I think I watched a video on it a long time ago and seem to recall that professors immediately break the chalk in half before using it. I don’t remember why.

I regularly use Hagoromo chalk on a personal chalkboard and I always break new sticks in half before use. For me, it feels more comfortable to use smaller pieces than large ones. It allows me to hold the chalk with more fingers, and avoids the awkward position where you have to hold the chalk like a pencil with a few inches of chalk resting on your hand. However, some friends swear by full, long sticks, so it's very much a matter of what you're used to.

Personally, I always was afraid of breaking the long chalk while writing, which would have caused the far end to fall to the floor and break into many small pieces. So why not be safe and write with a shorter stick.

I don't know anything about japanese chalk but in general you should break chalk in half because that makes the chalk not scratch with a high pitched sound when used.

As a Hagoromo-using professor: its biggest advantage to me is that it erases much more cleanly, so that after a few passes over the board you can still read the new writing instead of it looking like writing over half-erased writing.

That’s a great point that I forgot to mention. I’d reckon a large reason for this and its other benefits is how soft the chalk is.