School & District Management

Teaching Mathematics Requires Special Set of Skills

By Debra Viadero — October 12, 2004 6 min read
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As a young teacher in an East Lansing, Mich., elementary school, Deborah Loewenberg Ball realized teaching mathematics required a special kind of knowledge. Unfortunately, she didn’t have it.

Take long division, for example.

“There was no way of explaining to students what the procedure means, and what they’re really doing,” said Ms. Ball, now an education professor at the University of Michigan in Ann Arbor. “Kids forget that stuff all the time.”

Without the right explanations, she added, “all you can do is keep saying, ‘Bring down the zero.’ ”

Now, though, after 20 years of systematic study, Ms. Ball is beginning to get a handle on just what that special kind of knowledge is. With her university colleagues, she has defined what teachers need to know to teach math effectively and devised new ways of measuring whether they know it.

This is no mere exercise in academic hairsplitting. Ms. Ball’s work suggests that having what she calls “mathematical knowledge for teaching” matters for students’ learning. Her research shows that students whose teachers score high on the measures developed by Ms. Ball and her research partners learn more math over the school year than do students of low-scoring teachers.

“Deborah’s work really is the model that we look to to set the standard,” said Lee S. Shulman, the president of the Carnegie Foundation for the Advancement of Teaching, based in Stanford, Calif.

Having coined the term “pedagogical content knowledge” in the mid-1980s, Mr. Shulman is the godfather of the idea that teachers need to know their subject matter in special ways to teach well. But he said Ms. Ball has found concrete ways to elaborate, describe, and measure it in her own field of mathematics and to show that it matters.

“It’s what we should be doing in all the disciplines,” he said.

As Ms. Ball sees it, the knowledge good math teachers need consists of more than knowing math well or understanding how children think at particular developmental stages. It comes from knowing how to apply mathematical knowledge, quickly, in ways that make sense to students.

Reacting to Students

In universities that prepare teachers, she said, “it’s almost as though this kind of thing fell in the crack between the math department and the school of education.”

For example, when a student uses an unorthodox method for multiplying 25 times 35 and still gets the right answer, an effective teacher must determine in a split second whether that student’s approach is a genuine method that generalizes to other multidigit multiplication problems or pure dumb luck.

When the teacher, flummoxed, instead instructs the student to stick to the standard method, that can send the wrong message to the class, said Hyman Bass, a research mathematician who has been working with Ms. Ball since the late 1980s. “The message to the student in that circumstance is that mathematics is a subject where you’re not supposed to think,” he said.

Or suppose a teacher is teaching about polygons, the researchers write. A polygon is defined as “a closed plane figure consisting of straight lines,” according to Webster’s New World College Dictionary. To begin with, the teacher has to recognize whether students could even begin to understand the textbook definition of a polygon—and, if necessary, come up with another definition that is more user-friendly, but just as precise.

A teacher would also have to be able to pinpoint the sources of the errors their students make, to come up with word problems on the spot that illustrate key concepts, and carefully choose assessments that show whether students are “getting it,” not just making lucky guesses.

“This is a subset of knowledge that teachers need to know that mathematicians don’t even know,” Mr. Bass said.

In fact, the Michigan researchers have given some of their tests of mathematical knowledge for teaching to mathematicians. While the mathematicians answer most questions correctly, Ms. Ball said, they also get some wrong—and they often take much longer than experienced teachers do to puzzle out a correct response.

Deborah Ball’s work in defining teachers’ knowledge in mathematics grew out of earlier work showing that most teachers, like Ms. Ball herself at one point, didn’t know enough math. To better devise strategies for filling in the gaps, she decided to go back to the source: classrooms.

She and her colleagues studied a 3rd grade classroom intensively for a year, collecting videotape footage, student work, teacher journals, lesson plans, textbooks, and other “records of practice.” They collected videotapes from dozens of other classrooms as well and invited mathematicians and educators to sit with them to pore over all the material and make observations.

“You start realizing that no matter how you teach, kids make mistakes and teachers who can’t figure out what kids are doing are really ill-prepared to do the work,” Ms. Ball said.

From those real-life observations, the researchers have developed a bank of 200 test items, which can be used to evaluate professional-development, preservice, and other kinds of preparation programs for math teachers. However, the tests are not designed to evaluate the skills of individual teachers, said Heather C. Hill, an assistant research scientist who works closely with Ms. Ball on the assessments.

In addition, Ms. Ball, Ms. Hill, and colleague Brian Rowan gave a 30-item assessment to a total of 699 1st and 3rd grade teachers at 115 predominantly low-income schools across the country. The schools were drawn from the massive Study of Instructional Improvement, in which Ms. Ball is also participating.

The researchers also tested the students of those teachers twice—once in the fall and, later, in the spring of the same school year. The results showed that, while most students learned math over the course of the year, the learning gains were greatest for pupils whose teachers had also scored high on the 30-item “mathematical knowledge for teaching” test. In other words, the researchers figured, students got an extra one-third to one-half of a month’s worth of learning growth for every standard-deviation rise on their teachers’ test scores.

In addition, the researchers said that teachers’ scores on the mathematical-knowledge assessment seemed to matter more than how much time they spend teaching math during an average school day, whether they were certified, or whether they had taken extensive mathematics or math teaching courses.

Classroom Observations

To check against the possibility that the high-scoring teachers were simply smarter than their other colleagues, the researchers compared teachers’ scores on reading and language-skills tests with their scores on mathematical knowledge for teaching.

“If teachers scored high on both of them, then you begin to think maybe what you’re measuring has more to do with general intelligence,” Ms. Ball said.

But that did not turn out to be the case.

As an added check, the researchers also visited the classrooms of the highest- and lowest-scoring teachers.

“What if somebody scores great and you visit that teacher, and they’re really clumsy and you can’t understand what they’re saying,” Ms. Ball said. “Or what if someone scores badly, and they’re really skillful?”

“So far,” she added, “we’re seeing that there is some relationship between their scores and the way they teach.”

Ms. Ball said her findings are important because so few studies link what teachers know to their students’ math achievement.

The results also suggest that efforts afoot now to require teachers to take particular math courses—or to open up the field to individuals who might have degrees in math but no education training—could be on the wrong track.

“While those may be nice ideas,” Ms. Ball said, “these results suggest that might not be the most promising way to think about improving math instruction.”

Interesting ideas?
Send suggestions for possible Research section stories to Debra Viadero at Education Week, 6935 Arlington Road, Bethesda, MD 20814; e-mail:

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