To a certain extent, it is probably safe to say that many children experience a conflict between the public perception of their identity at school and the reality of their private lives. In Rodriguez's case, the public identity that he had to cultivate meant that he had to suppress his familial identity as a Mexican-American who spoke Spanish.
Like Rodriguez, there are children all over the country of varying ethnicities who speak one language at home and perhaps follow the other customs and traditions of another country, but behave differently at school. The reason the school, or public, identity is different is that it facilitates the kind of assimilation necessary for children to advance socially and academically. In America, speaking English is advantageous in schools, colleges, and in one's career because it is the dominant language. These days, however, being bilingual can be even more advantageous as demographics have begun to shift since the time when Rodriguez was in public school.
Another way that children might feel that they need to have a public identity that differs from the private is if they feel the need to conceal something they consider detrimental at home, such as financial struggles. Peers can be cruel to children who are unable to keep up with the clothes and possessions of their classmates.