What arguments did supporters of the public school movement make to support their cause

In a democratic society everybody should have the same possibilities. This is why if there are people who cannot afford private education, the government should provide a public education system. In that way all the citizen would have the same opportunities to study. Apart from that, more edycated citizens will make a stronger country.

its main object was the establishment of "settlement houses" in poor urban areas, in which volunteer middle-class "settlement workers" would live, hoping to share knowledge and culture with, and alleviate the poverty of, their low-income neighbors.