The books, with the no-nonsense title "Primary Mathematics," are published for the U.S. market by a small company in Oregon, Marshall Cavendish International. They are slim volumes, weighing a fraction of a conventional American text. They have a spare, stripped-down look, and a given page contains no material that isn't directly related to the lesson at hand.
Standing in an empty classroom one recent morning, Ramos flipped through two sets of texts: the Singapore books and those of a conventional math series published by Harcourt. She began with the first lesson in the first chapter of first grade.
For The Record
Los Angeles Times Tuesday, March 11, 2008 Home Edition Main News Part A Page 2 National Desk 1 inches; 47 words Type of Material: Correction
Math: An article in Sunday's Section A on the Singapore math curriculum said one of its advocates, Yoram Sagher, is a math professor at the University of Illinois at Chicago. Sagher works for Florida Atlantic University; he was formerly affiliated with the University of Illinois at Chicago.
In Harcourt Math, there was a picture of eight trees. There were two circles in the sky. The instructions told the students: "There are 2 birds in all." There were no birds on the page.
The instructions directed the students to draw little yellow disks in the circles to represent the birds.
Ramos gave a look of exasperation. Without a visual representation of birds, she said, the math is confusing and overly abstract for a 5- or 6-year-old. "The math doesn't jump out of the page here," she said.
The Singapore first-grade text, by contrast, could hardly have been clearer. It began with a blank rectangle and the number and word for "zero." Below that was a rectangle with a single robot in it, and the number and word for "one." Then a rectangle with two dolls, and the number and word for "two," and so on.
"This page is very pictorial, but it refers to something very concrete," Ramos said. "Something they can understand."
Next to the pictures were dots. Beginning with the number six (represented by six pineapples), the dots were arranged in two rows, so that six was presented as one row of five dots and a second row with one dot.
Day one, first grade: the beginnings of set theory.
"This concept, right at the beginning, is the foundation for very important mathematics," Ramos said. As it progresses, the Singapore math builds on this, often in ways that are invisible to the children.
Word problems in the early grades are always solved the same way: Draw a picture representing the problem and its solution. Then express it with numbers, and finally write it in words. "The whole concept," Ramos says, "is concrete to pictorial to abstract."
Another hallmark of the Singapore books is that there is little repetition. Students are expected to attain mastery of a concept and move on. Each concept builds upon the next. As a result, the books cover far fewer topics in a given year than standard American texts.
