My teacher told us to push a kid the highest on a swing, that you want to push on the bottom of the path. Im not sure why this is the case, obviously you wouldn't want to push against the kid. But it seems to me that as long as you were pushing on the downward swing it shouldn't matter. Because when you push the kid, you are adding energy to the system. it seems like the energy you add plus the total energy the kid had on the swing would be constant.