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In the Mathematics class, I sit next to John, who is captain of basketball team.

We don't use article with the name of a subject so going by this correct sentence should be

In Mathematics class, I sit next to John, who is captain of basketball team.

But what if I want to refer to a specific class then how will I say that ? We use article the to refer to a specific thing but since its incorrect to use the here how do I say that without using the with mathematics?

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When referring to classes, the is unnecessary because you only have one of that particular class, and you're not referencing a specific location/thing.

There are some examples when the would have to be used:

In the Math classroom, John's desk is right beside mine.

In the first-floor gym, there are trophies that John's team won last year.

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"In Mathematics class, I sit next to John, who is captain of the basketball team" is correct. The other form is clumsy English. You are correct about "the" being used to indicate specific things. However, they have to be uniquely individual and specific things. A mathematics class is a general thing that can be assigned individuality through context with possessives like "my," "mine," "yours," "his," etc.

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If you want to refer to a specific period of Mathematics class, you could say something like this:

In the first period Mathematics class, MATH 101A, I sit next to John, who is captain of basketball team.

Also, don't forget to include "the" before basketball team. We always include an article before this phrase, for whatever reason.

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