Singapore MathUsing the bar model approach, Singapore textbooks enable students

to solve difficult math problems—and learn how to think symbolically.

John Hoven and Barry Garelick

Here is a math problem you can solve easily:

A man sold 230 balloons at a fun fair in themorning. He sold another 86 balloons in theevening. How many balloons did he sell in all?

And here is one you can't:

Lauren spent 20 percent of her money on a dress. Shespent 2/5 of the remainder on a book. She had $72 left.How much money did she have at first?

In Singapore, where 4th and 8th grade students consis-tently come in first on international math exams, studentsleam how to solve both problems using che same bar modeltechnique. Students first encounter the technique in 3rdgrade, v here they apply it to very simple problems like thefirst one. In grades 4 and 5, they apply the same versatiletechnique to more difficult, multistep problems. By grade 6,they are ready to solve really hard problems like the secondone. With that solid foundation, students easily step intoalgebra. The bar modeling tool has taught them not only tosolve math problems but also to represent them symboli-cally—the mainstay of algebraic reasoning.

Bar modeling is a specific variant of the common Draw aPicture mathematics problem-solving strategy. Because Singa-pore Math uses this one variant consistently, students knowwhat kind of picture to draw. That's an advantage if the barmodel is versatile enough to apply to many complex prob-lems—and it is. It is especially useful for problems that involvecomparisons, part-whole calculations, ratios, proportions, andrates of change. It communicates graphically and instantly theinformation that the learner already knows, and it shows thestudent how to use that information to solve the problem.

Singapore's textbooks are used in more than 600 schools intbe United States and also by many homeschoolers. Thebooks were discovered and drew high praise when mathe-

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maticians and teachers investigated why Singapore scored sohigh on international math exams. Homeschoolers andteachers like them tor their simple and effective approach.Mathematicians like them for their logical structure, coherentcurriculum, and focus on the skills necessary for success inalgebra.

Scott Baldridge, a Louisiana State University mathemati-cian, uses the Singapore Math texts in math courses forpreservice teachers. He says,

Students are treated by the curriculum as future adults who willneed technical mathematics and the ability to do serious mathe-matical thinking in their careers.

Simple or Complex?Deceptively SimpleOpen a Singapore Math book to any page, and you may askyourself, "How can a child not learn this?" Each concept isintroduced with a simple explanation-—often just a few wordsin a cartoon balloon. Students with weak reading and mathskills benefit hugely from this direct simplicity

The first time 3rd graders seethe bar model technique, they seehow it's used in five demonstrationproblems.

The first two problems demon-strate the first basic variant of thetechnique—a single bar with twosections- This part-whole modelworks lor simple addition andsubtraction problems. (Part-wholerelationships are a constant themein Singapore Math, from 1stgraders learning to add to 6thgraders learning to divide frac-

— - , ^ lions.) For example,

rf^^^^^^^M Daniel and Peter have 450marbles.

Daniel has 248 marbles.

Mow many marbles does Peterhave?'

450

248 ?

The next two problems demonstrate the second variant ofthe problem-solving technique—two bars to represent twodifferent quantities. This comparison model works for prob-lems that are solved by subtraction.

Daniel has 248 marbles.

Peter has 202 marbles.

Who has more marbles? How much more?

248

202 ?

The final problem adds a side bracket joining the two bars.This variant works for two-step probletns. That statementdeserves emphasis: Singapore Math students are solving two-step problems in 3rd grade.

Mary had 120 more beads than Jill. Jill had 68 beads.

Step 1: How many beads did Mary have?

Step 2: How many beads did the two girls have altogether?

68 120:

So the textbooks appearance of simplicity is deceptive, Tnthis one lesson, Singapore Math students have already leamedthree basic variants of the bar model technique. They alsoapply these variants to problems that use a variety ofs>Tionyms for add and subtract. Discovering al! the differentways to express the idea of math terms like subtraction isimportant for all students, but especially for English languagelearners struggling with word problems on a year-endassessment.

Although the explanations are simple and direct, they chal-lenge students to think. When asked to find the differencebetween 9 and 6, for example, students understand that theyneed to use subtraction because of the part-whole relationshipthat has been used to teach them addition and subtraction.Because they know that 6 is part of 9, they easily understandhow to find the difference.

Practice problems in Singapore Math are designed to teachskills step by step. The first few practice problems in a typicallesson migbt provide the appropriate bar model, whereassucceeding problems require the student to construct it.

The texts are carefully crafted so that students are presented

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with hints and examples for applyingnew concepts and algorithmic tech-niques, thus providing the scaffoldingfor learning. For example, the decimaldivision problem 0.6 ^ 3 is made clearby showing six dimes (each equalingone 10th of a dollar) split into threeequal groups. There are two dimes ineach group, so six lOths dividedinto three groups equals two lOths:0.6 - 3 = 0.2. The lesson then asksstudents lo extend this concept to morecomplex decimal division problems,such as 2 - 4. Here students are givenanother hint—a cartoon characterthinking "2 is 20 lQths." The students,having been led to discover how 1 Othscan be divided into groups, can nowmake another discovery and expresswhole numbers in terms of lOths.

In-Depth MasterySmgapore Math has been used for thepast two years in grades 1-4 at theSouth River Primary School (gradesK-2) and South River Elementary

The grade 3 lesson shows howSingapore math textbooks presentmath concepts simply and directly.

School (grades 3-5) in New Jersey.The school district, concerned aboutyears of flat test scores in an areathat is largely low income, decidedto try Singapore's program afterhearing about it at a workshop.

South River Principal DorothyUnkel reports that teachers haddifficulty at first making the tran-sition to the new program:

Singapore's approach is veryteacher driven, much slowerpaced, and goes inio muchmore depth. Teachers aren'tused to that.

Singapore Math is able to teach at aslower pace and in more depth becauseit focuses instruction on the essentialmath skills recommended in theCurriculum Focal Points (NationalCouncil of Teachers of Mathematics,2006). As a result, students make morerapid progress in those essential skills;for example, they leam multiplication in1st grade. That surprising result—slowerpace resulting in more rapid progress—works for students who perform on,above, or below grade level.

South River Assistant SuperintendentMichael Pfister emphasizes that in theSingapore system, studenLs achievemastery, so schools do not need toreteach skills. That has implications forinstructional grouping in a typical U.S.classroom, where students' math skillsoften range from two years below gradelevel to two years above. Singapore Mathstudents should be grouped for instruc-tion with the textbook that is at theirlevel of understanding. Schools shoulduse extra resources to help low-achieving students leam appropriatematerial at an accelerated pace, not to

teach them material for which they havenot mastered the necessary prerequisiteskills.

Scaffolding the Way to AlgebraIn Singapore Math, 3rd graders begin toapply the bar model technique to multi-plication and division. By 4th grade,they are ready to apply it to fractions, asshown in the following problem:

A grocer has 42 apples. 2/7 of themare red, and the rest are green. Howmany of them are green?

42

red7 units = 42

green

1 unit = 42/7 = 6

5 units = 6 X 5 = 30

There are 30 green apples.

Note that students are encouraged tosee fractional "pieces" as units untothemselves. This viall become importantlater when students encounter fractionaldivision.

By 6th grade, students are solvingcomplex, multistep problems like theone presented at the beginning of thisarticle.

Lauren spent 20 percent of hermoney on a dress. She spent 2/5 ofthe remainder on a book. She had$72 left. How much money did shehave at first?

$723 units = $72 (in the bottom barmodel)

its = 5x$72/3 =remainder

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4 parts = $120 (in ihe top bar model)

5x $120/4-$150

Part-whole relationships are aconstant theme in Singapore Math.

By allowing students to identify theknowns and unknowns in a problemand their relation to one another, barmodeling sets the stage for the studentto move to algebraic representation, asfollows:

Amount Lauren had at first = x(length of the upper bar model)

After buying a dress, the remainder= r (length of the lower bar model)

From the lower bar model,

$72 = 3/5 of r = 3/5 X r

So 1/5 of r = $72/3 = $24. and

r = 5 x $ 2 4 = $120

From the upper bar model,

Amount spent on a dress = 0.20x= 1/5 of X

So R = 4/5 of X = 4/5 X x. whichimplies that

So 1/5 ofx = $120/4-$30, and

x = 5 x $ 3 0 =

For problems that are too complex tobe represented pictorially through barmodels, the compact conventiotis of alge-braic symbolism—to which the studenthas been sequentially and methodicallyled—can come to the rescue.

Solving Problems,Reinforcing ConceptsU would be a mistake to think that thebar model approach to solving problemscould be lifted out of Singapore Mathand used by itself. Although barmodeling provides a powerful tool torepresent and solve complex word prob-lems, it also serves to explain and rein-force such concepts as addition and

subtraction, multiplication and division,and fractions, decimals, percents, andratios. If not linked to the conceptsembedded in the lessons, the bar modelwould not necessarily be meaningful.The bar model and the basic skillsembedded in the mathematical prob-lems bootstrap each oiher.

The end result of the Singapore Mathprogram is that 6th graders can solvecomplex, multistep problems ihat mostU.S. students, even those in algebraclasses, would find challenging.According to a 2005 study by the Amer-ican Institutes for Research (AIR), Singa-pore Math 6th grade problems are"more challenging than the releaseditems on the U.S. grade 8 NationalAssessment of Education Progress"(p. xiii). AIR also found that

the Singapore texts are rich with problem-based development in contrast to tradi-tional U.S. texts that rarely get muehbeyond exposing students to themechanics of mathematics and empha-sizing the application of definitions andformulas to routine problems, (p. xii)

Singapore Math's trademark strategy—simple explanations for hardconcepts—works! 13

' Problems and bar models arereproduced from Singapore PnmaryMathematics Teacher's Guides, 3A (2003),4A (2004), and 6B (2006), (U.S. ed.).Oregon City, OR: SingaporeMath.com.Copyright by SingaporeMath.com.Used with permission. Available:www.singaporemath.com

ReferencesAmerican Institutes for Research.

(2005). What the United States cankamjrom Singapore's world-classmathematics (and what Singapore cankamjrom the United Stales): An

exploratory study- Washington, DC:Author. Available: www.air.org/news/documents/Singapore%20Report%20(Bookmark%20Version).pdf

National Council of Teachers of Mathe-matics. (2006). Curriculum focal pointsjor prekindergdrten through grade S mathe-matics: A quest foi coherence. Reston, VA:Author. Available: www.nctmmedia.org/cfp/ful l_document.pdf

John Hoven is an economist in theAntitrust Division of the U.S. Departmentof Justice, Washington D.C; jhoven©gmail.com. Barry Garelick is ananalyst for the U.S. EnvironmentalProtection Agency, Washington D.C;barrvg99@yahoo.com.

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