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I wrote my more unique perspective on the video in a comment there:

The poor teacher doesn't know how to deal with contradicting ideas, in a way that technically makes both correct. There are a few things that could technically be argued about in math (IIRC), but we use one system, that is mostly consistent with itself, to make it simpler. The kid isn't "wrong" per say (bear with me here, because it is practically the same as being wrong), but just using the wrong number system. Just as 1+1 equals 2 in the decimal system, it equals 10 in binary and perhaps 11 in his system. Different number systems can coexist; numbers just need to be converted to be compatible.

Now, for why his system is completely impractical. Sure, in his system, 4 might be labelled "22", but as the teachers don't know it, it can't be easily built upon for more complicated math, like multiplication. Plus, it's bad for communication in general – how are people supposed to know, if asked to "22" apples, to give 4 in the decimal system instead of 22 in the decimal system? Plus, the kid's system is simply impractical. Isn't it easier to add 1280+1280 when one only needs to write 2560 instead of 12801280? Add the same number to itself again, and one gets 5120 vs 1280128012801280. See how this alternative system is impractical? It's just not worth it.

Sure, kid, you are "right". But we teach in decimal, while your answers are in a different system. We cannot easily teach you more useful things using this system. You could try developing it yourself, but you will quickly find that it is completely impractical for larger numbers and more advanced (and useful) calculations, as well as for communicating and even using measurement tools, which use decimal. You can use it, but you will find it much much easier to function in the world, and probably even for personal use, if you use decimal instead. Put some effort into learning it, and it will be much more efficient in the long run.

I could probably come up with an even more tactful argument, but don't feel like going through the effort given that I am writing for non-existant loonies who somehow think adding 2+2 gives 22, then, even when shown them combining into 4 markers, proceed to still think it's 22. But with the right tactics, I believe it is 100% possible to say why "22" as an answer doesn't work, while still technically not calling it "wrong", therefore avoiding all this fictional backlash. It's all a just question of systems and terminology.

Exposure to and seriously considering different views (and also asking lots of questions about why math is the way it is) has allowed me to technically justify even something as crazy as this. It's fun to defend the existence of and tactfully argue views no one in their right mind would seriously consider believing, and it's good practice in dealing with any views one encounters at all, real or fictional, in my opinion.

I love seeing things from the side no one ever considers, provided that it's not something people will get too angry at me for.