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SingaporeMathUsing the bar model approach, Singapore textbooks enable studentsto solve difficult math problems-and learn how to think symbolically.John Hoven and Barry Garelick

Hre is a math problem you can solve easily:

A man sold 230 balloonsat 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, studentslearn how to solveboth problems using the same bar modeltechnique. Students first encounter the technique in 3rdgrade, where 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 reallyhard problems like the secondone. With that solid foundation,students easily step intoalgebra. The bar modelingtool has taught them not only tosolvemath problems but also to represent them symboli-cally-the mainstayof algebraicreasoning.

Bar modeling is a specific variantof thecommon Draw aPicture mathematicsproblem-solvingstrategy.Because Singa-

pore Math uses this one variantconsistently,students knowwhat kind of picture to draw. That's an advantage if the barmodel is versatileenough to apply to many complex prob-lems--and it is. It is especiallyuseful for problems that involvecomparisons,part-whole calculations,ratios, proportions, andrates of change. It communicatesgraphicallyand instantlytheinformation that the learner already knows,and it shows thestudent how to use that informationto solve the problem.

Singapore'stextbooksare used in more than 600 schools inthe UnitedStates and also by many homeschoolers.Thebooks were discovered and drew high praise when mathe-

maticiansand teachers investigatedwhy Singapore scored so

high on international math exams. Homeschoolersandteachers like them for their simple and effective approach.Mathematicianslike them for their logicalstructure, coherentcurriculum, and focus on the skills necessary for success inalgebra.

Scott Baldridge,a Louisiana State Universitymathemati-cian, uses the SingaporeMath texts in math courses forpreservice teachers. He says,

Students are treatedby the curriculumas future adultswho willneed technicalmathematicsand the abilityto do seriousmathe-maticalthinking in their careers.

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

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

The first two problems demon-strate the first basic variant of thetechnique-a singlebar with twosections. Thispart-wholemodelworks for simple addition andsubtraction problems. (Part-wholerelationships are a constant themein Singapore Math, from1stgraders learning to add to 6th

graders learningto divide frac-tions.) Fo r example,

Daniel and Peter have 450marbles.

Daniel has 248 marbles.

How many marbles doesPeterhave?'

450

248 ?

The next two problems demonstrate the second variantofthe problem-solvingtechnique-two bars to representtwodifferent quantities. This comparison modelworks for prob-lems that are solved by subtraction.

Daniel has 24 8 marbles.

Peter has 202 marbles.

Who has moremarbles? How much more?

248

202 ?

The final problem addsa side bracket joining the two barsThis variant works for two-step problems.That statementdeserves emphasis: Singapore Mathstudents are solving twostep problems in 3rd grade.

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

Step 1: How manybeads did Mary have?

Step 2: How manybeads did the two girlshave altogethe

?

68 120

So the textbook'sappearance of simplicityis deceptive. Inthis one lesson, SingaporeMath students have already learnthree basic variants of thebar model technique. They alsoapply these variants to problems that use a variety ofsynonyms for add and subtract. Discoveringall the differentways to express the idea of math terms like subtractionisimportant for all students, but especially for English languaglearners struggling with word problems ona year-endassessment.

Although the explanationsare simple and direct, theychalenge students to think. When asked to find the differencebetween 9 and 6, for example, students understand that theyneed to use subtraction because ofthe part-wholerelationshthat 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 typicalesson might provide the appropriatebar model, whereassucceeding problems requirethe student to construct it.

The textsare carefullycrafted so that students are present

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with hints and examples for applyingnew conceptsand algorithmic tech-niques, thus providing the scaffolding

for learning.For example,the decimaldivisionproblem 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 10ths dividedinto three groups equals two l0ths:0.6 - 3 = 0.2. The lessonthen asksstudents to extend this conceptto morecomplex decimal division problems,such as 2 + 4. Here students are givenanother hint-a cartoon character

thinking "2 is 20 10ths." The students,having been led to discover how lothscan be divided intogroups, can nowmake another discoveryand expresswhole numbers in terms of 10ths.

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

Th e grade 3 lesson shows howSingapore math textbooks present

math concepts simply and directly.

School (grades 3-5) in New JerseyThe schooldistrict, concernedaboutyears 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 thatteachers haddifficulty at first making the tran-sition to the new program:

Singapore'sapproach is veryteacher driven, much slowerpaced, and goes into 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 theCurriculumFocal Points (National

Council of Teachers of Mathematics,2006). As a result, students make morerapid progress in those essential skills;for example, they learn multiplicationin1st grade. That surprising result-slowerpace resulting in more rapid progress-works for students who perform on ,above, or below gradelevel.

South River AssistantSuperintendentMichael Pfister emphasizes that in theSingapore system,students achievemastery,so schools do not need to

reteach skills.That has implicationsforinstructionalgrouping in a typical U.S.classroom, where students' math skillsoften range from two years below gradelevel to two years above. SingaporeMathstudents should be grouped for instruc-tion with the textbook that is at theirlevel of understanding. Schools shoulduse extraresources to help low-achievingstudents learn appropriatematerial at an acceleratedpace, not to

teach them material for which they havenot mastered the necessaryprerequisiteskills.

Scaffolding the Way to AlgebraIn SingaporeMath, 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 encouragedtosee fractional"pieces" as units untothemselves. This willbecome importantlater when students encounter fractionaldivision.

By 6th grade, students are solvingcomplex,multistep problems like theone presented at the beginningof 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 ba rmodel)

5 units = 5 x $72/3 = $120 =remainder

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

5parts

= 5x $120/4

=

$150

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 algebraicrepresentation,asfollows:

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

After buying a dress, the remainder= r (lengthof the lower ba r model)From the lowerba r model,

$72=3/Sofr=3/5x r

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

r = 5 x $24 = $120

From the upper ba r model,

Amount spent ona dress = 0.20x= 1/5 ofx

So R = 4/5 ofX= 4/5 x x, whichimplies that

$120 = 4/5 ofx = 4/5 x x

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

x= 5 x $30 $150

For problems thatare too complex tobe representedpictorially through barmodels, the compact conventionsof alge-braic symbolism-to which the studenthas been sequentiallyand methodicallyled--can come to therescue.

Solving Problems,Reinforcing ConceptsIt would be a mistake to think that thebar model approachto solving problemscouldbe lifted out of Singapore Mathand used by itself. Althoughbarmodeling providesa powerful tooltorepresent andsolvecomplex word prob-lems, italso serves to explain and rein-

force such concepts as addition and

Part-whole relationships are a

constant theme in Singapore Math.

subtraction, multiplication anddivision,and fractions, decimals, percents, andratios. If not linked to the conceptsembedded in the lessons, theba r modelwould not necessarilybe meaningful.The ba r model and the basic skillsembedded in the mathematicalprob-lems bootstrap each other.

The end result of the SingaporeMathprogram is that 6th graders cansolvecomplex, multistep problemsthat mostU.S. students, even those in algebraclasses, would find challenging.According to a 2005 study by the Amer-ican Institutesfor Research (AIR), Singa-pore Math 6th grade problems are"morechallenging than the releaseditems on the U.S. grade 8 NationalAssessmentof EducationProgress"(p. xiii). AIR also found that

the Singapore textsare rich with problem-based development in contrast to tradi-tionalU.S. texts that rarelyget muchbeyond exposing studentsto themechanics ofmathematicsand empha-sizing the applicationof definitionsandformulas toroutine problems.(p. xii)

SingaporeMath's trademark strategy-simple explanationsfor hardconcepts-works! M

SProblemsand barmodelsare

reproduced from SingaporePrimaryMathematics Teacher's Guides, 3A (2003),4A (2004), and 6B (2006), (U.S. ed.),OregonCity, OR: SingaporeMath.com.Copyrightby SingaporeMath.com.Used with permission.Available:www.singaporemath.coin

ReferencesAmerican Institutes for Research.

(2005). What the United States canlearn rom Singapores wotld-classmathematics(andwhat Singapore canlearn rom the UnitedStates): An

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

National Councilof Teachers of Mathe-matics. (2006). Curriculumfocal points

for prekindergarten throughgrade 8mathe-matics:A questfor coherence.Reston, VA:Author. Available: www.nctmmedia.org/cfp/full_document.pdf

John Hoven is an economist in theAntitrust Division of the U.S. Departmentof Justice, Washington D.C.; [email protected]. Barry Garelick is ananalyst for the U.S. EnvironmentalProtection Agency, Washington D.C.;[email protected].

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COPYRIGHT INFORMATION

TITLE: Singapore Math: Simple or Complex?SOURCE: Educ Leadership 65 no3 N 2007

The magazine publisher is the copyright holder of this article and itis reproduced with permission. Further reproduction of this article inviolation of the copyright is prohibited.

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