MAP101 Fundamentals of Singapore Mathematics Curriculum

1

Singapore Mathematics: An IntroductionWhat is Singapore Mathematics?

There is no such thing as Singapore Mathematics in Singapore. What

has come to be known as Singapore Mathematics is the way students

learn Mathematics and the way teachers learn to teach Mathematics

in Singapore. This includes the curriculum, the textbooks, and the

corresponding teacher professional development.

The Singapore Mathematics curriculum is derived from an education

system that focuses on thinking and places a strong emphasis on

conceptual understanding and mathematical problem-solving. The

scope and sequence of the curriculum is well articulated and follows a

spiral progression. A pedagogy that is based on students progressing

from concrete to pictorial and then to abstract representations,

helps the majority of students acquire conceptual understanding of

mathematical concepts. Visuals are used extensively in textbooks.

Mathematics as a Vehicle to Develop Thinking SkillsSince the late 1990s, the Singapore education system has emphasized

on thinking skills as one of its pillars. Schools are encouraged to use

school subjects to help students to acquire good thinking skills and

develop good thinking habits. In this vein, the ‘Thinking Schools,

Learning Nation’ philosophy was introduced in 1997.

The latest version of the Singapore Mathematics (Primary) syllabus

states that Mathematics is “an excellent vehicle for the development

and improvement of a person’s intellectual competence in logical

reasoning, spatial visualisation, analysis and abstract thought” (Ministry

of Education of Singapore, 2006, p. 5).

The following Lesson 1 helps students develop visualization ability.

Lesson 2 helps students develop the ability to see patterns and

generalize the patterns. Visualization and generalization are examples of

intellectual competence that can be developed through Mathematics.

2

Allan puts some brown sugar on a dish.

The total weight of the brown sugar and dish is 110 g.

Bella puts thrice the amount of brown sugar that Allan puts

on an identical dish, and the total weight of the brown

sugar and dish is 290 g.

Find the weight of the brown sugar that Bella puts on the dish.

Lesson 1 In the Kitchen

3

Cheryl and David tried counting the letters in their names in a certain way.

For example, Cheryl counted the letters in CHERYL back and forth such

that the letter C is the 1st, H is the 2nd, E is the 3rd, R is the 4th, Y is the 5th,

and L is the 6th. Then she counts backwards such that Y is the 7th, R is the 8th,

E is the 9th, H is the 10th, C is the 11th. Using this way of counting, Cheryl’s

19th letter is E. When David counts his name DAVID, the 19th letter is V.

Find Cheryl’s and David’s 99th letter using this way of counting.

Lesson 2 Name Patterns

4

Mathematical Problem Solving as the Focus of Learning Mathematics Based on research from around the world, Singapore developed a

Mathematics curriculum in the late 1980s to enable students to develop

mathematical problem-solving ability. This was introduced to Primary 1

students in 1992.

Beliefs

Interest

Appreciation

Confidence

Perseverance

Numerical calculation

Algebraic manipulation

Spatial visualization

Data analysis

Measurement

Use of mathematical tools

Estimation Numerical

Algebraic

Geometrical

Statistical

Probabilistic

Analytic

Monitoring of one’s own thinking

Self-regulation of learning

Reasoning, communication and connections

Thinking skills and heuristics

Application and modelling

Mathematical Problem Solving

Attitudes

Metacognition

Pro

cess

esConcepts

Skills

In Lessons 3, 4, and 5, students do not have rules that they can

follow to solve the problems. Thus, students have to think of ways

to solve the problems and in this way, develop problem-solving

strategies. Lesson 3 helps to teach students a new skill. In Lesson

4, the problem is used to consolidate a skill, by allowing students to

practice. In Lesson 5, students have to apply a previously learned skill

to solve a problem involving the formula for a figure that is different

from the ones they have been taught, as students are not taught the

formula to find the area of a trapezium (also known as trapezoid) in the

Singapore primary curriculum.

5

Lesson 3 Sharing Three-Quarters

Share three-quarters of a cake equally among 4 persons.

What fraction of the cake does each person get?

6

Lesson 4 Make a Multiplication Sentence

Use one set of digit tiles to make correct multiplication sentences.

Resources: Digit Tiles (See Appendix)

×

7

Find the area of the figure.

5 cm

8 cm

4 cm 5 cm

Lesson 5 What is the Area?

8

Learning Theories A strong foundation is necessary for the students to do well in Mathematics.

In the Singapore textbooks, such a strong foundation is achieved through

the application of a few learning principles or learning theories.

Jerome Bruner The Concrete Pictorial Abstract Approach — the progression

from concrete objects to pictures to abstract symbols is recommended

for concept development. This is based on the work of Jerome Bruner

on enactive, iconic, and symbolic representations.

Students learn a new concept or skill by using concrete materials.

Bruner referred to this as the enactive representations of the concept

or skill. Later, pictorial representations are used before the introduction

of symbols (abstract representations). Reinforcement is achieved by

going back and forth between the representations.

For example, students learn the concept of division by sharing 12

cookies among 4 persons as well as by putting 12 eggs in groups of 4

before progressing to using drawings to solve division problems. Later,

they learn to write the division sentence 12 ÷ 4 = 3.

This is referred to as the CPA Approach.

79

Let’s Learn!

Division15

Sharing Equally 1 There are 12 cookies. Googol has 4 friends. He gives each friend the same number of cookies in a bag.

Each friend gets 3 cookies.

Indu Huiling Amil Weiming

Now I put 1 more cookie in each bag. I have no cookies left.

I try putting 2 cookiesin each bag. Then I have 4 cookies left.

079-083 MthsP1B U15.indd 79 6/27/06 6:14:37 PM

81

Let’s Learn!

How To Divide

Sharing

1

Googol has 6 mangosteens. He wants to divide the mangosteens into 2 equal groups. How many mangosteens are there in each group?

6 ÷ 2 = 3

There are 3 mangosteens in each group. Now he wants to divide them into equal 3 groups.

6 ÷ 3 = 2

There are 2 mangosteens in each group.

6 ÷ 2 = 3 and 6 ÷ 3 = 2 are division sentences.

We read 6 ÷ 2 = 3 as six divided by two is equal to three.

÷ stands for division.

How do I read6 ÷ 3 = 2?

MPH!Mths 2A U04.indd 81 7/7/06 8:25:15 PM

Pupil’s Book 1B p. 79 Pupil’s Book 2A p. 81

9

81

Let’s Learn!Let’s Learn!

Finding The Number Of Groups 1 There are 12 eggs. Put 4 eggs into each bowl. How many bowls do you need?

I need 3 bowls.

Do this until all the eggs are put into the bowls.

First put 4 eggsinto 1 bowl.

2 Crystal has 15 toy cats. She puts 3 toy cats on each sofa. How many sofas are needed for all the toy cats?

sofas are needed for all the toy cats.

079-083 MthsP1B U15.indd 81 6/27/06 6:15:11 PM

× 4 = 12

3 Googol has 12 snap cards.

He divides the cards equally among his friends. Each friend gets 4 cards. How many friends are there?

12 ÷ 4 =

There are friends.

4 Sulin has 18 cards. She gives the cards to some friends.

If each friend gets 3 cards, how many friends are there?

18 ÷ 3 =

There are friends.

× 3 = 18

Grouping

83

MPH!Mths 2A U04.indd 83 1/25/07 9:30:34 AM

Source: My Pals are Here! Maths (2nd Edition)

Textbook 1B p. 61

Textbook 2A p.94

Pupil’s Book 1B p. 81 Pupil’s Book 2A p. 83

The Spiral Approach — students revisit core ideas as they deepen their

understanding of those ideas. This is also one of Jerome Bruner’s theories.

For example, students learn to divide discrete quantities without the

need to write division sentences in Primary 1.

61

Divide 15 apples into 3 equal groups.

There are apples in each group.

I put 15 apples equallyon 3 plates.

PFP_1BTB_Chpt15.indd 61 2/2/07 4:41:01 PM

94

whole

part part

Divide 12 balloonsinto groups of 4.

Divide 12 balloons into3 equal groups.

part

2 Division

PFP_2A_TB_Chap5.indd 94 2/2/07 3:25:40 PM

In Primary 2, they revisit this idea and use division sentences to

represent the word problems.

62

I put 3 apples in a group.

Divide 15 apples into groups of 3.

There are groups.

PFP_1BTB_Chpt15.indd 62 2/2/07 4:41:06 PM

Textbook 1B p. 62

Textbook 2A p.95

95

1.

Divide 8 mangoes into 2 equal groups. There are 4 mangoes in each group. We write:

8 ÷ 2 = 4

This is division.We divide to find the number in each group.

Divide 8 by 2.The answer is 4.


97

4.

Divide 15 children into groups of 5. There are 3 groups.

We write:

15 ÷ 5 = 3 Divide 15 by 5.The answer is 3.

We also divide to findthe number of groups.

Exercise 5, pages 100-102


Textbook 2A p.97

Source: Primary Mathematics (Standards Edition)

10

2 4 furries shared 11 seashells equally among themselves.

a How many seashells did each furry receive?

b How many seashells were left?

a 11 ÷ 4 = ?

11 ones ÷ 4 = 2 ones with remainder 3 ones = 2 R3 Quotient = 2 ones Remainder = 3 ones

Each furry received 2 seashells.

b 3 seashells were left.

Divide the11 seashells into 4 equal groups.

4 × 2 = 88 is less than 11.

4 × 3 = 1212 is more than 11.

Choose 2.

2 R3 4 1 1 8 3

��

94

MthsP3A_U07(1 July) 94 7/5/06 2:39:28 PM

57

Let’s Learn!

Division By A 1-Digit Number

1 6381 sweets were given to the children at a fun fair. Each child received 3 sweets. How many children were there at the fun fair?

6 thousands � 3 = 2 thousands = 2000

3 hundreds � 3 = 1 hundred = 100

8 tens � 3 = 2 tens with remainder 2 tens = 20 with remainder 20

21 ones � 3 = 7 ones = 7

When 6381 is divided by 3, the quotient is 2127 and the remainder is 0.There were 2127 children at the fun fair.

Step 1

Divide 6 thousands by 3.

Step 2

Divide 3 hundreds by 3.

Step 3

Divide 8 tens by 3.

Step 4

Divide 21 ones by 3.

23 6 3 8 1 6

2 13 6 3 8 1 6 3 3

2 1 23 6 3 8 1 6 3 3 8 6 2

2 1 2 73 6 3 8 1 6 3 3 8 6 2 1 2 1 0

Th H T O

2 � 3

1 � 3

7 � 3

2 � 3

J43 4A CB U03 (57-70) 28Jul 57 7/28/06 2:56:07 PM

101

Let’s Learn!

Tens Ones

Division With Regrouping In Tens And Ones

1 Fandi and Farley went fi shing and caught some fi shes and crabs.

They shared the 52 fi shes equally between themselves. How many fi shes did each boy get?

52 ÷ 2 = ?

First, divide the tens by 2.

5 tens ÷ 2 = 2 tens with remainder 1 ten

Regroup the remainder ten: 1 ten = 10 ones Add the ones: 10 ones + 2 ones = 12 ones

Then, divide the ones by 2. 12 ones ÷ 2 = 6 ones

So, 52 ÷ 2 = 26.

Each boy got 26 fi shes.

22 5 2 4 1

��

22 5 2 4 1 2

��

2 62 5 2 4 1 2 1 2 0

��

Tens Ones

Tens Ones

MthsP3A_U07(1 July) 101 7/5/06 2:40:54 PM

Chapter 2: Whole Numbers (2)32

Division

7 a Divide 4572 by 36.

The answer is 127.

b What is 168 � divided by 16?

168 � divided by 16 is 10.5 �.

8 Carry out this activity.

Work in pairs to do these:

a 1065 � 97 b 13 674 � 7 c 1075 � 25

d 10 840 � 40 e 25 m � 48 m f 406 g � 28

Think of one multiplication and one division sentence. Get your partner to work them out using a calculator. Check that your

partner’s answers are correct using your calculator.

WB 5A, p 23Practice 1

Remember to press C before you start working on each sum.

C

4572

÷ 36

=

Press Display0

4 5 7 2

3 6

1 2 7

C

168

÷ 16

=

Press Display0

1 6 8

1 6

1 0 . 5

CB5A_U02(29-42).indd 32 9/7/07 10:26:28 AM

In Primary 3, the idea is extended to include the idea of a remainder.

They also learn to regroup before dividing 2-digit and 3-digit numbers.

In Primary 4, 4-digit numbers are used and in Primary 5, division of

continuous quantities are dealt with where 168 ÷ 16 = 10.5 rather than

10 remainder 8.

Pupil’s Book 3A p. 81

Pupil’s Book 4A p. 57 Pupil’s Book 5A p. 32

Pupil’s Book 3A p. 101


Jerome Bruner proposed the idea of spiral curriculum in 1960s.

A curriculum as it develops should revisit [the] basic ideas repeatedly,

building upon them until the student has grasped the full formal

apparatus that goes with them.

Bruner, 1960

Bruner recommended that when students first learn an idea, the

emphasis should be on grasping the idea intuitively. After that, the

curriculum should revisit the basic idea repeatedly, each time adding

to what the students already know until they understand the idea fully.

Bruner emphasized that ideas are not merely repeated but… revisited

later with greater precision and power until students achieve the reward

of mastery.

Bruner, 1979

11

Bruner explained that ideas that have been introduced in an intuitive

manner were then revisited and reconstrued in a more formal or

operational way, then being connected with other knowledge, the

mastery at this stage then being carried one step higher to a new level

of formal or operational rigour and to a broader level of abstraction

and comprehensiveness. The end stage of this process was eventual

mastery of the connexity and structure of a large body of knowledge.

Bruner, 1960

Bruner gave an example of how students learn the idea of prime

numbers and factoring.

The concept of prime numbers appears to be more readily grasped

when the child, through construction, discovers that certain handfuls

of beans cannot be laid out in completed rows and columns. Such

quantities have either to be laid out in a single file or in an incomplete

row-column design in which there is always one extra or one too few

to fill the pattern. These patterns, the child learns, happen to be called

prime.

Bruner, 1973

Zoltan Dienes Systematic Variation – Students are presented with a variety of tasks

in a systematic way. This is based on the ideas of Zoltan Dienes

(Dienes, 1960).

The above example shows mathematical variability. The variation is in

the Mathematics — addition without regrouping and with regrouping.

27

2

Let’s Learn!

Addition And Subtraction Within 1000

Simple Addition Within 1000

1 Add using base ten blocks. Use the place value chart to help you.

a 123 + 5 = ?

Tens OnesHundreds

123

5

First, add the ones.

1 2 3+ 5

83 ones + 5 ones= 8 ones

Then, add the tens.

1 2 3+ 5

2 82 tens + 0 tens= 2 tens

Lastly, add the hundreds.

1 2 3+ 5

1 2 81 hundred + 0 hundreds= 1 hundred

So, 123 + 5 = 128.


35

Let’s Learn!

First, add the ones.

3 14 7+ 1 2 9

67 ones + 9 ones= 16 ones

Regroup the ones.16 ones = 1 ten 6 ones

Then, add the tens.

3 14 7+ 1 2 9

7 64 tens + 2 tens + 1 ten = 7 tens

Tens OnesHundreds

Lastly, add the hundreds.

3 14 7+ 1 2 9

4 7 63 hundreds + 1 hundred= 4 hundreds

Addition With Regrouping In Ones

1 347 + 129 = ?

Tens OnesHundreds

347

129

476

So, 347 + 129 = 476.


Pupil’s Book 2A p. 27 Pupil’s Book 2A p. 35


12

The next example shows perceptual variability — the mathematical

concept is the same but students are presented with different ways to

perceive a 2-digit number.

Additional reading:

Read an article by Post (1988) which has a section on Dienes’ ideas of variability (http://www.cehd.

umn.edu/rationalnumberproject/88_9.html), as well as the six-stage theory of learning mathematics

(http://www.zoltandienes.com/?page_id=226)

Textbook 1B p. 30


The idea of multiple embodiment is to use different ways to

represent the same concept. In the above example, the concept of a

2-digit number such as 34 is represented in multiple ways — using

sticks, coins and base ten blocks. Is the representation more abstract

than another?

In the next example, 3-digit numbers are represented using base ten

blocks, number discs and digits.

The base ten blocks are shown proportionately using concrete

materials. For example, looks ten times as large as .

However, the number discs are non-proportionate.

For example, does not look ten times as large as .

This is another example of representing the same mathematical

concept in different ways, some are more abstract than others. It is im-

portant to provide students with these variations in a systematic way.

Multiplying by a 1-Digit Number

Represent numbers using place-value charts.

Le

sson

Lesson Objective• Usedifferentmethodstomultiplyupto4-digitnumbers

by1-digitnumbers,withorwithoutregrouping.

Le

arn

213canberepresentedintheseways.

Hundreds Tens Ones

Hundreds Tens Ones

Hundreds Tens Ones

2 1 3

Lesson 3.1 Multiplyingbya1-DigitNumber77

Student Book 4A

Source: Math in Focus: The Singapore Approach



Le

sson



Le

arn


Hundreds Tens Ones

Hundreds Tens Ones

Hundreds Tens Ones

2 1 3




Le

sson



Le

arn


Hundreds Tens Ones

Hundreds Tens Ones

Hundreds Tens Ones

2 1 3


13

Richard Skemp Richard Skemp (Skemp, 1976) provides Mathematics teachers with a

way to think about what constitutes understanding in Mathematics.

Skemp distinguished between the ability to perform a procedure,

(for example, long division), and the ability to explain the procedure,

(for example, explaining the rationale for ‘invert-and-multiply’ when

dividing a proper fraction by a proper fraction). He refers to the former

as instrumental understanding (or procedural understanding) and the

latter as relational understanding (conceptual understanding).

Singapore Mathematics curriculum expects instrumental understanding

to be accompanied by relational understanding. It is pointless to learn

a procedure without having a conceptual understanding.

“Although students should become competent in the various

mathematical skills, over-emphasising procedural skills without

understanding the underlying mathematical principles should be

avoided” (Ministry of Education of Singapore, 2006, p. 7).

Conventional understanding involves the ability to understand the use

of conventions. For example, it is a convention to use + as the symbol

for addition. Some conventions are not universal. For example, ÷ is

used as the symbol for division in some countries, but : is used as the

symbol for division in others. Conventions that are universal include

the order of operations. There are some facts, names, notations, and

usage which are universally agreed upon, and there are no particular

reasons for using those conventions.

12. Chelsea has 5 apple tarts. She cuts each tart into 12

. Find the number

of half-tarts Chelsea has.

5 ÷ 12

= 10

Chelsea has half-tarts.

13. 3 cakes are shared equally among some children. Each child gets

34

of a cake. How many children got 34

of a cake?

3 ÷ 34

= 4

children shared the cakes.

When you cut a whole into halves, you get 2 halves. So, in 5 wholes there are 5 × 2 halves.

When you cut a whole into three-quarters,

you get 1 13

three-quarters. So, in 3 wholes

there are 3 × 1 13

three-quarters.

66

6ATB_Unit 3.indd 66 4/24/09 5:10:13 PM

Additional reading:

Read the classic article originally published in Mathematics Teaching (1976) at http://www.grahamtall.

co.uk/skemp/pdfs/instrumental-relational.pdf


Textbook 6A

14

Find the value of 51 ÷ 3.

In some countries, this can also be written as 51 : 3.

Lesson 6 Long Division

51

15

Lesson 7 Bar Model

Marcus gave 14

of his coin collection to his sister and 12

of the remainder to his brother.

As a result, Marcus had 18 coins.

Find the number of coins in his collection at first.

In Lessons 6 and 7, we are able to explain the long division algorithms as well as the procedure to multiply fractions. We are said to possess relational understanding of these procedures.

16

References1. Bruner, J. (1960). The Process of Education. Cambridge, MA: Harvard

University Press.

2. Bruner, J (1966). On Knowing: Essays for the Left Hand. Cambridge,

MA: Harvard University Press.

3. Bruner, J. S. (1973). Beyond the information given: Studies in the

psychology of knowing, pp. 218-238. New York: W. W. Norton

& Co Inc.

4. Dienes, Z. P. (1960). Building up mathematics. London: Hutchinson

Educational Ltd.

5. Ministry of Education of Singapore. (2006). Mathematics Syllabus

(Primary). Singapore: Curriculum Planning and Development Division.

from http://www.moe.gov.sg/education/syllabuses/sciences/files/

maths-primary-2007.pdf

6. Post, T. (1988). Some notes on the nature of mathematics learning.

Teaching Mathematics in Grades K-8: Research Based Methods ,

pp. 1-19. Boston: Allyn & Bacon.

http://www.cehd.umn.edu/rationalnumberproject/88_9.html

7. Skemp, R. R. (1976). Relational and instrumental understanding.

Mathematics Teaching, 77, pp. 20-26.

http://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdf

17

Appendix

0 1

2 3

4 5

6 7

8 9

0 1

2 3

4 5

6 7

8 9

18

Notes