# What the United States Can Learn From Singapore’s World-Class Mathematics System

Excerpts from American Institutes for Research study found here:

INTRODUCTION

Singaporean students ranked first in the world in mathematics on the Trends in International

Mathematics and Science Study-2003; U.S. students ranked 16th out of 46 participating nations at

grade 8 (Mullis, et al., 2004). Scores for U.S. students were among the lowest of all industrialized

countries. Because it is unreasonable to assume that Singaporean students have mathematical abilities

inherently superior to those of U.S. students, there must be something about the system that

Singapore has developed to teach mathematics that is better than the system we use in the United

States.

This exploratory study compares key features of the Singapore and U.S. mathematics systems

in the primary grades, when students need to build a strong mathematics foundation. It identifies

major differences between the mathematics frameworks, textbooks, assessments, and teachers in

Singapore and the United States. It also presents initial results from four pilot sites that introduced

the Singapore mathematics textbook in place of their regular textbooks.

Analysis of these evidentiary streams finds Singaporean students more successful in

mathematics than their U.S. counterparts because Singapore has a world-class mathematics system

with quality components aligned to produce students who learn mathematics to mastery. These

components include Singapore’s highly logical national mathematics framework, mathematically

rich problem-based textbooks, challenging mathematics assessments, and highly qualified

mathematics teachers whose pedagogy centers on teaching to mastery. Singapore also provides its

mathematically slower students with an alternative framework and special assistance from an expert

teacher.

The U.S. mathematics system does not have similar features. It lacks a centrally identified

core of mathematical content that provides a focus for the rest of the system. Its traditional textbooks

emphasize definitions and formulas, not mathematical understanding; its assessments are not

especially challenging; and too many U.S. teachers lack sound mathematics preparation. At-risk

students often receive special assistance from a teacher’s aide who lacks a college degree. As a

result, the United States produces students who have learned only to mechanically apply

mathematical procedures to solve routine problems and who are, therefore, not mathematically

competitive with students in most other industrialized countries.

The U.S. mathematics system has some features that are an improvement on Singapore’s

system, notably an emphasis on 21st century thinking skills, such as reasoning and communications,

and a focus on applied mathematics. **However, if U.S. students are to become successful in these areas, they must begin with a strong foundation in core mathematics concepts and skills, which, by
international standards, they presently lack.**

EXPLORATORY METHODOLOGY

Carrying out in-depth analyses on systems as different as those in Singapore and the United

States poses serious methodological challenges. Singapore has a centralized mathematics system,

with detailed and consistent implementation procedures. This makes analysis of the separate

components of their system relatively straightforward. Characterizing the decentralized U.S.

mathematics system, in contrast, is difficult. We elected to rigorously study the components of the

U.S. system by selecting typical examples from the wide variety available in each component area:

• Standards: The United States has no national standards, but many states’ standards use

the National Council of Teachers of Mathematics (NCTM) framework as a model. We

used the NCTM standards in our analyses as a proxy for states that use a grade-band (e.g.,

K–2, 3–5), rather than a grade-by-grade structure in their standards. However, because

many states are currently shifting to a grade-by-grade structure in response to NCLB, we

supplemented our analysis by also examining standards from seven states (Exhibit A) that

organize content grade by grade. These states are home to approximately one-third of all

U.S. students.

• Textbooks: We limited our analysis to one traditional and one nontraditional U.S.

mathematics textbook.

• Assessments: We used sample assessment items from the federally supported National

Assessment of Educational Progress (NAEP) and from assessments from the same seven

states whose standards we examined in our comparative analysis.

• Teachers: For analyses of teacher quality in the United States, we drew from national

surveys on teacher education and from teacher preparation standards. We also examined

sample problems from teacher licensing exams.

Exhibit A. The Average Number of Topics per Grade in Selected U.S.

State Mathematics Frameworks Compared With Singapore’s

**Avg. No. of Topics per Grade
Singapore 15
California 20
Florida 39
Maryland 29
New Jersey 28
N. Carolina 18
Ohio 26
Texas 19 **

PREFERRED FEATURES OF THE SINGAPORE MATHEMATICS SYSTEMS

Our key findings show the advantages conferred by components of Singapore’s mathematics

system in comparison to similar components in the U.S. system.

Frameworks

A mathematically logical, uniform national framework that develops topics in-depth at

each grade guides Singapore’s mathematics system. The U.S. system, in contrast, has no official

national framework. State frameworks differ greatly; some resemble Singapore’s, whereas

others lack Singapore’s content focus.

Singapore’s framework, shown in Exhibit B, lays out a balanced set of mathematical

priorities centered on problem solving. It includes an emphasis on computational skills along with

more conceptual and strategic thinking processes. The framework covers a relatively small number

of topics in-depth and carefully sequenced grade-by-grade, following a spiral organization in which

topics presented at one grade are covered in later grades, but only at a more advanced level. Students

are expected to have mastered prior content, not repeat it.

The NCTM framework, while emphasizing higher order, 21st century skills in a visionary

way, lacks the logical mathematical structure of Singapore’s framework. It identifies content only

within broad grade bands (e.g., K–2, 3–5) and only in general terms, thus providing inadequate

content guidance to educators.

The seven state frameworks we examined exhibit varying degrees of focus, although none is as focused as Singapore’s. Exhibit A shows that three of the states, California, North Carolina, and Texas, have frameworks that are similar to Singapore’s, within 30 percent, in the average number of topics covered per grade. Two of these states, North Carolina and Texas, were praised in the 1990s as states where education reform had been particularly successful. Both states’ NAEP mathematics scores improved significantly. The similarity between these states and Singapore suggests a correlation between focused frameworks and good test performance.

By contrast, the frameworks of Florida, Maryland, New Jersey, and Ohio exceeded

Singapore’s average numbers of topics per grade by 70 to 160 percent. **If Singapore’s excellent test
performance is evidence that its curriculum exposes students to about the right number of topics per grade, then these states’ test performance suggests they cover too many topics and should reduce breadth of coverage and deepen topic instruction.**

Singapore recognizes that some students may have more difficulty in mathematics and

provides them with an alternative framework; the U.S. frameworks make no such provisions.

**S****ingapore’s alternative mathematics framework for lower performing students covers all the**

mathematics topics in the regular framework, but at a slower pace and with greater repetition.mathematics topics in the regular framework, but at a slower pace and with greater repetition.

Singapore also provides its slower students with extra help from well-trained teachers. NCTM and

the states we examined provide no alternative framework for slower mathematics students.

Moreover, such students are often unofficially tracked into slower mathematics courses, but unlike in

Singapore, these students are seldom taught all the required mathematics material. Evaluations have

shown that they frequently receive their extra help from teacher’s aides who lack college degrees.

Textbooks

Singapore’s textbooks build deep understanding of mathematical concepts through

multistep problems and concrete illustrations that demonstrate how abstract mathematical

concepts are used to solve problems from different perspectives. Traditional U.S. textbooks

rarely get beyond definitions and formulas, developing only students’ mechanical ability to

apply mathematical concepts.

There is a clear difference in how Singapore and traditional U.S. textbooks develop

mathematical concepts. The Singapore texts are rich with problem-based development in contrast to

traditional U.S. texts that rarely get much beyond exposing students to the mechanics of mathematics

and emphasizing the application of definitions and formulas to routine problems. **While such books
are filled with real-world illustrations, these seem to serve mainly to show students that mathematics concepts have real-world representations. The illustrations make virtually no contribution to helping students understand how to use the mathematics to solve real-world problems.**

The Singapore illustrations also feature a concrete to pictorial to abstract approach. Many

students who have difficulty grasping abstract mathematical concepts would benefit from visual

representations of mathematical ideas. As part of this approach, the Singapore illustrations

demonstrate how to graphically decompose, represent, and solve complicated multistep problems.

Another hindrance to the development of U.S. students’ mathematical understanding is the

U.S. texts’ lack of focus. Singapore’s textbooks follow its mathematically logical national

framework, but U.S. textbooks must serve multiple state markets. To do so, they find it necessary to

cover almost twice as many topics per grade so that all topics from many states’ frameworks can be

covered. Consequently, individual topic coverage in U.S. textbooks is much shorter and less comprehensive than what is found in Singaporean texts. **In fact, Singapore students are expected to
complete about one thorough lesson focused on a single topic per week, while U.S. students are
expected to complete about one lesson on a narrowly focused topic each day.**

**Finally, both Singaporean textbooks and U.S. textbooks “spiral” mathematical content –**

returning in successive years to the same concepts. However, while the spiral in U.S. textbooks

includes significant repetition and reteaching of the same content in two or three consecutive grades, the Singapore textbooks assume that what was previously taught was learned. In other words, Singapore textbooks do not repeat earlier-taught content, because students are taught to mastery the first time around.returning in successive years to the same concepts. However, while the spiral in U.S. textbooks

includes significant repetition and reteaching of the same content in two or three consecutive grades, the Singapore textbooks assume that what was previously taught was learned. In other words, Singapore textbooks do not repeat earlier-taught content, because students are taught to mastery the first time around.

Assessments

**The questions on Singapore’s high-stakes grade 6 Primary School Leaving Examination**

(PSLE) are more challenging than the released items on the U.S. grade 8 National Assessment

of Education Progress (NAEP) and the items on the grade 8 state assessments.(PSLE) are more challenging than the released items on the U.S. grade 8 National Assessment

of Education Progress (NAEP) and the items on the grade 8 state assessments.

Singapore’s grade 6 assessment contains almost double the percentage of constructed response

items as the U.S. grade 8 NAEP and a much higher proportion than that of state

assessments. This is an important difference because constructed-response questions generally are

more suitable for demonstrating students’ higher-level cognitive process in mathematics.

Overall, Singapore’s grade 6 assessment also contains a much greater percentage of items

that could be characterized as more difficult than either the U.S. 8th grade NAEP or any of the state

assessments we examined. Singapore’s most challenging questions are designed to help

Singapore identify the best students. These are more difficult than the most challenging questions on

the state grade 8 assessments as well as on NAEP.

Teachers

Singaporean elementary school teachers are required to demonstrate mathematics skills

superior to those of their U.S. counterparts before they begin teacher training. At every phase

of pre- and post-service training, they receive better instruction both in mathematics content

and in mathematics pedagogy.

Singapore’s teachers must take a stringent examination before being accepted to education

school, and while they are students, they are paid a teacher’s salary.

**By contrast, the SAT**

mathematics scores of entering U.S. elementary education majors are among the lowest of all college students.mathematics scores of entering U.S. elementary education majors are among the lowest of all college students.

After content-driven pre-service preparation, Singaporean teachers are encouraged to

continue to improve their knowledge and skills through 100 hours of required annual professional

training. U.S. education majors, in contrast, take fewer formal mathematics courses than the average

college graduate.

**T****he major U.S. teacher screening and licensing exams, the PRAXIS I and II,**

consist only of multiple-choice questions that, based on released items, appear far easier than items

from the exam that Singapore gives to 6th graders.consist only of multiple-choice questions that, based on released items, appear far easier than items

from the exam that Singapore gives to 6th graders.

AREAS OF STRENGTHS IN THE U.S. MATHEMATICS SYSTEM

COMPARED WITH SINGAPORE’S SYSTEM

Although the U.S. mathematics program is weaker than Singapore’s in most respects, the

U.S. system is stronger than Singapore’s in some areas.

The U.S. frameworks give greater emphasis than Singapore’s framework does to

developing important 21st century mathematical skills such as representation, reasoning,

making connections, and communication. However, to develop these skills in students, the U.S. frameworks need to do a better job of integrating them with rigorous mathematics content.

The U.S. places a greater emphasis on applied mathematics, including statistics,

probability, and real-world problem analysis.

The U.S. mathematics frameworks stress data analysis and probability, whereas the

Singapore framework treats statistics in a strictly theoretical way. Everyday Mathematics, the

nontraditional textbook we examined, uses a problem-based learning approach, which presents

multistep real-world mathematics problems. Such applications give students practice in

understanding how to apply mathematics in practical ways.

**However, the Everyday Mathematics**** lessons use real-world applications without providing the foundation of the strong conceptual topic development found in Singapore’s textbooks.** Even though Singapore’s textbooks would benefit from more real-world applications, their emphasis on conceptual development of mathematics and problem-based learning make them superior to U.S. textbooks overall.

**CONCLUSION**

Reform Options

Each component of Singapore’s educational system is designed to enhance the mathematical proficiency of students and their teachers. If the United States is to reform its mathematics system so that it more closely resembles Singapore’s successful system, the country needs to consider several options for improving each of the components of the system. The options are organized by how much change from current practice would be required and, hence, by how difficult it would be to gain

political acceptance for them.

Tinkering Options: Improve or extend existing reforms. States could revise their frameworks

to better match Singapore’s content grade by grade and strengthen implementation of NCLB reforms

for highly qualified teachers to ensure that teachers who meet the NCLB standards actually

demonstrate that they understand mathematics content and how to teach it. The federal government

could work with the states to produced a national bank of mathematics test items to encourage

greater comparability across the states.

Leveraging Options: Use market leverage to bring about improvement. Professional

organizations could develop an independent and objective textbook rating system that assesses the

depth of mathematics content in textbooks, much as the American Association for the Advancement

of Science has already piloted in the sciences.

Program Strengthening Options: Stay within the current U.S. education structure but

substantially strengthen the mathematical depth and rigor of the current components of the U.S.

mathematical system. U.S. textbooks could be reorganized so that they closely conform to the logical

topic organization, rich problem-based approach, and varied pictorial representations of

mathematical concepts found in Singaporean texts. Eighth-grade student assessments and teacherlicensing

exams could be strengthened so that, at a minimum, they are at least as challenging as

Singapore’s grade 6 student assessment.

Systemic Reform Options: Strengthen features of the U.S. mathematics system so that it more

closely resembles Singapore’s integrated, national mathematics system. Such steps might include

introducing a national mathematics framework, a national mathematics assessment, and value-added

accountability measures of school performance.

Further Validation of Exploratory Findings

Our exploratory results have identified key differences between the U.S. and Singapore

mathematics systems. These differences suggest potentially significant reforms that could improve

the U.S. mathematics system, but these findings require further validation from larger, more scientific studies. The suggested reforms need more thorough analyses and, ideally, small-scale

introduction prior to going to scale. Only through such further study can we build on our exploratory

findings to assess whether adopting the features that have produced a quality mathematics system for

Singapore would significantly improve the performance of the U.S. mathematics system and better

meet the challenging performance goals set by NCLB.