Crescent Walking vs. Direct Stepping
In crescent walking (C-walking, moonwalking) the stepping foot moves in a crescent shape when stepping in a stance, such as in a front stance. In direct stepping, the stepping foot moves straight forward. In crescent stepping, the stepping foot moves forward and inward until it is no less than a shoulder width from the base foot (to prevent double foot sweeps) and then it move back outward into the next stance. Crescent stepping shifts the body into a stable position so it may resist a pushing/pulling force at mid-step, but it takes longer than the direct step. The direct step is quicker, but the body is unstable throughout the step. In either case, the body weight must lag the step until the stepping foot has a firm footing. The crescent step is useful in street situations, since the stepping foot clears its path of objects. A direct step mat land on an unseen object causing a loss of balance.
In crescent walking, the accentuated movement of the moving foot toward and away from the stationary foot can makes the step more dynamic. With a greater overall movement, all else held constant, the larger range-of-motion may help make the movement stronger. It also lets you step inside an opponent's guard more easily. Also, when using kicks, such as a front kick, the moving foot must come toward the stationary leg anyway. Although it is less stable, direct stepping gives a wider base if things should go awry during the movement.
Taekwondo America stresses direct stepping.