Hi, I'm not familiar with the current crop of White Mountains. In the old days, you had to add ice and salt to the WM to provide the cold.
I have a Cuisinart with a freezable container. Cleaner, neater and easier to use. I'm very happy with the ice cream I can make in it. The downside is you have to freeze the container overnight before churning. But, you should make the mixture the day before as well.