# The Craft of Word Problems

## Using Algebra Word Problems to Explore Problem Space

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## A “Singapore Sling”

The Curriculum Planning & Development Division of the Ministry of Education in Singapore has produced a series of Primary Mathematics Texts, which we found useful for study in our Institute seminar. We read the U.S. Edition of
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Primary Mathematics, 6A and 6B
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(Times Media Private Limited, 2003). The Singapore curriculum series offers insight into how one Asian school district is presenting students with learning materials. Considering the comparative high success of Asian and Asian-American students in mathematics, it is instructive to read through these materials. Several immediate impressions are worth noting here:

- 1. The series is comprised of slim paperback texts and linked workbooks (copies of which we did not have on hand). The texts are two color printed, uncluttered (compared to U.S. textbooks) and filled with clear graphics and almost spare verbiage.
- 2. The main feature of the package is constant use of the self described “Concrete->Pictorial->Abstract” approach to problem solving. “Students are provided with the necessary learning experiences beginning with the concrete and pictorial stages, followed by the abstract stage to enable them to learn mathematics meaningfully. This package encourages active thinking processes, communication of mathematical ideas and problem solving.”11
- 3. The very first unit in 6A is “Algebra” (no shilly-shallying here!). Following units are “Solid Figures” (drawing and nets), “Ratio” (ratio and fractions, ratio and proportion, changing ratios), “Percentage”, and “Speed.” In 6B, the units are “Fractions”, “Circles”, “Graphs”, “”Volume”, Triangles and 4-sided Figures, and “More Challenging Word Problems.”

Overall, the texts seem to send a message that the study of mathematics can be both serious and quite engaging. The format does encourage students to think about problems, and the constant use of clear graphics prods students to think in both organized and visual-artistic ways. (In a very unscientific sampling, I tried some of the exercises with my younger high school students who usually struggle with text materials and found that they generally made a good effort, were not intimidated or belittled by the problem presentations, and, by not getting frustrated, were able to talk about solutions they were trying.)

Carpenter’s thesis in
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Thinking Mathematically
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stresses that “it is important for all children to learn that arithmetic and algebra make sense and that arithmetic and algebra are grounded in a basic collection of big ideas…and there is a small list of fundamental numeric properties that account for all symbol manipulation in arithmetic and algebra.”12

Reading Carpenter’s book while reviewing the Singapore curriculum for sixth grade provided an interesting pairing of thoughtful pedagogy with a specific example of textbooks grounded in the same outlook. Chapter by chapter in Primary Mathematics formulaic “
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rules
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” (with accompanying sample calculations so familiar in U.S. texts) are almost nonexistent. Instead, the text is presented more in the form of a Socratic dialogue with students, designed to lead them into investigation and exploration of mathematical and algebraic
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properties
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. For too many students (and adults), “arithmetic represents a collection of unrelated and arbitrary manipulations of numbers and symbols, and algebra is perceived as a separate collection of meaningless procedures that are only tangentially related to arithmetic.”13

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Thinking Mathematically
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and the Singapore Curriculum should be of great interest to math vertical teams--both emphasize the importance of generative learning (that which “serves as a basis for acquiring new knowledge”14). And both have the goal of students learning arithmetic in ways that naturally build a foundation of understanding to support the learning of algebra.