The Maya Software Renderer does not support the smooth preview mesh, however, Mental Ray does. Double check what render engine you're using because its not just a viewport thing (check the drop down menu at the top of the render view). I will say this, having a lot of geometry using smooth preview can start bogging down the viewport performance. It's mainly handy for reference and not a final smoothing solution.

Converting smooth mesh previews to polys can be problematic to work with even at the default smooth preview level of 2, the mesh density will skyrocket and its not the usual way of add geometry anyway. I don't know how your model looks now so I'm not getting a clear picture of your problem, it sounds like it can involve inserting edge loops, splits, merges and that may be complicating things and the instructor might not be heading in that direction. Save your work in regular phases, hopefully the instructor covers all this stuff. Also, post a screenshot