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Old English neowe, niowe, earlier niwe "new, fresh, recent, novel, unheard-of, different from the old; untried, inexperienced," from Proto-Germanic *newjaz (cf. Old Saxon niuwi, Old Frisian nie, Middle Dutch nieuwe, Dutch nieuw, Old High German niuwl, German neu, Danish and Swedish ny, Gothic niujis "new"), from PIE *newo- "new" (cf. Sanskrit navah, Persian nau, Hittite newash, Greek neos, Lithuanian naujas, Old Church Slavonic novu, Russian novyi, Latin novus, Old Irish nue, Welsh newydd "new").
The adverb is Old English niwe, from the adjective. New math in reference to a system of teaching mathematics based on investigation and discovery is from 1958. New World (adj.) to designate phenomena of the Western Hemisphere first attested 1823, in Lord Byron; the noun phrase is recorded from 1550s. New Deal in the FDR sense attested by 1932. New school in reference to the more advanced or liberal faction of something is from 1806. New Left (1960) was a coinage of U.S. political sociologist C. Wright Mills (1916-1962). New light in reference to religions is from 1640s. New frontier, in U.S. politics, "reform and social betterment," is from 1934 but associated with John F. Kennedy's use of it in 1960.