Must rent be pro-rated if lease does not start on the first of the month?

My lease began on December 20. The lease states that rent is due on the first of the month. Before I moved in, I paid two full months of rent for first and last month, plus security deposit equal to one month of rent. My landlord has treated my full month payment for the first month as applying to December, even though I was only a legal tenant for 12 days of the month. Should rent for December have been the full first month payment, or the pro-rated monthly amount for 12 days?