The paper presents a model that incorporates the physics of dynamic Jahn-Teller and double-exchange effects to explain the behavior of La$_{1-x}$A$_x$MnO$_3$ materials, which exhibit "colossal magnetoresistance." The model is solved using a dynamical mean field approximation. In the intermediate coupling regime, the interplay between these two effects reproduces the observed resistivity and magnetic transition temperature behaviors. The model captures the key physics of the materials, including the ferromagnetic-paramagnetic transition at an x-dependent temperature T$_c$(x) and the sharp drop in resistivity near T$_c$. The authors discuss the competition between electron itineracy and self-trapping, controlled by the ratio of Jahn-Teller self-trapping energy to an electron itineracy energy. They solve the model Hamiltonian using the dynamical mean field approximation and present numerical results, showing a phase diagram and the temperature and magnetic field dependence of resistivity. The theory is consistent with experimental observations, including the variation of T$_c$ and resistivity with x and the opening of a gap in the optical conductivity.The paper presents a model that incorporates the physics of dynamic Jahn-Teller and double-exchange effects to explain the behavior of La$_{1-x}$A$_x$MnO$_3$ materials, which exhibit "colossal magnetoresistance." The model is solved using a dynamical mean field approximation. In the intermediate coupling regime, the interplay between these two effects reproduces the observed resistivity and magnetic transition temperature behaviors. The model captures the key physics of the materials, including the ferromagnetic-paramagnetic transition at an x-dependent temperature T$_c$(x) and the sharp drop in resistivity near T$_c$. The authors discuss the competition between electron itineracy and self-trapping, controlled by the ratio of Jahn-Teller self-trapping energy to an electron itineracy energy. They solve the model Hamiltonian using the dynamical mean field approximation and present numerical results, showing a phase diagram and the temperature and magnetic field dependence of resistivity. The theory is consistent with experimental observations, including the variation of T$_c$ and resistivity with x and the opening of a gap in the optical conductivity.