I agree with the comments already posted.
In my personal experience the alum block will stop the bleeding from minor nicks, but not larger cuts. I've also found that a stiptic pencil won't stop bleeding from large cuts either. I've been told that the stiptic pencil is merely the same material as alum (potassium alum), but ground up and re-formed.
Cold water rinses probably help stop bleeding too.
When I get a real cut I soak up the blood with a paper towel, apply anti-biotic gel, and continue to blot until bleeding stops, then apply more anti-biotic. Lynn suggested this approach. It seems to help the cuts heal faster.
Good luck.