Equal Proficiency is Required in Both Languages
This myth had some support with the "Threshold Hypothesis" developed by Dr. Jim Cummins, well-published researcher in the area of childhood bilingualism and success in school. Dr. Cummins insisted that for children to enjoy the benefits of bilingualism, children's proficiency levels had to be at an developmentally-appropriate level in both languages. It should be noted, however, that Cummins' research was from the late 1970's.
Later research objected to Cummin's rule on empirical and theoretical grounds. In Carey Myles' book, Raising Bilingual Children, she contends that subsequent studies have shown that bilingual children who were significantly weaker in one language than the other, still achieved higher scores than monolingual children on various batteries of test.
It has been suggested that politics also played a role in the "which language should be taught first?" conflict.
The general consensus of the most linguists is that a child must be at grade level with one of the two languages. Their English, for example, isn't compromised by having less proficiency in another language.