Over the past 20 years educators have fought over the best way to teach numbers to kids. Advocates of traditional math tout the practice of algorithms and teacher-centered learning, whereas reform-math proponents focus on underlying concepts and student inquiry. In the face of continued declining scores in the U.S., these so-called math wars have heated up recently with the circulation of petitions, the release of contested curriculum guidelines and, in one case, the filing of a lawsuit. At stake is the ability of American high school graduates to perform everyday math tasks and compete in a global economy.
The war began in 1989, when the National Council of Teachers of Mathematics (NCTM) released a set of standards that reshaped a generation of instruction. Instead of having students memorize formulas and compute problems such as adding fractions, advocates of reform math encouraged students to develop their own visual representations of math concepts and use calculators to solve numerical tasks.
In recent years a détente between the two camps formed, one that emphasized a middle ground. But if there is a truce, it is an uneasy one—new volleys from both sides continue to appear. Last October, for example, the NCTM released yet another document, “Focus in High School Mathematics: Reasoning and Sense Making,” which calls for a new approach revolving around applications. “Our 15-year-olds cannot use math to address simple real-life situations,” explains Gary Martin, a professor of math education at Auburn University and chair of the committee that wrote the document. Martin says that the new guidelines teach students how to “apply mathematical reasoning in a variety of contexts” instead of simply “carrying out procedures in a rote way.” He cited as an example a problem that asks students to compare the relative fuel-efficiency gains in two pairs of vehicles. The answer varies depending on whether one considers relative fuel efficiency or the total number of gallons of gasoline saved.
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Although many educators have praised the report, critics say the document’s vague approach to mathematical analysis is reminiscent of the NCTM’s 1989 guidelines. “The sense is that all reasoning students attempt is valuable and should be celebrated,” says Stanford University mathematician Jim Milgram, who prefers a more traditional approach. “The trouble with this approach is that it is exactly status quo; we seem to have a mindset that, ‘Gee, Johnny reasoned’; it doesn’t matter that his actual reasoning is flawed.”
Vern Williams, a math teacher in Falls Church, Va., who has participated in several national math panels, says the high school guidelines downplay the link between reasoning and traditional procedures such as factoring polynomials. “Some of the most elegant math problems are deemed useless because they don’t involve real-world applications,” he says. Williams adds that many courses in geometry, “the one high school class that demands formal reasoning,” have already been “gutted” and are no longer proof-based. Instead students use algebraic tools to analyze geometric shapes, build three-dimensional models, and solve actual construction and design problems.
For his part, Martin says the document was not intended to define specific content; rather it shows how reasoning and sense making can be incorporated throughout the curriculum. But detractors of reform math do not seem to be ready to agree. In one notable example, a group of parents and educators in Seattle have filed a brief appealing the school board’s decision last May to adopt the Discovering Mathematics series, a reform-math high school text that uses student investigations as a means of discovering math principles—such as using toothpick models to derive recursive sequences. Citing declining test scores after a three-year pilot of the text, the suit claims the Discovering series is associated with a widening achievement gap between white students and minority and low-income students.*
*Erratum (2/25/10): The Discovering Mathematics series was not part of a three-year test pilot as mentioned in the story. The text should have read: "Citing declining test scores after a three-year pilot of a similar text, the suit claims that reform math is associated with a widening achievement gap between white and minority low-income students."
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“A good textbook is very important,” says Cliff Mass, a professor of atmospheric sciences at the University of Washington and one of three plaintiffs in the suit. He argues that the absence of “real math” in the Discovery series makes it very difficult for disadvantaged students to work through problems on their own.
Math experts have expressed other concerns about diluted instruction, including the decision by many school districts to require algebra I in eighth grade instead of the traditional ninth. Being younger, students tend to get a watered-down version of algebra I, but “then they are expected to continue on the trajectory of geometry and algebra II without a firm footing,” points out William McCallum, a math professor at the University of Arizona.
McCallum is working on Common Core, a 48-state initiative that aims to resolve some of the debates by producing standardized, nationwide guidelines. The effort, which is sponsored by the Council of Chief State School Officers and the National Governors Association, was scheduled to have been completed in February.
One sentiment unites almost all math professionals—after decades of wrangling, the system still isn’t working. According to the National Center for Education Statistics, U.S. high school students ranked in the bottom quarter in math performance, as compared with students of nations belonging to the Organization for Economic Co-operation and Development. In a future expected to depend even more on science, health and technology, that is bad news indeed.