math in focus - edl · pdf filepiaget’s cognitive learning theory that it is not...

Math in Focus Grades K‐5

Additional Online Resources

The History

The Underpinning Concept

Place Value

Model Drawing

Problem Solving

NEW PATHWAYS TO SUCCESS: A HISTORY OF SINGAPORE MATH

Background: History of the Development of Singapore Maths TextbooksFrom 1965 to 1979, many of the primary and secondary textbooks (and all mathematics textbooks) used in Singapore had been imported from other nations. But starting in 1980, Singapore began to take a new approach. The Curriculum Development Institute of Singapore (CDIS) was set up to develop primary and secondary textbooks for the New Education System. This initiative produced new developments in learning that propelled Singapore’s math students to the top of the international community.

Implementation of Primary MathsSingapore has a centralized educational system, meaning that the Ministry of Education sets all learning standards. As the CDIS began working to improve textbooks, the Ministry set new goals for math education. Many of the goals focused on problem solving and the heuristic of model drawing. The goals included:

acquisition and application of knowledge and skills •use of math language •foundation building•development of a positive attitude•appreciation of the power of math •

Primary Mathematics 1–6, the first Singapore Math program, was published in 1982. In 1992, the second edition of Primary Mathematics was revised to include more problem solving, with particular focus on using model drawing as a way of approaching solutions.

The new math curricula led to dramatic improvements in math proficiency. In 1984, Singapore scored 16th out of 26 nations in the Second International Science Study (SISS). Then in 1995, the nation’s students placed first in the Trends in International Mathematics and Science Study (TIMSS). This survey, conducted every four years, is designed to provide reliable data on how math and science achievement in the U.S. compares to that in other nations.

A Tighter Focus on ContentIn July of 1998, the Ministry of Education initiated another study of mathematics curricula, this time looking at the scope and sequence of their textbooks. The result of the study was a call for tighter content focus. Content removed or reduced from the subject syllabi included concepts that are not fundamental or that rely on plain recall, content covered at other levels or in other subjects, content that focused on technical details rather than conceptual understanding, and content no longer relevant in real-world practice.

More Emphasis on Problem SolvingIn 2006, Singapore once again revised the mathematics curriculum, this time placing greater emphasis on developing mathematical concepts and fostering the ability to apply them in mathematical problem solving situations. In addition, the new guidelines:

emphasize computational skills along with more conceptual •and strategic thinking.cover fewer topics in-depth and are carefully sequenced grade-by-grade.•cover concepts in one grade and in later grades at a more advanced level.•ensure that students master prior content, not repeat it.• encourage representing problems mathematically, using reasoning, •and communicating mathematical content.

Ministry of Education, Singapore 2006

More Choices for Teachers and StudentsIn 2001, Singapore privatized production of its primary level mathematics textbooks. Officials hoped that the collaboration with commercial publishers would produce quality textbooks at more affordable prices, allowing more choices for teachers and students. Singapore required all publishers to use the same focused syllabus developed in 1999.

The most widely used and highly rated new textbook series is My Pals Are Here! Maths for grades 1-6 published by Marshall Cavendish (Singapore) Pte Ltd. In 2007, Great Source— an imprint of Houghton Mifflin Harcourt Publishing Company, one of the leading educational publishers in the U.S., announced plans to develop a new, alternative elementary math program based on My Pals Are Here! Maths. The American version is titled Math in Focus™: The Singapore Approach1 , and will be available in 2009.

A Nation That Values LearningThanks to its remarkable mathematics program and the support of its people, Singapore has placed first on the TIMSS in 1995, 1999, and 2003, and in the top three nations in 2007. Those familiar with Singapore and its education system understand why. Anyone who visits Singapore or interacts with their educators learns quickly that the people of Singapore care deeply about their country and the education of the nation’s children. In Singapore the rule is, “No one owes me a job so everyone has to learn for himself.”

The government of Singapore devotes 20% of its Gross National Product (GNP) to education. Parents are actively involved in making sure every child is well-educated. From the Ministry of Education to the schools, the principals, the teachers, and the parents, everyone in Singapore believes that education is the answer to the preservation of the nation and its economy. Perhaps this is why the history of Singapore math from 1980 up to the present day is so celebrated, and why it has captured the interest of the world.

Patsy F. Kanter is an author, teacher, and international math consultant. She worked as the Lower School Math Coordinator and Assistant Principal at Isidore Newman School in New Orleans, Louisiana for 13 years. Patsy is the author of a number of mathematics programs, including Afterschool Achievers: Math Club (K–8), and Summer Success®: Math 2 (K–8), and co-author of Every Day Counts® Calendar Math and Partner Games (K–6) and Practice Counts3 (1–6). She is a consulting author on the U.S. version of My Pals Are Here! Maths, Math in Focus™: The Singapore Approach.

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1Math in Focus™ is a trademark of Marshall Cavendish (Singapore) Pte Ltd. 2Summer Success® is a registered trademark of Great Source, a Houghton Mifflin Harcourt Publishing Company 3Every Day Counts® is a registered trademark of Great Source, a Houghton Mifflin Harcourt Publishing Company

Math in Focus: The Singapore Approach The Underpinning ConceptBy Dr. Fong Ho KheongAssociate Professor,Head of Math and SciencesBahrain Teachers CollegeUniversity of Bahrain

IntroductionMath in Focus: The Singapore Approach is adapted from Singapore’s My Pals Are Here! Maths. The underpinning philosophy of both series is the same, and their aims and objectives are to ensure children’s ability to achieve mastery of mathematics concepts, computational skills, problem solving skills and application of mathematics to daily life activities. Math in Focus has also adopted both the Eastern and Western approaches to critical and creative thinking, thus preparing students to face challenges ahead of them. The Eastern style of teaching focuses on drill and practice to ensure mastery of facts, computation, and problem solving skills. The approach is related to Piaget’s cognitive learning theory that it is not sufficient to just assimilate ideas but also to accommodate ideas learned through drill, practice, and reflection. The Western approach complements the structured style of learning through peer interaction to stimulate thinking that leads to critical, enquiry-based, and creative thinking. In order to achieve these objectives, the author conceptualizes a framework for teaching and learning which were based on some well-proven teaching and learning theories, as well as research on how best children are taught certain mathematical concepts, computational skills, problem solving skills, and application of mathematics into daily life.

Singapore Math CurriculumMy Pals are Here! Maths is written to align closely to the Math framework developed by the Ministry of Education, Singapore. The Singapore Math curriculum has evolved over 20 years. In the 1980’s, the focus of the curriculum was on conceptual learning and problem solving. Towards the end of the 1990’s, based on the thinking school concept, the Singapore math curriculum expanded to develop different forms of thinking (creative, critical, and enquiry-based) through participation in math activities and solving mathematical problems. The framework of the Singapore math syllabus covers learning skills, concepts, processes, meta cognition, and developing students’ attitudes to learn and love mathematics. The Singapore Math curriculum is in line with the NCTM principles and standards.

Pedagogical and Theoretical BackgroundThe Math in Focus series adopted the constructivist approach to help children master mathematical concepts and skills. The following sequence of activities is used in the textbook: introduction of knowledge, informal assessment and reflection of knowledge, reinforcement of math concepts, and computational

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and problem solving skills through peer interaction in group activities and practice. Further activities are incorporated that empower critical and creative thinking in the curriculum, such as the investigative activities and Put on Your Thinking Cap problems. To help children construct mathematical knowledge and problem solving skills, presentation of learning concepts needs to be harmonious with children‘s ability to understand and construct the knowledge. Children need to see connections (emphasized by NCTM in the current standards and principles) between concepts that help them understand and master mathematical concepts, solve mathematical problems, and carry out investigation procedures. Math In Focus emphasizes connections so students see the links between concepts and topics, which helps them understand and solve problems such as finding the connection between geometry and numbers.

(i) Adoption of Dr. Jerome Bruner’s and Jean Piaget’s Theories

The most significant theory, which has been adopted for writing the Math in Focus series, is Bruner‘s theory on representations of mathematical concepts according to different levels of children‘s thinking. The representation based on the concrete, pictorial, and abstract (CPA) is adopted in the whole series. The way I interpret Bruner‘s theory is basically parallel to Piaget‘s stages of development theory. Children at a certain age in general can only conceptualize mathematical concepts depending on their levels of mental development. In this aspect, Bruner‘s idea was to emphasize concrete representation, which is in harmony with some children‘s ability to understand mathematical concepts at the early stages. Research shows that children could not depend too much on concrete representation as they need to move on to the next level so that they could conceptualize abstract (complicated) situations using pictorial representation such as the ‘model’ approach used in Singapore‘s My Pals are Here! Maths and Math in Focus: The Singapore Approach. Although not all challenging problems could be tackled using the ‘model’ method, it plays a significant role to help average and below average students solve the problems based on their levels of thinking at the concrete and semi-concrete operational stage.

(ii) Richard Skemp’s Theory of Understanding

The entire Math In Focus series also adopts Skemp‘s theory of instrumental and relational understanding. The gist is when a child understands in relation to other facts (relational understanding), the child can remember better than memorizing the facts without really understanding (instrumental understanding). Relational understanding also helps children to extend their knowledge in problem solving skills. Understanding this approach is related to the CPA approach. Using concrete and pictorial representation helps children to understand the concepts and skills presented.

(iii) Theory on Constructivism

The constructivist’s theory is broadly applied in this series. Mathematical ideas are conceptualized by constructing mathematical concepts and skills through simulation and accommodation. They are also enforced through various activities. Activities besides peer interaction involve continuous practice such as using journals for reflection and reinforcement. During peer interactions, the ideas taught are either reconstructed or reinforced to enable correction and understanding of ideas learned. Further constructivism activities are also emphasized to ensure mastery and student confidence in activities such as journal writing, which involves tremendous demand on student’s recall and reflection of the ideas they have learned.

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Strategies and MethodsMath In Focus is based upon the use of mental computation, model approach, and the heuristics which enhance teaching and learning. Mental computation is imminent if the children are to master mathematical skills and problem solving. Thorough speed and the ability to operate mentally helps children to excel in math. The program introduces not only the mental strategies, but also prepares students to perform mental sums through the use of number bonds and manipulatives like unit cubes and the math balance.

The use of model approach or bar modeling is based on the fact that children at the elementary stage would not be able to solve abstract problems. The introduction of the model approach helps children to visualize and be able to see connections between facts and information which embeds in the questions. The ability to see connections helps pupils to solve difficult and complicated problems. This is the cutting edge of the model method. In other words, the model method simplifies the problem solving situation and translates to a form which average and below average students can conceptualize.

The use of heuristics is also another strategy which helps average and below average students to tackle challenging questions. Based on Richard Gardner‘s theory, each child has his/her own talent to learn, and only by identifying the talent can you innovate the method to help them. This philosophy is adopted in this series. The use of diagrams, some manipulatives, games, and active participation are in accordance to Gardner‘s theory.

Creativity, Critical, and Enquiry ThinkingEducationists have advocated creativity, critical, and enquiry thinking. Achieving these thinking skills is the key objective of the Math in Focus series. Creativity could be trained if the children are put into an environment where their mental thinking is evoked. The series contains many activities where they are asked to create alternative solutions to a problem. Likewise, critical thinking is evoked through activities that require children to give suggestions on situations where variations are incorporated.

ConclusionThe ultimate objective of the Math In Focus series is to develop a program that enables children to empower their thinking and develop skills that will help lead them to a better future in society. This can be achieved through getting children involved in various activities (peer activities and interaction, further practice and journal writings), which helps them to develop creativity, critical thinking, and enquiring minds.

Math In Focus has adopted some useful research, teaching, and learning theories to develop the program, and led the author to write training materials which help students with mathematical concepts, skills, problem solving and mathematical investigations. The materials from Math in Focus were adapted from My Pals are Here! Maths, which was the most popular series used by Singapore schools. The My Pals are Here! Maths series has shown proven results in the 2007 TIMSS results. Children using this series were proven to excel in the international survey.

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About the AuthorDr. Fong Ho Kheong is an Associate Professor and Head of the Math and Science Department of the Bahrain Teachers College of the University of Bahrain in the Kingdom of Bahrain. He is also a former Associate Professor of the National Institute of Education, Nanyang Technological University, Singapore. He was involved in training Mathematics teachers in the National Institute of Education, Singapore, for 25 years. He also worked in the Education Testing Centre, University of New South Wales, Australia, dealing with assessment in primary Mathematics. He is the Founding President of the Association of the Mathematics Educators, Singapore. Dr. Fong obtained his Ph.D. from the University of London. He specializes in teaching high ability children and children who have problems in Mathematics. His research work includes diagnosing children with mathematical difficulties, teaching them to think to solve mathematical problems, and apply psychological theories for the teaching and learning of Mathematics. His experience in curriculum development has led him to innovate the use of the model drawing approach to tackle challenging problems. He has published more than 100 journal articles and research reports, as well as primary and secondary Mathematics books.

He is the consultant and principal author of Marshall Cavendish’s My Pals are Here! Maths series, which is currently being used by 80% of the primary schools in Singapore. He is the principal author of Math in Focus: The Singapore Approach, the United States Edition of My Pals are Here!, also published by Marshall Cavendish and distributed in the U.S. by Great Source, an imprint of Houghton Mifflin Harcourt.

ReferencesBruner, J. (2004). Toward a Theory of Instruction (Belknap Press).Cambridge, MA: Belknap Press of the Harvard University Press.Gardner, H. (1993). Frames of Mind: The Theory of Multiple Intelligences.New York: Basic Books.Gardner, H. (1999). Intelligences Reframed: Multiple Intelligences for the 21st Century. New York: Basic Books.Skemp, R. R. (1987). The Psychology of Learning Mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates.Slavin, Robert (2006). Educational Psychology: Theory and Practice. ISBN 0205373402

© Copyright Dr. Fong Ho Kheong 2009 1

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© Houghton Mifflin Harcourt Publishing Company. All rights reserved. Printed in the U.S.A. 1/10 4821 Z-1437071Math in FocusTM is a trademark of Times Publishing Limited

800.289.4490www.greatsource.com/mathinfocus

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Singapore Math: Place Value in Math in Focus™ There are several skills that are critical to mathematics success in the elementary grades. Among them are an understanding of number, number sense, and place value. Place value is the system of writing numerals in which the value of the digit is determined by its position, or relationship to the other digits. These values are multiples of a common base of 10 in our decimal system. In Singapore, where students consistently outperform American students in math, place value is considered a foundational skill for all mathematics learning.

The U.S. edition of Singapore’s math curriculum, Math in Focus, stresses mastery, coherence, and focus. Students master foundational math skills early and carry those skills with them as they progress through the grades. With each new concept learned, students build on what they have already learned. This systematic approach to mathematics is particularly evident in the program’s approach to teaching place value. From grade level to grade level, place value is developed, honed, and applied to operations. As a result, numbers are demystified and students excel in mathematics because, for them, numbers become tools for solving problems rather than obstacles that cause frustration.

Throughout Math in Focus, a concrete-pictorial-abstract approach encourages students to use place value blocks, chips, charts, and strips to create an association between the physical representation of numbers, the number symbol, and the number name, and later to perform arithmetic operations using these materials. These concrete and pictorial representations add meaning to numbers so that when students progress to the abstract, they have an understanding of what the numbers stand for. This way students experience problem solving as more than just a process, there is real meaning behind the elements of a problem.

In Grade 1, place value is developed in a coherent, focused manner to stress the importance of the number 10 and help students master it. Grade 1, Chapter 1 begins with the study of the set of numbers from 0 to 10. This is followed by an introduction to number bonds, showing the additive relationship between numbers less than 10 and, in later chapters, adding and subtracting up to 10. In these chapters, students come to understand 10 as an anchor number for future applications of sums and differences with 10 and multiples of 10.

1 50 Chapter 3 Addition Facts to 10

Add.Use number bonds to help you.

7 How many monkeys are there in all?

3

7

4

+ = + =

There are monkeys in all.

You can add in any order.

+ = +

4 added on to 3 is equal to 7.

added on to is

also equal to 7.

Gr1 TB A_Ch 3.indd 50 4/20/09 3:32:26 PM

Grade 1A, Chapter 3, page 50

In Grade 1, the study of place value concludes with the study of numbers to 100 with addition and subtraction. At this point, students are fully prepared to expand their understanding to three digits because of the gradual, strategic way in which it has been introduced, understood, and practiced. In addition, first-grade students are taught mental math in a chapter which calls on their knowledge of number bonds with 10 and multiples of 10 to solve mental computation problems.

In Grade 2, students study how to count, read and write numbers up to 1,000, and learn to add and subtract numbers to 1,000 with and without regrouping. Here place value is key to understanding why the algorithms work. Students learn how the place value ideas they learned in Grade 1 can be applied to the thousands place by using place value blocks. In addition, there is a chapter that teaches mental math strategies for adding and subtracting three-digit numbers without the use of paper and pencil. This solidifies students’ understanding of place value.

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Grade 1B, Chapter 14, page 138


138 Chapter 14 Mental Math Strategies

Find 23 + 10.

Group 23 into tens and ones.

Step 1 Add the tens. 20 + 10 = 30

Step 2 Add the result to the ones. 3 + 30 = 33

So, 23 + 10 = 33.

20

23

3

You can add tens mentally using the add the tens’ strategy.

Add mentally.

4 Find 15 + 20.

Group 15 into tens and ones.

Step 1 Add the tens. 10 + =

Step 2 Add the result to the ones. + =

So, 15 + 20 = .

5 Find 29 + 10.

Guided Practice

10

15

?

G1B_TB_Ch14.indd 138 12/31/08 9:52:08 AM

le

arn You can add tens to a 3-digit number mentally using the add the tens’ strategy.Find 213 + 50.Group 213 into tens, and hundreds and ones.

10

213

203

Step 1 Add the tens. 10 + 50 = 60

Step 2 Add the result to the hundreds 203 + 60 = 263 and ones.

So, 213 + 50 = 263.

tens

hundreds and ones

Add mentally. Use number bonds to help you.

10 Find 351 + 40. Group 351 into tens, and hundreds and ones.

?

351

301

11 237 + 50 = 12 613 + 70 =

Guided Practice

Step 1 Add the tens. + 40 =

Step 2 Add the result to the hundreds and ones.

301 + =

So, 351 + 40 = .

le

arn

12 Chapter 10 Mental Math and Estimation

MiF 2B_TB_Ch 10.indd 12 12/19/08 2:36:26 PM2

Place value is introduced with numbers to 20, in a chapter that examines all “teen” numbers (11 to 19) as “10 plus numbers.” This helps students master two major concepts that will play a pivotal role in their future mathematics learning:

1. Seeing all two-digit numbers as composed of tens and more2. Understanding expanded notation—10 + 1 or 1 ten plus 1, 10 + 2 or

one ten plus two,10 + 3 or one ten plus three, and so on.

Next comes a chapter focused on practice with addition and subtraction to 20, which helps to solidify students’ understanding of 10 as an anchor number, the place value of numbers to 20, and adding and subtracting of numbers to 20.

By the second half of Grade 1, students using Math in Focus have developed a solid understanding of place value with numbers to 20. As the year progresses, the concept of place value continues to be developed with a chapter devoted to numbers to 40, followed by addition and subtraction to 40. In this chapter students begin their study of regrouping, which is clearly based on their earlier study of place value up to 40. Previously, students learned to make a ten and learned how to use number bonds. These foundational skills enable them to begin working with multi-digit numbers. Focusing only on numbers to 40 allows students to associate numerals with easily managed quantities of physical materials while the place value concept is developed.


Lesson 1 Ways to Add 197

Guided Practice

Make a 10.Then add.Use number bonds to help you.

1 9 + 5 = 2 8 + 7 =

Let’s Practice

Make a 10.Then add.

1 9 + 4 =

2 7 + 9 = 3 9 + 8 =

4 8 + 3 = 10 + 5 6 + 8 = +

= =

on YoUR oWn

Go to Workbook A:Practice 1, pages 191–196

Gr1 TB A_Ch 8.indd 197 4/20/09 5:40:37 PM

Throughout Math in Focus, students develop fluency in understanding numbers in multiple place value representations; for example, two thousand five hundred is 25 hundreds or 250 tens.

In the later chapters of Grade 3, students explore strategies for mental math and addition and subtraction for numbers to 10,000. These are the final chapters exploring place value addition and subtraction. In Grades 4 and 5, students continue to explore place value, but now with an emphasis on multiplication and division, and decimals.

In Grade 4, students extend their understanding of the place value system to the hundred thousands place and decimals to hundredths. Again, fluency with place value is emphasized as students learn to understand. For example, students see 38.2 as 38 and 2 tenths, or 3 tens and 82 tenths, or 382 tenths, and can then apply their knowledge of place value to multiplication, division, and mental math.

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Multiplying with regrouping in hundreds, tens, and ones

Find1257.

Step 1Multiplytheonesby7.5ones735onesRegrouptheones.35ones3tens5ones

Step 2Multiplythetensby7.2tens714tensAddthetens.14tens 1 3tens17tensRegroupthetens.17tens1hundred7tens

Step 3Multiplythehundredsby7.1hundred77hundredsAddthehundreds.7hundreds11hundred8hundredsSo,1257875.

Step 3Multiplythehundredsby3.2hundreds 3 6hundreds

2 3 630 3 902003 600Total 696

1257

75

31

1257

5

3

1257

875

31

Hundreds Tens Ones

2003600 30390 236

72 Chapter 3 WholeNumberMultiplicationandDivision

Grade 4B, Chapter 7, page14

Express each of these as a decimal.

1 4100

oz5 oz 2 6100

in.5 in.

3 fivehundredths5 4 8hundredths5

Find the decimals that the shaded parts represent.

5 6

Find the decimal for each point on the number line.

7 0 0.10.01

Guided Practice

Find equivalent tenths and hundredths.Le

arn

10100

isequalto 110

or0.1.

10hundredths5 1tenth

Youcanregroup10hundredthsas1tenth.

Le

arn

Le

arn

0.03 0.05 0.06 0.08

Tenths Hundredths Tenths Hundredths

14 Chapter 7 Decimals


In Grade 3, the place value chart is extended to the ten thousands place. Predictably, the study of addition and subtraction in the ten thousands place follows. The depth of development of place value in Grade 3 directly correlates to the depth of development in Singapore Math. The chapter teaches the following skills:

1. Count, read, and write numbers to 10,000.2. Learn to count by ones, tens, hundreds, and thousands.3. Show representations of numbers up to 10,000 using place value

charts and place value strips.4. Identify the place value of each digit in the number and express the

number in standard, word, and expanded forms.5. Compare and verbally describe sets of numbers using the terms least

and greatest. 6. Write five-digit numbers in increasing or decreasing order.7. Apply the number and place value concepts to identify and complete

number patterns and find missing numbers on a number line.

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1 1,3295 thousand hundreds tens ones

2 1,3295 1 1 1

3 In1,329,

thedigit isinthethousandsplace.

itstandsfor .

itsvalueis .

Find each missing number or word.Use base-ten blocks to help you.

4 In2,548,

thedigit isinthehundredsplace.

thedigit4standsfor .

thevalueofthedigit8is .

5 In2,562,thevaluesofthedigit2are:

2,562

How many are there? Find the missing numbers.

Guided Practice

Thousands Hundreds Tens Ones

14 Chapter 1 Numbersto10,000

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ConclusionIn Math in Focus place value is presented in a systematic and focused manner. From the very beginning students understand numbers as a system of tens. This system lays the foundation for all future lessons including computation skills, problem solving, the decimal system, and more.

Mastery of math in the elementary grades is key for student success in the classroom and in the 21st century working world they will eventually enter. Since the 1980s, students in Singapore have consistently outscored U.S. students in math. Now, with Math in Focus, U.S. students can learn using the same effective manner with which students in Singapore have had such success, preparing them for a lifetime of deep mathematics understanding and real-world problem solving.

About the AuthorPatsy F. Kanter is an author, teacher, and international math consultant. She worked as the Lower School Math Coordinator and Assistant Principal at Isidore Newman School in New Orleans, Louisiana, for 13 years. Kanter is the author of a number of mathematics programs, including Afterschool Achievers: Math Club (K–8) and Summer Success®: Math (K–8), and co-author of Every Day Counts®: Calendar Math, Partner Games (K–6), and Practice Counts (1–6). She is a consulting author for Math in Focus, which is the U.S. version of Singapore’s My Pals Are Here! Maths. Both programs are published by Marshall Cavendish, Singapore (part of Times Publishing Limited).

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By Grade 5, students extend their understanding of the place value system to the millions place and the decimal places to thousandths. The focus is on division by two-digit numbers, decimal addition and subtraction, and multiplying and dividing decimals by whole numbers. As is the case throughout Math in Focus, place value is still the focus. For example, in learning how to multiply 0.6 by 8, students learn to think of the problem as multiplying 6 tenths by 8 to get 48 tenths, which is regrouped to form 4 and 8 tenths, or 4.8.

Divide decimals with one decimal place by a whole number with regrouping.

Divide0.8by5.

Le

arn

Step 1Dividetheonesby5.0ones50ones

Step 2Dividethetenthsby5.8tenths51tenthR3tenths

Regrouptheremainder3tenths.3tenths30hundredths

Step 3Dividethehundredthsby5.30hundredths56hundredths

05 0 . 8

0

0 . 15 0 . 8 0

0

8

53 0

0 . 1 65 0 . 8 0

0

8

5

3 0

3 00

Ones Tenths Hundredths



Ones Tenths

So,0.850.16.

Lesson 9.3DividingDecimals53


Singapore Math: A Visual Approach to Word ProblemsModel Drawing in Math in Focus™ Since the early 1980s, a distinguishing characteristic of the math taught in Singapore—a top performing nation as seen on the Trends in International Math and Science Study (TIMSS) reports of 1995, 1999, 2003, and 2007—is the use of “model drawing”. Model drawing, often called “bar modeling” in the U.S., is a systematic method of representing word problems and number relationships that is explicitly taught beginning in second grade and extending all the way to secondary algebra. Students are taught to use rectangular “bars” to represent the relationship between known and unknown numerical quantities and to solve problems related to these quantities.

In Singapore, 86% of primary schools use the math series My Pals Are Here! Maths. In Math in Focus, the U.S. edition of My Pals Are Here! Maths, students learn to use the bars to model problems that involve the four operations both with whole numbers, fractions, and ratios. The use of the rectangular bars and the identification of the unknown quantity with a question mark help students visualize the problem and know what operations to perform—in short, viewing all problems from an algebraic perspective beginning in early elementary grade levels.

The problems might be as simple as:

Jane has 10 cookies and Joe has 12, how many do they have altogether? (second grade)

Or as complex as:

Jessica and Lillian had the same amount of money. Jessica gave $1,140 to a charity and Lillian gave $580 to a different charity. In the end Lillian had 9 times as much money as Jessica. How much money did each girl have at first?

What makes model drawing so effective is less about the specific model—the rectangles—than the systematic and consistent way it is taught. Each grade level addresses distinct operations and number relationships—addition and subtraction in second grade, multiplication and division in third, fractions and ratios in fourth and fifth—so students can visualize and solve increasingly complex problems.

Typically, students in the U.S. are taught a variety of strategies for problem solving, including “draw a picture.” But usually this entails drawing objects, animals, or counters. It is not very efficient when you move to larger numbers. In Singapore, students learn to represent these objects with rectangles that enable them to see the number relationships, rather than focusing on the objects of the problems. Rectangles are used because they are easy to draw, divide, represent larger numbers, and display proportional relationships.

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2 3

Bar Modeling: Pictorial UnderstandingStudents are first introduced to model drawing in second grade to represent part/part whole situations that can be solved with addition or subtraction. The first problem, introduced in second grade, might be as simple as:

Helen has 14 breadsticks. Her friend has 17. How may do they have altogether?

Students would draw one bar, divided into two parts, one slightly longer than the other. In this problem the two parts are “known,” and the student must add to find the whole or the “unknown.”

But the next problem is:

There are 21 fish in a bowl. Fifteen are from students. The rest are from the school. How many are from the school?

Notice in this problem, the student knows the whole and one part, and can solve for the missing part either by adding up or subtracting, so students understand the relationship between addition and subtraction. Students solve for an unknown variable at a pictorial stage, which aids the transition into the abstract.

While part/part whole models can be used to represent many subtraction problems, they cannot be used to represent comparison problems—how many more or fewer is one quantity compared to another. Such a problem might be:

Grant buys 345 fruit bars. Ken buys 230 more fruit bars than Grant. How many fruit bars does Ken buy?

Notice how visually clear these comparison problems become when the two rectangles are drawn. Even when the problems become more complex—for instance asking students how much Ken and Grant have altogether—the visual representation helps students realize they must first figure out how many Ken has and then how much they have altogether. With just these two models, students can solve most multi-step, complex addition and subtraction problems.

Multiplication, Division, and FractionsAt the end of second grade and more thoroughly in third grade, students are taught to model problems that are solved by multiplication and division. Again, students distinguish between part/whole problems and comparisons. The first might be a problem like this: A box has 12 pencils. How many pencils are in 5 boxes? Students would draw a bar with five parts, each labeled as 12.

15 ?

21

?

14 17

?

Grant

Ken

345 fruit bars

345

230 fruit bars

The comparison problems might be as simple as:

Jim has $15. Tom has twice as much. How much does Tom have?

Or as challenging as:

The sum of two numbers is 36. The larger number is three times the smaller number. Find the two numbers.

Imagine drawing the smaller number as a rectangle. Then the larger number would be three of them and the sum of the two is 36.

The student quickly visualizes that the sum of the four bars is 36, and that 36 ÷ 4 = 9 for the smaller number and 27 for the larger one.

Students are gently led from simple problems with easily manipulated numbers to more complex ones that require more arithmetic and multiple steps. A comparable multiplication problem in fourth grade is:

Lisa had 1750 stamps. Minah had 480 fewer stamps than Lisa. Lisa gave some stamps to Minah. Now Minah has 3 times as many stamps as Lisa.

How many stamps did Minah have at first? How many stamps does Lisa have now?

4

Fractions and RatiosFinally, in fourth and fifth grade students use models to understand and solve problems that involve fractions and proportional thinking. Once again, some problems involve part/whole problems like this:

Vincent spent 4/7 of his money on a pair of shoes. The shoes cost $48. How much money did he have at first?

The comparison problem might read:

There are 3/5 as many boys as girls. If there are 75 girls, how many boys are there?

5

} 36

48

boys

girls

?

75

Lisa

Minah

1750

480 less

Minah = 1750 - 480 = 1270 at first

Minah + Lisa = 1270 + 1750 = 3020 total

Minah

Lisa

Lisa = 3020 ÷ 4 = 755 Minah = 3 x 755 = 2265

Lisa now has 755 stamps.

} 3020

2

2

5

must be 2

must be 3 (5 - 2)must be 3 because same as bottomdotted box and also because topmust be double the bottom

What is most exciting is that rather than using the usual student favorite—guess and check, or at least guess—students tackle word problems with efficient and strategic visual models that lead to generalizations. In summary, this puts them on the road to algebra and future success in higher-level mathematics.

The development of successful problem solving skills is a key part of mathematics learning. At the core of Math in Focus is the systematic development of these skills tied directly to the arithmetic. With Math in Focus and its model drawing approach, students gain the skills they need to tackle more and more complicated word problems from grade level to grade level. The program fosters both good number sense and the ability to solve complex problems. We can all agree these should be the goals of any good math program.

About the AuthorAndy Clark is a former elementary and junior high school teacher. He recently retired as the K–12 Math Coordinator for Portland Public Schools, an urban district that outperformed the state of Oregon and closed the achievement gap. Clark is coauthor of a number of math programs, including Every Day Counts®: Calendar Math (Pre-K–6), Algebra Readiness (6 & Up), Partner Games (K–6), and Math in Focus (K–5).

Conclusion: The Road to AlgebraBy now, you probably recognize that these rectangular bars, including the fractional units shown on page 5, are gradually leading students to the beginning of a concept of a variable and an unknown. These rectangles, which will become variable expressions in algebra, enable students to construct more abstract representations of problems as they continue in mathematics. Representing number relationships, comparisons, proportions, and changes becomes second nature as students do this from grade level to grade level.

By fifth grade, students may be trying problems as complex as:

A group of people pay $720 for admission tickets to an amusement park. The price of an adult ticket is $15, and a child ticket is $8. There are 25 more adults than children. How many children are in the group?

Or problems that involve changing quantities or situations that change:

Jane had $7 and her sister had $2. Their parents gave them each an equal amount of money. Then Jane had twice as much money as her sister. How much money did their parents give each of them?

6 7

Sister

Jane

Problem Solving In Singapore Mathwritten by Andy Clark

Ever since Singapore scored first in both 4th and 8th grades in the Trends in International Math and Science Study (TIMSS) comparison assessments in 1995, 1999, and 2003, and in the top three in 2007, math educators have been interested in the secret of Singapore’s success. While many factors have been catalogued – a coherent national curriculum, teacher training, a public belief in the importance of math to the national economy – without fail, all descriptions emphasize the importance of problem solving in the Singapore curriculum. 1(Leinwand 2008, AIR 2005, Lianghuo Fan & Yan Zhu 2007) This paper will consider how problem solving is taught in Singapore and why it has proven so successful.

A Greater Emphasis on Problem Solving Interestingly, the first Singapore math curriculum, which was written in the 1980s, did not emphasize problem solving. It was not until 1991, and the writing of a new curriculum in 1992 that Singapore began emphasizing problem solving in its curriculum. As described on the Singapore Ministry of Education’s Web site, problem solving is now the primary goal:

Mathematical problem solving is central to mathematics learning. It involves the acquisition and application of mathematics concepts and skills in a wide range of situations, including non-routine, open-ended and real-world problems.

The development of mathematical problem solving ability is dependent on five interrelated components, namely, Concepts, Skills, Processes, Attitudes and Metacognition. (CPPD 2006)

The Singapore Ministry of Education uses a graphic to represent their vision for mathematics teaching: a pentagon, with problem solving in the center and these five interdependent, necessary elements surrounding it. Textbooks, written specifically to address this structure, provide constant support for understanding all five elements. Students are encouraged to consider how they think, how they communicate, and how they solve problems, so they can apply their skills to subsequent problems. In its latest efforts, the Ministry is working to increase student communication skills and metacognition during problem solving.

1

If this sounds familiar to educators in the United States, it should. While math education in the 1970’s emphasized the importance of basic skills, in reaction to the “new math” of the fifties and sixties, the 1980’s and 1990’s saw the emergence of problem solving in national and state standards. In the 1980 Agenda for Action, and the 1989 and 2000 versions of Principles and Standards, the National Council of Teachers of Mathematics (NCTM) has similarly identified the centrality of problem solving to the math curriculum. (NCTM 1989, 2000) Most U.S. textbooks now include “problem solving strategies and problems.” Additionally, American researchers have conducted many studies on how to develop problem-solving abilities. (Schoenfeld 1985, 1992; Krulik and Rednick 1987; Qin, Johnson, Johnson 1995)

Differences in Performance LevelsDespite the increased emphasis on problem solving in the United States, students in Singapore continue to perform better in math. I would suggest there are five major reasons for this difference in performance:

1. Problem solving is embedded in Singapore texts, not as a separate activity but as central to every skill and concept discussion.

2. The problems that Singapore students work on are much more complex than those in typical American texts. Two- and three-step problems are the norm.

3. Non-routine, as well as routine problems are included in every grade level.

4. Students are taught specific problem-solving strategies in a carefully sequenced manner beginning in second grade. The most famous of these—model drawing—is used to solve word problems initially, but once acquired as a skill, it becomes useful for solving non-routine problems as well.

5. Student attitudes are addressed and supported.

Lets take a closer look at each of these critical differences.

Embedded Problem SolvingEach time a new concept is introduced in the Singapore textbooks, problem solving is central to the discussion. Consider the concepts of perimeter and area. Just as they do in the U.S., third graders in Singapore learn about these concepts. But in Singapore, students immediately consider them from several perspectives. So rather than just calculating the perimeter of a square with side length 3, students are asked:

A square of side 3 cm is formed using a piece of wire. The wire is straightened and then bent to form a triangle with equal sides. What is the length of each side of the triangle? (MIF 2nd grade)

When learning about area of rectangles, students don’t just calculate height times width, but are asked to find the side length of a square that has the same area as an 8 x 2 rectangle. In other words, whenever a skill or concept is taught, it is also applied in a problem setting. The problem may not even look like other problems that are modeled. Students learn that a single skill can be applied to a wide range of problems.

2

Complex Word ProblemsComplex word problems are the norm in the Singapore textbooks. Yet students begin with very simple problems.

There are 5 boxes of pencils. Each box has 12 pencils. How many pencils altogether?

By the end of the third grade chapter, however, they solve problems such as this one:

Shawn and Trish scored 36 goals in all. Shawn scored three times as many goals as Trish. How many goals did Trish score?

or

Flo saves 4 times as much money as Larry. Maria saves $12 less than Flo. Larry saves $32. How much does Maria save?

In other words, the Singapore textbooks take seriously the proposition that the purpose of math is to solve complex problems. Multiple-step word problems are introduced in the primary grades and become increasingly challenging in the higher grades. Thoughtfulness is displayed, though, in the simplicity of the initial problems. As students develop confidence, they tackle more complicated problems, including non-routine ones.

Solving Non-routine Problems and the Use of HeuristicsIn addition to complex word problems, Singapore curriculum emphasizes non-routine problems—those that go beyond the application of specific computation. Solutions to such problems often require a number of different strategies, or heuristics. Before looking at specific problems from the curriculum, let’s review the history of non-routine problems.

The NCTM (2000) defines problem solving as “engaging in a task for which the solution method is not known in advance” (p. 52). Johnson, Herr, Kysh (2004) define problem solving as “knowing what to do when you don’t know what to do” (p. 3). But perhaps the best definition may come from the math teacher who did the most in the last century to include problem solving in the math curriculum, George Polya. Polya (1965) says, “solving a problem means finding a way out of a difficulty, a way around an obstacle, attaining an aim which is not immediately attainable.”

Polya went on to describe problems, now called non-routine problems, which did not lend themselves to mechanical application of one or more straightforward algorithms. These are problems in which you have to figure out how to proceed, as well as to calculate successfully.

The Singapore textbook writers are very familiar with Polya’s work and include it in manuals designed for teachers. (Huat & Huat 2006) In particular they cite the 4-step model that Polya recommended: understand the problem, devise a plan, carry out the plan, and reflect on the solution. While this same structure is in many American textbooks as well, it is the special attention that Singapore math pays to steps 1 and 4 that may set it apart. In the first step, students connect a problem to prior problems. In the reflection stage students consider Would alternate solutions take a shorter time? Can the method be applied to other problems? (Huat 2006)

3

The Singapore curriculum and textbooks build on the work of Polya by teaching specific strategies for problem solving, what Polya called heuristics. Many of these—such as “look for a pattern,” “draw a picture,” “simplify the problem”, and “work backwards”, are included in American texts. But too often, American texts teach these strategies by “type.” That is, students learn “look for a pattern” and then they are presented with problems that can be solved in that way. In Singapore, students are instead encouraged to consider which strategy will work best for a particular problem. They are introduced to the strategies or heuristics and then given a variety of non-routine problems to solve.

Let’s look now at a third grade example.

Mr. King has a total of 19 geese, chickens and ducks on his farm. He has 3 more chickens than geese. He has 2 fewer ducks than geese. How many ducks does he have?

Try to solve this. Then imagine a third grade student solving it. Singapore students are taught to draw models to represent the situation. Students might use the most basic strategy of guess and check or they may try it more systematically. For instance, one student took 19 counters, assigning 1 to the duck 3 to the geese and 6 to the chickens. The student then added a counter to each row until all 19 had been used.

Other students used a “work backwards” strategy. They know that if they take away 5 chickens and 2 geese, then they will be left with three equal piles. Since there are 12 counters or animals left, this means there must be 4 of each, hence 4 ducks. Notice that many of these strategies lead to the generalization students will use later in algebra to solve the problem. Encouraging multiple approaches and evaluating their effectiveness is the essence of good problem solving.

Model DrawingThe most famous and developed of the heuristics (strategies) taught in Singapore is “model drawing,” sometimes called “bar modeling” in the U.S. Beginning in 2nd grade, students are taught to use rectangular shapes to model a word problem. These models:

• help students visualize abstract math relationships through pictorial representations.

• use rectangular blocks because they are easily divided.

• can be used before students know algebraic solutions, and can be used to model algebraic relationships.

Dr. Yeap Ban Har of Singapore’s National Institute of Education describes the model method this way:

The use of “model method” provides students with a means to handle information, deal with complexity, and at the same time, communicate their thinking through the use of visual models which they can manipulate.

4

These models begin quite simply and typically model word problems. In the Grade 2 problem at right, a rectangle, divided into two parts, models a simple subtraction situation. There are 20 eggs, 7 are duck eggs. How many are chicken eggs? Other models include simple multiplication and division. By 5th grade, however these models increase significantly in complexity. In fact, model drawing proves to be a powerful tool for non-routine as well as routine problems. Consider the following problem:

25% of the fish in a bowl are guppies. The same number of guppies as were originally in the bowl are then added. What percentage of the bowl is now guppies?

Initially students are confused because there do not appear to be enough numbers. Yet a simple diagram can help. Students draw a rectangle, color one-fourth as guppies. Then they add a length to the rectangle that is the same size as the guppies portion. The picture shows clearly that 2 of the 5 parts of the bowl are guppies or 40%.

Careful attention to the teaching of heuristics, to moving from simple to complex problems, and to sequencing the problems in such a way as to move from routine to non-routine problems is a hallmark of the textbooks. Still, problem-solving does not account for all the differences in performance between students in Singapore and the US. Attitudes toward mathematics and the development of metacognitive skills also play a role.

Building Enthusiasm and MetacognitionIn Singapore, efforts to develop positive attitudes and to improve metacognitive skills are evident in all aspects of mathematics learning, including classroom learning materials and the information sent home to students and families. Teachers are encouraged to solve mathematics problems themselves, so they develop a deeper understanding. Developing positive attitudes towards problem solving is a central focus—one that helps all students to feel better about math.

Ironically, at the same time as many states in the U.S. are depending exclusively on multiple choice tests, the Singapore assessment system is moving to incorporate more open ended assessment. Only 20% of the 6th grade placement test includes multiple-choice items. But Singapore’s educators believe that constructed response problems “require students to be able to read reasonably complex paragraphs. In addition students are expected to be able to communicate their thinking and methods by showing how they reason and arrive at the answer.” (Yeap 2008)

5

MPH!Mths 2A U03 p62.indd 62 11/18/08 2:38:16 PMMy Pals Are Here: Maths, Grade 2, page 62

When addressing the winners of the high school math prizes, Ms Grace Fu, Senior Minister of State, Ministry of National Development and Ministry of Education Learning of Mathematics told the winners:

Mathematics will imbue you with problem-solving skills such as the ability to understand a problem, identify the relevant information, look for relationships and patterns, make your own conjectures and apply mathematical knowledge and tools to solve or prove them. Mathematics, thus, provides good opportunities for the training of the mind to think critically and adapt to new situations, something that will be valuable to your future careers. (2008)

In Singapore, a country lacking natural resources and half the size of New York City, human capital is the most precious resource. The Singapore curriculum and textbooks recognize that developing problem-solving skills and creativity is a requirement for the 21st century.

This belief, in addition to careful attention to teaching and cultivating problem-solving skills, are lessons worth considering as we try to make our students competitive in the global marketplace in which they live.

6

Andy Clark is a former elementary and junior high school teacher. He recently retired as the K-12 Math Coordinator for Portland Public Schools, an urban district that outperformed the state of Oregon and closed the achievement gap. Andy is coauthor of a number of math programs, including Every Day Counts®: Calendar Math Pre-K—6, Algebra Readiness 6 & Up , Partner Games K-6, and Math in Focus™: The Singapore Approach K-5.

7

Sources

Fu, G. 2008. “Speech at the Singapore Mathematical Society Annual Prize Presentation Ceremony.” http://www.moe.gov.sg/media/speeches/2008/08/30/speech-by-ms-grace-fu-at-the-s.php

Ginsburg, Alan, and Leinwand, Steven; Anstrom, Terry; and Pollock, Elizabeth. What the United States Can Learn From Singapore’s World-Class Mathematics System.” American Institutes for Research. http://www.air.org/news/default.aspx

Huat, J. N. C., Huat, L. K. 2006. A Handbook for Mathematics. Singapore: Marshall Cavendish Education.

Johnson, K., Herr, T., and Kysh, J. 2004. Crossing the River with Dogs. California: Key Curriculum Publishing.

Krulik, S. and Rudnick, J. 1995. Problem-solving: A Handbook for Elementary School Teachers. Allyn & Bacon: Boston, MA.

Leinwand, Steven, and Ginsburg, Alan L. 2007. “Learning from Singapore Math,” Educational Leadership. November Volume 65, Number 3. 32.

Lianghuo Fan, and Yan Zhu, Representation of Problem-Solving Procedures: A Comparative Look at China, Singapore, and US Mathematics Textbooks” Educational Studies in Mathematics,

Springer Netherlands, 1573-0816 (Online) Published online: 31 March.Mathematics Syllabus Primary. 2006. Curriculum Planning and Development Division. http://www.moe.gov.sg/. 3.

National Council of Teachers of Mathematics (NCTM). 1989. Curriculum and Evaluation. NCTM: Reston, VA.

National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards. NCTM: Reston, VA.

Polya, G. 1957. How to Solve It: A New Aspect of Mathematical Method (2nd ed.). Princeton, NJ: Princeton.

Polya, G. 1965. Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (vol. 2). New York, NY: John Wiley & Sons, Inc.

Qin, Z, Johnson, D. W., and Johnson, R.T. 1995. “Cooperative Versus Competitive Efforts and Problem Solving.” Review of Educational Research. 65(2). 129 – 143.

Schoenfeld, A. 1985. Mathematical Problem Solving. New York: Academic Press.

Schoenfeld, A. 1992. “Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-making in Mathematics.” In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning. 334 – 370. New York: MacMillan.

Yeap Ban Har. “The Singapore Mathematics Curriculum and Mathematical Communication.”www.criced.tsukuba.ac.jp/math/apec/apec2008/papers/PDF/13.YeapBanHar_Singapore.pdf

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