Since we're talking about math: we all use Leibniz's notation for Calculus because it's just simpler than Newton's, even though they're equivalent. So that's "lazy"--in much the same way that functional polymorphism in OOP is "lazier" notation. But that laziness can make it *easier* to spot (some) errors, too, when the code is simpler to read.
Use each notation where it makes sense: dy/dx * dx/dt = dy/dt sure makes more sense when dealing with related rates than the alternative.
And lets not get into way in which the pre-cartesian convention (side-opp/hypotenuse, etc) for teaching trig can mess with people's minds.
For people who hate or can't understand algebra, the old school geometry method is the only way to go. For people who enjoy higher math, the old school method is incredibly confusing.