PRINCE WILLIAM County’s school system, with 85,000 students, is the second-largest in Virginia and among the 40 biggest in the United States; its annual budget is almost $1 billion. By contrast, the school system in Grayson County, in rural southwest Virginia, is one of the state’s smallest, with a budget of barely $20 million. The combined population at Grayson’s seven schools — just 1,763 students — easily would fit inside any of the dozen or so high schools in Prince William.

So why is little Grayson doing its utmost to provide pre-kindergarten classes to its most disadvantaged children while Prince William is doing so little? In fact, as a percentage of its school spending, Grayson is doing vastly more.

In Grayson, school officials allocate enough money to unlock all available state matching funds for the 33 children from low-income families who qualify for pre-K funding. The rationale for tapping those funds is sound: Pre-K classes improve learning skills, including reading readiness.

Meanwhile, in Prince William, the Republican-led school board has all but ignored low-income 4-year-olds, more than two-thirds of them minorities, who could benefit from the state-funded program. The county ponied up only enough to garner state funding for 72 children, just 4 percent of the 1,663 who were eligible for the program.

Incredibly, Prince William has found no more money in its $1 billion budget for disadvantaged 4-year-olds than Grayson has found in its $20 million budget.

In both cases, the state required only a modest local contribution to release matching funds that would cover all qualified children — a little over one-third of one percent of each county’s annual allocation for K-12 education. Grayson came up with the cash; Prince William mustered less than one one-hundredth of one percent of its overall budget.

Prince William’s median household income, about $100,000, is three times greater than Grayson’s. Prince William was a leader among big school systems in introducing all-day kindergarten in 2007 for 5-year-olds. And it’s true that the county has been struggling with rapid growth; classroom crowding is a problem. But Grayson faces its own challenges: With a dwindling student population, schools have closed and teacher salaries have been frozen. “We get every penny we can and utilize it to the best of our ability,” an official in Grayson told us.

The attitude in Prince William is different. Some county leaders and school board members barely acknowledge a problem; others are overtly hostile. The school board chairman, Milton C. Johns, has pleaded poverty, citing state cuts to education funding and arguing that the county’s priority is to maintain existing programs and services.

But Prince William’s stingy attitude is more a matter of priorities than budgets. Other large school districts around the state, both richer and poorer, do a much better job at tapping state dollars. Richmond, Norfolk and Newport News, to name just three, secure state funding for virtually all children eligible for pre-K. Prince William’s Northern Virginia neighbors, Fairfax and Loudoun counties, secured funding for slightly more than half.

No major school system in Virginia is even close to Prince William’s miserly participation rate of 4 percent in the pre-K program. In turning its back on its poorest children, the question for Prince William is this: What kind of place does it want to be?