We consider the inverse problem of identifying spatially dependent coefficients in linear, parabolic, partial differential equations with specified initial and boundary conditions. We primarily explore the recovery of the diffusivity coefficient in the heat equation using a very limited amount of data on a portion of the spatial boundary. In the process, we show that the coefficient dependent observation operator is injective. With this knowledge, we use the Backus-Gilbert approach, modified with an adjoint method, to construct an approximate solution to the problem. We show the results of numerical experiments for the single, spatial variable case and then apply the method to related problems.