The numerical quantization method (see [B.P.1, B.P.2, B.P.P.1]) is a grid method which relies on the approximation of the solution of a nonlinear problem (e.g. backward Kolmogorov equation) by piecewise constant functions. Its purpose is to compute a large number of conditional expectations along the path of the associated diffusion process. We give here an improvement of this method by describing a first order scheme based on piecewise linear approximations. Main ingredients are correction terms in the transition probabilities weights. We emphasize the fact that in the case of optimal quantization, a non neglectable number of correction terms vanish. We think that this is a strong argument to use it. The problem of pricing and hedging American options is investigated and a priori estimates of the errors are established.