This document summarizes a mathematics institute course held in Singapore by Dr. Yeap Ban Har. It provides an introduction to Dr. Har and his background, as well as an overview of the sessions to be covered in the course, including the Singapore approach to teaching mathematics, whole number operations, factors and multiples, lesson planning, word problem modeling and assessment. Case studies are presented in each section to illustrate the concepts.

1. Day 3 | 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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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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Open Lesson for Rising Fifth Graders |Session 3
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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Model Drawing |Session 4
Case Study 4 |
There are 440 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?