Perceptual decisions are ubiquitous and essential in life, and even in their simplest forms open a window into the neural mechanisms underlying cognition. However, the properties of decision mechanisms inferred in simple behavioral paradigms are not independent of the properties of employed stimuli and task. One key task property is the type of information provided by a single stimulus, which can be either stationary or dynamic. Stationary paradigms are based on evidence with fixed central tendency, often obscured by noise, whereas in dynamic paradigms the mean of the evidence can vary within the duration of the stimulus. Both types of evidence can be mapped on specific settings occurring in nature, but in scientific studies stationary paradigms have dominated. Here we introduce a novel dynamic, motion-direction discrimination task based on stimuli with rich, but controlled structure, which are well suited to characterize the dynamics of a decision process at fine timescales. Human participants were asked to report the prevalent direction of motion in a stimulus consisting of a random sequence of motion pulses (left or right) interleaved with static periods. We find that traditional models of evidence accumulation, although validated by decades of research with stationary paradigms, do not account for behavior in our task, even when augmented with non-linear mechanisms like boundaries, leakiness, or sensory adaptation. The key shortcoming of these models lies not in the nature of the accumulation process, but of the underlying evidence—only an accumulation process acting on a form of “relative” evidence, computed through a step of divisive normalization operating over time, closely captures the behavior, and explains otherwise puzzling, striking deviations from perfect accumulation. These findings point to a prominent, previously unrecognized, role for divisive normalization in evidence accumulation, which may prompt a re-interpretation of the mechanism of decision-making in stationary and dynamic paradigms.