This document contains information about a mathematics course taught by Dr. Yeap Ban Har in Singapore, including: - Contact information for Dr. Yeap Ban Har and background on his experience teaching mathematics. - An introduction to the Singapore approach to teaching mathematics, which uses Bruner's constructivist theory of a spiral curriculum and representations. - Examples of lesson plans and case studies for teaching various mathematics topics like fractions, ratios, proportions and algebra. - Discussion of instructional models, differentiated instruction, and holistic assessment approaches in mathematics. The document provides an overview of the Singapore mathematics teaching methods and includes examples of lessons and assessments that could be used when teaching a variety of math topics.

1. 9.10.11.12.13 June 2014
Singapore
Mathematics Institute
with Dr. Yeap Ban Har
coursebook

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Contact Information
 yeapbanhar@gmail.com
 www.banhar.blogspot.com
about yeap ban har
Dr Yeap Ban Har spent ten years at Singapore's National Institute
of Education training pre-service and in-service teachers and
graduate students. Ban Har has authored dozens of textbooks,
math readers and assorted titles for teachers. He has been a
keynote speaker at international conferences, and is currently
the Principal of a professional development institute for
teachers based in Singapore. He is also Director of Curriculum
and Professional Development at Pathlight School, a primary
and secondary school in Singapore for students with autism. In
the last month, he was a keynote speaker at World Bank’s READ
Conference in St Petersburg, Russia where policy makers from
eight countries met to discuss classroom assessment. He was
also a visiting professor at Khon Kaen University, Thailand. He
was also in Brunei to work with the Ministry of Education Brunei
on a long-term project to provide comprehensive professional
development for all teachers in the country.

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introduction
The Singapore approach to teaching and learning mathematics was the result of
trying to find a way to help Singapore students who were mostly not performing
well in the 1970’s.
The CPA Approach as well as the Spiral Approach are fundamental to teaching
mathematics in Singapore schools. The national standards, called syllabus in
Singapore, is designed based on Bruner’s idea of spiral curriculum. Textbooks are
written based on and teachers are trained to use the CPA Approach, based on
Bruner’s ideas of representations.
“A curriculum as it develops should revisit this basic ideas repeatedly, building
upon them until the student has grasped the full formal apparatus that goes with
them”.
| Bruner 1960
“I was struck by the fact that successful efforts to teach highly structured bodies
of knowledge like mathematics, physical sciences, and even the field of history
often took the form of metaphoric spiral in which at some simple level a set of
ideas or operations were introduced in a rather intuitive way and, once mastered
in that spirit, 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 1975
Bruner's constructivist theory suggests it is effective when faced with new material
to follow a progression from enactive to iconic to symbolic representation; this
holds true even for adult learners.
| Bruner 1966

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Open Lesson |
What do we want the students to learn?
Lesson Segment Observation / Question
How can we tell if students are
learning?
What help students who
struggle?
What are for students who
already know what we want
them to learn?
Summary

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Day 1 | Early Numeracy |Session 1
 Rational Counting
 Number Bonds
 Lesson Sequence
 Use of Literature
Lesson Sequence
 Anchor Task
 Guided Practice
 (Independent Practice)
Case Study 1 |
 Show 5 beans on a ten frame.
 Do it in another way.

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Case Study 2 |
Show the teacher five pieces of square tiles.
Make a shape using five square tiles.
There are some rules that we have to follow.

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Whole Number Addition and Subtraction |Session 2
 Materials
 Strategies
 Semantics
 Variation
Semantics
 Part-Whole
 Change
 Comparison
Case Study 2 |
Together, Jon and Kim have 32 coins.
Jon has 19 coins.
Find the number of coins that Kim has.

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Case Study 3 |
Lance has 10 coins more than Ming.
Together, they have 34 coins.
How many coins does Lance have?

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Case Study 4 |
At first, Lance had 10 coins more than Ming.
Then Ming gave Lance 6 coins.
Who had more coins in the end? How many more?

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Use of Activities for Math Learning |Session 4
Types of Lessons
 To develop basic ideas, concepts and skills
 To consolidate basic ideas, concepts and skills
 To extend basic ideas, concepts and skills
Case Study 5 |
Use the digits 0 to 9 not more than once to make an addition equation.

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Holistic Assessment for Young Learners |Session 5
Assessment Benchmarks
 Approaching Expectations
 Meeting Expectations
 Exceeding Expectations
Students should be able to perform rational counting.
Approaching Expectations The student is unable to count a plate of not more than ten
cookies.
 Can the student perform one to one correspondence?
 Can the student classify?
 Can the student rote count?
 Has the student grasp the principle of cardinality?
Meeting Expectations The student is able to count a plate of not more than ten
cookies.
 Also able to read the correct numeral
 Also able to read the correct number word
 Also able to write the correct numeral
 Also able to write the correct number word
Exceeding Expectations The student is able to count a plate of not more than ten
cookies. The student is also able to read and write the correct
numeral and number word.

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Day 2 | Differentiated Instruction |Session 1 and Session 2
 Remediation
 Enrichment
 Four Critical Questions
Four Critical Questions (DuFour)
 What do I want the students to learn?
 How do I know they have learnt it?
 What if they cannot learn it?
 What if they already learnt it?
Differentiated Instruction (Tomlinson)
 Content
 Process
 Product
 Affect

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Case Study 1 | Basic Idea Lesson
 Draw any triangle.
 How are the three angles in a triangle related?
Answer the four critical questions.
DI for Struggling Learners DI for Advanced Learners

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Case Study 2 | Basic Idea Lesson
Anchor Task | Mom baked two cakes.
After giving half of a cake to our neigbors, we ate
5
4
of a cake.
Answer the four critical questions.
DI for Struggling Learners DI for Advanced Learners

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Case Study 3 | Practice Lesson
Draw triangles and find the area of each.
Answer the four critical questions.
DI for Struggling Learners DI for Advanced Learners

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Use of Games in Math Learning |Session 4
Types of Lessons
 To develop basic ideas, concepts and skills
 To consolidate basic ideas, concepts and skills
 To extend basic ideas, concepts and skills
Case Study 4 |
Write expressions that include fractions and one of the four basic operations, one on each side
of the square such that the value of adjacent expressions are equal in value. Cut out the pieces,
mix them up and ask another group to arrange the pieces back again such that values of adjacent
expressions are equal.

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Journal Writing |Session 5
Case Study 5 | Problem-Solving Lesson
Let’s have a go at writing a math journal using this diagram as a stimulus.

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Day 3 | Whole Number Multiplication and Division |Session 1
 Strategies
 Semantics
Multiplication
 Group
 Array
 Area
 Rate
 Combination
Division
 Sharing
 Grouping
Case Study 1 |
Compare these three lessons on division of whole numbers
Anchor Task A | Try putting 14 children into 3 equal groups.

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Anchor Task B | Try putting 41 children into groups of threes.
Anchor Task C | Try putting 41 liters of water into 3 containers. Is it possible to
make sure each container contains the same amount of water?

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Case Study 2 |
X =
Given three digits, make two numbers, a 1-digit number and a 2-digit number,
so that the product has the largest possible value.

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Factors and Multiples |Session 2
 Jerome Bruner
 Zoltan Dienes
 Richard Skemp
Case Study 3 |
Use 12 square tiles to make a rectangle or square.

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Model Drawing |Session 4
Case Study 4 |
There are 40 children is Primary 3 Honesty.
19 of them are boys.
How many girls are there in Primary 3 Honesty?
Case Study 5 |
There are three times as many boys as there are girls in the soccer club.
There are 96 children in the soccer club.
Is this possible?

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Case Study 6 |
There is a group of people in a room.
A third of them are children.
A third of the children are boys.
There are 9 or 10 children in the room.
Which situation is possible?
For that situation, suggest questions that can be answered using the given
information.

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Holistic Assessment |Session 5
 Newman’s Procedure
o Read
o Comprehend
o Know Strategies
o Transform
o Compute
o Interpret
Approaching Expectations Unable to solve word problems that is required at the current
grade level. However, the student is able to handle single-step
word problems.
Meeting Expectations Able to handle typical word problem required at the current
grade level.
Exceeding Expectations Able to handle unusual word problems and / or complex word
problems.
Case Study 7 |
At first, the ratio of the number of students in Basketball to the number of
students in Soccer was 3 : 1.
When 18 students moved from Basketball to Soccer, the there were equal number
of students in both sports.
Find the number of students in these two sports.
What if the ratio is 4 : 1?

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Case Study 8 |
In a group of 96 students, a third of the boys and a fifth of the girls do not have
pets at home while 70 students have pets at home.
How many boys have pets at home?

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Fractions, Fractions, Fractions!
an in-depth study of the teaching of fractions
Day 4 | Two Fundamentals |Session 1
 Idea of ‘Nouns’
 Idea of Equal Parts
Case Study 1 |

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Show 2 equal parts.
What do you mean by equal parts?
Show 4 equal parts.
In lesson study, we might discuss why use squares. Why not circles? Why not
rectangles?

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Case Study 2 |
5
2
5
1

5
1
5
3

Equivalent Fractions |Session 2
Case Study 3 |
8
?
4
1

?
9
4
3


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A cake is cut into 6 equal slices.
Aaron and Ben share four slices.
Case Study 4 |
What fraction of the rectangle is shaded?

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Basic Operations |Session 4
Case Study 5 |
Mary has a blue ribbon that is
3
2
1 m long. She has a red ribbon that is
4
3
1 m long.
Source | Primary Mathematics (Standards Edition) 6A

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Case Study 6 |
There are 12 cupcakes left over. Alex takes
4
3
of them home.
How many cupcakes does Alex take?
There are
2
1
3 pies left over. Ali takes
4
3
of them home.
How many pies does Ali take?
Source | Primary Mathematics (Standards Edition) 6A

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Case Study 7 |
The longest side of a triangle is
4
3
2 times as long as the shortest side. The
shortest side is
3
2
in. Find the length of the longest side.
Source | Primary Mathematics (Standards Edition) 6A

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Practice and Problem Solving |Session 5
 Instructional Models
o Teaching through Problem Solving
o Teaching for Problem Solving
o Teaching of Problem Solving
Case Study 8 |
A total of 325 boys and girls attended a performance in the school hall.
5
4
of the
boys and
4
3
of the girls left the hall after the performance ended. There were 29
more boys than girls who remained in the hall. How many girls attended the
performace?
Source | Catholic High School (Primary) Primary 6 Examination

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Case Study 9 |

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Day 5 | Ratio and Proportion |Session 1
 Problem-Solving Approach
 Three-Part Lesson Format
A total of 325 boys and girls attended a performance in the school hall.
5
4
of
the boys and
4
3
of the girls left the hall after the performance ended. There
were 29 more boys than girls who remained in the hall. How many girls
attended the performace?
Source | Catholic High School (Primary) Primary 6 Examination

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Case Study 1 |
Find the area of a polygon with one dot inside it.
How does the area vary with the number of dots on the perimeter of the polygon?

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Find the area of a polygon with four dots on the perimeter.
How does the area vary with the number of dots inside the polygon?

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Advanced Bar Model Method |Session 2
Case Study 2 |
Four friends, Ravi, Johan, Meng and Emma, shared the cost of a present.
Ravi paid 50% of the total amount paid by the other three friends. Meng paid
60% of the total amount paid by Johan and Emma. Johan paid ½ of what Emma
paid. Ravi paid $24 more than Emma.
How much did the present cost?
Source | Primary Six Examination in a Singapore School

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Case Study 3 |
At a swimming meet, School A had 18 more swimmers than School B and 6 fewer
swimmers than School C. The ratio of the number of boys to the number of girls
from the three schools was 1 : 3.
The ratio of the number of boys to the number of girls in School A, School B and
School C were 1 : 3, 1 : 5 and 2 : 5, respectively.
Find the total number of swimmers from the three schools.
Source | Primary Six Examination in a Singapore School

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Teaching Algebra |Session 4
 Ideas Development
o Variable
o Expression
 Simplify
 Expand
 Factor
o Equation
 Linear
 Quadratic
 Others
Case Study 4 |
Solve 7 – x = 4.
Source | Primary Mathematics (Standards Edition) 6A
Case Study 5 |
There are three times as many boys as there are girls in the soccer club.
There are 96 children in the soccer club.
Number of boys
Number of girls

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Case Study 6 |
(a) Find the value of 3s – 1 when s = 4.
(b) Solve 3s – 1 = 11.
Source | Primary Mathematics (Standards Edition) 6A
Case Study 7 |
Is it possible to factor 252 2
 xx into linear factors?

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Is it possible for 252 2
 xx = 0?
Case Study 8 |
Use algebra tiles to show 522
 xx and 142
 xx .
In each case try to rearrange the tiles to form a square.

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Holistic Assessment |Session 5
 Skemp’s Types of Understanding
o Instrumental
o Relational
o Conventional
Approaching Expectations Student is unable to solve typical systems of linear equations.
The source of difficulty is likely to be
 knowing the meaning of ‘solve’ (conventional)
 knowing how to read algebraic expressions (conventional)
 knowing how to do arithmetic manipulation (instrumental)
 …
Meeting Expectations Student is able to solve typical systems of linear equations.
Exceeding Expectations Student is able to solve typical systems of linear equations.
There is also evidence that the student is able to extend his/her
understanding to less common situations.
Case Study 9 |
Solve 171
2
1
3
1
3
1
2
1
 yxyx .