You can increase the gauge by attending lessons. When the gauge displayed on the screen is filled, it triggers a Special Lesson featuring Dire Crowley, which grants extra EXP of every kind. 

 

The Special Lesson will continue until a character raises his hand 5 times during the lesson.

However, if a lesson ends before the character can do so, the remaining number of times will be carried over to the next lesson.
Note: You can check the times remaining either at the top of the screen during a lesson or on the lesson selection screen.

 

 

Notes: The mark below the number in the center reflects the number of Special Lessons remaining.

If the times remaining are carried over to the next lesson, they will not be counted as part of a Special Lesson.

 

Note: Special Lesson before the remaining times are carried over is counted as one lesson in Character's Mission.

 

Example:
If a Special Lesson occurs with 4 or less times remaining of raising the hand, a Special Lesson will be carried over to the next lesson, but Character's Mission will only progress in the first lesson.