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 1712

1

3

1

3

1

2

1 yxyx .