So, after some googling and research on how to turn the hard little stones into vibrant green plants, I grabbed my toothpicks and started stabbing away. Apparently, the trick is to insert 3-4 toothpicks about 1/2" into the widest part of the stone. This creates a way to prop the stone up and not submerge it too deeply into the water.
Once the toothpicks are in place, you place the stone in a glass with water just barely covering the bottom. After placing it in a sunny spot, you wait. After a few weeks, the stone will split and root will shoot forth. Just to be safe, I am trying to grow two stones. I've left them in my bathroom window because that is often the warmest and sunniest spot in the house.
I'll be posting more photos as this (hopefully) successful process.