As Darrell Feltmate has said, “life is too short to turn ugly wood.†http://tinyurl.com/y7eyr6
That piece of wood was destined to fail before you started turning it. The pith everyone is talking about is the center of those branches or dark spots on your piece. When you are first learning to turn a bowl, look for wood with the least amount of potential problems. In other words look for hardwood that is clear, has no cracks, has a straight grain with no voids.
Bill Grumbine http://www.wonderfulwood.com/ has a wonderful video on turning bowls. It contains everything from choosing a log, to cutting it, mounting it on a lathe and all the way to apply a finish. Well worth the money