Final answer:
Public high schools are the type of educational institution that first appeared as a result of Horace Mann's common schools movement, reflecting a significant expansion in public education and aiming to provide broad access to education for all children.
Step-by-step explanation:
The educational institution that appeared for the first time as a result of Horace Mann's common schools movement is public high schools. Before Mann's reforms, education in the early 1800s was very basic, often conducted in one-room rural schoolhouses, or through private tutors for the wealthy. Public schools, before the common school movement, mostly did not exist. Horace Mann, as the head of the Massachusetts board of education, championed the establishment of public schools that taught academic subjects along with morality and discipline. These schools were funded by taxes and sought to provide education for all children, aiming to reduce the gaps in society between the rich and poor and prepare youth for their social and civic duties as adults. Public education rapidly expanded, with the number of students attending public schools tripling from seven million in 1870 to twenty-one million by 1920, and the number of public high schools growing from one hundred in 1860 to over six thousand by 1900.