While educators and policymakers debate whether 8th graders can be readied to learn algebra, Sigrid B. Frawley sits in front of her kindergartners with a magic bag.

She puts three tokens in the bag and pulls five out. Then she asks the students: “What’s the rule?”

“Add two,” is the answer.

Next, she puts four tokens in and pulls six out. “What’s the rule?” she asks.

The 5-year-olds don’t know it, but they’re talking about an algebraic equation: x+2=y.

Students at Walter Stillman Elementary School in this suburb of New York City, in fact, get doses of algebra starting in kindergarten and lasting throughout their careers here. By the time they reach middle school, one-third of them will be ready for algebra in the 7th grade. Almost all take the course in the 8th grade.

“The answer to, ‘How do you get them there?’ is: You give it to them early,” said Principal William B. Greene.

American middle schoolers are increasingly being moved into algebra courses. In what was once a rite of passage in high school, now middle schoolers—some as early as 7th grade— are now expected to learn about the mathematics of variables and quadratic equations. In California, for example, all students are expected to learn algebra in the 8th grade, according to the state’s standards.

In a 1998 survey, 95 percent of high school graduates had taken algebra, a 14 percent increase from eight years earlier, according to the Council of Chief State School Officers.

To get students ready for that leap, the consensus among mathematicians and educators is that students need to be introduced gradually to algebraic concepts throughout the elementary school years. The only debate is how to teach it.

“The student of algebra need not begin with a formal course in the subject,” the National Research Council concluded in a January report on teaching mathematics. “From the earliest grades of elementary school, students can be acquiring the rudiments of algebra.”

“Elementary school really is the critical place for fixing America’s algebra problem,” said James Kaput, a professor of mathematics at the University of Massachusetts Dartmouth. “We’re only slowly coming to terms with that.”

### Proper Functioning

At Stillman and at Tenafly’s other three elementary schools, teachers follow the *Everyday Mathematics* curriculum and textbook series developed at the University of Chicago.

In addition to her magic bag, Ms. Frawley integrates equations with variables into her arithmetic lessons. She’ll write the equation 3+__=5, and ask her students to fill in the blank.

Second graders are encouraged to draw graphs of equations similar to the ones created from the objects Ms. Frawley draws from her bag. If the rule is to add 2 to every number, the students are asked to find a place for everything on the x and y axis.

By 5th grade, students are completing word problems with introductory algebraic reasoning. For example, they’re told that mules travel six miles per hour. They’re asked to calculate how far the mules will have traveled after one, two, and six hours. Then, they’re asked to figure out how many hours the animals took to walk 30 miles and 48 miles.

If they saw the equation as 6x=y, as they would in an algebra class, they might not understand it. But with the concrete examples in front of them, about 85 percent of the students can solve the equations, said Mireille Bany, one of the school’s 5th grade teachers.

“We’ve done it so many times that they know how to do it,” she said.

Such preparations are important, math experts say, because middle schoolers often have a difficult transition into algebra.

After years of solving problems using basic arithmetic skills, they struggle with the mathematical reasoning and manipulations required to succeed in algebra.

Students who learn arithmetic come to think of the equals sign as a function, much the same way the plus sign instructs them to add, according to Thomas P. Carpenter, the director of the National Center for Improving Student Learning and Achievement in Mathematics at the University of Wisconsin- Madison.

The equals sign, however, “expresses a relationship rather than a command to do something,” he said. If students don’t understand that, they can’t solve such beginning algebraic equations as 3x+5=15.

“Fifteen isn’t the answer,” Mr. Carpenter said, “but that’s how they are thinking about it in arithmetic.”

Simple equations, such as the fill-in-the-blanks in Ms. Frawley’s kindergarten class, help students understand what the equals sign means in algebra.

### How, Not Why

While there’s wide agreement that algebra should be introduced in the early grades, differences remain over how to teach it.

California’s math standards—which follow the traditional method of teaching the subject—emphasize mastering addition, subtraction, multiplication, and division in the early grades. Algebraic concepts are introduced as ways to apply the algorithms.

“You should be doing certain kinds of variable equations all the way along,” said Wayne W. Bishop, a professor of mathematics at California State University-Los Angeles and one of the authors of the California standards. “You can’t just assume you’re going to dump everybody into 8th grade algebra if you haven’t done some preparation with elementary linear equations.”

Teachers should mix simple word problems into arithmetic as early as 1st grade, Mr. Bishop said. They should even use “x” as a variable as part of the teaching, he said.

By contrast, the National Council of Teachers of Mathematics’ standards downplay the algorithms and treat algebraic principles more fully. In addition, teachers are encouraged to use real-life examples to illustrate the mathematics. One example in the standards has students filling in the blanks of a chart projecting the cost of bunches of balloons. The chart says that one balloon costs 20 cents, two cost 40 cents, three cost 60 cents, and four cost 80 cents. Students are expected to complete the series that extends to a bunch of seven balloons. Using variables such as “x” is “too abstract,” contends Lee V. Stiff, the NCTM’s president.

“When it comes to doing algebraic manipulations, [students] can see how they did it with numbers, and then they can do it with variables,” said Mr. Stiff, a professor of mathematics education at North Carolina State University in Raleigh.

### Teaching Teachers

But even though young students are able to learn simple algebra, the ability of many elementary teachers to help them learn it is open to question.

“You have elementary school teachers who do not know what algebra is about, so they’re not in the position to think about how the arithmetic they’re teaching will mesh with algebra later,” said Roger Howe, a professor of mathematics at Yale University and a member of the panel that wrote the National Research Council report.

Elementary school teachers generally don’t excel in mathematics in high school and often aren’t expected to take high-level courses in college. They often come to their first jobs afraid that they are unable to teach the subject, said Ms. Frawley, the kindergarten teacher here in Tenafly.

Mr. Carpenter’s center at the University of Wisconsin is working with teachers in Los Angeles, Phoenix, and San Diego as part of a larger project to get teachers thinking about how they can introduce variables into their grade school math curricula.

Good teachers need to “build algebra eyes and ears,” said Mr. Kaput of the University of Massachusetts, so that they can take advantage of any opportunity to teach about the subject. Mr. Kaput and his colleagues are working with 350 teachers in Fall River, Mass., to help them engage young students in algebraic thinking.

“What we do is teach the teachers how to take a problem and build a series of problems off of it,” Mr. Kaput said. “It helps build [students’] computational skill and builds on the deeper understanding that we’re after.”

Here in Tenafly, new teachers attend a one-week seminar on *Everyday Mathematics* before they enter the classroom. They then meet with mentors once a month to learn more about the goals of the program.

The key to making the experience work, Ms. Frawley said, is to avoid the term “algebra” and focus on the real-life applications of the mathematics of the subject.

“To say to them, ‘You’re teaching algebra in a kindergarten class,’ that would throw them,” she said of the new teachers. “They think of that as a high school class.”

But teachers have been able to pick up on the program and have fully integrated algebraic thinking into the early grades in a matter of three years, Ms. Frawley said.

Sometimes, they’re teaching the subject without even thinking about it.

One morning early this month, Terry Moore, a 3rd grade teacher at Stillman Elementary School, is filling out the March calendar on a whiteboard. He writes the numbers symbolizing the first seven days of the month in their boxes. He calls on a student to tell him the number of every Wednesday of the month. The simple answer is to add seven to Wednesday the 7th.

As Principal Greene watches, he points out that the exercise is a simple algebraic equation: x+7=y.

Later, Mr. Moore is asked if he thought of the exercise as an algebra lesson.

“Only since you mentioned it,” he said. “It wasn’t set up to do that, but you could make a function of it.”