The detection of image motion is fundamental to vision. In many species, unique classes of retinal ganglion cells selectively respond to visual stimuli that move in specific directions. It is not known which retinal cell first performs the neural computations that give rise to directional selectivity in the ganglion cell. A prominent candidate has been an interneuron called the 'starburst amacrine cell'. Using two-photon optical recordings of intracellular calcium concentration, here we find that individual dendritic branches of starburst cells act as independent computation modules. Dendritic calcium signals, but not somatic membrane voltage, are directionally selective for stimuli that move centrifugally from the cell soma. This demonstrates that direction selectivity is computed locally in dendritic branches at a stage before ganglion cells.