Yeap Ban Har, Ph.D.
National Institute of Education
Nanyang Technological University
Singapore
banhar.yeap@nie.edu.sg
A multi-racial, multi-cultural, multi-religious and
multi-lingual society. Gained independence in 1965.
Population of 4 million with ½ million students in
about 350 schools, all public.
Presentation Outline
 Singapore Education in a Nutshell
 Singapore Mathematics Curriculum
 National Tests in Singapore
 Mathematics Teaching in Singapore
 Mathematics Teachers & Professional Development
 Achievements of Singapore Students
 Questions & Discussion
Development & Focus
Moulding the Future of the Nation
Thomas L. Friedman 2006
The World Is Flat
Thomas L. Friedman 2006
The World Is Flat
Nurturing Knowledge Workers
through Ability-Driven Education
Beaton, Mullis, Martin, Gonzalez, Kelly & Smith 1996
The Singapore mathematics curriculum
states that “an emphasis on mathematics
education will ensure that we have an
increasingly competitive workforce to
meet the challenges of the 21st century”
Schools around the world are forced to move
forward if they want to make a difference to the
future of their students.
Agenda for Action (USA) and Cockcroft Report
(UK) led to the introduction of a problem-solving
curriculum in 1992.
Singapore Mathematics Curriculum Framework
Beliefs
Interest
Appreciation
Confidence
Perseverance
Numerical calculation
Algebraic manipulation
Spatial visualization
Data analysis
Measurement
Use of mathematical tools
Estimation
Monitoring of one’s own thinking
Self-regulation of learning
Mathematical
Problem
Solving
Concepts
Numerical
Algebraic
Geometrical
Statistical
Probabilistic
Analytical
Reasoning,
communication &
connections
Thinking skills &
heuristics
Application & modelling
This initiative was introduced in 1997. It focuses on
explicit teaching of thinking through school
subjects.
It was revised in 2001 to give more emphasis on a
range of heuristics.
This initiative was introduced in 2003. It focuses on
inculcating good habits of minds amongst
students.
This call was made in 2004 to encourage the
development of strong foundation knowledge.
Sample Test Item
3 cm
5 cm
3 cm
7 cm
Find the area of the figure shown.
3 cm
5 cm
3 cm
7 cm
Find the area of the figure shown.
3 cm
5 cm
3 cm
7 cm
Find the area of the figure shown.
3 cm
5 cm
3 cm
7 cm
Find the area of the figure shown.
This call has been made since 2005. Every child
should be developed to maximize their potential
using different approaches.
Technological advances have compelled educators
to re-consider what is considered significant in
school curriculum.
Kaput, Noss & Hoyles 2002
It was revised in 2007 for this reason.
It places greater emphasis on visualization and
number sense. With the introduction of calculator,
mental strategies are also emphasized.
PSLE: Emphasis on Visualization
The figure below is made up of 2 identical squares, 4
identical rectangles and 3 identical semi-circles. What is the
area of the figure?
70 cm
PSLE: Emphasis on Number Sense
Each of the three cards shown is printed with a
different whole number.
The smallest number is 23.
When these numbers are added two at a time, the
sums are 61, 71 and 86.
What is the largest number on the cards?
23
It is recognized that mathematics
is “an excellent vehicle for the
development and improvement of
a person’s intellectual
competence in logical
reasoning, spatial visualization,
analysis and abstract thought”
Ministry of Education Singapore 2006
Emphasis on Basics & Problem Solving
National Tests in Singapore
 Grade 6 Primary School Leaving Examination PSLE
 Has always been under the control of Singapore
Ministry of Education
 Singapore Examinations and Assessment Board
(SEAB) was set up recently to handle national tests in
Singapore
 Grade 10 GCE Ordinary Level
 Has always been offered by Cambridge Examination
Syndicate until 2008.
 Grade 12 GCE Advanced Level
 Has always been offered by Cambridge Examination
Syndicate until 2007.
PSLE Paper 1 50 minutes
MultipleChoice Items
10
5
1m
2m
10%
10%
Short-Answer
Items
10
5
1m
2m
20%
PSLE Paper 2 1 hour 40 minutes 
Short-Answer
Items
Structured
&
Long-Answer
Items
5
2m
10%
13
3m
4m
5m
50%
Knowledge 25%
 Recall specific
 mathematical facts,
 concepts,
 rules, and
 formulae.
 Perform straight-forward computations
Comprehension 35%
 Interpret data
 Use mathematical concepts, rules, and formulae
 Solve routine or familiar problems
Application & Analysis 40%
 Analyse data
 Apply mathematical concepts, rules and formulae in a
complex situation
 Solve unfamiliar problems
Basic & Routine Skills
 Example 1
 Find the value of 12.2 ÷ 4 .
 Example 2
 A show started at 10.55 a.m. and ended at 1.30 p.m.
 How long was the show in hours and minutes?
Basic & Routine Skills
 Example 3
 Find <y in the figure below.
70o
70o
y
70o
Basic & Routine Skills
 Example 4

 Cup cakes are sold at 40 cents each.
 What is the greatest number of cup cakes that can be
bought with $95?
 Example 5
 From January to August last year, Mr Tang sold an
average of 4.5 cars per month, He did not sell any car in
the next 4 months. On average, how many cars did he
sell per month last year?
Students are expected to handle novel situations i.e. those not
in the textbooks or previous examinations.
Problem Solving
1 + 2 + 3 + 4 + 5 + … + 95 + 96 + 97
The first 97 whole numbers are added up.
What is the ones digit in the total?
Problem Solving
Rena used stickers of four different shapes to make a
pattern. The first 12 stickers are shown below. What
was the shape of the 47th sticker?
            ………?
1st
12th
47th
Students are expected to handle multi-step problems with some
degree of complexity.
Problem
Solving
In Figure 1, Tank A is completely filled with water and Tank B is
empty. Water is poured from Tank A into Tank B without spilling.
The heights of the water level in the two tanks are now equal as
shown in Figure 2.
What is the height of the water level in Tank A in Figure 2?
Problem Solving
Siti started saving some money on Monday. On each
day from Tuesday to Friday, she saved 20 cents more
than the amount she saved the day before. She saved a
total of $6 from Monday to Friday. How much money
did she save on Monday?
Problem Solving
Siti started saving some money on Monday. On each
day from Tuesday to Friday, she saved 20 cents more
than the amount she saved the day before. She saved a
total of $6 from Monday to Friday. How much money
did she save on Monday?
$6
Problem Solving
Siti started saving some money on Monday. On each
day from Tuesday to Friday, she saved 20 cents more
than the amount she saved the day before. She saved a
total of $6 from Monday to Friday. How much money
did she save on Monday?
$4
Such visual methods are given emphasis in place of the more
abstract algebraic methods.
Berm and Mei saved B800 altogether.
A quarter of Berm’s savings was B65 more than a fifth of Mei’s
savings. How much was Berm’s savings?
Problem Solving
 David and Michael drove from Town A to Town B at
different speeds. Both did not change their speeds
throughout their journeys. David started his journey
30 minutes earlier than Michael. However, Michael
reached Town B 50 minutes earlier than David. When
Michael reached Town B, David had travelled 4/5 of
the journey and was 75 km away from Town B.
4 fifths
A
75 km
B
Problem Solving
 David and Michael drove from Town A to Town B at
different speeds. Both did not change their speeds
throughout their journeys. David started his journey
30 minutes earlier than Michael. However, Michael
reached Town B 50 minutes earlier than David. When
Michael reached Town B, David had travelled 4/5 of
the journey and was 75 km away from Town B.
 What was the distance between Town A and Town B?
 How many kilometers did David travel in 1 hour?
 What was the time taken by Michael to travel from
Town A to Town B?
Problem Solving
4 fifths
75 km
B
A
1 fifth  75 km
4 fifths  4 x 75 km = 300 km
The distance between the two towns is 375 km.
50 min  75 km
10 min  75 km ÷ 5 = 15 km
60 min  15 km x 6 = 90 km
David traveled 90 km in 1 hour.
90 km  1 h
360 km  4 h
15 km  10 min
375 km  4 h 10 min
David took 4 h 10 min.
Michael took ..............
In Grades 5 and 6, students who have not mastered foundation
knowledge follows a Foundation Mathematics program which
ensures students master enough mathematics to continue with
program in Grade 7 and beyond.
Problem Solving in Foundation Mathematics
 Aini used 1-cm square tiles to make rectangles as
shown below. She recorded the area and perimeter of 3
such rectangles in the table shown.
Problem Solving
 Complete the table above for Pattern 4 and Pattern 5.
Pattern
Area of the
rectangle in cm2
Perimeter of the
rectangle in cm
Pattern 1
3
8
Pattern 2
8
12
Pattern 3
15
16
Pattern 4
24
Pattern 5
24
Problem Solving
 She made a rectangle of area 80 cm2.
 What is the pattern number of this rectangle?
 What is the perimeter of this rectangle?
Problem Solving for High-Achievers
Grade 10
 A fly, F, starts at the point with the position
vector (i +12j) cm and crawls across the
surface with a velocity of (3i + 2j) cm s-1. At the
instant that the fly starts crawling, a spider, S, at
the point with position vector (85i + 5j) cm, sets
off across the surface with a velocity (-5i +kj) cm
s-1, where k is a constant. Given that the spider
catches the fly, calculate the value of k.
Students are expected to make sense and interpret the
mathematics they perform.
Features of SingaporeMath
The use of
appropriate
manipulatives. These
are common
manipulatives that
have inexpensive
alternatives
Focus on key
concept – what it is,
and what it is not
Definitions are
developed through
induction, starting in
an informal fashion
Opportunities to
make connections to
previously taught
concept - half
Mathematical terms
are explained.
Designed for
learners of English
language …
.. providing other
learning
opportunities for
learners fluent in the
language
Opportunities for immediate
formative assessment
Teachers know the content that
they are assessing.
Teachers have opportunities to
learn content knowledge. The
program was designed for
teachers who do not necessarily
have strong mathematics
background.
Opportunities for the majority to
reach proficient level …
.. opportunities to engage
advanced learners
Provides independent
practice
Provides opportunities for
student-constructed
responses
Practice for the key concept
Comparing Areas (Biggest)
Even as a new concept is
introduced …
.. learners are given practice on
recently-learnt materials
Comparing Areas (Smallest & Greatest)
Comparing Areas (Same Size)
Building upon previously
developed concepts
Non-standard unit  Standard unit
Concrete  Pictorial  Abstract
Opportunities to develop abstract ideas
Providing
challenging
materials for
advanced learners
while the rest reach a
level of mastery for
basic materials
Non-standard unit  Standard unit
Focus on key concept –
what it is, what it is not
Many opportunities to learn
More formal definitions are
provided if appropriate
Students are finding area of
figures (previously learnt
content) even as they find
the perimeters (new content)
Opportunities to help struggling
students to reach mastery
Opportunities for proficient
students to not develop
common misconceptions,
e.g. confusion between area
and perimeter
Opportunities for advanced
students to make new connections
A collection of tasks is usually
not a random but purposeful
collection
Opportunities to help
struggling students to reach
mastery
Visuals are prevalent in mathematics eaching and learning in
Singapore textbooks.
Example
Example
Are these four-sided shapes?
Example
Base Ten Blocks
Example
Example
Explain who is correct.
Write in your notebook.
Example
At first, Shop A had 156kg of rice and Shop B had 72kg of rice. After each
shop sold the same quantity of rice, the amount that Shop A had was 4
times that of Shop B. How many kilograms of rice did Shop A sell?
156 kg
72 kg
Designing Practice
Practice is not repetition.
Practice provides variation.
Practice for intellectual development.
Professional Development
http://www.nie.edu.sg
Teacher Training
Initial Teacher Training
Diploma in Education (2 years)
for non-graduates
Postgraduate Diploma in
Education (1 year) for
graduates
Bachelor of Science / Arts
(Education) for non-graduates
(2 years)
Qualifications of Teachers
Level
Primary
Secondary
Junior
Colleges
NonGraduates
Graduates
5 928
6 669
1 070
10 602
Education Statistics Digest 2007
11
2 524
Teacher Training
In-service Teacher Training
Teachers are entitled to 100
hours of professional
development per year.
A range of professional development
program is available – in-house or by NIE,
MOE, leading to certification or
otherwise, professional development leave
and conferences
Teacher Training
Initial Teacher Training
Diploma in Education (2 years)
for non-graduates
Postgraduate Diploma in
Education (1 year) for
graduates
Bachelor of Science / Arts
(Education) for non-graduates
(2 years)
TIMSS 2003
International Benchmark Grade 4 Mathematics
Performance
91%
73%
64%
Singapore
38%
International Average
33%
8%
Advanced
High
Intermediate
International Benchmark Grade 8 Mathematics
Performance
93%
77%
51%
44%
Singapore
International Average
24%
6%
Advanced
High
Intermediate
Comparison between Grade 4 and Grade 8
Mathematics Performance
93%
91%
77%
73%
64%
51%
44%
38%
High
33%
24%
8%
Grade 4
Grade 8
Singapore
6%
Grade 4
Advanced
Grade 8
International
Intermediate