What were some of the ideas used in the "sweeping transformation" of the public education system in the 1850s?
The educational reforms of the 1840s and 1850s were based around the idea that a high quality education should be available to all. The person most commonly connected with this idea is Horace Mann.
Before this time, there was very little public education (and, in truth, public education was still rather scarce by 1860 with only 300 public high schools in the country). The schools that did exist were taught by teachers who essentially had no training.
With Mann and other reformers, this changed. They believed that the government should provide free education and so they worked to create public school systems. They believed that the teachers should be well trained and so they created "normal schools" specifically for this purpose.
Their main ideas, then, were that education should be of a relatively high quality and should be provided for free by the government.