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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philomath (n.)

"a lover of learning," 1640s, from Greek philomathēs, from philo- "loving" (see philo-) + mathos "learning," from manthanein "to learn," from PIE root *mendh- "to learn." 

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cun (v.)

"to learn to know, inquire into," from Old English cunnian "to learn to know," ultimately from the same ancient root as can (v.1) and compare con (v.3). Surviving into 17c. and perhaps later in dialects. Also compare cunning.

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opsimathy (n.)

"education late in life," 1650s, from Greek opsimathia "late learning," from opse "late, after a long time" (related to opiso "backward," opisthen "behind," from opi, a variant of epi "on it, at it;" see epi-) + manthanein "to learn" (from PIE root *mendh- "to learn"). Related: Opsimath (n.).

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apprentice (n.)
"one bound by legal agreement to an employer to learn a craft or trade," c. 1300, from Old French aprentiz "someone learning" (13c., Modern French apprenti, taking the older form as a plural), also as an adjective, "unskilled, inexperienced," from aprendre "to learn; to teach" (Modern French apprendre), contracted from Latin apprehendere "take hold of, grasp" mentally or physically, in Medieval Latin "to learn" (see apprehend). Shortened form prentice, prentis long was more usual in English.
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polymath (n.)

"person of various learning," 1620s, from Greek polymathēs "having learned much, knowing much," from polys "much" (from PIE root *pele- (1) "to fill") + root of manthanein "to learn" (from PIE root *mendh- "to learn"). Related: Polymathy "acquaintance with many branches of learning" (1640s, from Greek polymathia "much learning"); polymathic.

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chrestomathy (n.)

"collection of literary passages" (especially from a foreign language), 1774, from French chrestomathie, from Latinized form of Greek khrestomatheia "desire of learning; book containing selected passages," lit. "useful learning," from khrestos "useful" (verbal adjective of khresthai "to make use of," from PIE root *gher- (2) "to like, want") + manthanein "to learn" (from PIE root *mendh- "to learn"). Related: Chrestomathic.

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mathematic (n.)

"mathematical science," late 14c. as singular noun, mathematik (replaced since early 17c. by mathematics, q.v.), from Old French mathematique and directly from Latin mathematica (plural), from Greek mathēmatike tekhnē "mathematical science," feminine singular of mathēmatikos (adj.) "relating to mathematics, scientific, astronomical; pertaining to learning, disposed to learn," from mathēma (genitive mathēmatos) "science, knowledge, mathematical knowledge; a lesson," literally "that which is learnt;" from manthanein "to learn," from PIE root *mendh- "to learn."

As an adjective, "pertaining to mathematics," from c. 1400, from French mathématique or directly from Latin mathematicus.

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*mendh- 

Proto-Indo-European root meaning "to learn." It forms all or part of: chrestomathy; mathematic; mathematical; mathematics; opsimathy; polymath.

It is the hypothetical source of/evidence for its existence is provided by: Greek menthere "to care," manthanein "to learn," mathēma "science, knowledge, mathematical knowledge;" Lithuanian mandras "wide-awake;" Old Church Slavonic madru "wise, sage;" Gothic mundonsis "to look at," German munter "awake, lively."

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experience (v.)
1530s, "to test, try, learn by practical trial or proof;" see experience (n.). Sense of "feel, undergo" first recorded 1580s. Related: Experienced; experiences; experiencing.
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