The song that Geordi sings is the opening lines of the Major General's Song from the W.S. Gilbert and Arthur Sullivan opera 'The Pirates of Penzance': "I am the very model of a modern Major-General, I've information vegetable, animal, and mineral."
There is a humorous reference to this episode in Star Trek: Deep Space Nine, "Accession". Worf becomes alarmed at hearing that Keiko is pregnant again, and hastily plans a vacation to Earth for when she is due, so he does not have to assist with the birth again.
For the scene in which La Forge and Crusher evacuate the air from the cargo bay, miniature barrels were constructed and attached with pins to a vertical miniature floor. The pins were then removed, with the barrels falling against a blue screen. By shooting at a high frame rate, the desired effect was achieved.
La Forge and Doctor Crusher almost become the victims of a design flaw in the USS Enterprise-D's cargo bay. The console that raises the exterior cargo bay door and lowers the atmospheric-containment force field apparently does not contain controls to reventilate the bay once the force field is restored and the exterior door is closed. Those controls are located on a separate wall-mounted panel next to the interior cargo bay doors.
One of the children suggests singing a song called "The Laughing Vulcan And His Dog." Some have speculated that the fictional song title was inspired, or perhaps a reference to the character Sybock from Star Trek V The Final Frontier. Writers have said however, this was not the case.
The trivia item below may give away important plot points.
At the end, the children present Picard with a commemorative plaque with their names on it. Their last names are the same names as their own names, although the character of Paterson wrote his last name as Supra when in actuality the actor's last name is Supera. This, however, could be intentional and a reflection of the character's immaturity.