France 90+Uk 180+ France 90 is this legal???

[homeschoolmama]

Hello, I am a homeschool mama from the US and I would like to spend a year in Europe with my 6 children to study the cultures of various countries, history, art, architecture, etc. We would like to rent a flat for 90 days in Fance, 180 days in UK and 90 days again in France and travel Europe from our respective residence. As I understand it we can be in France without a visa for 90 days and again for 90 days after exiting for 180 days which is perfect because the UK short stay limit is 180 days and no visa required for US citizens. It is my understanding that I would not have to get long term stay visas if we could move around like this and we wish to focus on these two countries primarily anyway. I have not had success with the UK border patrol -- asked the question and got the royal runaround passing the buck and noone will answer. I do not want to violate the law and we do wish to stay in Europe for a year. In case this is not possible, I have made appointments with the French Consulate in LA for the end of this month to apply for a long stay visa but they have been very contrary and difficult to deal with already refusing to answer questions that are not addressed on their wesite ( also LA is a long trip for us and VERY expensive--we want to save our $ for our European trip, not spend it on taking a chance of getting a visa with them). Any advice or sugestions for us??? Thank you so much in advance for taking the time to answer me!