April 1, 2009

The second grader, out for an afternoon with her dad, manages to knock down a number

of pins at the bowling alley but is uncertain how to keep score. There are three pins left standing, and she tries to figure out how many she has knocked down. She knows there were 10 pins to begin with, and she could have simply subtracted three from 10. Instead she comes up with an equation: 10-x=3.

Younger children don’t exhibit the difficulties ascribed to older children who struggle with algebra, says Bárbara Brizuela, who leads the Tufts Early Algebra Project and has been successfully teaching the subject to Somerville and Boston children. Photo: Courtesy of Bárbara Brizuela

To solve the problem, she used algebra, which moves students from basic arithmetic functions to more complex reasoning. It’s necessary not only to progress to higher forms of math in school, but it develops logic, problem-solving skills and the ability to see patterns by teaching a familiarity with unknown numbers, as well as how to form equations and solve them.

Algebra is usually taught in middle school or high school with mixed results. For many students it’s about as palatable as Brussels sprouts: they know it’s good for them but they would just as happily skip it, given the choice. Teachers sigh, figuring that for some students, algebra is just too difficult.

But that’s not so, says Bárbara M. Brizuela, an associate professor in the education department. For the last decade, Brizuela and her colleagues have been teaching algebra to Somerville and Boston children in grades two through five with positive results, finding that young children are able not only to grasp algebraic concepts, but that learning these ideas earlier enhances their overall understanding of mathematics.

The Early Algebra Project began in 1998 and today has been expanded beyond the classroom to include a summer camp, which was held last year in Roxbury. A new session is scheduled for this summer in a partnership with the Center for Engineering Education and Outreach at Tufts.

Joining Brizuela at the Early Algebra Project are Analúcia D. Schliemann, a professor of education at Tufts, and David W. Carraher, a senior researcher at TERC, a nonprofit education research organization in Cambridge, Mass., along with other faculty, staff and graduate students from the education department, and collaborators from other universities. The project is funded by grants from the National Science Foundation. Most of the children in the project come from minority and first-generation immigrant families.

Algebra helps students generalize from arithmetic operations with specific numbers to more abstract ideas that consider functional relationships between values. Mathematics educators have long argued that exposure to algebra concepts and representations in elementary school will prepare students to learn more advanced mathematics in middle and high school.

According to Brizuela, younger children don’t necessarily exhibit the difficulties ascribed to older children who struggle with algebra. “It’s an important lesson,” she says, “because it leads us to realize the difficulties are not in the children, but instead that there is something wrong with how we have traditionally been teaching mathematics.”

The literature on teaching algebra says adolescents have difficulties thinking about variables and unknowns, yet the children in the Early Algebra Project are able to understand these ideas.

“In general, we’ve found that children are able to generalize, which is what you’re looking for when you’re introducing algebra in middle school,” says Brizuela, who is on a Fulbright Fellowship this semester researching algebra and lecturing on its potential for early learners at the Universidad Nacional del Comahue in Bariloche, Patagonia, Argentina. “We’re looking for the children to appropriate a language with which they can make generalizations about relationships and quantities. Very young children are able to do this.”

She points out that from a very young age a child can form generalizations. “It’s a basic ability,” she says. “When a two-year-old calls every four-legged animal a dog, that’s a generalization. Not only are there algebraic concepts that young children can understand, says Brizuela, but teaching them at a young age is similar to starting a foreign language when young: it’s simply easier.

Because algebra takes time to learn, it’s easier to start in elementary school and continue throughout high school than to start in middle school. “If all you do is focus on arithmetic and problems that always have the same structure, by the time students are in sixth grade, you’ve basically thwarted their possibilities,” she says.

The Early Algebra Project classrooms are lively as children are encouraged to brainstorm to figure problems out on their own. Then they are asked to show their ideas about a problem and their suggested solutions. Teachers encourage them to represent their ideas in different ways, including algebra notation. They might, for example, draw a graph and a table for the same function, learning that relationships and rules can be expressed in different ways.

By working with children for more than a decade, including following the same group of students for six years, from third through eighth grade, the project members have learned there is a long-term impact of students’ early experiences with algebra. In assessments done last year with sixth and seventh graders, children who had been through the Early Algebra Project when they were in elementary school scored higher on a range of mathematics questions than children who had not participated in the project. For example, the children successfully solved a problem given to tenth graders on the Massachusetts Comprehensive Assessment System (MCRS) test.

While convinced of their success and the positive impact of their methods, the staff of the Early Algebra Project say they are realistic, not expecting to rewrite school curricula overnight but instead using teacher education to help teachers learn how to introduce new concepts in the classroom.

Toward that end, Brizuela, Schliemann and Carraher have published *Bringing Out the Algebraic Character of Arithmetic: From Children’s Ideas to Classroom Practice* (Lawrence Erlbaum Associates, 2007), and the program’s website, http://www.earlyalgebra.org, offers learning activities for teachers. “We have sample activities that we hope will open up for teachers the possibility of thinking that they can teach, and that elementary school children can learn a lot more than just arithmetic,” says Brizuela.