Dielectric and rheological measurements on polymer melts have demonstrated that rates of dynamical processes on short length scales decrease more rapidly than relaxation rates associated with longer length scales as the melt is cooled towards its glass transition temperature. We demonstrate here that the dynamically disordered Rouse model is consistent with these observations and provides a simple physical picture of molecular motions in supercooled melts. In this coarse-grained model of the dynamics of a single molecule, the beads of a harmonic chain are characterized by mobilities that fluctuate between zero and a finite value at a rate that is determined self-consistently. We present calculations of two observables whose temperature dependence near the glass transition has revealed the inadequacies of the Rouse model: J(e), the steady-state compliance, and omega(s)/omega(n), the ratio of segmental and normal mode peak frequencies in the dielectric spectrum of a polar polymer.