This document contains notes from a mathematics institute course with Dr. Yeap Ban Har in Singapore. It provides biographical information about Dr. Har and his background in education. It then outlines the key concepts to be covered in the course, including the Singapore approach to mathematics education, early numeracy, addition and subtraction strategies, lesson planning, and holistic assessment for young learners. Examples and case studies are provided to illustrate the various mathematical concepts and techniques.

1. Day 1 | 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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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 coind 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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Open Lesson for Rising Second 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 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 unable 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 unable 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.