The first Korean classes taught to foreigners were in the late 19th century, when a Russian university started offering Korean as a second language classes in 1897.
Since, a large number of people started learning Korean. The first Korean language Institute in Korea was actually in North Korea, established some time after independence from Japan, and the first Institute offering Korean language classes to foreigners in South Korea was the Yonsei University Korean Language Institute (KLI), established in 1959.
While in North Korea, most people learning Korean were Russian, Chinese and Eastern European diplomats and military officers, in South Korea, Japanese and Western journalists, diplomats, military officers and mostly missionaries were learning Korean. As of today, while most Korean language institutes in North Korea have ceased to function, in the South, more there are large numbers of hagwons and universities offering Korean classes to foreigners, with around 90% of the students being from mainland China.
With the sudden inflow of Chinese students learning Korean in order to attend university or get a job in Korea, hagwons and language institutes have been caught by surprise. There was suddenly a high demand for Korean as a Second language teachers, textbooks, and other materials, meaning that there was very little time for such institutions to carefully elaborate such materials and to design a policy for teaching Korean.
One should know that South Korea is one of those countries with what is called "diglossia", meaning that there is a "high form" and "low form" of language. By high form I don't mean jondeamal or "polite speech", but differences between how to language is written and spoken in front of a crowd and how it is spoken when having conversations.
Korean institutions have issued official dictionaries and grammar books on how language should be spoken and written "properly". Therefore, Korean language institutes teach that form of language rather than the more colloquial form.
Why? With large numbers of Chinese students seeking to attend university or get a job in Korea and who have actually very little interest in learning the language, as well as many other students learning the language in order to get a job, there is a strong demand to learn "written" rather than "spoken" Korean.
Korean authorities have shown disappointment at the way Korean as a Second language is being approached by foreigners. The Korean government wants "rich" people who learn Korean in order to make important financial investments in Korea, not students from developing countries seeking economic opportunities (in other words, stealing jobs away from Koreans) to learn the language.
Indeed, surveys indicate that while a few years back most Korean language students were overseas Koreans, Japanese, Europeans and North Americans, today, most students are Chinese and Southeast Asian students. The Korean government is desperately trying to attract students from developed countries to learn the language, with little success.
The Korean government is faced with a dilemma that it may arise in the future: while those studying Korean do so in order to stay in Korea. Indeed, languages like English are preferred in the countries of those who are learning Korean, and learning Korean will only limit their opportunity to get a job in their home country. Korean language students therefore choose to marry Korean citizens and stay in Korea or do their best to get a job in Korea. While it is bringing a lot of cultural diversity to Korea and helping solve the demographic problems Korea is facing, they are "economic immigrants", not "cultural immigrants", and will therefore do very little efforts to fit in Korean society.