It is well know that homotopy groups of spheres are extremely difficult to compute.

One way to compute these groups is using Postnikov approximation or Whitehead tower and Serre's spectual sequence argument. This method is explained in detail in Chapter 18 of Bott and Tu's book Differential Forms in Algebraic Topology.

But this method involves complicated constructions like loop spaces and Eilenberg-Maclane spaces which makes it hard (at least to me) to see what maps between spheres realy generate the homotopy groups.

So I want to know:

1.Is there any method to construct maps between spheres which generate the homotopy groups of spheres computed in this manner?

2.How do the explicit generators relate to the constructions in this method?

qualitativeaspects of these functions is in some sense more useful than an actual formula. Tilman's answer about the Thom-Pontrjagin construction gives such a qualitative description, by providing the framed bordism type of the preimage of a regular value under $f$. – Craig Westerland Apr 5 '11 at 12:24