の間(あいだ)にWhile, During, Between, Period

とき expresses the time of an action or event, like the English 'when'. It can be used with nouns as in 雨の時 'when it rains', verbs as in 父が帰ってきた時 'when Dad came home', or adjectives as in 暑い時 'when it's hot'. In contrast, の間に is used with nouns, and describes a period within which something happens, similar to the English 'while' or 'during', e.g. 冬休みの間に ('during winter break').

間(あいだ) means 'period' or 'space', and in addition to its literal meaning, it is used in expressions meaning 'while' or 'between', e.g. 1月と4月の間に 'between January and April'. In the ているあいだに form, the focus is on something happening during another ongoing action. For example, 勉強しているあいだに電話がかかってきた ('The phone rang while I was studying.').

ながら attaches to verb stems to express performing one action simultaneously with another, e.g. 音楽を聴きながら走る 'listen to music while running'. In contrast, 間に can follow various words or phrases to express doing something while a certain state or condition persists, e.g. 若いうちにマラソンを走りたい ('I want to run a marathon while I'm young.').

ているところだ conveys that an action is in progress at that moment, with the nuance that the speaker is occupied with it, e.g. 映画を観ているところだ ('I'm in the middle of watching a movie.'). In contrast, の間に is used to express that something else happened at some point during the duration of that action, e.g. 映画を観ている間に父が帰ってきました ('My father came home while I was watching a movie.').