However...

Many of my students in the Vancouver area study hard for the grammar section of the SAT, however they miss the problem with this sentence.
So what's wrong with the above sentence?
It's called a comma splice -- two sentences joined by a comma.
Students study (subject verb)
they miss (subject verb)
You may counter that the two sentences are joined by "however."
But the word "however" is not a conjunction, so it cannot join two sentences.
If a student sees "however" in a sentence, the word should raise a red flag. He or she immediately should see if "however" connects two separate sentences.
The error can be fixed by joining the sentences with a semicolon or period.
Or the word "although" can be substituted for "however," as "although" is a conjunction.
Examples of the corrected sentence:
Many students study hard for the grammar section; however, they miss the problem with this sentence.
Many students study hard for the grammar section. However, they miss the problem with this sentence.
Many students study hard for the grammar section, although they miss the problem with this sentence.