Summary: We consider the inverse problem of reconstructing the diffusion coefficient in a quasi-linear parabolic differential equation in divergence form from measurements of the solution at a finite number of points in the interior of the domain. An equation error method is developed which transforms the inverse problem into a system of linear operator equations for the diffusion coefficient and which can be solved by the conjugate gradient method in a very efficient and stable manner. A detailed error analysis relates the required number of measurements with their accuracy. Numerical results illustrate the performance of the method.