Step 1 :The family's gross annual income is $44,400.
Step 2 :To find the gross monthly income, we divide the annual income by 12. This gives us $3700.
Step 3 :The maximum monthly mortgage payment is 28% of the gross monthly income. So, we calculate 28% of $3700, which gives us $1036.
Step 4 :\(\boxed{1036}\) is the maximum amount the family should spend each month on a mortgage payment.