Education was sporadic in the frontier army. Some considered the experience of frontier living and the regular interaction with Indians an education in itself. But many wanted their children to have an education where they could learn the fundamentals of reading, writing, and arithmetic. Some left their families behind; some sent their children off to school when they were old enough. In some cases, officers and their wives would home school their own children.
Councils of administration (which consisted of three or four officers) were authorized to raise funds for a school so in many cases there were post schools, often run by the army chaplain. In the absence of a chaplain, a sergeant, another educated enlisted man, or a civilian might run a school.
At Fort Scott, there was no chaplain until 1850, when Reverend David Clarkson arrived. He not only saw to the spiritual needs of the men, he also served as the schoolmaster. It is not known if there was a school at Fort Scott prior to the arrival of the chaplain.
For more information on schools, read pgs 133-135 of The Old Army: A Portrait of the American Army in Peacetime, 1784-1898by Edward M. Coffman, copyright 1988 by Oxford University Press, Inc.