One of the largest grammatical differences between English and Korean concerns formality. Hierarchy is so important in Korean culture that the language enshrines it as a fundamental element of grammar. The grammar of Korean speech is determined by the relative ranks of the speaker and audience within the prevailing hierarchy. Think of the way a Romance language conjugates its verbs to encode subjects, then imagine adding another layer of inflection to encode the speaker's rank relative to the audience. It's complicated and Wikipedia can tell you all about it if you like.
All you need to know to follow the rest of my story is that the least polite paradigm is banmal, which is basically Korean stripped of formality. Superiors use banmal when speaking to their inferiors, who use a more polite form in reply. Close friends use banmal amongst themselves. Typically, Koreans begin a relationship with more polite forms and move towards a mutual banmal consensus as the relationship deepens. Using banmal inappropriately denies a person the respect they are due and is a social insult that won't likely pass unnoticed or unpunished, especially if a feisty ajumma with a big flowery carpetbag and the low outrage threshold typical of her kind is on the receiving end of the presumption.
So let's get back to this niece: why doesn't she respect her foreign teacher? As her aunt explains it, it starts with the language. English has nothing akin to Korean's byzantine formalities. It offers only rudimentary honorifics (Mr., Mrs., etc.) and its politeness is usually a matter of word choice, never a basic point of grammar. The niece, being Korean, is used to encoding politeness and respect with grammar. Not finding any equivalent tool in English, she intuits that it isn't a polite language and that no special respect is due her foreign English teacher, a refugee from a rude land where people don't even respect their elders--they use the second person to address everyone! (Polite Korean speech refers to superiors in third person.)
This unwarranted generalization isn't confined to children; adult Koreans make similar inferential mistakes. Some Koreans go straight to banmal when speaking Korean with strange foreigners, though they wouldn't be so rude to their countrymen. They feel the lack of formality they perceive in English allows them to ignore the rules of Korean when speaking to waygooks. The perception of English speakers as completely informal is encouraged by the entertainment industry: subtitles of English dialogue always use banmal.
In conclusion, here's a random picture of winter near Insadong: