The language taught in schools common to China, Japan, and Korea (excluding English)

It is Classical Chinese.
Classical Chinese is the written form of Chinese from the Spring and Autumn/Warring States period to the Han Dynasty,
and it served as the official written language of the East Asian world until the early 20th century.
It was also a mandatory subject in the public education systems of
Korea, China, Japan, Taiwan, Hong Kong, Macau, and North Korea.
Japan, Joseon, and Vietnam recorded official documents in Classical Chinese.
Although it had already become a language detached from spoken Chinese by the Tang Dynasty,
Classical Chinese remained the official language until the very beginning of the 20th century,
and writings that used spoken Chinese were treated as inferior, referred to as "Baihuawen" (Vernacular Chinese).
However, due to China's language reforms in the 1920s,
Vernacular Chinese pushed out Classical Chinese to become the official written language.
Nevertheless, it remains an essential vocabulary for studying ancient East Asian scholarship,
and a significant portion of the words we use today were constructed using Classical Chinese grammar.