The Ministry of Education is seeking to reintroduce hanja (한자, 漢字) to part of the elementary school curriculum. Korean students are in elementary school until about age 12 or 13, at which time they move on to middle school. For a long time, Chinese characters (漢字) were the sole means to write the Korean language. If the truth be told, they didn’t write in Korean, but in classical Chinese. Especially towards the 19th century, the trend became to mix the Chinese characters with the Korean alphabet. In other words, Chinese characters were used for words derived from Chinese (about 70% of the vocabulary). The Korean alphabet was used for native Korean words and for grammatical particles, of which Korean has a superabundance.
In North Korea, they were banned and the Korean alphabet alone was used. However, the North Koreans found that the students were having troubles with the language. Therefore, about 2,300 Chinese characters were introduced. While everything is still printed only in the alphabetic script in North Korea, the North Koreans have found that having a firm grounding in the Chinese characters only helps comprehension in reading all alphabetic script, mostly because so much vocabulary comes from the Chinese language.
South Korean language policy has gone through a number of wildly different shifts. In the beginnings of the Republic of Korea, most everything was printed in Chinese characters. Thus, if you were take a glance at the newspapers from the period, you will notice that with a few exceptions, everything is written with Chinese characters. They were later on banned, only to be brought back, banned from education again, and finally brought back as an optional course. Middle school students were taught 900 Chinese characters. High school students studied an additional 900. This made up the official list of Chinese characters.
The trend in publishing became to use Chinese characters only when necessary. They were especially used in newspaper headlines, as Chinese writing favours terse sentences. Thus, you can pack a lot of information in a smaller space. Nowadays, the trend is not to really mix the characters with the text. Rather, perhaps to highlight that they are of foreign origin, the characters are placed in parenthesis next to the possibly ambiguous word.
Needless to say, proponents of all alphabetic writing are not pleased and have lodged protests with the government. You can read about that here.
The late president Kim Dae Jung (金大中) was very much a proponent of education in Chinese characters. He was educated in the Chinese classics. In fact, republishing some of his work has proven a bit tricky, as he made frequent use of Chinese characters in his writing. In the days following his funeral, publication of last journals was delayed, mostly because they used so many Chinese characters and thus required some editing.