synthetic method. … Construct another proof based on an analytic method. Also, explain the didactic advantages of the analytic method when we teach proof.”

achievement. Although we cannot simply conclude from this that Korean (or East Asian in general) teachers must have profound knowledge in the subject matter and are competent in pedagogy, it is reasonable to expect that these teachers are more competent in performing their roles than in many western countries where there is an acute shortage of teachers. This aspect of teachers’ competency will be discussed later in more detail. 4. The TIMSS Video 1999 Study 4.1. Background In parallel with the TIMSS 1995 study, a video study that examined instructional practices in eighth-grade mathematics was conducted for the three countries of Germany, Japan and the Unated States. The results of this study show that the classroom teaching in Japan is substantially different from that in Germany and the United States (Stigler & Hiebert, 1999), and the results have attracted much attention in the mathematics education community and the public at large. Prompted by the success of the TIMSS 1995 video study, a similar but larger scale study was conducted along with TIMSS-R in 1999. In the TIMSS 1999 Video Study, seven countries (Australia, Czech Republic, Hong Kong SAR, Japan4, Netherlands, Switzerland, United States) were involved in the area of mathematics, and five countries (Australia, Czech Republic, Japan, Netherlands, United States) were involved in the area of science. The main goal of the study is to describe and compare eighth-grade mathematics and science teaching across the countries concerned, and to: • • •

discover alternative ways to teach mathematics and science, examine teaching in one’s own country with fresh eyes, and create digital library of public use videos for teacher professional development.

The results of the mathematics component of this video study were released in March 2003 (Hiebert et al, 2003), but the science results have not been published yet. So in the following section, classroom teaching in East Asia will be discussed based on the results of the mathematics study only. 4.2. Sampling and Data Collection The TIMSS 1999 Video Study utilized a national probability sample of a target of 100 eighth-grade lessons per country. As such, it claims to be a video survey, the first of its kind, and the results should be representative of the teaching in the countries concerned. But because of the high cost involved in videotaping, only one lesson per teacher was videotaped. The idea of the study is not to conclude about the performance of individual teachers, but to get an overall impression of the teaching in a certain country. Since different countries cover different content in their eighth grade, it has not been possible to choose a common topic for video-taping. Instead, the lessons were sampled across the 4

The 1995 Japanese data were re-analyzed using the 1999 methodology in some of the analysis.

school year to get a comprehensive coverage of the content taught in the eighth grade in each of the countries. Videotaping in different countries followed standardized camera procedures, with one camera focusing on the teacher and another camera focusing on the class as a whole. Altogether 638 lessons were videotaped for mathematics, ranging from 50 lessons in Japan and 140 lessons in Switzerland. 4.3. Data Coding and Analysis An international team of coders from the participating countries developed a set of 45 codes and applied them to the video data. In addition to this quantitative analysis, a number of more qualitative analyses were performed on the data (Hiebert et al, 2003). One such analysis was performed by an expert panel, known as the Mathematics Quality Analysis Group, comprising mathematicians and mathematics educators at the postsecondary level. The group reviewed a randomly selected subset of 120 lessons (20 lessons from each country except Japan5) based on detailed descriptions of the lessons prepared by the international video coding team. These descriptions of the lessons were examined â&#x20AC;&#x153;country-blindâ&#x20AC;?, with all indicators that might reveal the country removed. That is, when the group analyzed the contents of the 120 lessons, they did not know which lesson table corresponds to a lesson in which country. 4.4. Results From the results of the analysis of the codes, a number of distinctive features were identified in the two East Asian countries in the study, Hong Kong and Japan.

4.4.1. Dominance of teacher talk In all of the countries studied, the teacher did a lot more talking than their students. For the classrooms in the two East Asian countries, the video data shows that they differ in the amount of teacher talk. Hong Kong teachers were among the most talkative teachers in the seven countries (second only to the American teachers), while the Japanese teachers spoke the least number of words among the teachers in all the seven countries. Figure 5 shows that Hong Kong teachers on average spoke nearly 5800 words per lesson6, while Japanese teachers spoke only 5148 words per lesson, compared with the average of 5533 words across the six countries.

Since this is the same group of experts who performed a similar analysis to the 1995 TIMSS Video data, including the Japanese data, 4 years ago, the Japanese data was not included in this analysis. 6 Since lesson duration varies across country, the lesson time reported here is standardized to 50 minutes of lesson time.

Figure 5. Average number of teacher and student words per lesson 7000 6000

5536

5902

5798

5452

5148

5360

5000 4000 3000 2000 824

810

1000

1016

766

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1018

0 AU

Average number of teacher words

Average number of student words

On the other hand, students in both Hong Kong and Japan were the least talkative among the students of the countries in the study. They spoke the least number of words in the classroom compared with their counterparts in other countries. If we look at the ratio of the average number of words spoken by students during the lesson versus those spoken by their teachers during the lesson, the verbal dominance of the East Asian teachers is even more evident. For every word that a student uttered, the Hong Kong teacher spoke an average of 16 words and the Japanese teacher spoke an average of 13 words, compared to the minimum ratio of 1:8 (US), and an average of 1:11 (see Figure 6).

Number of Teacher Words Per 1 Student Word

Figure 6. Average number of teacher words to every one student word per lesson 20 16 16 13 12 9

10 8

8 4 0 HK

4.4.2. More opportunities to learn new content The TIMSS 1999 Video Study classified the purposes of lesson segments according to whether they are for “reviewing” (addressing content introduced in previous lessons), “introducing new content” (content that students has not worked on in an earlier lesson), or “practicing new content” (students practicing or applying content introduced in the current lesson). The average percentages of lesson time devoted to each of the three purposes are shown in Figure 7 below.

Figure 7. Average percentage of lesson time devoted to various purposes 10000% 16

Percent of Lesson Time

8000%

6000%

4000% 58

2000%

36 24

0% AU

CZ Review

Introduce New

Practice New

We can see that more time in the East Asian classroom is spent on dealing with new content compared to the other countries, and we can infer from this data that East Asian students learn more subject matter than their counterparts in other countries. 4.4.3. Problems more complex What is the nature of the subject matter that these students learn? In the Video Study, it was found that the lesson time in all the seven countries were dominated by students working with mathematical problems, and thus one of the major units of analysis in the study is the mathematical problems. Two aspects of the characteristics of the problems presented in the lessons are their procedural complexity and the kind of reasoning that is involved. These are related to how long each of the problems dealt with is worked on. Figures 8 and 9 below show the average percentage of problems of different levels of procedural complexity and the average percentage of problems that were worked on for more than 45 seconds. From the two Figures, we can see that in general, East Asian students have more opportunities to work on procedurally more complex problems which required a longer duration to solve. Figure 8. Average percentage of problems per lesson that were worked on for longer than 45 seconds 98

Percent of Problems

100 78

80 60

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40 20 0 AU

Figure 9. Average percentage of problems per lesson at each level of procedural complexity 8

Percent of Problems

100

6 12

60 40

20 17 0 AU CZ Low Complexity

HK JP NL SW US Moderate Complexity High Complexity

4.4.4. Problems set up using mathematical language Another characteristic of the problems solved by these two East Asian countries is that the majority of the problems were also set up using the mathematical language and symbols, and in contexts unrelated to the real life. (In this regard, East Asian classrooms are similar to those in Czech Republic.) This is evident from the data shown in Figure 10 below. Figure 10. Average percentage of problems per lesson set up with a real life connection or with mathematical language or symbols only

Percent of Problems

100 80

9 27

15 25

60 40

89 71

20 0 AU

Set-up contained a real life connection Set-up used mathematical language or symbols only

4.4.5. More proof Probably related to the last point, the problems that East Asian students work on involved more proof than those worked on by students in other countries, as shown in Figure 11 below.

Figure 11. Percentage of lessons that contained at least one proof

nt of Lessons

100

80 60 39 40

4.4.6. Summary From the discussions above, we can see that although East Asian students do not talk a lot in the classroom, they are exposed to more instructional content. The mathematics problems they worked on were set up mainly using mathematical language, and were procedurally more complex. Compared with the problems solved by students in other countries, they take a longer duration to solve and they involve more proof. This finding is rather consistent with the judgment of the Mathematics Quality Analysis Group, who finds the content covered in Hong Kong7 (and Czech Republic) lessons relatively more advanced than that in other countries.

Percent of Sub-sampled Lessons

Figure 12. Percentage of lessons in sub-sample at each content level 100 15

0 20

30 35

Advanced

40 40

Moderate

Elementary/Moderate

45 20

Moderate/Advanced

40 0

5 0

Elementary

In addition to judging the how advance the content covered in the lesson, the Mathematics Quality Analysis also judged the overall quality of the mathematics in the lessons along 4 dimensions8: 1. 2.

coherence (defined by the Group as “the (implicit and explicit) interrelation of all mathematical components of the lesson”). presentation (“the extent to which the lesson included some development of the mathematical concepts or procedures”).

The 1995 Japanese lessons were not included in this analysis, and so the only East Asian country discussed here is Hong Kong. 8 Given that only 20 lessons were chosen from each country, the results in this section of the paper need to interpreted with care.

3. 4.

engagement (whether students in that lesson were likely to be “actively engaged in meaningful mathematics”). overall quality (“the opportunities that the lesson provided for students to construct important mathematical understandings”).

The results of their judgment are show in Figure 13 below. As can be seen, according to the mathematics quality analysis group, the Hong Kong lessons are more coherent, the mathematics presentations are more fully developed, and the students are more likely to be engaged in the lessons. In short, the overall quality of the teaching in this East Asian country is judged to be high. Figure 13. General ratings for each overall dimension of content quality of lessons 5.0 4.0

HK SW AU NL CZ US

3.0

HK SW CZ AU NL US

HK CZ SW AU NL US

2.0

HK CZ SW AU NL US

AU CZ HK NL SW US

1.0 0.0 Coherence

Presentation

Student engagement

Overall quality

4.6. The need for In-depth Case Study From the TIMSS Video Study, it can be seen that one of the striking differences between the East Asian classroom and that in the West is that students in the former are exposed to more advanced and higher quality instructional content than students in the latter. East Asian students learn more content at a higher level of complexity and abstraction, and this may be one of the important factors in accounting for their high achievement in international studies. However, the quantitative analysis of the video data may fail to review the subtlety of the complexity of classroom teaching. Thus, there is a need for indepth case studies to supplement the information obtained from quantitative studies. Case study does not only yield valid data, but it also allowed intention of the participants to be made known. Such case studies of course have their limitations. For example, in interviews with teachers, what we can obtain are merely reported (rather than observed) teaching, and may reflect more the teachers’ intention rather than their performance. Notwithstanding these limitations, case studies provide an important supplement to the kind of video study data reported above. Taking the results of these studies together, it may yield a more accurate picture of what East Asian teaching is like. 5. Replication of Ma’s study

A recent important comparative study between the East and the West which looked into teacher’s reported teaching was conducted by Ma (1999). Ma interviewed 23 elementary school mathematics teachers in the US and 72 Chinese teachers in Shanghai using the four Teacher Education and Learning to Teach Study (TELT) tasks (Ball, 1988) to probe into teachers’ understanding of the mathematics and the related pedagogy. She found that the Shanghai teachers in her study are more competent than the US teachers in terms of a “profound understanding of fundamental mathematics” and the related pedagogy. Shanghai or China however did not participate in TIMSS or PISA, and hence we cannot link Ma’s finding concerning teachers and their teaching with the achievements scores of these international studies directly. In an attempt to relate Ma’s findings to other parts of East Asia and hence relate the findings with the TIMSS and PISA results, a small-scale replication of Ma’s study was carried out in Hong Kong and Korea (Leung & Park, 2002). Nine elementary school mathematics teachers from each of Hong Kong and Korea were interviewed using the TELT tasks that Ma used in her study. For each task, following Ma’s interviewing method, a classroom scenario in which a particular mathematical idea plays a crucial role was presented to the teachers. Teachers were asked how they would handle the classroom situation in the given scenario, and their understanding of the mathematics and the related pedagogy were probed. Part of the results of this study will be reported below (in the episodes below, “I” stands for the interviewer, and “T1, T2 etc. stand for the teachers). 5.1. Conceptual versus Procedural Understanding In the interviews, the Hong Kong and Korean teachers in our sample were asked how they would deal with the kind of mistakes presented below: Task 1:

Multi-digit Number Multiplication

Some sixth-grade teachers noticed that several of their students were making the same mistake in multiplying large numbers. In trying to calculate 123 × 645_ the students seemed to be forgetting to “move the numbers” (i.e., the partial products) over on each line. They were doing this: ×

123 645__ 615 492 738 1845

instead of this: 123 × 645_ 12

615 492 738 79335 While these teachers agreed that this was a problem, they did not agree on what to do about it. What would you do if you were teaching sixth grade and you noticed that several of your students were doing this? Teachers’ responses to this scenario expose both their knowledge of multi-digit number multiplication and their strategies in teaching the topic. The results are juxtaposed with the results in Ma’s study and shown in Tables 2 and 3 below. Table 2. Teachers’ knowledge of multi-digit multiplication algorithm

Procedural Conceptual + procedural

US Shanghai Hong Kong 14 (61%) 6 (8%) 0 (0%) 9 (39%) 66 (92%) 9 (100%)

Korea 1 (11%) 8 (89%)

Table 3. Reported teaching strategies

Procedurally directed Conceptually directed

US Shanghai Hong Kong 16 (70%) 9 (13%) 7 (78%) 7 (30%) 63 (88%) 2 (22%)

Korea 6 (67%) 3 (33%)

As can be seen from Table 3, the reported strategies of the Hong Kong and Korean teachers were very procedural. The following excerpt illustrates one such procedural strategy: I:

What would you do if you find your students are making this kind of mistakes?

T1: I’ll teach them that if you use the hundreds place times the unit place, your answer should be written under the “6”, align with this “6”. Now the textbook leaves two empty spaces there, and I’ll tell them if you have empty spaces, it is easy for you to align wrongly. So I’ll teach them after you get the product, say six times three equals eighteen, you should immediately put down two zeroes in the two empty spaces. If you have something there, you won’t align wrongly. So the first thing is to align, … (It is) the same with the second number. It is the tens place, so you should put your answer under the tens place, and you put a zero at the unit place … However, when probed into the reasons for the procedures of “aligning” and “putting in zeroes”, it is found that the teacher actually fully understood the rationale for “moving the numbers” (i.e. the partial products). I:

Why are students making this kind of mistakes?

T1: It’s a problem with place value. In fact the 6 here stands for 600. I can change 645 into 600 + 40 + 5, and then do it step by step, dividing (the multiplication) into 123 × 600 + 123 × 40 + 123 × 5. Then I will address the problem of aligning the numbers (i.e. the partial products). I will tell them that you need to add in zeroes … The responses of the teachers in our sample to this task show that they possessed conceptual as well as procedural understanding of the algorithm behind multi-digit number multiplication. However, the majority of the teaching strategies reported by these teachers were procedurally rather than conceptually directed. Compared with the results of Ma’s study, it seems that in terms of reported teaching strategies, the Hong Kong and Korean teachers are more akin to the reported practice of the US teachers, but in terms of their understanding of the mathematics concepts, they are more akin to the Shanghai teachers. 5.2. Competence in Mathematics and Pedagogy Teachers in the study were not explicitly tested on their competence in the subject matter that they teach, but their response to the following task revealed their competence in mathematics: Task 2:

Division by Fractions

People seem to have different approaches to solving problems involving division with fractions. How do you solve a problem like this one? 1 3/4 ÷ 1/2 = Imagine that you are teaching division with fractions. To make this meaningful for kids, something that many teachers try to do is to relate mathematics to other things. Sometimes they try to come up with real-world situations or story-problems to show the application of some particular piece of content. What would you say would be a good story or model for 1 3/4 ÷ 1/2 ? All the Hong Kong and Korean teachers in our interviews were able to calculate the division correctly (as were the Shanghai teachers in Ma’s study), and at least half of the teachers in each of Hong Kong, Korea and Shanghai understood the division by fractions concept well enough to be able to generate a representation of the problem by providing a story. Compared with their American counterparts, the East Asian teachers are clearly more competent in the subject matter that they teach. Table 4. Teachers’ knowledge of division by fractions

Correct answer Correct story

Shanghai

9 (43%) 1 (4%)

72 (100%) 65 (90%) 14

Hong Kong 9 (100%) 6 (67%)

Korea 9 (100%) 5 (50%)

In responding to this task in the interview, some Korean teachers in our study reported the use of the following strategy in teaching division of fractions, despite the fact that the Korean textbook uses another approach. Teacher T2:

This is the explanation provided in the (6th grade) textbook:

7 × 2 ÷ 4 × 1 = ( 7 × 2 ) ÷ ( 4 × 1) = 13 ÷ 1 = 7 ÷ 1 = 4 4 ×2 ×2 2 4 2 4

7×2 = 7 × 2 1 4 ×1 4

But I don’t use this formula because it is very hard for sixth graders to understand… The students have already learned the rule of fraction multiplication which is basically the same. I’m afraid if I introduce this principle, students might wrongly generalize this to addition of fractions like (a/b + c/d = (a + c)/(b + d)). … I prefer to use the previously promised rule. At the beginning of teaching division by fractions, I introduce the example “1 ÷ 1/7”. Assuming you are dividing a 1m rope into 1/7 m units, this results in 7. So, 1 divided by 1/7 is the same as 1 multiplied by 7. At this point, I set up a rule that “division by fraction is the same as multiplication by the reciprocal of the fraction”. I’ll make my students recall this rule, and then use it here. So again we can see that these East Asian teachers fully understood the concepts behind the procedures of division of fractions, but they deliberately taught in a rather procedural manner for pedagogical reasons. This can be attributed to a kind of mentality which I would described as “East Asian Pragmaticism”, a point that I will return to later in this paper. 5.3. Guiding Students to Explore Mathematics Task 3: Relationship between Perimeter and Area Imagine that one of your students comes to class very excited. She tells you that she has figured out a theory that you never told the class. She explains that she has discovered that as the perimeter of a closed figure increases, the area also increases. She shows you this picture to prove what she is doing: 4 cm 4 cm

8cm 4 cm

Perimeter ＝ 16 cm Area ＝ 16 square cm How would you respond to this student?

Perimeter ＝ 24 cm Area ＝ 32 square cm

When presented with this task, none of the Hong Kong teachers and only one Korean teacher we interviewed thought that the statement is correct. But only two of them knew that the statement is definitely wrong from the very beginning and were able to provide a counter-example. Most of them were unsure whether the statement is correct or not at the beginning. This situation provides a golden opportunity for teachers to guide students

explore the mathematics behind this problem, and indeed this was what many Shanghai teachers did in Ma’s study. But for this study, none of the Hong Kong and Korean teachers made use of the opportunity to guide students to explore the relevant mathematical ideas. Following the criteria used in Ma’s study, the different responses of teachers are classified in Table 5 below: Table 5. Teachers’ reactions to the incorrect statement

Accepted the claim Not sure Explore using problematic strategy Explore using correct strategy

Shanghai

2 (9%)

6 (8%)

18 (78%) 0 (0%) 2 (9%) 16 (22%) 1 (4%)

50 (70%)

Hong Kong 0 (0%) 2 (22%) 1 (11%) 6 (67%)

Korea 1 (11%) 0 (0%) 2 (22%) 6 (67%)

Here it should be pointed out that the figures above correspond to the end results of the interview on this task. So “accepted the claim” means a teacher accepted the claim at the end of the interview; “not sure” means the teacher was still unsure whether the statement is true or not at the end of the interview; “explore using problematic strategy” means some exploration activities were conducted, but the end result was incorrect; and “explore using correct strategy” means some exploration which ended with the correct conclusion. The last category does not imply that a systematic and mathematically sound approach was used in the exploration. Actually, for the Hong Kong and Korean teachers, usually a correct result was obtained after some unsystematic trial-and-error type of exploration. For example, below is an excerpt of the interview with a Hong Kong teacher: I:

How would you react to the student’s finding?

T3: I would say: “yes, it works for your diagram, but is it always true? Have you tried more examples to see whether it works or not? What will happen if you change it to another diagram? Does it only hold true for squares and rectangles? Have you tried other diagrams? I haven’t thought about it, let’s explore it together.” At this point, one might expect that this Hong Kong teacher will help students explore the mathematics behind this task systematically, as many of the Shanghai teachers in Ma’s study did. So the interviewer continued to probe into what the teacher’s next step would be: I: If you have some time to explore the problem with the student, how would you explore it?

T3: Frankly speaking, I have no idea how to explore it. I’ll try (more examples) with the student, try different diagrams. May be first change the sides (of the rectangles), then change for another diagram, and then change the length and width, to see what will happen (to the area). We will explore together, but I have no idea what directions to take. (And after trying some diagrams, she by chance found a counter-example.) So we can see that although exploration was mentioned by the teacher, she actually had no idea how to go about exploring the mathematics with students. In sum, we can see that most Hong Kong and Korean teachers had a good grasp of the underlying concepts of elementary mathematics, and they were proficient in the calculation procedures. But they were weak in their ability to guide students in genuine mathematical investigations, and their reported teaching was very procedural.

6. Discussion One of the most important factors that emerges from the two studies is that of the teacher. It seems that East Asian students are taught by teachers who are relatively competent in the subject matter, albeit using rather traditional teaching method involving mostly procedural teaching and a lot of teacher talk. An obvious question that arises is that if these East Asian teachers are competent in the subject matter, why are they teaching in a procedural rather than a conceptual manner? Is it because they are unable to access their conceptual knowledge in their teaching or is it a matter of choice? 6.1. East Asian Pragmaticism One explanation of the procedural teaching of East Asian teachers can be offered by the pragmatic mentality in the East Asian culture. Within the classroom, this is manifested in the so-called Topaze Effect (Brousseau, 1997), where teachers may feel a kind of social contract in the role of a teacher and consider it a duty to efficiently deliver the content in a given time period. Also, from the interviews of the replication study, it is found that some of the East Asian teachers believed that for elementary school, there is no need to teach in a conceptually rich manner. It would be inefficient or even confusing for elementary school children to be exposed to rich concepts instead of clear and simple procedures. As one Korean teacher explained using the example of multiplication of fractions: Teacher T4: When I teach multiplication of three fractions in the sixth grade, I usually emphasize that multiplication should be done from left to right. For example, when I teach 4/7×5/3×1/2 , I always ask my students to multiply 4/7×5/3 first. The associative law holds for multiplication, so in fact there is no need to highlight this fact. You can say that I have taught a wrong method to students. However, I stress that fact because mentioning the order is helpful when students perform mixed operations with multiplication and division. Consider the operation 4/7÷5/3×1/2. Here, the order is

important. If students do the multiplication first, they will get the wrong answer. I ask students to follow the rule of order which is not strictly correct mathematically, because I consider the operation which will be dealt with in the next chapter So in their East Asian pragmaticism, these teachers choose to teach in a procedural manner which they think will work better as far as students’ performance in examinations is concerned. 6.2. Repetitive Learning If East Asian students are exposed to such procedural teaching, why have they been doing so well in international studies? This is not an easy question to answer, but some preliminary thoughts will be offered below. Recent literature on the East Asian learner seems to suggest that the so-called “procedural teaching” in the East Asian classroom does not necessarily imply rote learning or learning without understanding. Actually, understanding is “not a yes or no matter, but a continuous process or a continuum” (Leung, 2001). The process of learning often starts with gaining competence in the procedure, and then through repeated practice, students gradually gain understanding. There is however a pre-requisite to this argument. The procedures to be practiced must be embedded in a well-designed curriculum based on sound concepts. Given a curriculum with a set of practicing exercises that vary systematically, repeated practice becomes an important “route to understanding” (Hess & Azuma, 1991, quoted in Biggs, 1996). In the East Asian culture, repetitive learning is not rote learning, but “continuous practice with increasing variation” (Marton, 1997). 6.3. The Competence Cycle How did the East Asian teachers develop their competence? Ma (1999) found that the Shanghai teachers developed their mathematics competence when they were school students while their pedagogical competence was mainly acquired during their teaching career. In the replication study, however, the teachers in the sample thought that both their mathematics and pedagogical competence were mainly acquired while they were in school. Thus, quality teaching seems to be “inherited”. East Asian students, taught by competent teachers, acquire competence in mathematics. When they graduate and join the teacher force, they in turn become competent teachers, and once a good cycle starts, the positive effects cumulate and increasingly reproduce themselves. Unfortunately, this also holds true for a vicious cycle as well. How did the good cycle start in the first place? Research thus far is not able to give a clear answer to this question. But one possible explanation may be the cultural values underlying these East Asian countries. It should be noted that the East Asian countries we have been referring to all share a common culture – that of Confucianism, referred to by some scholars (Biggs, 1996) as the Confucian Heritage Culture (CHC). The values under CHC that are of relevant to this discussion include:

• • • • •

a strong emphasis on the importance of education high expectation for students to achieve attribution of achievement more to effort than to innate ability a serious attitude towards study the ideal of a scholar teacher

These values are also reinforced by a long and strong tradition of publication examination, which acts as a further source of motivation for learning. 6.4. The Scholar-Teacher Of the values mentioned above, the most important seems to be that of the ideal of a scholar teacher. In the traditional Chinese culture, the image of the teacher is that of an expert or a learned figure in the subject matter. Skills in teaching are of course also important, but no teacher will be respected if he is not an expert in the area that he teaches. This image of the scholar-teacher may provide incentives for East Asian teachers to strive to attain competence in the subject matter and in teaching. And probably it is this prevailing cultural value of the emphasis on education and the scholarteacher that starts and keeps the good cycle of competent teachers-competent students in East Asia. The discussion above points to the potential of cultural factors as the underlying reasons for classroom practice and student achievement in East Asia. The cultural factors discussed touch upon fundamental values such as the value of education, the nature of teaching and learning, and the understanding on the role of the teacher. These may be the key to understanding classroom practice and student competence in East Asia. 6.5. What can be learned from the East Asian classroom? If the East Asian instructional practices and the resulting high achievement of East Asian students are so much related to the underlying culture, there are clear implications as far as educational policies are concerned. First, simple transplant of educational policies from high achieving countries to low achieving ones would not work. Since the teachers and their teaching are so much influenced by the underlying cultural value of the place, one cannot transplant the practice without transplanting the culture as well. But is the culture itself transplantable? Culture by definition evolves slowly and stably with the passage of long periods of time, and there is simply no quick transformation of culture. What we can learn from another culture through comparative studies is to identify not only the superficial difference in educational practice, but the intricate relationship between the educational practice and the underlying culture of other countries. Through studying these relationships in different cultures, we may then begin to understand the interaction between educational practices and culture, and through identifying the commonality and differences of both the educational practices and the underlying cultures, we may then determine how much can or cannot be borrowed from another culture.

7. Further Research Because of the limitation of what can be included in one research paper, this paper merely relies on two particular studies of the East Asian classroom, and it happens that the classrooms are confined to the subject of mathematics only. Further research on a wider scope is therefore needed, especially on whether what was found for the mathematics classroom of the two studies discussed in this paper applies for the science classroom as well. However, as pointed out above, further research should not focus only on student achievement and instructional practice. Effort should be put in trying to relate the educational practice with the underlying cultural value. Without a full appreciation of the influence of culture on the educational practice, all attempts to try to apply the results of comparative studies to improve educational policies may prove to be futile.

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Appendix Figure 1. TIMSS-R results: Mathematics

Figure 2. TIMSS-R results: Science

Figure 3. PISA results: Mathematics 24

Figure 4. PISA results: Science