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Love and Math I

(Center for Comprehensive School Reform and Improvement, Issue Brief, March, 2006.)

By Craig D. Jerald

Some elementary teachers might have breathed a sigh of relief when it became clear that the new "highly qualified teacher" rules in No Child Left Behind (NCLB) would not require much in the way of subject-matter knowledge. Those rules asked middle and high school teachers to demonstrate proficiency in the subjects they teach, either by earning a college major or passing a test in those subjects, in addition to having a bachelor's degree and a teaching certificate.(1) However, although the law made a nod to elementary teachers knowing their subjects too, for K-5 teachers in most states, it required nothing new.

Of course, that didn't seem odd to most educators and administrators. After all, how much mathematics does a first- or second-grade teacher really need to know in order to teach arithmetic to 6- or 7-year-olds? While experts occasionally harrumph about teachers with weak math skills flocking to the lower grades, the issue certainly hasn't generated the outraged response that reports on "out-of-field teaching" in middle and high schools has during the last 10 years.

Indeed, a longstanding tenet of American education-one built into the very fabric of teacher training and licensure-is that elementary teachers need only general teaching skills and that having a caring personality is sometimes more important than how much math (or science or history) a teacher knows. In the lower grades, so the reasoning goes, even teachers who have weak math skills themselves simply need to be sure they know enough to teach a given concept or skill. And they can do that by staying one step ahead of the children in the curriculum-for example, by brushing up on the rules for multiplying three-digit numbers the week before they begin a unit on it.

Added Value of Specialized Knowledge

But an important new study published last year suggests that this conventional wisdom is very, very wrong. Researchers tested the mathematical knowledge of first- and third-grade teachers and then examined the impact of that knowledge on student gains in mathematics over the course of a year. The study revealed that a teacher's own mathematical knowledge has a substantial impact on student learning even at the first-grade level, a finding that left even the researchers themselves "modestly surprised."(2)

That impact remained even when researchers accounted for other factors that might influence student learning gains, including family background (the number of parents in the home, parent education levels, and family income), how often students were absent, teacher certification and experience, and how much time teachers spent on math lessons. And the impact was quite large: Children taught by educators who have high-level math knowledge gain the equivalent of about two or three weeks of extra instruction compared with students whose teachers have average math knowledge and skills. In fact, teacher knowledge had as big an impact on math learning as student poverty or race.

Because of the so-called "math wars" now raging in some states, it's important to note that the study wasn't conducted by fringe researchers with an ideological axe to grind or published in some obscure academic journal. The research team included Deborah Lowenberg Ball, dean of the School of Education at the University of Michigan, and the findings were published in the American Education Research Journal, the flagship publication of the nation's oldest and largest association of education researchers. Moreover, the study was part of a large-scale, longitudinal research project called the Study of Instructional Improvement, which has been tracking comprehensive reform and improvement in 120 U.S. schools for the past several years. (Visit for more information.)

The researchers found that two kinds of teacher mathematical knowledge are important for student learning-"common" and "specialized." Common math knowledge includes basic mathematical concepts and procedures for solving basic problems, such as figuring out percentages or multiplying and dividing by fractions. In other words, common knowledge includes what anyone who was fortunate enough to benefit from good elementary and secondary math education would know.

But students benefited most when teachers also had an extra "fluency" in thinking about and dealing with basic mathematical concepts and problems-a "specialized knowledge" that goes beyond common math skills. Such teachers are good at puzzling out where a student went wrong in multiplying two three-digit numbers, for example, or knowing whether an unusual solution to a fractions problem would work for other kinds of problems the student might encounter later in the curriculum.

For example, three students might solve the problem 6 x 87 in three different ways, two getting the answer wrong and the other getting the answer right. Their teacher has to be able to analyze the work of the two who arrived at incorrect answers to understand-and help them understand-what they did incorrectly. If the third used an unusual approach to arrive at the correct answer, the teacher must signal whether that approach will always work with similar problems the student might encounter later rather than simply marking the problem correct or saying "good job."

Ball and her colleagues take pains to emphasize that this specialized math knowledge is still math knowledge rather than simply knowing how to teach well-in other words, mathematical reasoning rather than simply instructional reasoning. For example, "Appraising non-standard solution methods is not a common task for adults who do not teach. Yet, this task is entirely mathematical, not pedagogical: to make sound pedagogical decisions, teachers must be able to size up and evaluate the mathematics of these alternatives-often swiftly and on the spot."(3)

What's new about this particular study isn't the concern that some elementary teachers might lack important mathematical knowledge and skills, but the evidence that it matters so much even in the earliest grades. Indeed, a book by Liping Ma, a onetime student of Ball's now at the Carnegie Foundation for the Advancement of Teaching, kicked off an intense debate among math educators in 1999. Ma studied a group of elementary school teachers in China and another in the United States and documented that Chinese elementary school teachers had deeper knowledge of mathematics than their American counterparts.(4)

That book has been used to advocate for changes to preservice training, which, of course, might be part of the solution. However, Ma found that while Chinese teachers do enter teacher education with strong math skills, they developed "profound understanding of fundamental mathematics" during their teaching careers. Indeed, in China, those who teach math to elementary school students teach only math.

Solving for X

The new research adds up to trouble for school staff, district administrators, and assistance providers who have gone about the business of school reform and improvement without questioning the longstanding assumption that teaching elementary math requires only basic math skills.

For example, efforts to improve math achievement often involve implementing new math programs. Leaders and assistance providers routinely consider the resources teachers will need to make a program work such as curriculum, textbooks, instructional materials, and time. They also consider how much professional development to provide to teachers. This research suggests they should pay special attention to whether teachers need professional development that addresses math skills.

There's no doubt the topic will be an uncomfortable one for many teachers. It's far easier to discuss how much training teachers will need to understand and use a new instructional technique, especially one that comes "bundled" in a new math program, than to talk about whether teachers have adequate mathematical knowledge and skills. Given teacher sensitivity to this issue, are there politic, non-threatening ways to broaden the scope of the "needs assessment" stage of the school improvement process to allow for such questions?

Such efforts will be particularly important for closing achievement gaps. Ball and her colleagues found that low-income and minority students were more likely to have teachers with low-level math knowledge in elementary school-a fact they believe will make it hard for schools to help such students keep pace with their peers. Indeed, the researchers were surprised to find that the inequities were biggest for minority students: "We find these results shameful."(5)

Moreover, assistance providers, education leaders, and policy makers need to begin asking how they can help schools assess whether teachers have sufficient math knowledge and identify those who do not. Guesswork alone will not do. Nor will teacher self-identification. Good school improvement efforts minimize uncertainty and base action on knowledge. Unfortunately, the assessments developed by the University of Michigan have not been converted into general use instruments that can be used to make judgments about individual teachers. (However, the researchers have released a fairly extensive set of sample items.(6))