Public schools: Elites vs. the middle class

Another reason to favor school choice over the public school monopoly, from Wikipedia:

Randall Collins contributed the idea of credentialism to the study of class-based differences in educational attainment. Collins maintains that public schools are socializing institutions that teach and reward middle class values of competition and achievement. Anglo-Protestant elites are selectively separated from other students and placed into prestigious schools and colleges, where they are trained to hold positions of power. By teaching middle-class culture through the public education system, the elite class ensures a monopoly over positions of power, while others acquire the credentials to compete in a subordinate job market and economy. In this way, schools of medicine, law, and elite institutions have remained closed to members of lower classes.

Thus, as I interpret this paragraph, the hypothesis says prestigious private schools teach how to lead, while the public school monopoly teaches how to serve.