"여자아이가 엄마의 머리를 빗으로 빗는 것을 좋아한다."
Translation:The girl likes combing her mom's hair.
My impression is:
(1) 그녀는 엄마의 머리빗기를 좋아합니다 = She likes to comb her mother's hair.
--> the use of "-기" to nominalize the verb puts the focus on the type of activity described by the verb
(2) 그녀는 엄마의 머리를 빗는 것을 좋아합니다 = She likes the combing of her mom's hair = She likes combing her mom's hair
--> the use of "-는 것" to 'nominalize' a clause (similar to the use of the English gerund) puts the focus on the process of the activity.
Note: Of course, I might be wrong. But it's worth writing this out here in the hope that someone could set me on the right track.
I am not an expert in Korean grammar. Maybe you are right. I hope one of Duo's moderators sees this and comments! I do know that in English grammar "She likes combing her mother's hair" = "She likes to comb her mother's hair." Both verbals are used as the direct objects and both are equally correct, in English.
I reckon it is the same in Korean. "-는 것" and "-기" are mostly (not always) interchangeable depending on the main verb in use (좋아하다, to like, in this case).
I guess DLG just want to draw Learners's attention to the 2 forms of nominalization of verbs akin to the English "gerund" and "infinitive".
You are not wrong. As there can never be an exact translation from one language to another, for consistency DLG adopt their own convention (set of terminology and style) which could be quite rigid at times; amongst those, 어머님 (mother, Hon.), 어머니 (mother, familiar), 엄마 (mom, intimate).
It is worth pointing out however that "엄마" is usually used to address (and sometimes, to refer) to one's own mother, unlike the English mom.
Its use here gives the impression that the Korean sentence is a direct reverse translation of the English one?