Most parents try to keep their children away from sharp objects. Robert Harris buys his girls swords.

The sport of fencing has taken over the Harris household. Mr. Harris, 45, his wife, Maia, 42, and daughters Stella, 15, and Emma, 18, train together and compete at a national level. Six years ago, a coupon for youth fencing lessons arrived in the mail. “I thought, ‘Great, another after-school activity,’ ” he recalls.

When...