
Caitlin K. answered 05/31/19
English Instructor with 10+ Years of Tutoring Experience
The punctuation for conjunctive adverbs (or adverbs that connect ideas, like conjunctions do) like however depends on their usage in the sentence. A semi-colon and comma will be used if there are two independent clauses being connected. For example:
It was raining outside; however, she still took the dog for a walk.
In this case, It was raining outside is an independent clause, as is she still took the dog for a walk. In other words, each of these clauses can stand on its own. Semi-colons, like periods, are used to connect two independent clauses.
On the other hand, if the conjunctive adverb is being used as a transition word in a sentence, it will be set off with commas. For instance:
It was raining outside. Jill was, however, dry on her walk because she brought an umbrella.
In this case, however is functioning as an interrupter rather than a connector of independent clauses. In this case, we surround it with commas to indicate the interruption.
In the example you've provided, then, the correct punctuation would be as follows:
Contrary to first opinions, however, the school was public and cost nothing to enter.
This is because however is functioning as an interrupter, as in the second example.