How is school district tax calculated when moved out of the district?-

The amount turbo tax says we owe for our ohio school tax is basically  the same as we paid last year. This year our income is slightly lower and we moved out of the district in May. Shouldn't our school tax be calculated either based on earned income while living in that district or prorated to the date we moved? Why we would we owe the same tax though we lived there less than half the year?