Pieri's formula describes the intersection product of a Schubert cycle
by a special Schubert cycle on a Grassmannian. We present a new geometric
proof, exhibiting an explicit chain of rational equivalences from a suitable
sum of distinct Schubert varieties to the intersection of a Schubert variety
with a special Schubert variety. The geometry of these rational
equivalences indicates a link to a combinatorial proof of Pieri's formula
using Schensted insertion.