'Goody, goody gumdrops' has the sound of a typical, if archaic, British children's expression of delight. There is, however, a strong case to be made for the phrase being an American coinage. 'Goody, goody' has been used in the US since at least the late 18th century to express pleasure; for example, in the early American ballad opera, The Disappointment, circa 1760:
"Oh! goodee, goodee, oh! we shall see presently."
'Gumdrops' themselves are certainly an American invention. They are a confection of sweetened and flavoured gum and have been on sale in the US under that name since the mid 19th century. The Illinois State Chronicle ran an advertisement in August 1859 for the confectioner George Julier, which offered:
"Fresh GumDrops, assorted flavor wholesale or retail"
A 'drop' is, incidentally, a common name of various sweets and cakes, for example, acid-drops, lemon-drops, dropped scones. The name derives from the method of making the sweets by dropping balls of the ingredients onto a surface to set.
The first citation of 'Goody, goody gumdrops' that I can find comes in a cartoon by the American humorist Carl Ed (pronounced 'Eed'). Ed's 'Harold Teen' cartoon strip ran for many years in the USA and was syndicated in several newspapers, notably The Oakland Tribune, where this cartoon was printed in November 1936:
It isn't clear whether Ed coined the phrase or whether he had heard it elsewhere.
The phrase is still used, usually with an awareness of its archaic tone. The British press began to use it from 2002 onward, in various punning headlines relating to the Big Brother contestant, Jade Goody. She had become an unpopular figure in the UK, following accusations of her racist bullying of an Indian contestant on the show in 2007 - to newspaper headlines along the lines of "Baddy Goody drops herself in it".
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