In discussions of the mathematics achievement—or lack thereof—of U.S. students, algebra is often the focus of attention. The reason, experts say, is simple: Algebra is so central to mathematics that, if students do not learn it, their future options are severely limited.
Algebra is "the first `cut' place," says Mary Lindquist, president of the National Council of Teachers of Mathematics (NCTM). Without it, students are cut off from future mathematics study and from careers in technology and the sciences. "Algebra has been identified as a gatekeeper, not only for math but for science and other higher learning," agrees Dorothy Strong, manager of mathematics support for the Chicago Public Schools.
Concern over algebra's role as a filter is leading educators to change the way they teach the subject, to make it more accessible to more students, experts say. The city of Chicago and the state of Louisiana have even made algebra a requirement for all students, a step Mississippi plans to take in 1995.
In the past, however, only a minority of students have done well in algebra. One reason for this limited success is the "abrupt jump" between algebra—which typically begins in 8th or 9th grade—and the math that has preceded it, explains Christian Hirsch, a professor of mathematics and statistics at Western Michigan University. After a steady diet of concrete work with numbers, many students find the formal, abstract work with symbols they encounter in algebra foreign and unpalatable.
Moreover, algebra instruction has focused on how to transform symbolic expressions from one form to another—how to factor x2 + 5x + 6 into (x + 2)(x + 3), for example—without conveying what these expressions mean, Hirsch says. Algebra has not been tied to real-life contexts that students would find more understandable and motivating.
Examining Data
Experts consider this emphasis on symbol manipulation misguided. Teaching these skills without ever applying them to a real problem is "as if you teach French and only focus on the grammar and never have a real conversation," says Zalman Usiskin, director of the University of Chicago School Mathematics Project.
To make algebra lessons more authentic, teachers are shifting away from symbol manipulation to a new emphasis on "modeling," Lindquist says. Modeling calls on students to examine data, represent it mathematically, and verify their work to the real world.
This change reflects a reordering of priorities to highlight the concepts that most students should learn—not just the future calculus students, experts say. "There are ideas in algebra that all kids need to be familiar with and have some confidence using, whether they're going to end up on the shop floor or in the board room," Hirsch says. This essential knowledge has less to do with symbol manipulation, he explains, than with using math to represent real situations and answer questions about them.
Materials developed by the Core-Plus Mathematics Project, which Hirsch directs, use real-world situations, often involving data, to pose interesting questions. In this approach, "x has meaning," he says. "It's the height or weight of something, and not just `24.'"
Word problems are also changing, Usiskin says. Contrived puzzle problems ("Mary is half as old as her father was when he was three times the age of ...") are being replaced by problems that resemble real-life applications of algebra. For example, students might be asked to graph data, from 1910 to the present, on the world record for running the mile. They would discover that the result is almost a straight line, Usiskin says, and they could describe the line with an equation and then make some predictions.
These new word problems ask students to use algebra to examine data, "which is what people in business, government, and statistics do all the time," Usiskin says. Ironically, these problems tend to be easier because students have some context; they can use intuition as a guide. "The old problems were devised so that you couldn't use intuition," he notes.
Importance of Graphing
The ready availability of graphing calculators is also changing algebra, experts say. The study of functions—the relationship between two variables—is being moved much earlier, even into first-year algebra. Now students can see a pictorial representation of functions, such as a change in population over time or the change in mailing cost relative to the weight of a parcel. And they can readily see the effect of changing either variable or both. "Technology enables you to play with these things and explore them," Usiskin says.
In the Chicago schools, study of algebra begins with graphing, Strong says. When students first encounter an equation such as x2 - 3x + 1 = y, for example, they try different values for x and graph the results. Graphing gives students "an opportunity to have a picture of what this equation is talking about," Strong says. "Then we build everything else on that."
Algebra lessons must be relevant to students, Strong emphasizes. In the past, "we put lessons before them that were totally foreign," she says. Now teachers make explicit links to students' daily lives. For example, they ask students to lay out the city of Chicago on a coordinate grid, placing (0,0) on the spot where they live.
Another new development in the teaching of algebra is an effort to lay a better groundwork during the K–8 years. According to the NCTM's math standards, "By integrating informal algebraic experiences throughout the K–8 curriculum, students will develop confidence in using algebra to represent and solve problems."
"Algebra is the language for describing patterns," says Usiskin. "The earlier you [teach] it, the easier it is for kids to learn." For example, an elementary teacher could discuss the equation 5 + 9 = 9 + 5, asking students to describe the general pattern. Students might say, "You can switch the numbers." The teacher could then suggest expressing the pattern as a + b = b + a.
Requiring all students to take algebra can be "an excellent thing," Strong believes—but only if instruction is improved first. "If all we do is require the same abstract algebra not related to anything that they couldn't learn before, then we do [students] a disservice," she asserts.