Sometimes easy geometric pictures have awkward seeming algebraic descriptions.
On pages 6 and 7 of Concise I gave examples where I both gave a geometric picture
and explicit formulas to make the idea of such translation clear.  In other cases,
(as in cofiber homotopy equivalence) I just found it quick and easy to write down 
the homotopies (in terms of other homotopies). Sometimes it is just way to laborious
to draw the pictures, other times it is too laborious to write the homotopies out.
One should learn to be happily eclectic and absorb all techniques available.