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1.
When a baby's teeth first appear, what is really cut is the gums as the teeth push through them. The Oxford English Dictionary gives "cut" in this sense, with "tooth" or "teeth" as the object, in quotations going back to 1677.

Babies and puppies chew on things when they are "cutting teeth." The way I've heard the expression is in reference to how one gained experience. "I 'cut my teeth on' writing funeral notices for a newspaper."

Other toothsome expressions: "to cut one's eyeteeth" or "to cut one's wisdom teeth." Mr. Funk says the first means: "To acquire wisdom; to learn the ways of the world. An eyetooth is a canine tooth, the third from the center in the upper jaw. The expression is somewhat literal, for the implication is that by the time a person has got his permanent set of canine teeth, has reached the age of twelve or fourteen, he has passed out of babyhood and has reached years of discretion. This working of the saying appears to have originated in the United States, first recorded in 1870 by the essayist, Ralph Waldo Emerson, though undoubtedly long in use before that date. The British version, dating to the early eighteenth century, is 'to have one's eyeteeth' or 'to have one's eyeteeth about one.' The usage differs slightly, carrying the implication of alertness against chicanery; that is, to use one's knowledge and experience in one's dealings." From "2107 Curious Word Origins, Sayings & Expressions from White Elephants to a Song and Dance" by Charles Earle Funk (Galahad Book, New York, 1993). A second source says: "If wisdom does come with age, 'to cut one's wisdom teeth' is more appropriate, for these are cut at the ages of 17 to 25 and up to age 50! These molars have been known as 'dentes sapientiae,' 'teeth of wisdom,' since the time of Hippocrates." From "Encyclopedia of Word and Phrase Origins" by Robert Hendrickson (Facts on File, New York, 1997).
1) Though a fine officer in the marines, General KT first cut his teeth in leadership and character as an Eagle Scout. 2) I cut my teeth on making homemade rockets while I was a high school student in the engineering club.
by Laniidae March 16, 2008
 
2.
gain early experience
The legendary German fighter pilot, the Red Baron, cut his teeth when accompanying his father in various hunting trips during his childhood

I cut my teeth on the leadership when I was elected as the president of the classes in middle school
by vanderpol August 02, 2010