That's a toughie. Now note that you'll only see that link in the blue bar along the top if the visitor is logged in. If your students will only be reading the blog, then there's nothing to worry about. Blogger's "Next Blog" link shows up for all viewers while WP.com's "Next Blog" link only does if your logged in.
My concern though is that the category links within each post is going to their global listings and there's no control over those. For example, the post that you've made on your blog is listed under Uncategorized which, if you click on it goes to this page which is a listing of other blogs using that same category and you have no control over it.
Normally in this case, I'd just throw up a blog for you on my servers, offer support, and away we go but I can't do that as it's against the TOS here for me to point you at my company (I do paid webhosting myself and just help out here. I have no problem with offering free hosting in a case like this.) and I'm not taking any new clients as I'm overloaded myself until I can get QWest to get me some more space.
I'm guessing that your local school district doesn't offer any webspace or anything for you to use? If they do, I could throw up a wordpress blog for you in about five minutes. Just let me know.
Going to offer this as a suggestion and I'm doing this as this is a special case. (Don't everybody else do this.) Contact Automattic directly, explain your situation to them and see if they can just throw up a seperate blog for you on their servers somewhere. They've done it for others in the past but, again, it's been for specific clients. They might be able to do something for you in this case.