Teachers do teach students to recognize root words--I see my students working on their English assignments with root words when they should be learning French ;-) I teach French and German. I always point out relationships between words in the target language and English, and with French, it is always a Greek or Latin root. When they learn the word 'porte' (door) or 'porter' (wear or carry) I point out import, export, airport, seaport, portal, porthole, report, deport, deportation, portable, port-a-potty... When they learn durer (to last) I point out, or ask for related words (duration, endure, durance vile...the latter few students know, so I encrease their vocabulary and knowledge of roots while teaching French.
The very best way to increase a child's vocabulary is wide reading. If you read to a child, you can introduce literature and words which are far beyond their free-reading level, as your tone of voice and interpretation of the story will provide the meaning of the word--and you are right there to explain the word that they did not catch the meaning of!
There is also some research showing that vocabulary retention and spelling can be improved by free reading that does not focus on vocabulary acquisition or spelling, but on enjoyment. Eventually most people learn to recognize which spelling looks ro sounds right, but even well-educated adults who know many roots still have their bugaboos.