Transitive / intransitive verbs

Solution 1:

I don't think it is obvious whether "decide" is transitive, in the example, but whether it is probably has nothing to do with a negation associated with "against". The most straightforward way to take the question is as a choice between two possible constituent structures:

1. The teacher [V decided] [PP against [NP doing karaoke]]
2. The teacher [V decided against] [NP doing karaoke]

Structure 1. is intransitive, since the V is not followed by a tree-sister NP, but 2. is transitive, since it is.

If structure 2 is correct, because "against" is a particle, it should be possible to move it to after the direct object to get 3.:

3. *The teacher decided doing karaoke against.

If structure 1 is correct, it should be possible to prepose the the prepositional phrase:

4. ?Against what did the teacher decide?

The unacceptability of 3. and the (relative) acceptability of 4. is evidence that 1. and not 2. is the correct structure.

On the other hand, we could take the following to argue the other way:

5. *Against doing karaoke, the teacher decided.
6. Doing karaoke was decided against by the teacher.