Mathematics: Counting Them In
When it comes to learning to read, researchers and educators have long known that all children do not come to school equally prepared. Some, born into middle-class homes filled with books and newspapers, may start school already knowing how to read many words or at least understanding that sentences must be read from left to right. Children from families without the same advantages, on the other hand, may enter kindergarten without a clue about how to decipher the printed word.
The fact that the same kinds of differences can also affect young children's mathematical abilities, however, is less well-known.
It's not that kindergarten children from disadvantaged backgrounds cannot recite numbers as well as their more privileged peers. What they lack, says researcher Sharon Griffin, are intuitive understandings about numbers and what they represent. "They may be unable to tell what number comes after a certain number, such as seven, or what number comes two numbers after seven,'' writes Griffin, an assistant education professor at Clark University in Worcester, Mass. "They cannot answer questions like, 'Which is bigger: six or eight?' ''
And, if asked, "How much is nine and two?'' these children might answer, "92,'' she says, "because they have no idea of the magnitude or that 92 is a long way from two.''
So Griffin and her research partner, Robbie Case, director of the University of Toronto Institute for Child Study, are now perfecting a mathematics program aimed at helping disadvantaged children fill in that knowledge gap. Called "Teaching Number Sense,'' the program has been tested over five years in eight schools in the United States and Canada with good results.
In some of those studies, the researchers say, poor children who started school with little "number sense'' were able to catch up to peers from more affluent households. Moreover, the gains that kindergartners made in the program were maintained a year later--even for children who had gone on to conventional 1st grade math programs. And students were able to apply the knowledge they had gained in contexts they had never before encountered.
The researchers don't yet know for sure whether those results will fade as children grow older, but they say they are optimistic that will not occur. "What we're teaching,'' Griffin says, "is the specific conceptual structure that we think is the foundation for all higher learning in mathematics.''
When Griffin and Case began working together more than six years ago, the prevailing view in the field suggested that children's early mathematical understandings were universal. Studies had shown, for example, that American children entered school with a grasp of numbers on a par with that of Japanese children and those of other developed nations. At the time, the two researchers were crafting a test to gauge children's cognitive development in mathematics. They maintained that children approach tasks in different ways at ages 4, 6, 8, and 10. At age 5 or 6, they believed, youngsters undergo a key transformation that enables them to make sense of problems involving number and quantity.
As a "fluke,'' Griffin and Case gave the test to a group of Portuguese-immigrant kindergartners in inner-city Toronto in 1987. The results showed, to the researchers' surprise, that those children lagged as much as two years behind middle-class children. They grasped math, in other words, at the level of 3- and 4-year-olds.
The researchers spent the next several years testing other ethnic and racial student groups, all of them disadvantaged: whites, Hispanics, Southeast Asians, and African Americans, among others. Regardless of the students' ethnic origins, the results were the same: Children from low-income households lagged two to three years behind their middle-class counterparts. Moreover, that gap grew wider as students moved through school.
Griffin and Case are not sure why that lag occurs, but, based on interviews with parents and other studies, they do have a couple of strong suspicions. They point to the lack of board games in lower-income homes as one explanation. "Board games,'' Case says, "are really an excellent way to learn about numbers.'' They also suggest that more hypothetical talk about numbers takes place in middle-class homes. "Say Suzie is going to have a birthday party,'' Griffin says of a middleclass child. "Her parents may ask, 'How many friends do you want to invite?' and 'How many favors will we need?' '' Such conversations, she says, are less likely to occur in lower-income homes.
The Teaching Number Sense program sets out to introduce experiences like these into the lives of disadvantaged children. At the kindergarten level, for example, the program comprises 30 interactive number games that, like board games, help students deepen and construct their understanding of numbers in a tangible way.
In one such game, for example, small groups of children gather around a board with a number line for each player. To play, children roll the dice, add the numbers together, and then ask the "banker'' for that sum in poker chips. Each player places the chips along his or her own number line and then moves a playing piece the same number of spaces, counting aloud until the piece rests on the last chip. As the children count out their chips and squares, other children learn by watching to make sure no one makes a mistake. The first child to reach the "winner's circle'' beyond the 10th square on the number line wins. Along the way, players can speed--or slow--their progress by drawing "chance'' cards that direct them to move back or ahead one space.
As soon as the children become adept at these simple counting tasks, they're encouraged to begin assessing quantity: Who is closest to winning? How do you know?
Griffin and Case first tested their program in a series of studies with some 200 disadvantaged kindergartners, most of whom had already failed the researchers' number-knowledge test. Across all the studies, half the students were given the Teaching Number Sense units, and half received traditional math instruction. At the end of three to four months of the special training, the researchers found that 53 percent to 87 percent of the students in the Teaching Number Sense program passed the test. Of the students in the control groups, some of whom had also received small-group instruction and special attention, only 14 percent to 37 percent passed.
Moreover, fewer program students were making wild guesses in response to questions, and more were using strategies judged to be reasonable. They were also better able to take what they had learned and "transfer'' it to solve unfamiliar tasks.
A smaller group of program students--ones who had gone on to a traditional 1st grade math program the next year--continued to hold the gains they had made in kindergarten. At the end of 1st grade, those students were still performing better than a control group of students with similar demographic characteristics. Their teachers, who had no idea that their pupils had received special training the year before, also rated them as good students.
Since then, the researchers have developed similar programs for the 1st and 2nd grades and have begun a longitudinal study to track the progress of another 260 students in the program over three years. So far, evidence suggests that by the end of 1st grade, students who were exposed to Teaching Number Sense units in both kindergarten and 1st grade did as well on the number-knowledge tests as middle-class students who were not in the program. The performance of a control group of disadvantaged students, in comparison, had already begun to slide by then.
Such results have caught the eyes of curriculum publishers who are seeking to market the program. But the most pressing question for researchers now is how to ensure that teachers can teach the program, since it works best when children learn in small groups. "When we go in to do it, we are an extra one or two pairs of hands,'' Griffin says. "When we leave, teachers revert to teaching in ways they feel most comfortable with.''
Griffin plans to seek foundation funds to develop a more intensive, systematic kind of teacher-preparation effort. (Thus far, the James McDonnell Foundation has funded the Teaching Number Sense experiments.) She hopes to structure a program that would encourage teachers themselves to explore higher-level mathematics through hands-on experience with number games. This, she says, will help them reinvent their own beliefs about how children learn.
"What we've been forced to conclude,'' Griffin says, "is that these programs require radical changes that are much harder for teachers to make than we thought.''--Debra Viadero