1. Day 2 | 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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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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Open Lesson for Rising Sixth 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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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.