Schubert calculus is a great source of beautiful identities which really ought to have bijective proofs. For instance, Macdonald (1991) proves, non-bijectively, an identity for a weighted sum over the reduced words for a permutation pi. I'll give a algorithmic bijective proof in the simplest case: when pi is a dominant permutation, the sum evaluates to l(pi)! As a result, we get a Markov growth process for the associated probability distribution, in which reduced words for large permutations are "grown" randomly, one transposition at a time. My bijection uses a novel application of David Little's generalized "bumping" algorithm and, with care, can be implemented quite efficiently on a computer.