First let's make sure we have the correct terminology. The 255.255.0.0 is the subnet mask. You are not subnetting the subnet mask. You subnet a network address.
With that being said, let's look at the example 1A on page 159 that you are referring to.
The Class A default mask is 255.0.0.0. In this example, he is going to use the mask 255.255.0.0. So, he has taken 8 bits away from the host portion to use for network addressing. These are your 8 subnet bits. So he does the calculations based on that.
Hope this helps.
Edit: Just realized this thread is a year old. Oops.