Paid Maternity Leave- Can my employer dictate when I take it?

Hi Everyone!

I have just begun talks with my employer regarding maternity leave.  My husband and I are both teachers, and very intentionally planned for a summer baby, having watched all of our teacher friends at schools throughout the country do the same (and seeing their paternity/maternity leave begin at the end of the summer).  My employer is telling me that my six weeks paid maternity leave must start the day the baby is born.  My baby is due July 20th, meaning my paid leave would end right as summer does, and it's back to work (unless I take unpaid leave, which I don't really think we can afford to do).  

I am furious, as I am an academic year employee.  Contractually, I do not have work obligations during the summer.  I feel my paid maternity leave should begin when I want it to (the first day of teacher training) assuming I can fulfill all work obligations (which I will, seeing as I have none until the first day of teacher training at the end of summer).  

Have other teacher Moms or Dads experienced something similar?  I intend to fight my employer tooth and nail.  They are telling me it is "so generous" because before they offered no paid leave.  My attitude?  Thanks for paying me for "leave" when I already had the paid time off. 

Also, if it helps, we are at a private Catholic school, with no union. 

Thanks in advance!