敎其子以齊語"Du's Handbook" has explained that 以 (이) can be used as a direct object marker, which seems to be how it was translated here. 齊(제) was an ancient kingdom of China during the Spring and Autumn Period. And 語 (어) means "language." Therefore, the sentence seems to make sense, but couldn't the sentence also be translated as follows?
Taught his son the language of Qí.
以 (이) can mean "to use," "to take," "with," "in order to," and a few other things, so I am suspicious of 以 being used simply as "a marker."敎其子以齊語Taught (敎) his (其) son (子) using (以) the Qí (齊) language (語).
I do not know the story from which the sentence was taken, but if the person had wanted to express the meaning "Taught his son the Qí language" instead of "Taught his son using the Qí language", I think he would have written it one of the following two ways:
敎其子齊語Taught (敎) his (其) son (子) the Qí (齊) language (語).
以齊語敎其子Took (以) the Qí (齊) language (語) [and] taught (敎) [it to] his (其) son (子).
The first sentence simply uses the standard Chinese "indirect object"-then-"direct object" order to clarify the direct object. The second sentence clarifies and emphasizes the direct object by moving it to the front of the sentence and using 以 with its meaning of "to take," which in Korean is 가지다. Therefore, in Korean, the second sentence would translate as follows: "제어를 가지고 (以齊語) 그의 아들에게 가르쳤다 (敎其子)." The only problem with the second sentence is that the meaning could still be unclear since 以 could also be translated as "to use," which means the sentence could be translated as "Used (以) the Qí (齊) language (語) to teach (敎) his (其) son (子)."
Taught (敎) his (其) son (子) to use (以) the Qí (齊) language (語).