Mediocre Is Not Good Enough

Issues Forum: Math Instruction

August 11, 1997

DIALOGUE: Creative Math, or Just Fuzzy Math?

Mediocre Is Not Good Enough

By THOMAS ROMBERG

ADISON, Wis. -- For the past quarter-century it has been obvious that Americans need a better understanding of mathematics, science and technology if our society is to prosper.

Although some schools, teachers and parents are satisfied with the way math is being taught, a recent study of math and science education in 41 countries described most math curriculums in the United States as "a mile wide and an inch deep." Given the fragmented, unchallenging material, taught by teachers who have little opportunity for professional development, it is no wonder that student performance is mediocre.

Answers and 'Solutions Techniques'

The "standards-based" mathematics curriculum, as its proponents call it, or "whole math," as its critics describe it, encourages students to invent their own ways to solve problems.

In one widely used curriculum, called MathLand, teachers are told how to guide sixth graders through the following problem:

"I just checked out a library book that is 1,344 pages long! The book is due in 3 weeks. How many pages will I need to read a day to finish the book in time?"

The old way to solve the problem would be to use the algorithm for long division: 1,344 divided by 21. The curriculum guide calls for a new approach, explaining that "division in MathLand is not a separate operation to master, but rather a combination of successive approximations, multiplication, adding up and subtracting back, all held together with the students' own number sense."

Teachers are advised to ask students to write down how they would solve the problem. When they are finished, the guide suggests, "Encourage a variety of solution techniques that go beyond just the numeric answer to descriptions and convincing sketches."

Mathematics is not just another subject to be covered in class. It is a human activity involving the ability to represent quantitative and spatial relationships in a broad range of situations, express those relations using the language of mathematics and use various techniques to carry out numerical procedures. Mathematics is also about making predictions, and interpreting results.

That is the view of the National Council of Teachers of Mathematics, which in 1989 drew up new standards for teaching math and evaluating student performance. The council carefully considered changes in mathematics and its applications in our increasingly technological society. Council members also examined how math was taught in other countries, and recent work of psychologists on how children learn.

The council maintains that the systems of signs and symbols of mathematics must be learned and experienced as genuine languages -- languages everyone needs to communicate, reason, compute, generalize and formalize 20th-century experience, and to serve 21st-century needs.

Moreover, systematic logical forms of reasoning and argument must be learned through personal and social experiences.

Yes, students need to develop certain skills. They need to be able to construct, follow and judge arguments. They need to be able to deduce and induce from context and formulate counterexamples. They need to be able to apply spatial, proportional, algebraic and graphic reasoning, and to construct proofs.

The standards the council created to help students learn these skills have received wide support from mathematicians and math educators. These standards consist of 50 brief statements about what all students should learn in math classes, organized into three levels through 12th grade.

At each level, there are four general standards -- problem-solving, communication, reasoning and connections -- and specific content standards about topics like algebra, geometry and statistics. The council supports introducing some of these topics in earlier grades (algebra in the middle grades, for example), and putting more emphasis on subjects like statistics.

The council has provided schools and teachers with examples of lessons and instruction guidelines. Though the rhetoric in the council's recommendations argues for balance between proven procedures and new activities, the examples tend to emphasize the new activities.

For instance, students should see the question of what kind of calculation procedure to use in a numerical problem as a decision that depends on the problem at hand. Although the council wants all students to learn to estimate and use mental calculation and to be equally comfortable using paper and pencil, calculators and computers, nowhere does it argue that students do not need to memorize multiplication facts. Nor does the council say that students should use a calculator for all computations.

Similarly, the examples the teachers council has produced often portray students working on real-world problems, providing oral or written explanations of how a problem was solved, or collaborating with other students. But the council never meant to imply that such approaches are appropriate for every lesson.

Many horror stories cited by the standards' critics, if accurately portrayed, are indeed bad. Sometimes teachers initially put too much emphasis on the general standards without considering mathematical content. For example, they may teach problem-solving without having students learn algebraic procedures.

Sometimes honest attempts by well-meaning teachers to try a new method of instruction, like having students work in groups, have led them to initially give up class discussions or individual assignments. Sometimes publishers' claims that their texts met the council's standards were mere puffery.

Sometimes material that in an author's mind seemed wonderful proved to be less than wonderful in a classroom. And sometimes even appropriate material is ineffective because the teacher using it doesn't have enough math background or the right training to turn the recommended activities into a learning experience for students.

Despite these problems, test scores tell us that teaching math by using the council's recommendations makes sense. Teachers and teacher trainers know that we need a new approach to math instruction, and most defend the standards. And while criticism is essential to refine the suggested techniques, it's unfair to attack the entire program because of initial missteps and isolated examples of misapplied guidelines.

Unless we reform math education so that our children can be prepared for the immense technological changes already occurring, our nation will lose -- and so will our children.

Thomas Romberg, a professor of mathematics education at the University of Wisconsin, was chairman of the commission on standards of the National Council of Teachers of Mathematics from 1986 to 1995.