Algebra in grade school

Algebra in elementary school? No sweat
by Marjorie Howard

For many students, algebra is not the high point in their careers. It's no wonder. Algebraic notation is mystifying, its purpose unclear. Just when they are becoming comfortable with numbers, students are asked to work with letters that sometimes stand for particular, unknown numbers, sometimes for sets of possible values, and sometimes for nothing at all.

For a group of Somerville, Mass., elementary school students, however, algebra is already a natural part of their math lessons, and the idea of using letters to represent variables makes perfect sense.

In a pilot program, Somerville, Mass., elementary school children are using algebra to solve problems that older students might find challenging.

The students are part of a pilot project run by Tufts educators and a Cambridge-based group called TERC, a nonprofit education research and development organization. Funded by a $1 million grant from the National Science Foundation, the three-year project, "Bringing Out the Algebraic Character of Arithmetic," aims to show that it is possible to successfully introduce algebra in elementary school. The study began in 1999 with 60 second-graders who are now in fourth grade. The hope is to continue the study as the children advance through school.

Higher standards
"There is widespread concern in the United States among math educators about the kind of math we're teaching in the early grades," said Analœcia Dias Schliemann, professor of education. "The National Council of Teachers of Mathematics has established standards aimed at making sure children do not struggle with algebra when they reach high school, but the standards do not show how this can be done."

Schliemann and her colleague, B‡rbara Brizuela, assistant professor of education, together with David Carraher of TERC and their research team hope to help children think algebraically. The team is also documenting classroom discussions so that others can learn from and build upon their work. Their videos and analyses are put onto CD-ROMs and DVDs using videopaper technologies. Videopapers offer both text and a video demonstration side by side on a computer screen.

So far, so good
So far, their approach seems to be working. The children are using algebra to solve problems that older students might find challenging. In a pilot study carried out in 1998-99, the students' MCAS scores (Massachusetts now requires students to pass achievement tests in math and English to graduate from high school.) were significantly better than those of their peers at district and state levels. While the program did not "teach to" MCAS, said Schliemann, the children benefited from the "high standards set while aiming for understanding."

A research team that includes two Tufts faculty members is working to strengthen the way mathematics is taught in the early grades.

Algebra is the branch of mathematics in which variables and functions are used instead of specific numbers and computations. An arithmetic question about addition might ask, "If you have five pieces of candy and someone gives you three more, how many do you have?" The children add the two numbers together and answer eight.

The research team approaches an arithmetic operation differently. For example, a teacher might tell the class that there are two closed boxes of candy, and each has the same amount of candy. One box is John's; the other is Mary's. Mary also has three more candies on top of her box. The children are asked to consider possible values that could be contained in the boxes and to show their answers on paper.

The children use tables to systematically consider the possible answers. If they say there could be two candies in each box, they would represent the boxes as 2+2+3, noting the two boxes plus the additional candy that Mary has. After considering a series of possible values, they are encouraged to use a letter to represent any amount of candy that could be in the each box. From this the children suggest that the total amount of candies will be N+N+3 or 2N+3.

The better deal
In another problem, children were told that Raymond has an unknown amount of money. His grandmother offers him a choice of two deals. In the first deal, she will double his money; in the second, she will triple his money but take away $7. The question is, which is the better deal?

The children realize they do not know how much money Raymond has and won't be able to find out. Using tables and graphs that they themselves devise, the children conclude that if Raymond has less than $7, he should take the deal that offers to double his money. But if the amount is more than $7, the second deal is a better one. They also conclude that if Raymond has $7, it doesn't matter which deal he accepts.

Some of the research team's findings have been presented at conferences in the United States, Japan and Holland. The team also has signed a contract with Lawrence Erlbaum and Associates to publish a book called Bringing Out the Algebraic Character of Arithmetic: From Children's Ideas to Classroom Practice.

Schliemann and Brizuela hope to get funding to make their findings accessible to a wide range of mathematics teachers and to gradually strengthen the teaching of early mathematics.