A mixed construction is a sentence with incompatible elements that begins with one type of structure and shifts to another type of structure. In these sentences, the speaker sets out to say one thing and abruptly switches to something else, resulting in confusion.
A sentence that is logical has a subject and a predicate. When a subject is introduced in a sentence, an expectation is set up about the grammatical direction the sentence is going in, and when that expectation is not met, the sentence does not sound right. Take this mixed construction example:
Teachers, a noble profession, involves a lot of patience.
Teachers is not a profession; teaching is. When teachers was introduced as the subject of the sentence, it created the expectation that the rest of the sentence would describe something teachers do or are. The predicate involves a lot of patience takes the sentence in a different grammatical direction, making it a faulty predicate. We could rewrite the mixed sentence this way and it would be grammatically correct:
Teachers have a lot of patience.
However, the original sentence clearly intended teaching, as a profession, to be the subject, and the predicate was intended to show that teaching does indeed require a lot of patience.
Teaching, a noble profession, involves a lot of patience.
Teaching is a noble profession that involves a lot of patience.
Of course, it would also be correct, if a little less elegant, to simply divide the sentence in two.
Teaching is a noble profession. Teachers have a lot of patience!
Sentences with mixed constructions can often be found in first drafts of writing; the writer sets out with a stream of ideas that sound misconnected on a second reading. This is just one of the reasons reading over and editing your writing is always a great idea.