The math wars and the case for problem-centered math

The theme of this series of articles is problem-centered math, an approach to mathematics education based on problem solving. We could just as easily have called this a student-centered approach, or, to use the more formal term, constructivist; it follows the theory that learning occurs when students construct their own knowledge. In problem-centered math instruction, students construct their own understanding of mathematics through solving problems, presenting their solutions, and learning from one another’s methods. This is very different from traditional methods of teaching math, in which the teacher specifies a procedure from the start and students practice that procedure until they are comfortable with it.

The move toward problem-centered math is supported by the National Council of Teachers of Mathematics and backed by a great deal of research, which has shown that students are better able to perform basic mathematical operations when they develop their own strategies and procedures. Nevertheless, like any reform movement, it has generated a great deal of controversy. Reformers argue that mere computation, in an age of calculators and computers, should not be the goal of math education; they want to produce citizens who can use mathematics to solve problems and to understand the world around them. Traditionalists call the new approach "fuzzy math," arguing that children must develop basic math skills before they can understand concepts and that the most effective way to learn skills is through memorization and practice.

There are, unfortunately, extremists on both sides of the issue, and the conflict between them has erupted into a series of "math wars" in California, where the State Board of Education’s decision to adopt a computationally driven mathematics curriculum led to particularly strong controversy. Extremists for the cause of reform can make group work and creative problems the end rather than the means, giving the impression that there is no "right answer" or that getting the right answer does not matter. On the traditionalist side, many parents, teachers, and interested citizens insist that if drill and practice in computation was good enough for them, it is good enough for today’s children — ignoring the fact that they themselves often never mastered the basic math skills in question.

Yet both reformers and traditionalists have legitimate points to make. At bottom the controversy is about the nature and purpose of mathematics. Most people see mathematics as a set of rules, skills, and facts to be learned, a static body of knowledge; students, therefore, need to be taught these rules and facts. The mathematicians who discover those rules and facts for the first time are special; that process of discovery, once completed, need not be repeated. Others, however, see mathematics as a body of ideas, ever-changing and constructed by humans to help them understand the world around them. That process of discovery is one that anyone learning mathematics can, and should, participate in. On the one hand, mathematics is about skills; on the other, it is about concepts.

We are advocating the problem-centered approach because research has shown, and we agree, that children harness the power of mathematics when they devise their own procedures for solving problems and that children who develop their own mathematical understanding more effectively learn skills and are better able to apply those skills to new problems than children who are given formal procedures from the beginning. And yes, those results even translate into higher scores on standardized tests.

Are there drawbacks to problem-centered math? It requires, certainly, a radical shift in how most teachers run their classrooms. It requires "letting go" — allowing students to make mistakes, discover those mistakes themselves through discussion and debate, and develop their own strategies for solving problems. It requires great patience to nudge students with questions; it’s tempting simply to tell them the answer or how to find it. And, most significantly, it requires greater mathematical understanding on the part of the teacher. Instead of relying on a single procedure for solving a problem, the teacher has to be comfortable with multiple approaches and be guided by his or her own mathematical sense.

But we believe the benefits of problem-centered math are easily worth the trouble. In this series of articles, we’ve compiled a set of professional development and classroom resources to help you understand what problem-centered math is really about, why it’s necessary, and how to implement it in your own classroom. Based on a workshop we sponsored at the North Carolina Center for the Advancement of Teaching in May 2001, the articles include video, text, and illustrations that show problem-centered math in action as well as ready-to-use problems and lesson plans to put it into practice. Because the eighth-grade gateway has become so important for students, we have chosen eighth-grade math — and particularly the curriculum’s geometry strand, which causes students the most trouble — as the focus of our efforts.

We invite you to take your time with these resources. Test the waters with a simple activity such as QuickDraw. Using the clinical interview as a guide, observe your students and try to understand how they are thinking about mathematics. Then use the other resources to find a way to help your students develop a stronger mathematical "sense" that they can apply to real-world problems. We promise the results will be worth the trouble.