The term "normal school" was used to refer to schools for training teachers. The term is no longer in use. These schools were not created until well after the colonial period ended. Normal schools were not created until the 1820s (for a private normal school) or the 1840s (for publicly funded normal schools).
The creation of public normal schools was part of the reform movement of the 1840s. During this time, there were many sorts of social reform movements in the United States that were meant to improve the general society. The abolitionist and women's rights movements were among these, as was the movement to improve education. Normal schools were created so as to train teachers who would, presumably, be better able to educate their students. This would improve society because more educated people would be better able to participate in society and to improve the economy.