I have a question about grants...

I ran the following command, assuming it would allow me to make any changes to the tables in the public schema:

grant all on all tables in schema public to User1;

However, when I try to add a new column I still get the error message below:

ERROR:  must be owner of table Tab1

I guess I'm just a little misinformed on the purpose of the grant all on all command, and how can I refine this to allow a user to add columns to a table without changing ownership?