Camarath
Pale Master Tarrasque
It seems clear to me that "at one-half speed" means that tumbling movement counts against your speed twice as much non-tumbling movement does (i.e. every 5ft of tumbling movement is worth 10ft of non-tumbling movement). I think that as "part of normal movement" means that tumbling may be freely interspersed with normal movement. For example I think a character with 30ft movement could move 5ft normally (5), tumble 5ft (5+10=15), move 5ft normally (15+5=20), and tumble 5ft again (20+10+30) as their move action. I see no limit on how much of your movement may be used for tumbling other than your speed and your ability to make the increasingly difficult tumble checks.