blake institute june 2014 complete

9.10.11.12.13 June 2014

Singapore Mathematics Institute

with Dr. Yeap Ban Har

coursebook

2 | P a g e

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.

3 | P a g e

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

4 | P a g e

Open Lesson |

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

5 | P a g e

Day 1 | 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.

6 | P a g e

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.

7 | P a g e

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.

8 | P a g e

Case Study 3 |

Lance has 10 coins more than Ming.

Together, they have 34 coins.

How many coins does Lance have?

9 | P a g e

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?

10 | P a g e

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.

11 | P a g e

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 able 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 able 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.

12 | P a g e

Day 2 | 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

13 | P a g e

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

14 | P a g e

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

15 | P a g e

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

16 | P a g e

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.

17 | P a g e

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.

18 | P a g e

Day 3 | 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.

19 | P a g e

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?

20 | P a g e

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.

21 | P a g e

Factors and Multiples |Session 2

Jerome Bruner

Zoltan Dienes

Richard Skemp

Case Study 3 |

Use 12 square tiles to make a rectangle or square.

22 | P a g e

Model Drawing |Session 4

Case Study 4 |

There are 40 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?

23 | P a g e

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.

24 | P a g e

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?

25 | P a g e

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?

26 | P a g e

Fractions, Fractions, Fractions!

an in-depth study of the teaching of fractions

Day 4 | Two Fundamentals |Session 1

Idea of ‘Nouns’

Idea of Equal Parts

Case Study 1 |

27 | P a g e

Show 2 equal parts.

What do you mean by equal parts?

Show 4 equal parts.

In lesson study, we might discuss why use squares. Why not circles? Why not

rectangles?

28 | P a g e

Case Study 2 |

5

2

5

1

5

1

5

3

Equivalent Fractions |Session 2

Case Study 3 |

8

?

4

1

?

9

4

3

29 | P a g e

A cake is cut into 6 equal slices.

Aaron and Ben share four slices.

Case Study 4 |

What fraction of the rectangle is shaded?

30 | P a g e

Basic Operations |Session 4

Case Study 5 |

Mary has a blue ribbon that is 3

21 m long. She has a red ribbon that is

4

31 m long.

Source | Primary Mathematics (Standards Edition) 6A

31 | P a g e

Case Study 6 |

There are 12 cupcakes left over. Alex takes 4

3 of them home.

How many cupcakes does Alex take?

There are 2

13 pies left over. Ali takes

4

3 of them home.

How many pies does Ali take?

Source | Primary Mathematics (Standards Edition) 6A

32 | P a g e

Case Study 7 |

The longest side of a triangle is 4

32 times as long as the shortest side. The

shortest side is 3

2 in. Find the length of the longest side.

Source | Primary Mathematics (Standards Edition) 6A

33 | P a g e

Practice and Problem Solving |Session 5

Instructional Models

o Teaching through Problem Solving

o Teaching for Problem Solving

o Teaching of Problem Solving

Case Study 8 |

A total of 325 boys and girls attended a performance in the school hall. 5

4 of the

boys and 4

3 of the girls left the hall after the performance ended. There were 29

more boys than girls who remained in the hall. How many girls attended the

performace?

Source | Catholic High School (Primary) Primary 6 Examination

34 | P a g e

Case Study 9 |

35 | P a g e

Day 5 | Ratio and Proportion |Session 1

Problem-Solving Approach

Three-Part Lesson Format

A total of 325 boys and girls attended a performance in the school hall. 5

4 of

the boys and 4

3 of the girls left the hall after the performance ended. There

were 29 more boys than girls who remained in the hall. How many girls

attended the performace?

Source | Catholic High School (Primary) Primary 6 Examination

36 | P a g e

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?

37 | P a g e

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?

38 | P a g e

Advanced Bar Model Method |Session 2

Case Study 2 |

Four 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

39 | P a g e

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

40 | P a g e

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

41 | P a g e

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?

42 | P a g e

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.

43 | P a g e

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 .