Your school did it wrong or you are describing it wrong. The box 1 amount should be the total tuition (and fees) paid, regardless of whether it was paid by scholarship or out of pocket (or by loans, 529 plans, parents or grandparents).
If you are sure that you are correct, then yes the box 1 amount (actually shown on your 1098-T) is the amount to be used to claim the credit (plus the cost of any books and a computer). You'll have to make adjustments when you enter the 1098-T.
