When you walk in our front door our school "table" is visible, but I usually don't mind :) We had a separate school room (our third bedroom) before we moved a year ago, but now the boys have separate rooms and {althought they're sharing the same room for fun right now} we need the spare room for friend and family guests. So we moved school to the kitchen. Once I got fed up with having to move EVERYTHING off the table for lunch and dinner I pulled out one of our card tables. The height is better for our boys (our kitchen table is bar-height) and we have just the right amount of space for math and language times. We usually take science and history to the living room, anyway. If we're in the middle of something it just stays on the table and we move on to the meal, but I do pick up finished work each day to file away. It also comes in handy when we have guests~a "kids" table is already set up and ready to go...as soon as I clear off the remaining school stuff!
As to tidyness...I like a tidy house, too, but I don't stress over a busy school table/counter~it means we've been working. I did not like it all over our kitchen table, though.