The kids – Prince, 13; Paris, 12; and Blanket, 8 – were spotted this week on their quick family getaway in Maui with Katherine, a security team and friends.
Paris, who has had the most trouble privately coping with the loss of their famous father, was photographed swimming at the hotel pool with a solemn expression on her face.
Says a source close to the Jackson family: "Early on, Paris didn't think she could live without her father … she just didn't know how she was going to function. But [Katherine] has been so supportive of her that Paris is much better.
Her brothers, meanwhile, appeared more carefree on the trip, often smiling and playing in the waves with other children on the beach.
"You have to remember that everybody in that family was suffering from grief, so Michael's kids never felt alone in it," the source continues. "That helped them to become stronger."
Clearly, the children have come a long way in the past year. Says the source, "They are much more open, and playful. They have matured."
On Friday's actual anniversary of Jackson's death, the family is scheduled to head to their late father's hometown in Gary, Indiana, where Katherine is expected to reveal a large monument of the pop icon outside of the home where the Jacksons grew up nearly 50 years ago.
A memorial and candlelight vigil is due to follow, concluding with Jackson's hit "We Are The World."
Reporting by CHAMP CLARK