Just a quick question on the て form, as what I am about to say isn't really
made clear by textbooks. Is the tense of a verb in て form dependent on what
follows? for example if someone was to say 教えてあげた, would the 教えて become past
tense (or completed, as at doesn't have to take place in the past)?

Thanks is advance.
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て-form
The tense of the て form of verbs
In this case the tense of the verb will be taken from あげた.

あげた also tells you who did for whom. In a conversation where someone uses
教えてあげた with an implied 私は, the speaker "I told/taught" someone.

Additionally, the common counter part for あげた would be もらった. In a conversation
where someone said 教えてもらった with an implied 私は, the speaker was saying "I was
taught/told ...".
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The tenses in `V1て+V2` depend on what V1 and V2 are.

**General conjunction**

When V1 and V2 are both just normal verbs, such as:

> ジョンは走って食べた。  
>  John-TOP run-TE ate-PAST  
>  "John ran and ate."

the tense of V2 applies to V1 as well.

**V2 modifying V1**

There are some V2s, such as いる、おく、しまう、もらう、くれる、あげる、ほしい which simply modify V1.
(You can detect that you are working with one of these because you cannot
insert adverbs after V1て.) These cases require more care.

> 電話番号を教えてほしかった。  
>  phone-number-OBJ teach-TE want-PAST  
>  "I wanted you to tell me the phone number."

Here, 教えて has no tense (and neither does "tell" in English).

For your specific example:

> 生徒に数学を教えてあげた。  
>  students-IOBJ math-OBJ teach-TE give-PAST  
>  "I taught the students math for their benefit."

As you can see in the English, changing the tense of `V1てあげる` ("doing V1 for
someone's benefit"), clearly changes the tense of V1.

You can basically intuitively tell how tense markings affect V1 once you learn
the meanings of the various V2s which fall into this category.
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