Apples and Oranges: A Discussion of the Use of International Comparisons in Mathematics Education Reform

Introduction

In 1983, A Nation at Risk (National Commission on Excellence in Education, 1983) was released, heralding in a new wave of reform in mathematics education. This report, commissioned by the Secretary of Education at the time, T.H. Bell, had as its objective to review the quality of education in the United States and make recommendations based on its findings. A Nation at Risk was an alarmist document, including the now famous statement:

"If an unfriendly power had attempted to impose on America the mediocre educational performance that exists today, we might well have viewed it as an act of war. As it stands, we have allowed this to happen to ourselves. We have even squandered the gains in achievement made in the wake of the Sputnik challenge. Moreover, we have dismantled essential support systems which helped make those gains possible. We have, in effect, been committing an act of unthinking, unilateral educational disarmament." (National Commission on Excellence in Education, 1983, p.5)

One of the first indicators of risk that this report cites is the poor performance of American students on international comparisions of student achievement a decade before. While the report doesn't specify what comparisons it is referring to, the First International Mathematics Study (FIMS) and the First International Science Study were carried out during the sixties, and many of the results were published in the early seventies, about a decade before A Nation at Risk (Medrich & Griffith, 1995). It might be safe to assume, therefore that these international comparisons of math and science achievement, which showed American students performing comparatively poorly, were fuel for the report's fire.

In the 1981 school year, the Second International Mathematics Study (SIMS) was carried out. As with the first study, American students faired poorly. In both the FIMS and the SIMS, the US mean score was within the lower half of all participating countries (Medrich & Griffith, 1995). These results helped to further catalyze reform efforts in mathematics education. One of the results was the first version of the National Council of Teachers of Mathematics (NCTM) standards documents, published from 1989 through 1995 in three separate volumes. Following much public reaction and critique, the second version was published in 2000 as Principles and Standards for School Mathematics (NCTM, 2000).

In 1995 the Third International Mathematics and Science study was completed. The US again came out low in the rankings, again fueling heated reactions in the world of math reform. In 2001, the Mathematics Achievement Partnership under Achieve Inc., released a draft version of standards for middle school mathematics, Foundations for Success: Mathematics for the Middle Grades (Mathematics Achievement Partnership, 2001). While these two approaches to developing standards for mathematics education in the US differ in some ways, they are both part of the reform efforts which have been motivated to a significant degree by international comparisons of mathematics achievement.

NCTM Standards

The NCTM Principles and Standards document does not directly discuss the influence that international comparisons, such as the FIMS-TIMMS series, had on its develoment. When discussing the need for standards, however, the document makes mention of studies which indicate that American students are faltering in their mathematics achievement, specifically citing TIMSS. In the preface to the Principles and Standards document, the authors mention that research concerning the teaching and learning of mathematics was collected through commissioned papers and a research conference sponsored by the National Science Foundation. The research used is slated to appear in a research companion volume (NCTM, 2000). At the same time, the NCTM's Research Advisory Committee has been made responsible for collecting and making the research papers available through a research website. While this site is not yet fully developed, it is interesting to note that one of only a handful of links on this developing site is to a summary of research on mathematics education in Japan (see http://www.nctm.org/about/committees/rac/). What is significant here is that, while not directly spelled out in the standards documents, there can be little doubt that the international comparions in mathematics teaching and student achievement, as well as related research that was sparked by these comparisons, have influenced the formulation of the NCTM standards.

Achieve Standards

The connection between the Achieve standards and international comparisons is quite clear and directly mentioned in the document itself. In the introduction, the authors discuss TIMSS and state that it presents "a sobering picture of how U.S. students perform when compared with their peers around the world" (Mathematics Achievement Partnership, 2001). They mention that Achieve undertook its own analysis of TIMSS data, reaffirming that the concepts that US students encounter in eighth-grade is not on par with what other nations are doing. Repeatedly throughout these introductory sections, the document reiterates that, while the standards they are presenting might seem daunting, other countries (presumable particularly those who scored highest, though the document doesn't mention any countries specifically) succeed in reaching those standards. It is surely not coincidental that one of Achieve's Senior Associates, William Schmidt, was the first author of A Splintered Vision (Schmidt, W., C. McKnight, et al., 1997), a report which analyzed data concerning curricula of TIMSS participating countries, which was collected in conjunction with the study. In sum, the Achieve standards are very directly based on international comparisons, particularly on analyses of TIMSS.

About This Discussion

Given that so much of the recent math reforms have been motivated by international comparisons, and that some of the standards that are being developed come straight out of those comparisons, it is important to really examine what these comparisons consist of and to consider what they can, and cannot, tell us about mathematics education in the US. This paper will endeavor to contribute to that process by considering some of the literature of international comparisons, looking at how these international comparisons have been drawn upon for these two standards documents, and thinking about what this could mean for US mathematics education. Comparitive studies that have been undertaken on three countries in particular will be the focus of this exploration, because these countries came in at the top of the TIMSS study, and are now often referred to as the top achievers in mathematics: Japan, Singapore, and South Korea.

Because a great amount of emphasis in both standards documents, but particularly in the Achieve standards, is placed on graded curriculum, this exploration will begin with what international comparisons have said about curriculum. The main question to be raised here is, do these top performing countries really follow the challenging curriculum that Achieve claims they do? After looking at these curricular issues, I will then move on to look at other factors that may influence students' mathematics achievement. The question I will be asking then is, what other reason for the top performance of these countries have been identified? Finally, I will look more in depth at one area which has been identified as extremely important in a number of these international comparative studies, charactersitics and preparation of teachers, asking, what difference does the teacher make? The final discussion will tie these questions back to the standards documents to consider how the international comparisons are being used and whether this use is appopriate.

Curriculum and Textbooks

The standards developed by Achieve rest almost entirely on the notion that curricula in the countries that are outperforming the US cover more complicated material earlier. In order to discuss this orientation in an informed way, it is necessary to consider these curricula and associated textbooks. A number of studies have specifically undertaken international comparisons of these items, while larger comparisons have included curricula and textbooks as one aspect. It will be informative to look at what these studies have found and, in particular, whether the top performing countries really follow the challenging curriculum that Achieve claims they do.

One well known comparative study of curricula that came out of the TIMSS, and which has already been mentioned here, is A Splintered Vision (Schmidt, W., C. McKnight, et al., 1997). This study examined the data that was collected from textbooks and curriculum guides, focusing on the US curriculum. Examining official curricula (state curricula in the US, often national curricula in other countries), the authors report that on average US curricula included more topics through middle school than 50-75 percent of the other countries. They found that topics were added at a similar rate, but that no topics were dropped, making the length of time each topic remains in the curriculum longer than in most other countries. The conclusion drawn, and now often cited, is that the US curriculum is broad but shallow, that we touch on topics again and again, but never deeply. Other countries, Germany and Japan being most often cited in this study, are able to move to more complicated material earlier, because they approach the curriculum in a narrower, but deeper way. The basic idea is that they are able to gain mastery by concentrating on a smaller number of topics and then moving on. In addition, the performance expectations implied in US curricula and textbooks centers on routine procedures and discrete knowledge, which hampers conceptual understanding.

In The Learning Gap, Harold Stevenson and James Stigler (Stevenson & Stigler, 1992) make some of the same observations made by Schmidt and his colleagues. While the study as a whole reflects numerous aspects of Japanese and American education, their description of curricular and text comparisons adds depth to this discussion. The authors note that there is little agreement in the US on one national curriculum, which starkly contrasts with the highly centralized curriculum of Japan. They note that, in Japan in any particular grade that the same material is expected to be covered on roughly the same day throughout the country. The contrast is obvious, when US teachers may not even be covering the same topic as the teacher in the next room.

The authors also discuss the broad coverage in American textbooks (Stevenson & Stigler, 1992). Japanese textbooks are a lot smaller than American books, with and quite a bit more plain. The authors surmise that part of the reason for the difference in textbook coverage relates to the curricular demands. Because the US curriculum is so broad, teachers cannot hope to cover all the topics -- they have to make do, part of a practice Schmidt et al refer to as "satisficing" (Schmidt, W., C. McKnight, et al., 1997, p. 78) . Because teacher's have ultimate control over which topics are covered, there is no way to know what topics are being covered from year to year. Thus, textbooks have no choice but to continue wide coverage, both to satisfy differing curricula and possibly different teacher choices.

Another significant difference in texts is the detail offered. Japanese texts are less explicit and carry fewer practice and review problems. The discussion and elaboration of the content is left more to the teachers. The American textbooks tend to include more examples of logarithms and present explicit explanations, or perhaps 'directions' is a better term, for topics covered.

Other studies that have looked at curricula in Japan and the United States add more to this picture. Mayer, Sims, and Tajika (Mayer et al, 1995) articulate additional differences in textbooks. Examining three Japanese texts and four American texts, the authors found that the Japanese texts devoted more space to explanations, relevant illustrations, and worked-out examples, while US texts devote more space to unsolved exercises and eye-cathing, but irrelevant, illustrations. They also found that the Japanese texts were more likely to include multiple representations and inductive reasoning. An examination of a Japanese teacher's guide, by Gill and McPike (1995), resonates with all of these descriptions, pointing out that, in the US, the topic of weight is taught in grades one through four, mixed in with length, volume, and other types of measurement. The Japanese texts cover length and volume in depth in grades one and two and then begin weight only in grade three, but devote a significant period of time to it to cover it in depth.

While comparative studies of Korean and Singaporean curricula and textbooks appear to be less common (likely because Japan participated and did well in all three International Mathematics Studies). However, there are indications that they contrast with US curricula and textbooks in some of the same ways as those of Japan. One comparative study by Hy Kim (1993) looked at the topics of measurement and geometry in first through eighth grades in the US and South Korea. As in the Japanese studies Kim notes the repetition of these topics throughout the grades. In addition, he finds that the Korean texts integrate topics more and, significantly, concepts and skills are more closely intertwined. In addition, the Korean texts include more advanced topics than the American texts. The one study I was able to locate concerning Singapore curriculum mentioned that teacher choice was fairly open and that the national curriculum, as developed in 1992, focuses on problem solving skills through "meaningful activities, competence in basic skills, mathematical communication through oral work, group discussion and presentation, investigative work and mathematical thinking" (Seng 2000).

The curricula and textbooks of the three countries focused on here, as well as the many others examined in the TIMSS study, clearly have differences between them. However, the comparisons that have been done between them and the US point to some significant issues. The theme of a repetitive American curriculum appears again and again. While such a spiral curriculum does have advantages, theoretically reinforcing students' understanding and allowing for connection of topics, it appears that the practice may not be living up to the theory. More often, it seems, coverage may end up being simply shallow and repetitive because of the wide range of topics to be addressed each year, and the need for teachers to "satisfice". Another noticeable theme is the lack of deep explanation or inductive approaches in US texts, which could help support deep understanding of the mathematical topics addressed. While not all the studies specifically address a comparison between the complexity or level of material covered at one level, there does appear to be an implication, in some cases a direct research finding, that the curricula of countries examined are able to move to more difficult material more quickly. This comes from both more focused curricula which allow for more in depth development of topics and from an orientation towards conceptual understanding rather than procedural knowledge.

Culture and Context

While a number of studies look at the differences in curricula and textbooks and how these may put American students at a disadvantage, a larger body of work examines the many variables that may affect student outcome, curricula and textbooks being only one aspect. The assumption made a number researchers undertaking international comparisons is that it is not possible to look at one aspect of mathematics education, such as the curriculum, in a vaccuum. They believe that many other factors, one might refer to them collectively as culture and context, also contribute. In order to think about the orientatons of Achieve, which rests largely on curriculum and testing, and NCTM, which addresses other factors more completely, it will be useful to examine what else could be contributing to the comparatively poor performance of the US on international achievement tests such as the FIMS-TIMSS series.

Attitudes Toward School

One significant difference reported in a number of studies (Wu, 1999; Beaton et al, 1996; Sorensen, 1994; Stevenson & Stigler, 1992) is that the Asian students (Korean, Singaporean, and Japanese) see school as more central in their lives than do American students, and that their schooling is more strongly supported by their parents. Stevenson and Stigler (1992), who focused on China and Japana, found that, while American elementary and junior secondary students did clock slightly more hours on non-academic extra-curricular activities, the difference was not as great as sometimes assumed. However, they did note that homework was more likely to take precedence over other activities. Television watching might take up as much time as in the US, but only after homework was completed. In China and Japan it is common for parents to purchase workbooks for their children, which are aligned to the national curriculum (Stevenson & Stigler, 1992).

In South Korea and Singapore, authors describe students spending far more time on academics than their American counterparts (Sorensen, 1994; Ellinger & Carlson, 1990; Menon, 2000). In South Korea, Sorensen mentions that students at the middle school level are likely to attend after-school 'cram schools' which reinforce their mathematics and, particularly, give them an opportunity to practice routine procedures (while these schools exist in Japan, Stevenson and Stigler find that they are more often attended by older students). Numerous authors mention that South Korea and Singapore, in particular, are shaped to a large degree by high stakes exams, and that students spend a great deal of time working on exam-oriented skills. Because educational achievement is highly valued by the students and their families, a lot of this time is spent outside of class, so that in-class time is more available for conceptual-oriented work (Sorensen, 1994; Stevenson & Stigler, 1992; Menon, 2000; Wu, 1999).

There are a number of different explanations for why students in the Asian countries mention here seem to see school, and mathematics, as perhaps more central than American students do. Some have claimed that this is due to a Confucianist emphasis on family structure, frugality, hard work, and stress on education (Sorensen, 1994). Others point out that in these countries students can expect to find a fairly consistant link between their level of educational attainment and future economic success (Sorensen, 1994; Meno, 2000). In addition, the high competition coming from high-stakes testing and related tracking may also serve as motivation for students to take their studies more seriously (Ellinger & Carlson, 1990; Menon, 2000). Whatever the explanation, the consistent observation remains that students in the top three TIMSS countries appear to make schooling and mathematics more central in their lives, both inside and outside the classroom, than do American students.

Other Contextual Issues

A number of other factors influence student acheivement, both in and out of school. Some of these school/teacher-related factors will be addressed in the next section. Before moving on to these, however, a compilation of some other factors that could be affecting achievement levels as seen on international comparisons should also be mentioned. For example, we might consider Singapore, which placed at the top of the rankings on TIMSS. Singapore is a relatively small country, with a population of about 4 million (about the size of Atlanta). The population of Singapore is fairly diverse, with about 20% being immigrants and non-residents from East and Southeast Asia (Singapore Department of Statistics). The United States, on the other hand, has a total population of almost 300 million. While a smaller percentage of US residents were foreign born than in Singapore, their countries of origin span the entire globe, with an appreciable proportion coming from every continent (http://www.census.gov/). The difference is obvious and important. Aside from the difficulties in comparing two such different contexts, we must consider the very different issues that must be dealt with in these two education systems. One example of such differences was touched on by Gerald Bracey in a Phi Delta Kappan editorial (Bracey, 1997). Here he points out that in Singapore many of the lowest skilled, lowest paying jobs are taken by Malaysians who commute from nearby Malaysia. As a result, Singapore does not have the responsibility of educating the children of these workers. As Bracey puts it, Singapore is able to "outsource its poverty and its low-achieving students" (p. 656). Ironically, this is almost the opposite of Washington DC's problem.

Similar examples could be developed for South Korea and Japan as compared with the United States, though they would clearly point to different issues Both of these countries are also less diverse than the United States, in terms of geographic ancestry and language at the very least. In addition, all three countries have highly centralized education systems. While one may wish to argue that a more centralized curriculum is exactly what standards such as Achieve and the NCTM documents are moving towards, the reasons for the US decentralized system are important and are highly political, and will not soon change. Fundamental cultural, economic, political, and philosophical differences exist between the United States and the countries with which we are comparing ourselves in terms of mathematics achievement. These differences must imply very different advantages, disadvantages, challenges, and issues to be addressed within the education system. These clearly move well beyond questions of curriculum and textbooks.

The Teacher

One significant factor that influences student achievement, as is recognized by almost all of the researchers looking at international comparisons, is the teacher. Many studies have focused on teachers, or included teachers as one focus, when undertaking comparative analysis. One important observation that can be made in light of these studies is that the teaching profession in the three countries of focus here, Japan, Singapor, and South Korea, cannot be assumed to have the same characteristics. In fact, it will be useful here to consider some of the ways that teachers in these three countries seem to be similar, as well as the ways in which they differ.

Similarities Among the "Top Three"

A number of characteristics seem to be common among teachers, and/or the context of their teaching, in the three countries being examined, many of which appear to reflect the culture in which they are embedded and philosophies concerning education. One significant element that the these three countries appear to share is a strong level of parental support for education and for the teacher. In each of these countries, as mentioned previously, parents encourage students to spend time outside of the classroom on academic work. This appears to be especially salient in South Korea and Singapore, where students are exposed to high stakes testing and competitive pressure throughout their schooling. In both these countries extra-curricular tutoring and 'cram schools' are quite common, all of which come at a high cost that parents are willing to pay (Menon, 2000; Seng, 2000; Sorensen, 1994; Ellinger, 1990).

In all three countries, there is a clear notion of the parents and teachers working together towards the education of their children. Whereas American teachers are often in the position to think of parents as their clients, and there are sometimes adverserial relations between them, Japanese parents show high levels of respect for teachers and are highly unlikely to side with their child against a teacher (Stevenson & Stigler, 1992). In South Korea, this working relationship may be thought of as the teacher and parent working together to help the students through the hurdles of exam (Sorensen, 1994). Regardless of why, there is clearly a strong and positive relationship between the parents and the teachers.

In each of these three countries, there is also strong support within the education system for teachers and for teacher development. Perhaps one of the most important elements of this is that in all three countries, teaching is a highly respected and fairly well paid profession. Teachers in Japan and South Korea have group preparation time built into their schedules. In South Korea, for example, teachers are in school 8 hours per day, but teach only three or four hours. During the remaining time, teachers prepare together with other teachers in common rooms (Grow-Maienza et al., 1996). Similarly, Japanese teachers commonly work together in order to improve their teaching, as well as attending external teacher development activities. In Japan, the practice of lesson study is quite common, where teachers of a common subject come together in order to design, develop, implement, and then evaluate and improve a lesson (Stigler & Hiebert, 1999). Teachers in Singapore are also expected to attend in-service trainings, for which they are not paid (Menon, 2000).

Teaching preparation is also somewhat similar, not only among these three countries, but also with the US. In Japan, teachers at the elementary-school level complete four years of university, including a short student-teaching stint and a challenging qualifying exam. South Korean and Singaporean teachers must also complete four years of university, though they must take more mathematics credits (Wagner, 1999; Menon, 2000; Shimahara & Sakai, 1995) . There might be differences among these countries and between them and the US concerning specific course material required, but I was unable to find pertitent information. Perhaps the significant difference between these three countries on the one hand, and the US on the other, is that in both Japan and Singapore, beginning teachers are considered novices, and in Japan they are assigned a mentor teacher for their first year teaching. In both these countries they are also required to attend a fairly substantial amount of in-service training during their first few years teaching, guided by their mentors and other master teachers (Stevenson & Stigler, 1992; Shimahara & Sakai, 1995; Kinney, 1998).

Difference Between the "Top Three"

The main, and very important, area of difference among these three countries is teaching style. The approach used in Japanese classrooms has become quite well known in mathematics reform in the US. This approach includes open questions and classes centered around concrete problems. Students often work in groups and are encouraged to develop alternative strategies. Japanese teachers are oriented more towards thinking of the class of the whole rather than a collection of individuals, and students are encouraged to learn from each other (Stigler and Hiebert, 1999; Shimahara & Sakai, 1995; Stevenson & Stigler, 1992; Wu, 1999).

The picture drawn of teaching in South Korea, however, is very different. Researchers examining teaching in South Korea found that teaching there is highly teacher-centered, relying greatly on choral recitation, whole class teaching and lecturing, and memorization. Teachers typically use an Instruction/Practice/Evaluation structure in their classes, using real-world problems, and numerous modes for the presentation of these problems. The teacher leads the class through procedures using guiding questions in a systematic way and, during the evaluation phase, prompts students to help evaluate each others' work (Grow-Maienza, 1996).

While I was unable to find conclusive research depicting teaching in Singapore in detail, at least one author claims that teaching is highly memorization-oriented (Menon, 2000). Another author has written that instruction in this country has become highly oriented towards problem-solving (Seng, 2000). In either case, we might imagine that teaching in Singapore is just as unique to that country as teaching in the US, Japan, and South Korea are to those countries.

While the differences in teaching style are apparent, there is also an important common element among the three East Asian countries considered here. Researchers discussing teaching in these countries discuss the coherence of lessons as a significant element. There is a greater sense of connections throughout the lesson, as well as between conceptual understanding and skill. In addition, outside distractions are much less common (Seng, 2000; Grow-Maienza, 1996; Stevenson & Stigler, 1992; Stigler & Hiebert, 1999). Thus, while we cannot identify one particular teaching approach in all of these three countries to identify as the source of high achievement, we should take note of this common element: lesson coherence.

Discussion

Having looked at a number of studies involving international comparisons, there are some general observations we can make before looking back at the NCTM and Achieve standards, the point at which we started. One very clear point to note is that their are a wide range of variables addressed in the studies, only some of which were examined here. Researchers have identified a number different factors that may contribute to student achievement. There might be individual factors that have greater influence, or the factors may work in combination -- there is no way to know for sure. What we can say with certainty is that we cannot identify one particular factor and claim that it is entirely responsible for student achievement.

Another interesting observation that can be made from reviewing this literature is that the presentation and discussions of findings appear to be heavily influenced by the preconceptions or agenda of the authors/researchers. Different authors writing about the same institutions may draw very different conclusions depending on the lense through which they are interpreting it. Thus, some authors speak very positively about elements of education in Singapore (Seng, 2000), while others undertake a rather negative critique (Menon, 2000). These differences seem to have some connection to the authors' views concerning a particular education system or element therein as a possible model for the United States to emulate.

The most important lesson to be learned from a review of research literature such as this is that the factors contributing to student achievement are, indeed, complex, and that each element is deeply embedded within a larger cultural, social, political, and economic context. We cannot simply lift a piece of an education from one system and transfer it to another. Or, if we try to do that, we cannot assume that it will have the effect that we had hoped, because it will now be placed in an entirely different context.

NCTM and Achieve Standards

Having considered some of the literature on international comparisons, we can now look again at the NCTM and Achieve Standards. Beginning with NCTM, it seems fair to say that they are not advocating that we take any one particular aspect of another country's education system and implement it in the United States. Their standards have been informed by thinking about what the international assessments tell us about our own students (such as that there is a large achievement gap) and what insights math education in other countries may give us. The standards themselves, however, appear to be home-grown. The descriptions of the need for standards, how the standards were conceived, and how they were developmed indicate that they respond to the needs of students in the United States and come from ideas about how best to address these needs within this context. They include philosophical positions and recommendations concerning a broad set of topics related to mathematics education, set quite distinctly in the United States (NCTM, 2000). In short, NCTM appears to have put great consideration into the context and to recognize the complex and important interplay of a multitude of factors which can contribute to student achievement.

The Achieve standards, on the other hand, rest largely on the international comparisons of curricula and textbooks. While they touch on teacher training, this area does not receive nearly as much emphasis as curriculum, testing, and comparing national and international standards or benchmarks. This standards document makes very clear the fact that it is based on curricula of other countries, what standards students in other countries are able to reach. There is no discussion of other factors in those countries that might work to support such standards (Mathematics Achievement Partnership, 2001). It seems as though Achieve has attempted to identify an element of mathematics education in other countires which they feel contributes most greatly to student achievement and to transfer it to the US. There does not seem to be much consideration of the context in which it was embedded or into which it will be transferred.

In closing, while the mathematics standards Achieve is recommending might very well be attainable, we are not told how to get there. The developers of these standards appear to have simply transferred these standards from countries which they have identified as doing well in mathematics education. NCTM, on the other hand, seems to have developed standards based on the needs of US students and embedded in the American context. As such, the NCTM standards appear to be built on a stronger foundation.

References

Beaton, A., I. Mullis, et al. (1996). Mathematics Achievement in the Middle School Years. IEA's Third International Mathematics and Science Study (TIMSS). Chestnut Hill, MA: International Association for the Evaluation of Educational Achivement; TIMSS International Study Center. (ERIC Document Reproduction Service No. ED406419)

Bracey, G. (1997). "More on TIMSS." Phi Delta Kappan 78(8), 656-657.

Ellinger, T. & D. Carlson (1990). "Education in Korea: doing well and feeling bad." Foreign Service Journal 67(6), 16-18.

Gill, A. and McPike, L. (1995). "What we can learn from Japanese teachers' manuels." American Federation of Teachers 19(1), 14-15.

Grow-Maienza, J. (1996). Mathematics Instruction in Korean Primary Schools: An International Collaborative Study. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago, IL, March 26, 1996. (ERIC Document Reproduction Service No. ED420490).

Kim, H. (1993). "A comparative study between an American and a Republic of Korean textbook series' coverage of measurement and geometry content in first through eighth grades." School Science and Mathematics 93(3), 123-126.

Kinney, C. (1998). "Building an excellent teacher corps: How Japan does it." American Educator 21(4), 16-23.

Mayer, R., V. Sims, et al. (1995). "A comparison of how textbooks teach mathematical problem solving in Japan and the United States." American Educational Research Journal 32(2), 443-460.

Medrich, E. and J. Griffith (1992). International Mathematics and Science Assessment: What Have We Learned? Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement. (Report No. CNES 92-011). Retrieved December 13, 2001 from http://nces.ed.gov/spider/webspider/92011.shtml.

Menon, R. (2000). "Singapore's education system." Mathematics Teaching in the Middle School 5(6), 345-347.

Mathematics Achievement Partnership (2001). Foundations for Success: Mathematics for the Middle Grades. Consultation Draft.

National Commission on Excellence in Education (1983). A Nation at Risk. US Department of Education. 2001. Retrieved December 10, 2001 from http://www.ed.gov/pubs/NatAtRisk/.

NCTM (2000). Principles and Standards for School Mathematics. Reston, VA, National Council of Teachers of Mathematics.

Schmidt, W., C. McKnight, et al. (1997). A Splintered Vision. Boston, MA: Kluwer Academic Publishers.

Seng, S. (2000). Teaching and Learning Primary Mathematics in Singapore. Paper presented at theAnnual International Conference and Exhibition of the Association for Childhood Education International, Baltimore, MD. (ERIC Document Reproduction Service No. ED439812).

Shimahara, N. and A. Sakai (1995). Learning to Teach in Two Cultures. New York, NY: Garland Publishing, Inc.

Singapore Department of Statistics (2001) Singapore Population. Singapore Dept. of Statistics, May 2001. Retrieved December 15, 2001 from http://www.singstat.gov.sg.

Sorensen, C. (1994). "Success and education in South Korea." Comparative Education Review 38(1): 10-35.

Stevenson, H. and J. Stigler (1992). The Learning Gap: Why Our Schools Are Failing and What We Can Learn From Japanase and Chinese Education. New York, NY: Summit Books.

Stigler, J. and J. Hiebert (1999). The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom. New York, MY: The Free Press.

Wagner, S., H.-J. Lee, et al. (1999). A Comparative Study of the United States, Turkey, and Korea: Attitudes and Beliefs of Preservice Mathematics Teachers Toward Mathematics, Teaching Mathematics, and Their Teacher Preparation Programs. Paper presented at the Annual Meeting of the Association of Mathematics Teacher Educators, Chicago, IL. (ERIC Document Reproduction Service No. ED445907).

Wu, A. (1999). Japanese Education System: A Case Study Summary and Analysis. Research Today. International Studies. Washington, DC, National Institute on Student Achievement, Curriculum, and Assessment. (ERIC Document Reproduction Service No. ED426958).