If you're interested in a business, rather than a non-profit concept, then finance it for them.
What do I mean by that? Do not give it away for free.
The app comes with a one or two year introductory period (whatever you can stomach), in which the schools can use it at no cost. After that, it's $x per month per student (or $$$ flat for the whole school per year). Hook them, get them dependent on it, and then charge them for your useful product. Keep it inexpensive, so that if it is in fact useful, they will find the money for it when the time comes rather than have to abandon their solution.
Constantly stay in touch with the schools and the board of education. You have to sell your way into their budget planning and you have to do it far ahead of time. The no cost introductory period exists solely to provide you time to sell your way into their budget plans, because that can take a lot of time and effort. That is as much a political navigation process as it is business. Constantly solicit feedback, they have to feel like you're part of their education system to justify expenditures.
So it goes like this.
1) Free for a year or two. Get your product adopted and embedded into the system.
2) Solicit constant feedback. Get to know everyone involved in determining the budgets.
3) With enough lead time, sell your way into their budgeting plans. Keep your product inexpensive.
4) Scale it.