Etymology
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teach (v.)

Old English tæcan (past tense tæhte, past participle tæht) "to show, point out, declare, demonstrate," also "to give instruction, train, assign, direct; warn; persuade," 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." Related to Old English tacen, tacn "sign, mark" (see token). Related: Taught; teaching.

Lemonade Vendor (Edgar Kennedy), enraged: I'll teach you to kick me!
Chico: you don't have to teach me, I know how. [kicks him]

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.

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teaching (n.)
Old English tecunge "act of teaching," verbal noun from teach (v.). As "that which is taught" from c. 1300.
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teacher (n.)
"one who teaches," c. 1300; agent noun from teach (v.). It was used earlier in a sense of "index finger" (late 13c.). Teacher's pet attested from 1856.
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reteach (v.)

also re-teach, "to teach over, teach again or anew, supply with new teachings," 1640s, from re- "back, again, anew" + teach (v.). Related: Retaught; reteaching.

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taught 
past tense of teach (v.), from Old English tahte, past tense of tæcan. For the unrelated adjective meaning "stretched or pulled tight," see taut.
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teachable (adj.)
late 15c., "capable of being taught" (of persons), from teach (v.) + -able. Of subjects, from 1660s. Teachable moment, attested from 1917, not common until after c. 1960, extends the sense to "appropriate for instruction." An Old English word for it was leorningende. Related: Teachableness.
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beteach (v.)
Old English betæcan "give up to, impart, deliver; appoint, set apart, dedicate," from be- + teach (v.). Form and sense confused with betake. Meaning "impart, teach" is from c. 1300; the word was obsolete or archaic from 16c. Related: Betaught; beteaching.
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*deik- 
Proto-Indo-European root meaning "to show," also "pronounce solemnly," "also in derivatives referring to the directing of words or objects" [Watkins].

It forms all or part of: abdicate; abdication; addict; adjudge; apodictic; avenge; benediction; betoken; condition; contradict; contradiction; dedicate; deictic; deixis; dictate; diction; dictionary; dictum; digit; disk; ditto; ditty; edict; Eurydice; index; indicate; indication; indict; indiction; indictive; indite; interdict; judge; judicial; juridical; jurisdiction; malediction; malison; paradigm; policy (n.2) "written insurance agreement;" preach; predicament; predicate; predict; prejudice; revenge; soi-disant; syndic; teach; tetchy; theodicy; toe; token; valediction; vendetta; verdict; veridical; vindicate; vindication; voir dire.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit dic- "point out, show;" Greek deiknynai "to show, to prove," dike "custom, usage;" Latin dicere "speak, tell, say," digitus "finger," Old High German zeigon, German zeigen "to show," Old English teon "to accuse," tæcan "to teach."
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propaedeutic (n.)

"an introduction to an art or science," 1798, from Greek propaideuein "to teach beforehand," from pro "before" (see pro-) + paideuein "to teach," from pais (genitive paidos) "child" (see pedo-). By 1849 as an adjective, "pertaining to the introduction to any art or science."

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lere (v.)

OE læran, Kentish leran "to teach" (cognate with Old Frisian lera, Old Saxon lerian, Dutch leeren, Old High German leran, German lehren "to teach"), literally "to make known;" see lore). From early 13c. as "to learn."

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