BBS April 2010 Singapore Math in Indonesia by BBS Maths Consultant Dr Yeap Ban Har

Bina Bangsa Schoolprofessional development

for mathematics teachers

pedagogy for engagementproblem-based learning

and

www.mmepdpm.pbworks.comYeap Ban Har

What are the goals of this workshop?

To see examples of using problems to organize lessons so that students are engaged.

To use lesson study structure to deepen our understanding of student engagement

To provide platforms for learning to mastery – that means every student must have mastered the key focus of a lesson by the end of the lesson, and if not the next lesson should be designed for that

To provide constant opportunities to develop 21st Century skills

Example One

Primary Mathematics Standards Edition (Grade 6)

problemsfor students to learn new ideas

Area > 2r2

Area < 4r2

2r2 < Area < 4r2

2r2 < Area < 4r2

2r2 < r2 < 4r2





Learning through Inquiry


problemsfor students to apply what they learn










Example Two

Task 1 Find the longest distance between two points on a circle.Fuchun Primary School, Singapore: Lesson Study

Learning through CollaborationFuchun Primary School, Singapore: Lesson Study

Learning through CollaborationFuchun Primary School, Singapore: Lesson Study

Task 2 Measure circumference and diameter of four circles.Fuchun Primary School, Singapore: Lesson Study

A teacher observing student engagement during the lesson.Fuchun Primary School, Singapore: Lesson Study

Task 3 Draw a circle given that its circumference is 6 metres.Fuchun Primary School, Singapore: Lesson Study

Used the relationship between circumference and circumference

Did not use the relationship between radius and circumference

Learning by Doing

Professional LearningRatio 2.8 3.0 3.1 3.2 3.3 3.5 3.8

Frequency 1 3 5 7 18 4 1

Teachers discussing what they saw in the lesson, talking about how students can be taught to appreciate the significance of π and that its value is approximately 22/7.

The circles were of radii 4 cm, 6 cm, 8 cm and 10 cm. Why would a circle of radius 7 cm be included?

Singapore teachers doing PLC spend 2 hours a week on such activities to deepen their learning from workshops they have attended.

Example Three

Learning by Collaborating

Fuchun Primary School, Singapore

North Vista Primary School, Singapore

Example Four

This problem is about a game between two players.

n beans are placed between two players. They take turn to remove either 1 or 2 beans. They cannot remove any other number of beans. The winner is the one who takes the last 1 or 2 beans. If this cannot be done then the game ends in a draw.

You are given about 50 beans to investigate the winning strategy for this game.

Write a short essay that includes:1. a description of how the game is played2. a detailed description of how one can win a game3. an opinion on the statement ‘the player who starts will win the game’.

[12]

Anglo Singapore International School Bangkok 2010 Lower Secondary SA1

Learning by Reflecting

Example Five

All whole number can be written as a sum of consecutive whole numbers.

Investigate this statement.

Learning through Inquiry

Example Six

Da Qiao Primary School, Singapore

Problem: What is half of one half? What is half of one third?

Example Seven

1 ÷ ¾ = ?

12 ÷ 4 = 3

1 ÷ ¾ = ?

12 ÷ 4 = 3

1 ÷ ¾ =

3

11

2 ÷ ¾ =

3

112

3 ÷ ¾ =

3

113

n ÷ ¾ = 3

11n

12 ÷ 4 = 3

3 ÷ ¾ = 4

makingconnections

Primary Mathematics Standards Edition Grade 6

How to make sure the butterfly cannot fly

How do you get a butterfly?

First there is the egg which hatches into a caterpillar. The caterpillar eats and grows. At the right time, it makes a cocoon out of its own body. While in the cocoon, the caterpillar changes into a butterfly.

When the butterfly is ready, it starts to break through the cocoon. First a hole appears. Then the butterfly struggles to come out through the hole. This can take a few hours.

If you try to "help" the butterfly by cutting the cocoon, the butterfly will come out easily but it will never fly. Your "help" has destroyed the butterfly.

The butterfly can fly because it has to struggle to come out. The pushing forces lots of enzymes from the body to the wing tips. This strengthens the muscles, and reduces the body weight. In this way, the butterfly will be able to fly the moment it comes out of the cocoon. Otherwise it will simply fall to the ground, crawl around with a swollen body and shrunken wings, and soon die.

If the butterfly is not left to struggle to come out of the cocoon, it will never fly.

We can learn an important lesson from the butterfly.

Lim Siong Guan Head, Civil Service

Example Eight

Example Nine

The newspaper report stated that a draw would have sent Singapore to the Asian Cup Final. A team needed to finish first or second in their group to qualify for the Asian Cup Finals.

Figure 1

The table shows the number of each team had played (P), won (W), drew (D), lost (L) as well as the numbers of goals it scored (F) and the numbers of goals it conceded (A). A team got 3 points for a win, 1 point for a draw and no point for a loss. The total number of points is shown in the last column (Pts).

The results of the last two matches in each group are shown in Figure 3.

Use only the information available to explain why ”a draw would have sent Singapore to the Asian Cup Finals”.

Figure 1

Figure 3

Learning in and about the Real World

Example Ten

Best Chance to Score

• A soccer player is on a breakaway, dribbling the ball downfield, parallel to a sideline. From where should he shoot to have the best chance to score a goal?

PSLE2010 Problems

Cup cakes are sold at 40 cents each. What is the greatest number of cup cakes that can be bought with $95?

$95 ÷ 40 cents = 237.5

Answer: 237 cupcakes

Basic Skill Item

Basic Application Item

4 puffs for $2.80

$50 ÷ $2.80 = 17.86 (to 2 decimal

places)17 x $2.80 = $47.60

(17 x 4 + 3) puffs = 71 puffs

$50 - $47.60 = $2.40 3 puffs

Learning in and about the Real World

Engaging Students

• Learning through Inquiry• Learning by Doing• Leaning through Collaboration• Learning through Reflection• Learning in and about the Real World

Student Engagement in Research Lessons at BBS

Student Engagement in Research Lessons at BBS

PSLE2010 Problems

Application Item

210

5

1

8

3

210

5

1

8

3

21040

7

5

1

8

3

3040

1

96040

32

5

4 Mrs Hoon sold 960

cookies.

Bina Bangsa Schoolprofessional development

for mathematics teachers

problem-based learning


Example Eleven

Problem: Find the distance between points A (2, 3) and B (10, 9).Students working collaboratively on a problem to learn the coordinate geometry topic on distance between points. Traditionally, students were given a formulae to do this.

Every group was able to use their previous learning (Pythagoras Theorem) to solve the main problem)

Later they were asked to find the distance between the points (5, 1) and (9, 4). They were also asked to find points where the distance is a whole number.

53

4

10

6

8

15

12

12

Problem-Based Lesson

• Find the distance between points A (2, 3) and B (10, 9).

• Find the distance between points C (-2, -2) and D (2, 1).

• Find the distance between points E (2, 5) and F (x, 2).

• Find two points where the distance is a whole number.

Group Task

• Design one double-period lesson based on a single problem with colleagues teaching the same level.

• Present your lesson to include– The Problem– Expected Solutions from Students– Is the problem-based lesson to teach a new topic,

for drill-and-practice or for applying knowledge?– Please submit your problems via email

[email protected] or upload on www.mmepdpm.pbworks

mailto:[email protected]

http://www.mmepdpm.pbworks/

Problem

What is the topic or learning

outcome?

What are the anticipated

responses or difficulties?

• Inquiry• Doing• Collaboration• Reflection• Real World

• Content• Process• Skills 21

Example Twelve

Teachers solved the problem of converting a square into an equilateral triangle of the same area. Three suggestions were forwarded.

Change the square into an equilateral triangle of the same area.

Suggestion 1

Suggestion 2

Suggestion 3

Example Thirteen

A whole number that is equal to the sum of all its factors except itself is a perfect number.

Find perfect numbers.

The ancient Christian scholar Augustine explained that God could have created the world in an instant but chose to do it in a perfect number of days, 6. Early Jewish commentators felt that the perfection of the universe was shown by the Moon's period of 28 days.

The next in line are 496, 8128 and 33 550 336. As René Descartes pointed out perfect numbers like perfect men are very rare.

Example Fourteen

Problem: How is the circumference and diameter of a circle related?Fuchun Primary School, Singapore: Lesson Study

Professional LearningRatio 2.8 3.0 3.1 3.2 3.3 3.5 3.8

Frequency 1 3 5 7 18 4 1

Teachers discussing what they saw in the lesson, talking about how students can be taught to appreciate the significance of π and that its value is approximately 22/7.

The circles were of radii 4 cm, 6 cm, 8 cm and 10 cm. Why would a circle of radius 7 cm be included?

Singapore teachers doing PLC spend 2 hours a week on such activities to deepen their learning from workshops they have attended.

Example Fifteen

Rectangle 1

Rectangle 2

Problem: Make Rectangle 10.

Problem-Based Lesson

• Find the number of square tiles in Rectangle 10.

• Find the number of square tiles in Rectangle n where n is any large number.

• Which rectangle has 63 square tiles?

Example 16

Find the area bounded by the curve

the x-axis and lines x = 1 and x = 4.

)1ln( xy

Problem to teach trapezium rule.

Example 17

Two forces 3 N and 4 N are applied on an object. Find the resultant force on the object.

Problem to teach vector addition.

Example 18

Problem to teach measurement of volume of liquids.

PCF Kindergarten Pasir Ris West, Singapore

Example 19

How many non-congruent quadrilaterals can be made? Each quadrilateral is made using all the tangram pieces.

“Children are truly the future of our

nation. “Irving Harris

Education

BBS April 2010 Singapore Math in Indonesia by BBS Maths Consultant Dr Yeap Ban Har