We formulate an approach to the A + A --> products reaction that is based on a reaction-diffusion equation frequently used for the A + B problem but requires an appropriate generalization for the A + A problem. Starting from this reaction-diffusion equation, we construct the first equations in a moment hierarchy whose first two members are the global density of A particles and the pair correlation function. We terminate the hierarchy via an approximation that relates the three-particle correlation function to two-particle correlation functions and thereby obtain a set of coupled equations that turns out to be linear and hence analytically tractable. This approach leads naturally to the proportionality of the rate of the reaction to the pair correlation function evaluated at r = a, where a is the diameter of the reacting particles. In other words, the reaction rate is proportional to the probability that two A particles are sufficiently close. In the more traditional approach based on the Smoluchowski theory for trapping phenomena, the reaction rate is instead proportional to the gradient of the pair correlation function. We discuss the differences between these paints of view and their consequences. We also present numerical simulations in one and two dimensions in order to check our predictions. We confirm the well-known anomalous rate law in one dimension (the anomalies are marginal in two dimensions) and the proportionality of the reaction rate to the two-particle correlation function. Our simulations show that the rate of the reaction is indeed determined entirely by the spatial distribution of a very small shell of particles around a given reactant particle. Anomalous kinetics is a direct reflection of the deviation of the spatial distribution of this small shell from a random configuration. We also present simulation results that confirm the predicted distance and time scaling of the pair correlation function in one dimension.