Math can be hard enough, but imagine the difficulty when a teacher is just one chapter ahead of the students.
It happens, and it happens more often to poor and minority students. Those children are about twice as likely to have math teachers who don't know their subject, according to a report by the Education Trust, a children's advocacy group.
Studies show the connection between teachers' knowledge and student achievement is particularly strong in math.
"Individual teachers matter a tremendous amount in how much students learn," said Ross Wiener, who oversees policy issues at the organization.
The report looked at teachers with neither an academic major nor certification in the subjects they teach.
Among the findings, which were based on Education Department data:
-In high-poverty schools, two in five math classes have teachers without a college major or certification in math.
-In schools with a greater share of African-American and Latino children, nearly one in three math classes is taught by such a teacher.
Math is important because it is considered a "gateway" course, one that leads to greater success in college and the workplace. Kids who finish Algebra II in high school are more likely to get bachelor's degrees. And people with bachelor's degrees earn substantially more than those with high school diplomas.
The teaching problem is most acute in the middle grades, 5-8, the report said. That's a crucial time for math, said Ruth Neild, a research scientist at Johns Hopkins University.
"This is a time when kids are making a really important transition from arithmetic to mathematics," Neild said. "It takes careful instruction, and if kids can't get that, and really get it, they're not going to succeed in math in high school."
Yet it can be tougher to find qualified teachers for middle schools, especially in low-income areas, said Neild, who studied the problem in Philadelphia public schools. She did not work on the Education Trust report.
Teachers should not be blamed for out-of-field teaching, the report said. It can happen anywhere there is a teacher shortage in a particular discipline. It can also happen where there is no shortage but where school administrators have planned poorly.
Congress tried to fix the problem in the sweeping 2002 No Child Left Behind Law. The law insisted that all teachers in core academic subjects be "highly qualified" by 2006.
But the most well-known aspect of No Child Left Behind is its requirement for annual state tests in reading and math, and the penalties it imposes on schools that fail to make progress.
The teacher requirement is less well-known, and also less onerous. States were allowed to come up with their own definitions of "highly qualified." As a result, most teachers in the U.S. today are deemed highly qualified.
When it comes to out-of-field teaching, state officials may be understating the problem, the report said.
Researchers compared two different sets of Education Department data, reports from state officials and a survey of teachers themselves. Teachers said out-of-field teaching happens far more often than states reported.
For example, in Arizona in 2004, the state said 94.4 percent of core classes were taught by highly qualified teachers. But Arizona teachers told the federal government in 2004 that 58.4 percent of core classes were taught by someone certified in the subject he or she was teaching. That was the most recent year in which the teacher data was available.
The report found a similar gap in 16 other states.
The report also called attention to places where people are trying to fix the problem.
Boston and Chicago have teacher residency programs much like medical residencies, with aspiring teachers working alongside mentor teachers before they are assigned their own classrooms.
The University of Texas at Austin, the University of North Carolina system and the university system of Georgia all are trying to develop strong teachers who will teach in local schools.
Louisiana is overhauling its teacher-preparation programs. And Denver and Guilford County, N.C., schools offer financial incentives to attract the best teachers to schools and subjects that are hard to staff.
Wiener, the Education Trust official, said teaching is the key to fulfilling the goal of No Child Left Behind - that every student will be able to read and do math on grade level by 2014.
"We cannot meet our goals for increasing student achievement unless and until we focus on improving teaching quality and the effectiveness of teachers in front of the classroom," Wiener said.