Many mistaken ideas still plague our grammar books and lessons today. Here, we present the case of prepositions. As early as the 1700s, a few grammarians had realized a deep flaw in the way we think about these words. The fundamental flaw, applying across most frameworks, is in the idea that a preposition must have an object NP as its complement. Our reanalysis, based on that of the Cambridge Grammar of the English Language, vastly reduces the number of subordinators and also significantly cuts the number of words like "before" which have wrongly been said to be adverbs here, prepositions, there and subordinators elsewhere. This understanding allows teachers and students to leverage strong commonalities that apply across all PPs, commonalities that we set out. We acknowledge that this shift in thinking can be tricky for teachers who may expect it to be confusing for students too. We clearly show that this system is in fact much simpler, both to teach and to understand.