State and county education officials in the D.C. area have been sparring with increased intensity recently over who has the best ideas for educating students, in bureaucratic battles that have put millions of dollars at stake and, some say, compromised children’s learning. Students in Fairfax County could miss out on subjects like science and social studies to comply with Virginia’s plan to increase physical education — and it would cost the school system at least $18 million.

In Montgomery County, school officials passed up at least $12 million because they didn’t agree with Maryland on how teachers should be evaluated.