Middle English tēchen, from Old English tæcan (past tense tæhte, past participle tæht) "to show (transitive), point out, declare; demonstrate," also "give instruction, train, assign, direct; warn; persuade."

This is reconstructed to be from Proto-Germanic *taikijan "to show" (source also of Old High German zihan, German zeihen "to accuse," Gothic ga-teihan "to announce"), from PIE root *deik- "to show, point out." It is related to Old English tacen, tacn "sign, mark" (see token). The notion is "to show how to do something by way of information or instruction." Related: Taught; teaching.

By mid-14c. as "disseminate" a system of belief. By c. 1200 as "indicate" how something is to happen; used by 1560s in threats, "make known to one at a cost."

Enraged lemonade vendor (Edgar Kennedy): I'll teach you to kick me!

Chico: you don't have to teach me, I know how. [kicks him]

["Duck Soup," 1933]

The usual sense of Old English tæcan was "show, declare, warn, persuade" (compare German zeigen "to show," from the same root); while the Old English word for "to teach, instruct, guide" was more commonly læran, source of modern learn and lore.

"one who gives regular instruction in schools," by 1839, from school (n.1) + teacher.

"housing provided for teachers by a school," 1916, from teacher (n.) probably on model of parsonage.

"office, function, or position of a teacher," 1846, from teacher (n.) + -ship.