Pour the rum into a cocktail shaker. Tear the mint leaves to pieces and place them in the shaker as well. Add in the lime juice. Use the back of a spoon and muddle the contents of the shaker for 1 minute.
Add in the simple syrup and ice. Cover the shaker and shake for 30 seconds. Pour in a tall glass over ice. Top the mixture with seltzer water and stir with a swizzle stick. Garnish with lime wedge and mint leaves.