1200 sf? That is plenty of space!
To fix the problem, try this:
1. Buy a small shed.
2. Move into the shed.
3. Realize you had plenty of space before you moved into shed.
4. Move back into the large house.
Remember the old story where someone asked a wise old person what to do about their house being too small. I think he told them to fill it with all kinds of animals, goats, chickens, pigs, to where he could hardly stand to live there. Every time he returned for advice the wise person told him to put more stuff in his house. Then he finally got told to get rid of the stuff, and voila, his house was large enough.
And problem solved!
I have three kids and currently have usable living space of 400 sf and we are happily working towards having more space but could live like this forever if necessary.
However if you are able to expand there's nothing wrong with that! I remember my parents planning to convert their garage to make more space, but we were all grown up by the time it happened! Now the house seems HUGE....would have been nice to have all that space back then!
Good luck, and uber kudos on homeschooling your kids.