Lee Ngan HoeSchool 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

Singapore – A Brief Introduction The Singapore Mathematics Curriculum Reflections Concluding Comments Questions and Discussion

SINGAPORE:A BRIEF INTRODUCTION -

ISLAND, CITY, STATE, COUNTRY

Source: Singapore Department of Statistics - http://www.singstat.gov.sg/

Founded in : 1819Gained independence in : 1965

People : Mainly migrantsNational Language : Malay

Official /Working Language : English

A Summary

• 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 StudiesThat 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

SINGAPOREMATHEMATICSCURRICULUM

The Singapore Mathematics Curriculum –

Basically a national curriculum:

Textbooks must be approved before being adopted in schools.

Context

THE CURRICULUM• 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?

OTHER THAN CONTENT …

SINGAPORE MATHEMATICS CURRICULUM FRAMEWORK

SINGAPORE MATHEMATICS CURRICULUM FRAMEWORK

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.

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.

Mathematical knowledge PerformanceSense-making Connection

Facts AutomaticityProcedures FluencyApplications Flexibility

The Content – Multi-faceted:

Problem SolvingTeaching for Problem SolvingTeaching of Problem Solving

Teaching through Problem Solving

The Pedagogy – Multi-faceted

Promotion OfEffectiveness

EfficiencyElegance

The Performance – Multi-faceted

What is the value of 12 + 14 + 18 + 116 + 132 + 164 + . . . ?

n

1 2 0.5 0.52 4 0.25 0.753 8 0.125 0.8754 16 0.0625 0.93755 32 0.03125 0.968756 64 0.015625 0.9843757 128 0.0078125 0.99218758 256 0.00390625 0.996093759 512 0.001953125 0.998046875

10 1024 0.000976563 0.99902343811 2048 0.000488281 0.99951171912 4096 0.000244141 0.99975585913 8192 0.00012207 0.9998779314 16384 6.10352E-05 0.99993896515 32768 3.05176E-05 0.99996948216 65536 1.52588E-05 0.99998474117 131072 7.62939E-06 0.99999237118 262144 3.8147E-06 0.99999618519 524288 1.90735E-06 0.99999809320 1048576 9.53674E-07 0.99999904621 2097152 4.76837E-07 0.99999952322 4194304 2.38419E-07 0.99999976223 8388608 1.19209E-07 0.99999988124 16777216 5.96046E-08 0.9999999425 33554432 2.98023E-08 0.9999999726 67108864 1.49012E-08 0.99999998527 134217728 7.45058E-09 0.99999999328 268435456 3.72529E-09 0.99999999629 536870912 1.86265E-09 0.99999999830 1073741824 9.31323E-10 0.99999999931 2147483648 4.65661E-10 132 4294967296 2.32831E-10 133 8589934592 1.16415E-10 134 17179869184 5.82077E-11 135 34359738368 2.91038E-11 136 68719476736 1.45519E-11 137 1.37439E+11 7.27596E-12 138 2.74878E+11 3.63798E-12 139 5.49756E+11 1.81899E-12 140 1.09951E+12 9.09495E-13 141 2.19902E+12 4.54747E-13 142 4.39805E+12 2.27374E-13 143 8.79609E+12 1.13687E-13 144 1.75922E+13 5.68434E-14 145 3.51844E+13 2.84217E-14 146 7.03687E+13 1.42109E-14 147 1.40737E+14 7.10543E-15 148 2.81475E+14 3.55271E-15 149 5.6295E+14 1.77636E-15 150 1.1259E+15 8.88178E-16 151 2.2518E+15 4.44089E-16 152 4.5036E+15 2.22045E-16 153 9.0072E+15 1.11022E-16 154 1.80144E+16 5.55112E-17 155 3.60288E+16 2.77556E-17 156 7.20576E+16 1.38778E-17 157 1.44115E+17 6.93889E-18 158 2.8823E+17 3.46945E-18 159 5.76461E+17 1.73472E-18 160 1.15292E+18 8.67362E-19 161 2.30584E+18 4.33681E-19 162 4.61169E+18 2.1684E-19 163 9.22337E+18 1.0842E-19 164 1.84467E+19 5.42101E-20 165 3.68935E+19 2.71051E-20 166 7.3787E+19 1.35525E-20 167 1.47574E+20 6.77626E-21 168 2.95148E+20 3.38813E-21 169 5.90296E+20 1.69407E-21 170 1.18059E+21 8.47033E-22 171 2.36118E+21 4.23516E-22 172 4.72237E+21 2.11758E-22 173 9.44473E+21 1.05879E-22 174 1.88895E+22 5.29396E-23 175 3.77789E+22 2.64698E-23 176 7.55579E+22 1.32349E-23 177 1.51116E+23 6.61744E-24 178 3.02231E+23 3.30872E-24 179 6.04463E+23 1.65436E-24 180 1.20893E+24 8.27181E-25 181 2.41785E+24 4.1359E-25 182 4.8357E+24 2.06795E-25 183 9.67141E+24 1.03398E-25 184 1.93428E+25 5.16988E-26 185 3.86856E+25 2.58494E-26 186 7.73713E+25 1.29247E-26 187 1.54743E+26 6.46235E-27 188 3.09485E+26 3.23117E-27 189 6.1897E+26 1.61559E-27 190 1.23794E+27 8.07794E-28 191 2.47588E+27 4.03897E-28 192 4.95176E+27 2.01948E-28 193 9.90352E+27 1.00974E-28 194 1.9807E+28 5.04871E-29 195 3.96141E+28 2.52435E-29 196 7.92282E+28 1.26218E-29 197 1.58456E+29 6.31089E-30 198 3.16913E+29 3.15544E-30 199 6.33825E+29 1.57772E-30 1100 1.26765E+30 7.88861E-31 1

12 + 14 + 18 + 116 + 132 + 164 + . . .= 1

Using Calculator / Computer

Effective?

n

1 2 0.5 0.52 4 0.25 0.753 8 0.125 0.8754 16 0.0625 0.93755 32 0.03125 0.968756 64 0.015625 0.9843757 128 0.0078125 0.99218758 256 0.00390625 0.996093759 512 0.001953125 0.998046875

10 1024 0.000976563 0.99902343811 2048 0.000488281 0.99951171912 4096 0.000244141 0.99975585913 8192 0.00012207 0.9998779314 16384 6.10352E-05 0.99993896515 32768 3.05176E-05 0.99996948216 65536 1.52588E-05 0.99998474117 131072 7.62939E-06 0.99999237118 262144 3.8147E-06 0.99999618519 524288 1.90735E-06 0.99999809320 1048576 9.53674E-07 0.99999904621 2097152 4.76837E-07 0.99999952322 4194304 2.38419E-07 0.99999976223 8388608 1.19209E-07 0.99999988124 16777216 5.96046E-08 0.9999999425 33554432 2.98023E-08 0.9999999726 67108864 1.49012E-08 0.99999998527 134217728 7.45058E-09 0.99999999328 268435456 3.72529E-09 0.99999999629 536870912 1.86265E-09 0.99999999830 1073741824 9.31323E-10 0.99999999931 2147483648 4.65661E-10 132 4294967296 2.32831E-10 133 8589934592 1.16415E-10 1

Unit SquareUnit Length

Unit Length12

12 +

14

14

18

18

116

116

132

132

164

164+ + + + + . . . = 1

Using Paper Cutting

Efficient?

S = 12 + 14 + 18 + 116 + 132 + 164 + . . . 2S = 1 + 12 + 14 + 18 + 116 + 132 + 164 + . . . 2S = 1 + S

S = 1

Using Generalizable Pattern –An Algebraic Approach

Recognized that it is a Geometric Progression (GP), with first term a = and common ratio r = , and the required answer is the sum to infinity, which exists since r < 1. This 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.

Elegance?

THE KEY APPROACHES:

• THE SPIRAL CURRICULUM (CURRICULUM APPROACH)

• THE CONCRETE-PICTORIAL-ABSTRACT (C-P-A) DEVELOPMENT OF CONCEPTS (PEDAGOGICAL APPROACH)

• LEARNING EXPERIENCES (2013) (LEARNING APPROACH)

THE SPIRAL CURRICULUM

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.

THE C-P-A DEVELOPMENT OF CONCEPTS

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

LEARNING EXPERIENCES

http://www.flashlightcreative.net/swf/mindreader/

THE KEY APPROACHES THROUGH THE CURRICULUM, TEACHING, AND LEARNING – LEARNING IS ABOUT MAKING CONNECTIONS :• THE SPIRAL CURRICULUM (CURRICULUM APPROACH) –

CONNECTING TO EXTEND EXISTING KNOWLEDGE AND SKILLS

• THE CONCRETE-PICTORIAL-ABSTRACT (C-P-A) DEVELOPMENT OF CONCEPTS (PEDAGOGICAL APPROACH) – CONNECTING TO MAKE SENSE OF LEARNING

• LEARNING EXPERIENCES (2013) (LEARNING APPROACH) – CONNECTIONS TO REALISE THE CURRICULUM

REFLECTIONS…

• Eclectic approach towards teaching and learning of mathematics in a centralised system

• Consistency-based and integrated approach towards curriculum changes

Curriculum

An ironical curriculum?

• Value on education and mathematics• Exposure to eclecticism, striking for balance,

strive for unity

An “rojak” culture?

Geographical

• Size• East or West?

A blessed nation?

AdvantagesIn

International Benchmarking Studies

• Every school is a good school concept: Raising the average

• National examinations• Emphasise on strategic use of Information

and Communication Technology

AreInternational Benchmarking Studies

Useful?

What have we gained?

• Economically• Educationally

CONCLUDING COMMENTS

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 self-improvement. 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-bad-at-math/280914/

Some Thoughts …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.

QUESTIONS & DISCUSSION

You may continue to contact me at: leenganhoe@gmail.com

Thank You & All the Best!

It has been great being part of this community.

S = a + ar + ar2 + ar3 + ar4 + ...

𝑆𝑟 = 𝑎𝑟 + a + ar1 + ar2 + ar3 + ar4 + ...

Sr = ar + S

S (1r – 1) =

ar

S (1 − 𝑟r ) = ar

S = a1 − r

Recognized that it is a Geometric Progression (GP), with first term a = and common ratio r = , and the required answer is the sum to infinity, which exists since r < 1. This approach is generalizable to any GP with r < 1.

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.