I've listened to a few coaches discuss catching lately. One of them tells his catchers on a pitch just off the plate to maintain their stance but sway their body left or right in order to catch the pitch in the center of their bodies. Another coach said not to do that - just reach out with thier arm and catch it. The reason the second coach gave was that the swaying movement signals to the umpire that the pitch was a ball because the catcher had to move to get it. He also said umpires don't like the extra movement by the catcher and it sometimes obstructs thier view of the pitch. I know some coaches have debated the topic elsewere on this website but I wanted to get an umpires perspective.
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