By school teaching is to down gear and engine brake plus pedal brakes. However, I believe these are from the era where cars have all drum brakes. As drum brakes will lose their effectiveness as it gets hotter, the idea was to use as much of the engine braking as possible so as to maintain the temperature of the drum brakes. As a novice, it is always good to know how to do engine braking as there might be times your brakes fail, this is where engine braking becomes useful.
However, there is another school of thought where engine and transmission is for moving and brakes are for braking. Would you like to wear out your engine / transmission for braking and subsequently an expensive replacement or simply wear out the brake pads which are relatively cheap. This problem might be more prominent elsewhere, where their cars can drive until the end of life, as opposed to only 10 years here. Also depends on how often you go for change of engine oil, as engine braking would definitely increase the number of engine rotations as opposed to idle plus brake.
braking would wear out your gears and other parts.