For those of you souls that are not familiar with this ingenious hybrid of an eating utensil, you're in for a treat. The spork is a cross between a spoon and a fork. It has the base of a spoon which gives it the "bowlish" scooping shape with up to four tines protruding from the end which gives it the picking characteristics of a fork. It has been rumored that the US Military was the first to patent this invention at the start of WWII for the soldiers. KFC and Taco Bell has helped spread its popular use by offering sporks instead of spoons and forks for their meals.
When the US Military was stationed in Japan after World War II, most of the schools in Japan had adopted the spork and chopsticks as their eating utensils of choice - probably due to its usefulness and its easy production and cost effectiveness. I personally would consider most Asian countries to be very traditional - you know - honoring ancestors, having certain rituals and customs that are passed down from generations to generations, etc. However, Japan to me, seems to be the most modern and willing to change for the better, while still encouraging its past tributes and customs (such as the Japanese gardens, tea ceremonies). However, supposively, starting in the 1980s, the sporks started to lose support in Japan because it favored bad eating habits - so they started re-introducing the delicate use of chopsticks.
This is apparent in the sixth and final episode of FLCL. In the beginning, the teacher or "sen-sei" Miya-jun tries to (and fails to) teach her students about practicality of chopsticks. She tries to pick up a mushroom (in a pathetically vain manner) and then drops it. She seems obviously fustrated and to redeem herself and encourage class participation she clicks or taps her chopsticks in the air while chanting "hai!".

Honestly, I don't really see a significance of the FLCL director and crew to include this two minute clip of a failed chopstick lesson unless it had some more importance to it. Perhaps it suggests a particular change: maybe the change to chopsticks reflects how society as a whole desires improvement or a change from the norm, much like how the town of Mabase, once thought to be "boring" was turned upside down by the recklessness of Haruko Haruhara. And Japan seems to almost be a melting pot of cultures on it's own now, perhaps sort of mimicing how the United States is, and wants to reach out and spread on the chopstick supremacy. You'd be surprised at how different cultures view chopstick etiquette (Korea, China, Japan all have different "proper" and "improper" ways.)

Maybe the chopstick situation bears no significance at all, perhaps it's just a simple reference to something happening in Japan at the time and was put in comedic fashion much like the other references such as South Park and Lupin III.
Perhaps unbeknownst to you, I did a little bit of research on proper chopstick etiquette from an expert that has been perfecting this skill for eighty years thus far. She gave me a few pointers and then a few "do not's." Supposively, in the Chinese culture, your palm is never to face upwards when handling the chopsticks with food. The back of your hand (the nuckle side) should be facing up. Another proper thing to do is to use the blunt end of your chopsticks when picking up food when multiple people are sharing. And you aren't suppose to stab or poke your food with the finer end. I think the Japanese culture would probably view some of the same etiquettes. I'm tired, the end!