You can only get a credit for education expenses paid in the tax year. Therefore, your qualified education expense for 2015 would be the expenses you actually paid in 2015. You would not include the amount paid in Dec. 2014 on your 2015 tax return. You would include Spring 2016 tuition if you paid it in 2015.
You will need to adjust the amount on your 1098T to reflect the amount actually paid for the year. To do this, click on the link titled "What if this is not what I paid to this school" under box 2 on the 1098T screen. You can enter the amount that was actually paid here.