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Dave does, indeed, feel as though he lives in two worlds. The world of his family, as personified by Pa, is poor, filled with hard physical labor, and includes minimal formal education. Dave's other world, based at the school and led by Professor Herbert, is filled with ideas and approaches to life and learning that are completely different from anything Pa experienced.
When Pa goes to the school with Dave to confront Professor Herbert, Dave was afraid to think about what might happen. "It was hard to tell what he would do. I could tell Pa that school had changed in the hills from the way it was when he was a boy, but he wouldn't understand." Dave feared Pa might shoot Professor Herbert, and Dave was afraid Pa would embarrass himself and Dave, too. On the other hand, Dave realized that
Professor Herbert really didn't know how much work I had to do at home...He had never lived in the hills. He didn't know the way the hill boys had to work so that they could go to school.