I agree with the above - in its current form, I think you'd attract early adopters who, even if they love your product, might have trouble convincing others to sign up and fork out for it. With an institutional model, said early adopters (presumably those in charge of educational technology spending) can try and push it out to teachers in their school.
Along these lines, perhaps consider some way to allow those who use and love your product to bring others in with minimal friction - let them try out the product in an administrative and student role, without having to sign up and learn on their own. Most teachers I know won't try something out until they really see the value in it (i.e. students engaged, fulfills some kind of need).
We're working on adding institutional accounts. There are some complexities involved, mainly involving legal issues and various whitepapers that need to be written, but I wholeheartedly agree with your idea: early adopters are a great way to get institutions to adopt a product.
As for your second suggestion, do you think a way for current users to send potential users an invitation to try a demo account (a sandbox) would do the job?
Along these lines, perhaps consider some way to allow those who use and love your product to bring others in with minimal friction - let them try out the product in an administrative and student role, without having to sign up and learn on their own. Most teachers I know won't try something out until they really see the value in it (i.e. students engaged, fulfills some kind of need).