The straight answer is that most commonly (this doesn't mean all) mortgages in the US will compound monthly, you will not likely see a running tally of how the interest is growing over the course of one month. In the mortgage process you hopefully got an amortization schedule which details each principal and interest payment, should the payments be made exactly matching the loan term you were granted. should you make an additional payment and apply it to principal, your payment for the amortization schedule will stop matching what you were initially given, and the interest amount should go down slightly in comparison to the schedule. The servicer should at least be able to show you clearly how much each payment applies to principal, escrow and interest on your statement. You will still have a mortgage payment every month for the same amount, but you will pay off the loan earlier since you are increasing the amount allocated to principal.
This is the way to verify your extra payments are being applied properly, compare against your amortization schedule & track your principal balance. If the allocation amounts aren't differing from the original schedule, as they should after at least a month after the first extra payment hits principal, something is wrong and you need to call/speak in person with someone at your lender/servicer.
I understand what you're talking about on the student loan data, you see the interest growing daily up to the day you make a payment. You knock out the current interest amount and then the rest hits principal.
