Karen Budd is one of those parents who school administrators try to avoid.

To begin with, she understands math. She earned a math degree and did some graduate work in engineering at the University of Pittsburgh.

Every school superintendent who has ever attended a PTA meeting knows that the math-savvy parents can be difficult. They often have complaints about the way teachers are handling the subject. And the many administrators who successfully avoided taking any math after high school, as I did, know they are ill-equipped to defend themselves.

Adding to the troublesome nature of Budd is her early and unhappy experience with Everyday Mathematics, one of the modern conceptual textbook series favored by many experts in mathematics education. It is distressing to her that Fairfax County, where she lives, has added Everyday Math to some of its schools.

I sympathize with those who have to answer Budd's letters. But I think parents like her are an important part of the effort to make schools better, and we ought to remember that they are becoming even more influential as we hand them a new generation of weapons -- the results of those achievement tests the state and federal governments are making the schools give to children.

Take as an example what Budd has done with some new testing data. She looked at two large Washington area school districts, Fairfax and Anne Arundel, which are using different math programs. She says she has found a remarkable discrepancy in results.

Like all such statistical arguments, this math gap is full of qualifications, limits and confusing twists -- many educators insist no such gap exists -- but I want to display her conclusions as a sign of what other parents might want to try. Her work is also a warning to administrators that they are going to have to pay more attention to such comparative figures themselves, if only for self-protection.

Budd's research was motivated by what she says happened to her children when they encountered Everyday Math at a private school. "At first blush," she said in a letter to a superintendent who was using the program, "my instincts told me it was not rigorous enough, nor content rich. However, I decided at the time to be deferential to the educators and accepted the program assuming that educators knew more about mathematics education than I."

"I rue the day that I accepted it without question and that I did not investigate further. My children's days were to be filled with counting on a scroll to 1 million and playing games. Addition and subtraction facts were not emphasized properly. Fractions in the fifth grade were glossed over. These should have been major red flags to me as these skills are all very critical to success in algebra. It should have occurred to me that whoever wrote this program had a significant deficit in understanding or remembering advanced mathematics and in knowing what it takes for children to be successful in advanced mathematics," the letter said.

Since then, she said, she has spent many summers helping her older daughter, Katie, fill the mathematical void left by Everyday Math. For her daughter, there is a happy ending -- she headed off last month to Grove City College to major in mechanical engineering. Budd, however, said she is concerned about other children whose parents have neither the education to tutor nor the means to have their children tutored.

Budd pulled her younger daughter Kelsey out of the school that was using Everyday Math when she was in second grade. Budd said Kelsey's mathematical gaps were much less severe and easily filled in by Fairfax County public schools' more traditional math curriculum at the time.

When Budd discovered Fairfax was planning to install Everyday Math in at least five schools, she decided to take her concerns to school officials. She looked for examples to buttress her beliefs. Although she had not lived in Anne Arundel, she was impressed with the progress it made using Saxon Math, a more traditional textbook series that has been successful in many schools.

Her research appears to show Saxon raising scores substantially in Anne Arundel. One of Budd's comparison charts shows what seems to be less improvement in Fairfax's lowest performing elementary schools over four years. Those schools did not use Everyday Math, which has just been adopted in the county, but they also did not use Saxon, despite pleas from Budd and other parents.

Tim McEwen, president and chief executive officer of Austin-based Harcourt Achieve, which sells the Saxon series, said, "It is not surprising that Anne Arundel County saw a dramatic increase in test scores." John Saxon, a former military officer who created the textbooks, "discovered a way that builds skills in an incremental manner, with frequent repetition, leading to an understanding of mathematical concepts and the connections between them," he said.

April Hattori, vice president for communications at New York City-based McGraw-Hill Education, publishers of the Everyday Math series, strongly disagreed with Budd's analysis. She said the data in Budd's comparison charts are distorted and misleading.

"Everyday Math has achieved significant documented results in school districts across the country," she said.

In Dallas, for instance, the percentage of students passing the state math test rose in all elementary schools after Everyday Math was introduced, she said, with the biggest gain in fourth grade, from 65 percent passing in 2000 to 75 percent passing in 2001.

Kitty Porterfield, spokeswoman for Fairfax County public schools, said the Anne Arundel and Fairfax data used by Budd are measuring different things, percentile rank in Anne Arundel and percent scoring at or above the 50th percentile in Fairfax. She said if Fairfax's data were arranged the same way as Anne Arundel's, its students would also have shown "an impressive gain."

Anne Arundel County officials said schools there had used Everyday Math before Saxon was introduced but only in a few classes. Jonathan Brice, a spokesman for Anne Arundel schools, said the county is "guardedly optimistic" about the success of its math courses and is looking forward to improving the learning of students in all subjects.

Budd said that when she displayed at a Fairfax County math curriculum advisory council her chart showing the gain in computation skills among Anne Arundel students using Saxon, the reaction from the teachers in the room was that that was only computation. "So I showed the problem-solving charts and then, guess what . . . we heard how testing is not authentic," she said.

"I do this," she said, "because having become so involved with my own kids and having done exhaustive research, mostly because I never believed my own instincts, I now feel responsible to bring these results to light for the kids it would benefit and for the taxpayers who are being squeezed by this county for pretty average results."

See what I mean? Educators do not like hearing parents talk that way, but they ought to get used to it. These arguments used to be just opinions, tossed back and forth like water balloons on a college weekend. Now, with so much testing going on, and so much information on the Internet, both parents and teachers have facts to present, and everyone is going to have to listen more carefully.

Karen Budd, shown above with daughter Kelsey, compared student test scores from Fairfax and Anne Arundel counties because of her concerns about Fairfax's decision to use the Everyday Mathematics textbook series.