It's "recklessly". "reck" is a very old word meaning "to care, heed, have a mind, be concerned about"; so "reckless" means "without taking heed".
I actually thought it was directly related to "reckon" (meaning "to think or calculate"), but when I looked it up it turned out not to be the case (except much further back in the etymological tree).
<pedantic>
It's "recklessly". "reck" is a very old word meaning "to care, heed, have a mind, be concerned about"; so "reckless" means "without taking heed".
I actually thought it was directly related to "reckon" (meaning "to think or calculate"), but when I looked it up it turned out not to be the case (except much further back in the etymological tree).
</pedantic>