math in focus a singapore approach … in focus a singapore approach overview richard bisk ph.d. ......

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Math In Focus A Singapore Approach

Overview

Richard Bisk Ph.D. Professor Emeritus

Mathematics Department Worcester State University

[email protected] https://sites.google.com/site/singmathproject/

What’s Most Important?

• Standards/Frameworks

• Curriculum

• Facilities

• Textbooks

• Assessments

• Teachers

Singapore Books in the U.S.

Two versions, all published by Marshall Cavendish Education:

a. Primary Math – US Edition, California Edition,

Common Core Edition.

b. Math In Focus – HMH Adaptation of My

Pals are Here. I am a consulting author for this series.

Singapore

• City-State of 5 million off the southern tip of the Malay peninsula.

• Third highest per capita income in the world.

• Before independence in 1965, its per capita GDP was $511.

Questions

1. What is the most important question in a

math class?

2. Why the interest in Singapore Math?

3. Why am I interested in Singapore Math?

4. What’s different about Singapore Math?

What are its key characteristics?

1. What is the most important

question in a math class?

Answer: “WHY?”

The habit of understanding must begin at a young age. Students who develop the view that the mathematics is about rules without meaning have difficulty when the math becomes more complex.

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Why the interest in Singapore?

a. TIMSS Studies - 1995, 1999, 2003, and 2007, 2011.

b. National Math Panel Report - 2008

c. Common Core State Standards Initiative (CCSSI) - 2010

TIMSS – 2011

South Korea

Singapore

Taiwan

Hong Kong

Japan

Russia

Israel

Finland

United States

England

International

613

611

609

586

570

539

516

514

509

507

500

Grade 8

Singapore

South Korea

Hong Kong

Taiwan

Japan

Northern Ireland

Belgium

Finland

England

Russia

International

606

605

602

591

585

562

549

545

542

542

500

Grade 4

Dr. Richard Bisk - [email protected]

Instruction in Singapore is in English

TIMSS – 4th Grade Problem Trends in International Math and Science Studies

Peter bought 70 items and Sue bought 90 items. Each item costs the same and the items cost $800 altogether. How much did Sue pay?

Singapore: 83%

Germany 57%

United States: 23%

National Math Panel

• Even in elementary school, the U.S. is not among the world leaders; only 7% of U.S. fourth-graders scored at the advanced level in TIMSS, compared to 38% of fourth-graders in Singapore, a world leader in mathematics achievement. (page 4)

• In elementary school textbooks in the United

States, easier arithmetic problems are presented far more frequently than harder problems. The opposite is the case in countries with higher mathematics achievement, such as Singapore. (page 26)

Common Core Standards

The composite standards [of Hong Kong, Korea and Singapore] have a number of features that can inform an international benchmarking process for the development of K–6 mathematics standards in the US.

(Second paragraph of introduction)

3. Why am I interested in Singapore

Math?

Professional Development

• “I never realized that I do not understand math until I had to teach mathematics from the Singapore textbooks.”

Teacher quoted in A.I.R. report prepared for U.S. DOE

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How do you teach math when you don’t know it well?

You focus on rules, procedures and memorization; or on manipulatives, games and activities that you can’t readily connect to concepts.

The Big Issue “How vs. Why”

In my professional development programs and in my classes for prospective teachers, I use the texts to help teachers develop a deep understanding of math so that they can teach for understanding.

4. What’s different about Singapore

Math? What are its key

characteristics?

• Depth emphasized over breadth: More time is spent on each topic. Fewer topics are covered in a year. Greater focus on mastery.

• Concrete-Pictorial-Abstract Approach: Abstraction gives math its power. But abstraction must be grounded in understanding.

• Problem Solving Emphasis: Model drawing diagrams are used to promote understanding of word problems and provide a bridge to algebraic thinking.

• More Multi-Step Problems: Problems often require the use of several concepts.

• Coherent Development: Topics are introduced with simple examples and then incrementally developed until more difficult problems are addressed.

• Mental Math: Techniques encourage understanding of mathematical properties and promote numerical fluency.

• Teacher and Parent Friendly: Since mathematical content is clear, it is often easier for teachers to plan lessons. Parents can read the books and help children.

• Minimal Explicit Review: Topics are taught for mastery. Prior material is constantly reused.

• A high level of expectation is implicit in the curriculum.

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Stress on Developing Conceptual Understanding: Students and teachers

learn to focus on “why” not just “how.”

The curriculum teaches for mastery

through understanding and practice.

Concepts from MIF

• CPA Approach

• Use of Anchor Problems/Activities

• Place Value

• Operations on Whole Numbers

• Mental Math

• Model Drawing / Bar Diagrams

• Operations on Fractions

• Algorithms

C→P→A

Concrete: ?

Pictorial: ││││││││

Abstract: 8

Key Idea

• Introduce Topics at a Conceptual Level

• Learning math facts and algorithms come after developing meaning.

• More examples as we discuss specific topics.

Anchor Tasks

• Start a lesson with a single task or problem;

• Often before being shown by the teacher how to solve the problem.

• Encourages problem solving.

• In MIF, these are labeled as “Learn.”

8 cm

3 cm

5 cm

Question 1 – Class in SG The figure shows two identical

right-angled triangles overlapping

each other. Find the area of the shaded

part.

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Place Value

We use the base 10 place value system.

In this system, we form bundles.

If we combine 8 straws and 5 straws, we get

IIIIIIII + IIIII = IIIIIIIIII + III

= 1 ten + 3 ones

Making Ten

part 7

10 whole

part 3


Ten Frame


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Addition and Subtraction

As we move through the elementary grades, we want a consistent meaning of these operations, regardless of whether we are dealing with whole numbers, fractions or integers.

Structure of Math

• Definitions

• Applications – Models for Using

• Models for Understanding

• Rules (aka Theorems)

3 + 2 is the number we get when we start at 3 and count 2 steps.

3 → 4 → 5

a + b is the number we get when we start at a and count b steps.

What is Addition?

Set Model – combine two sets of objects.

Count the number of objects.

The objects or nouns must be the same.

3 toys + 2 toys = 5 toys

Measurement Model – Combine objects and measure their total weight, length etc.

First I jumped 3 feet and then I jumped 2 feet. How many feet did I jump altogether.

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Number Line Model: 3 + 2

Start at 3 on the number line and take 2 steps to the right.

0 1 2 3 4 5

Compare this to the definition. We will address

this again when we discuss fractions

Computation Thinking Strategies

• Adding +1, +2

19 + 2 (by counting on)

• Doubles

4+4, 5+5, 6+6. 7+7, … (practice)

• Tens Combinations

3+7, 4+6, 2+8, …. (practice)

• Adding 10

6+10 (point out and use)

• Related to doubles

7+8 = (7+7)+1 (Mental math practice)

• Compensation

29 +13 = 30+12 (Mental math practice)

Meaning of “=“

The number to the left has the same value as the number to the right.

Good example: 9+6=10+5

Bad example: find 3+5+9+2

3+5=8+9=17+2=19

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Subtraction

How do we define subtraction?

What are the different types of applications of subtraction that are seen in the early grades?

What are some tools or models for helping students understand subtraction?

Definition: Subtraction is the _____________ addition. So 13 – 5 is the numbers that fits in the blank:

Number Bonds and Four Fact Families

7

4

3

Applications of Subtraction

• Take Away

• Part-Whole

• Comparison

Take Away

Bill has 7 cookies. He eats 3 cookies. How many cookies remain?

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Part - Part - Whole

Bill and Maria share 15 cookies. Bill gets 9 of them? How many does Maria get?

Comparison

Maria has 10 cookies.

Bill has 3 cookies.

How many more cookies does Maria have?

1A 1A

2A 1A

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Models for teaching

• Manipulatives – set model

• Number Lines - measurement model

• Counting Down

13 – 6 = 13 – 3 – 3 = 10 – 3 =7

• Counting Up on a Number Line

15 – 8 = 2 + 5 = 7

0 8 10 15

Grade 1 – Common Core

• CCSS.Math.Content.1.OA.B.4 Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.

• CCSS.Math.Content.1.OA.C.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.

Use strategies such as making ten ( 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14);


Anchor Task - Number Bonds and CPA Number Bonds

Number Bonds Grade 2 – Common Core

• CCSS.Math.Content.2.OA.B.2 Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers.

http://www.corestandards.org/Math/Content/1/OA/B/4

http://www.corestandards.org/Math/Content/1/OA/C/6

http://www.corestandards.org/Math/Content/2/OA/B/2

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MULTIPLICATION

Definition – Multiplication of whole numbers is repeated addition:

3 x 5 = 5 + 5 + 5

In this example 3 and 5 are factors; 15 is the product.

Three Models of Multiplication

Set model: 3 x 5 is interpreted as 3 groups of 5 objects.

Measurement model

3 x 5 is interpreted as a bar diagram with 3 sections of length 5. Or using a number line, start at 0 and take 3 hops of length 5.

Rectangular array model

3 x 5 is interpreted as 3 rows of 5 objects. The objects could be dots, stars or boxes.

While this is often called the area model, you should not use this term with

students until you have introduced the concept of area in 3rd or 4th grade.

4. Distributive Property

3 x (12) = 3 x (10 +2) = 3 x 10 + 3 x 2

This can be explained nicely using rectangular arrays.

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10 2

3

Distributive Property

6 x 7 = (6 x 5) + (6 x 2)

6 x 32 = (6 x 30) + (6 x 2)

6 * (y + 2) = 6y + 12

Important

These properties should be taught as principles about the ways numbers behave, illustrated with models and practiced with mental math exercises and word problems. They are not simply rules to memorize.

DIVISION

Definition – Division is defined by missing factors. The number 56 ÷ 8 is the unique missing factor in ____ x 8 = 56.

Multiplication and Division are opposite operations.

Interpretations of 20÷4

• Partitive Model: “20 is 4 groups of what unit?” (Sharing)

• Quotative Model: “20 is how many groups of 4?” (Grouping)

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Number of Objects sharing

groupin

g

It is important to use both models.

Which model works for division by a fraction?

Example: 12 pounds of flours is put into bags weighing 1 ½ pounds each. How many bags are there?

Clear Definitions are Important

They help us answer difficult questions.

Here’s one that confuses many math majors. But with clear definitions, it isn’t that hard.

Division by Zero

10 ÷ 0 = ?

0 ÷ 0 = ?

Why is it undefined?

KEY POINTS

Stress on Developing Conceptual Understanding: Students and teachers

learn to focus on “why” not just “how.”

Depth emphasized over breadth: The

curriculum teaches for mastery through

understanding and practice.

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MENTAL MATH

Emphasizes an understanding of place value and the distributive, commutative and associative properties

Compensation

7 + 8


3 + 5

More Compensation - Addition

99 + 28 =

43 + 78 =

Subtraction Compensation

72 – 59 =

Or 72 – 59 =

Richard Bisk - Worcester State University - [email protected]

Multiplication Compensation

50 x 18 =

Multiply by 25

48 x 25

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Division Compensation

2300 ÷ 50 =


Multiplying or Dividing by 4

• Multiply by 4

• Divide by 4

Richard Bisk - Worcester State University -

[email protected]

Multiplying or Dividing by 5

• Multiply by 5

• Divide by 5

Multiplying by 9

Multiply by 10 and subtract the original number:

35 x 9 =

Distributive Property (again)

13x 98

Various Properties that let us rearrange

2 x 387 x 5 =

256 x 4 =

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Mental Math Problems - I

1. 95 + 47 = _____

2. 81 - 38 = _____

3. 268 - 69 = _____

4. 362 – 98 = _____


Mental Math Problems - II

5. 50 × 42 = _____

6. 25 × 64 = _____

7. 20 × 11 = _____

8. 50 × 19 = _____


Mental Math Problems - III

9. 60 ÷ 25 = _____

10. 321 ÷ 50 = _____

11. 11 ÷ 20 = _____

12. 7 ÷ 25 = _____


Mental Math Problems

13. 15 × 9 = _______

14. 15 × 19 = _______

15. 99 ¼ × 12 = _________

16. 152 + 15 × 85 =__________


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Questions

What is a fraction?

Is it one number or two?

How would your students answer these questions?

C –> P -> A

• Conceptual First

• Formalize Last

Unit Fractions

Is this 1 unit shown 3 times? Or 1 unit broken into 3 pieces? 1/3 is the number that multiplied by 3 gives 1

It takes 3 “one thirds” to make 1.

2/3 means 2 of “what it takes 3 of to make 1.”

A fraction is a point on a number line.

0 1/3 2/3 3/3 4/3 5/3 6/3 7/3 8/3

Common Core – Grade 3

Understand a fraction 1/3 as the quantity formed by 1 part when a whole is partitioned into 3 equal parts; understand a fraction 2/3 as the quantity formed by 2 parts of size 1/3.

MIF – 2B

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Common Core – Grade 3

Understand a fraction as a number on the number line; represent fractions on a number line diagram.

2/3

The denominator tells us our noun - what we have.

The numerator tells us our adjective - how many we have.

2 dogs + 3 hats = ???

2 sevenths + 3 fourths = ????

Addition and Subtraction

4 sevenths + 2 sevenths = 6 sevenths

7 ninths - 5 ninths = 2 ninths

If we have the same fractional unit, we count, add and subtract fractions like whole numbers

2B

How does this compare to whole number addition?

4 one + 2 ones = 6 ones

7 tens – 5 tens = 2 tens

2 + 3 = 23 ????

2 tens + 3 ones = 23

Addition and Subtraction Different fractional units

Convert to equivalent fractions with like units.

Students must understand why this must be done, before they learn the algorithm.

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Equivalent Fractions Key Idea

MIF – 3B : p 121

3B – P124 Draw a sequence of (bar) diagrams to

show how to add

1/2 + 1/3

(Parker/Baldridge: page 140 example 2.4

1 2

2 5

MIF 5A

Ready for the rest of the page?

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What would you say to a student who does the following:

1/3 + 2/5 = 3/8

Multiplication of Fractions

What does it mean to multiply fractions? How can we define multiplication of fractions in a way that is consistent with our definition of multiplication of whole numbers?

6 x ½ or ½ x 6

We want them to have the same value.

We can think of this as 6 groups of ½.

We often read ½ x 6 as 1/2 of 6.

Divide it into 3 equal horizontal pieces

and shade one them. This is 1/3 of the

square.

Divide the square vertically into 2 equal pieces. Shade ½ of the 1/3.

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Division of Fractions

Yours is not to reason why, just invert and multiply. (O.P.)

If you believe in things that you don’t understand, then you suffer (S.W.)

Division of Fractions

How did we define division of whole numbers?

6 ÷ 2 = ?

2 x ? = 6

2

5

3

7 ?

? 3

7

2

5

DIVISION OF FRACTIONS The Teaching Sequence 1. Review: Whole Number ÷ Whole Number

2. Teach: Fraction ÷ Whole Number 3. Teach: Whole Number ÷ Fraction 4. Teach: Fraction ÷ Fraction

Example 1

Two children share 5 oranges equally.

Draw two models for 5 ÷ 2

Note relationship to multiplying by reciprocal.

Example 2

Four children shared 2/3 of a pie equally. What fraction of a pie did each get?

Draw models for 2/3 ÷ 4

a. pie b. bar model c. number line


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Example 3

Five oranges are cut into half-orange pieces. How many piece are there?

Draw models for 5 ÷ ½


Example 4

Find 6/7 ÷ 2/7

Draw a bar diagram.


Example 5

Find 5/7 ÷ 2/7

Draw a bar diagram.


Example 6

Find 5/6 ÷ 1/3

Draw a bar diagram.


What problems would you pose next?

Summary

• Fractions are numbers too!

• Same properties as whole numbers.

• Important to teach whole number arithmetic so that the transition to fractions and integers is seamless!

math in focus a singapore approach … in focus a singapore approach overview richard bisk ph.d. ......

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