This document contains notes from a mathematics institute in Singapore led by Dr. Yeap Ban Har. It includes information about Dr. Har, introductions to teaching ratio and proportion, the advanced bar model method, algebra, and holistic assessment. Sample mathematics problems and lessons are provided to illustrate various concepts. The document appears to be notes that were given to participants of the institute on teaching approaches and content in Singapore mathematics.

1. Day 4 | 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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Ratio and Proportion |Session 1
 Problem-Solving Approach
 Three-Part Lesson Format
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 |
Three 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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Open Lesson for Rising Seventh 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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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 .