stopped caring for my son one week before the end of school meaning I had to pay extra for additional childcare.
Did they give notice prior to ceasing providing their service? Was that notice of the length as described in the service agreement?
As I work term time only I have always paid her over a full 52 weeks.....
Why? How is that defined in the agreement? This could be a very important aspect, as as others have said you may be paying for care, or you may be paying for care plus a retainer fee, and the payment may or may not have been split over a full year to make budgeting easier.
the contract runs from September 14 to September 15.
Does it say that the agreement is an annual agreement which is renewed annually? They run their own business so can have their own terms of business but what was actually agreed? Read the contract you have, see if you can find anything about length of the contract, renewing the contract.
They gave you notice, not you giving them notice, so I would expect them to be reasonable in calculating any outstanding payment. They will need to clearly explain how the payment is calculated. It could well be that you pay a certain figure for childcare and that the figure is split over 12 months/52 weeks. Without seeing your contact it is impossible to know.
Have you calculated what you feel is owed and are you able to set out why you feel that amount is owed? If so, then pay that. Then leave it up to your ex-childminder to decide what to do - they can take it through small claims court procedure but they will need to prove that the money is owed.