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

June 20, 2010

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.
Singapore’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.

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.

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.

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.

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. The 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.

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.

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