What is the solution to Holling's teacher, Mrs. Baker, not hating him?
Stating a solution to Holling's teacher, Mrs. Baker, hating him is implicitly agreeing that she did hate him. One of the things that happens between Holling and Mrs. Baker is that in return for coaching him in running, he coaches her in how to be more welcoming and, in connection with this, we and Holling learn she was an Olympic runner. The significance of these two things is that Mrs. Baker is not a warm-and-fuzzy person; she is highly disciplined focused individual. Those two qualities joined together could very well add up to someone who seems like they could eat students for breakfast. In other words, it may only have appeared that Mrs. Baker hated Holling--on top of being annoyed that her time to herself was now shared time with someone she felt she needed to keep busy.
That being said, when Mrs. Baker started reading Shakespeare to Holling, that is the beginning of when we see the tide changing in their relationship. After he lands the role of Ariel in The Tempest and performs well, they have a bond on a shared experience through Shakespeare (and since Shakespeare's plays were all about relationship between people, this is a very fitting place to begin a bond with someone--that may be a good first date idea...). So it seems the solution to changing the relationship between Holling and Mrs. Baker was to find common ground that could reveal them and unite them through their intellects, achievements, and feelings such as admiration, gratitude and pride (and relief on Holling's part).