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How to think about Map[ ] ?


I can understand that  

   Map[f,{a,b,c}] --> {f[a], f[b], f[c]}

But would someone want to give a little tutorial on how to understand 
the (what seem to me) bizarre results I get when I try various
combinations like

   Map[f, a + b + c]

   Map[f, a + b - c]

   Map[f, a * b * c]

   Map[f, a / b / c]

and other more complex forms.  (I'm particularly confused by the
difference between the * * and the / / cases.  I'm not acquainted 
with LISP, and have never encountered any other form of mathematics 
that operates as  Map  seems to.)

Thanks...

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