Lee Ngan Hoe
School of Education, College of Health and Human Services, St. Ambrose University, USA
LeeNganH@sau.edu
Mathematics & Mathematics Education, National Institute of Education, Nanyang Technological University, Singapore
nganhoe.lee@nie.edu.sg
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Singapore – A Brief Introduction
The Singapore Mathematics Curriculum
Reflections
Concluding Comments
Questions and Discussion
Source: Singapore Department of Statistics - http://www.singstat.gov.sg/
Founded in : 1819
Gained independence in : 1965
People : Mainly migrants
National Language : Malay
Official /Working Language : English
• A small country – island, city, state, country
• Warm and humid
• Generally safe from natural disasters and
crime
• Known for shopping and eating
• Common use of English
International Benchmarking Studies
That Places Singapore In The Limelight:
• TIMSS – the Trends in International
Mathematics and Science Study or as previously
known Third International Mathematics and
Science Study
• PISA – Programme for International Student
Assessment
• TEDS-M – Teacher Education and Development
Study in Mathematics
Context
The Singapore Mathematics Curriculum –
Basically a national curriculum:
Textbooks must be approved before being
adopted in schools.
•
•
The Ministry of Education Primary Mathematics
Syllabus Document (Year of implementation – 2007)
(http://www.moe.edu.sg/education/syllabuses/sciences
/files/maths-primary-2007.pdf)
The Ministry of Education Primary Mathematics
Syllabus Document (Year of implementation – 2013)
(http://www.moe.edu.sg/education/syllabuses/sciences
/files/maths-primary-2013.pdf)
• Framework – What to address?
• Approaches – How to address?
The Framework for the Singapore Mathematics Curriculum,
developed in 1990, for example, survived, with minor modification,
the major curriculum review for the 2000 syllabuses which took into
account the three new Initiatives. One of the key reasons for the
Framework’s survival is its rigour and robustness in presenting the
philosophy and principles underlying decisions made about what
mathematics education should equip our students with.
Lee, N.H. (2008). Nation Building Initiative: Impact on Singapore Mathematics
Curriculum. In Niss, M. (Ed.) 10th International Congress on Mathematical
Education Proceedings (CD). Copenhagen: Roskilde University.
Mathematical knowledge
Sense-making
Facts
Procedures
Applications
Performance
Connection
Automaticity
Fluency
Flexibility
Problem Solving
Teaching for Problem Solving
Teaching of Problem Solving
Teaching through Problem Solving
Promotion Of
Effectiveness
Efficiency
Elegance
What is the value of
1
2
+
1
4
+
1
8
+
1
16
+
1
32
+
1
64
+ ...?
n
1
1
1
1
1
1
+ +
+
+
+
+ . . .= 1
2
4
8
16
32
64
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
2
4
8
16
32
64
128
256
512
1024
2048
4096
8192
16384
32768
65536
131072
262144
524288
1048576
2097152
4194304
8388608
16777216
33554432
67108864
134217728
268435456
536870912
1073741824
2147483648
4294967296
8589934592
0.5
0.25
0.125
0.0625
0.03125
0.015625
0.0078125
0.00390625
0.001953125
0.000976563
0.000488281
0.000244141
0.00012207
6.10352E-05
3.05176E-05
1.52588E-05
7.62939E-06
3.8147E-06
1.90735E-06
9.53674E-07
4.76837E-07
2.38419E-07
1.19209E-07
5.96046E-08
2.98023E-08
1.49012E-08
7.45058E-09
3.72529E-09
1.86265E-09
9.31323E-10
4.65661E-10
2.32831E-10
1.16415E-10
0.5
0.75
0.875
0.9375
0.96875
0.984375
0.9921875
0.99609375
0.998046875
0.999023438
0.999511719
0.999755859
0.99987793
0.999938965
0.999969482
0.999984741
0.999992371
0.999996185
0.999998093
0.999999046
0.999999523
0.999999762
0.999999881
0.99999994
0.99999997
0.999999985
0.999999993
0.999999996
0.999999998
0.999999999
1
1
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
2
4
8
16
32
64
128
256
512
1024
2048
4096
8192
16384
32768
65536
131072
262144
524288
1048576
2097152
4194304
8388608
16777216
33554432
67108864
134217728
268435456
536870912
1073741824
2147483648
4294967296
8589934592
17179869184
34359738368
68719476736
1.37439E+11
2.74878E+11
5.49756E+11
1.09951E+12
2.19902E+12
4.39805E+12
8.79609E+12
1.75922E+13
3.51844E+13
7.03687E+13
1.40737E+14
2.81475E+14
5.6295E+14
1.1259E+15
2.2518E+15
4.5036E+15
9.0072E+15
1.80144E+16
3.60288E+16
7.20576E+16
1.44115E+17
2.8823E+17
5.76461E+17
1.15292E+18
2.30584E+18
4.61169E+18
9.22337E+18
1.84467E+19
3.68935E+19
7.3787E+19
1.47574E+20
2.95148E+20
5.90296E+20
1.18059E+21
2.36118E+21
4.72237E+21
9.44473E+21
1.88895E+22
3.77789E+22
7.55579E+22
1.51116E+23
3.02231E+23
6.04463E+23
1.20893E+24
2.41785E+24
4.8357E+24
9.67141E+24
1.93428E+25
3.86856E+25
7.73713E+25
1.54743E+26
3.09485E+26
6.1897E+26
1.23794E+27
2.47588E+27
4.95176E+27
9.90352E+27
1.9807E+28
3.96141E+28
7.92282E+28
1.58456E+29
3.16913E+29
6.33825E+29
1.26765E+30
0.5
0.25
0.125
0.0625
0.03125
0.015625
0.0078125
0.00390625
0.001953125
0.000976563
0.000488281
0.000244141
0.00012207
6.10352E-05
3.05176E-05
1.52588E-05
7.62939E-06
3.8147E-06
1.90735E-06
9.53674E-07
4.76837E-07
2.38419E-07
1.19209E-07
5.96046E-08
2.98023E-08
1.49012E-08
7.45058E-09
3.72529E-09
1.86265E-09
9.31323E-10
4.65661E-10
2.32831E-10
1.16415E-10
5.82077E-11
2.91038E-11
1.45519E-11
7.27596E-12
3.63798E-12
1.81899E-12
9.09495E-13
4.54747E-13
2.27374E-13
1.13687E-13
5.68434E-14
2.84217E-14
1.42109E-14
7.10543E-15
3.55271E-15
1.77636E-15
8.88178E-16
4.44089E-16
2.22045E-16
1.11022E-16
5.55112E-17
2.77556E-17
1.38778E-17
6.93889E-18
3.46945E-18
1.73472E-18
8.67362E-19
4.33681E-19
2.1684E-19
1.0842E-19
5.42101E-20
2.71051E-20
1.35525E-20
6.77626E-21
3.38813E-21
1.69407E-21
8.47033E-22
4.23516E-22
2.11758E-22
1.05879E-22
5.29396E-23
2.64698E-23
1.32349E-23
6.61744E-24
3.30872E-24
1.65436E-24
8.27181E-25
4.1359E-25
2.06795E-25
1.03398E-25
5.16988E-26
2.58494E-26
1.29247E-26
6.46235E-27
3.23117E-27
1.61559E-27
8.07794E-28
4.03897E-28
2.01948E-28
1.00974E-28
5.04871E-29
2.52435E-29
1.26218E-29
6.31089E-30
3.15544E-30
1.57772E-30
7.88861E-31
0.5
0.75
0.875
0.9375
0.96875
0.984375
0.9921875
0.99609375
0.998046875
0.999023438
0.999511719
0.999755859
0.99987793
0.999938965
0.999969482
0.999984741
0.999992371
0.999996185
0.999998093
0.999999046
0.999999523
0.999999762
0.999999881
0.99999994
0.99999997
0.999999985
0.999999993
0.999999996
0.999999998
0.999999999
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Unit Square
Unit Length
1
4
1
2
Unit Length
1
8
1
2
+
1
4
+
1
8
+
1
16
+
1
32
+
1
16
1
32
1
64
1
64
+ ...
= 1
S
=
1
2
+
1
4
+
1
8
+
1
16
2S
=
1 +
1
2
+
1
4
+
1
8
2S
=
1 + S
S
=
1
+
+
1
32
1
16
+
+
1
64
1
32
+ . . .
+
1
64
+ . . .
Recognized that it is a Geometric Progression (GP), with first term a =
1
1
2
and common
ratio r = , and the required answer is the sum to infinity, which exists since r < 1. This
2
approach is generalizable to any GP with r < 1.
Lead to problem posing – rather than consumers of mathematics, students are
encouraged to be creators of mathematics.
Each topic is revisited and introduced in increasing depth from one level to the
next to enable students to consolidate the concepts and skills learned and to
develop these concept and skills further.
It is not just about representing an idea in different forms, it is about connecting
the various representation to make sense of the mathematics to be learnt
http://www.flashlightcreative.net/swf/mindreader/
• Eclectic approach towards teaching and
learning of mathematics in a centralised
system
• Consistency-based and integrated approach
towards curriculum changes
• Value on education and mathematics
• Exposure to eclecticism, striking for balance,
strive for unity
• Size
• East or West?
Advantages
In
International Benchmarking Studies
• Every school is a good school concept:
Raising the average
• National examinations
• Emphasise on strategic use of Information
and Communication Technology
Are
International Benchmarking Studies
Useful?
What have we gained?
• Economically
• Educationally
One way to help Americans excel at math is to copy the approach of the Japanese,
Chinese, and Koreans. In Intelligence and How to Get It, Nisbett describes how the
educational systems of East Asian countries focus more on hard work than on inborn
talent:
1. “Children in Japan go to school about 240 days a year, whereas children in the
United States go to school about 180 days a year.”
2. “Japanese high school students of the 1980s studied 3 ½ hours a day, and that
number is likely to be, if anything, higher today.”
3. “[The inhabitants of Japan and Korea] do not need to read this book to find out that
intelligence and intellectual accomplishment are highly malleable. Confucius set that
matter straight twenty-five hundred years ago.”
4. “When they do badly at something, [Japanese, Koreans, etc.] respond by working
harder at it.”
5. “Persistence in the face of failure is very much part of the Asian tradition of selfimprovement. And [people in those countries] are accustomed to criticism in the
service of self-improvement in situations where Westerners avoid it or resent it.”
We certainly don’t want America’s education system to copy everything Japan does (and
we remain agnostic regarding the wisdom of Confucius). But it seems to us that an
emphasis on hard work is a hallmark not just of modern East Asia, but of America’s past
as well. In returning to an emphasis on effort, America would be returning to its roots, not
just copying from successful foreigners.
Source: The Atlantic – The Myth of ‘I’m Bad at Math (2013) http://www.theatlantic.com/education/print/2013/10/the-myth-of-im-badat-math/280914/
Education is embedded in a sociocultural
context. Curriculum development
should be approached from a
integrative rather than additive manner,
reflecting and refining the aspiration of
the people. International benchmark
studies is but just one way to better
understand the gaps that exist in
curriculum, teaching and learning. A
balanced and holistic approach is
needed for curriculum development.
Thank You & All the Best!
It has been great being part of this community.
You may continue to contact me at:
leenganhoe@gmail.com
Recognized that it is a Geometric Progression (GP), with first term a =
1
1
2
and common
ratio r = 2, and the required answer is the sum to infinity, which exists since r < 1. This
approach is generalizable to any GP with r < 1.
S
a + ar + ar2 + ar3 + ar4 + ...
=
𝑆
𝑎
=
𝑟
S
𝑟
a
=
r
r
1
a
r
r
S ( – 1) =
S(
1− 𝑟
S =
r
)=
a
1−r
a
r
+ a + ar1 + ar2 + ar3 + ar4 + ...
+ S
Let AB be the 2-digit number.
So, A = 1, 2, 3, 4, ..., 7, 8, 9 and B = 0, 1, 2, 3, 4, ..., 7, 8, 9
The value of AB is: 10 × A + B.
The value of sum of the two digits is: A + B.
So, your answer is: 10 × A + B – (A + B) = 9 × A
In other words, your answer is 9, 18, 27, 36, 45, 54, 63, 72, 81.