Here is a math problem you can solve easily:A man sold 230 balloons at a fun fair in the morning. He sold another 86 balloons in the evening. 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. She spent 2/5 of the remainder on a book. She had $72 left. How much money did she have at first?

*bar model*technique. Students first encounter the technique in 3rd grade, where they apply it to very simple problems like the first one. In grades 4 and 5, they apply the same versatile technique to more difficult, multistep problems. By grade 6, they are ready to solve really hard problems like the second one. With that solid foundation, students easily step into algebra. The bar modeling tool has taught them not only to solve math problems but also to represent them symbolically—the mainstay of algebraic reasoning.

Scott Baldridge, a Louisiana State University mathematician, uses the Singapore Math texts in math courses for preservice teachers. He says,Students are treated by the curriculum as future adults who will need technical mathematics and the ability to do serious mathematical thinking in their careers.

## Deceptively Simple

*not*learn this?” Each concept is introduced with a simple explanation—often just a few words in a cartoon balloon. Students with weak reading and math skills benefit hugely from this direct simplicity.

*part-whole model*works for simple addition and subtraction problems. (Part-whole relationships are a constant theme in Singapore Math, from 1st graders learning to add to 6th graders learning to divide fractions.) For example,<BQ>Daniel and Peter have 450 marbles.Daniel has 248 marbles.How many marbles does Peter have?</BQ>

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The next two problems demonstrate the second variant of the problem-solving technique—two bars to represent two different quantities. This comparison model works for problems that are solved by subtraction.Daniel has 248 marbles.Peter has 202 marbles.Who has more marbles? How much more?

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?

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*add*and

*subtract*. Discovering all the different ways to express the idea of math terms like

*subtraction*is important for all students, but especially for English language learners struggling with word problems on a year-end assessment.

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## In-Depth Mastery

South River Principal Dorothy Unkel reports that teachers had difficulty at first making the transition to the new program:Singapore's approach is very teacher driven, much slower paced, and goes into much more depth. Teachers aren't used to that.

## Scaffolding the Way to Algebra

In Singapore Math, 3rd graders begin to apply the bar model technique to multiplication and division. By 4th grade, they are ready to apply it to fractions, as shown in the following problem:A grocer has 42 apples. 2/7 of them are red, and the rest are green. How many of them are green?

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By 6th grade, students are solving complex, multistep problems like the one presented at the beginning of this article.Lauren spent 20 percent of her money on a dress. She spent 2/5 of the remainder on a book. She had $72 left. How much money did she have at first?

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By allowing students to identify the knowns and unknowns in a problem and their relation to one another, bar modeling sets the stage for the student to move to algebraic representation, as follows: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 ofr= 3/5 ×rSo 1/5 ofr= $72/3 = $24, andr= 5 × $24 = $120From the upper bar model,Amount spent on a dress = 0.20x= 1/5 ofxSoR= 4/5 ofX= 4/5 ×x, which implies that$120 = 4/5 ofx= 4/5 ×xSo 1/5 ofx= $120/4 = $30, andx= 5 × $30 = $150

## Solving Problems, Reinforcing Concepts

The end result of the Singapore Math program is that 6th graders can solve complex, multistep problems that most U.S. students, even those in algebra classes, would find challenging. According to a 2005 study by the American Institutes for Research (AIR), Singapore Math 6th grade problems are “more challenging than the released items on the U.S. grade 8 National Assessment of Education Progress” (p. xiii). AIR also found thatthe Singapore texts are rich with problem-based development in contrast to traditional U.S. texts that rarely get much beyond exposing students to the mechanics of mathematics and emphasizing the application of definitions and formulas to routine problems. (p. xii)Singapore Math's trademark strategy—simple explanations for hard concepts—works!