Classical models of perceptual decision-making assume that animals use a single, consistent strategy to integrate sensory evidence and form decisions during an experiment. Here we provide analyses showing that this common view is incorrect. We use a latent variable modeling framework to show that decision-making behavior in mice reflects an interplay between different strategies that alternate on a timescale of tens to hundreds of trials. This model provides a powerful alternate explanation for "lapses" commonly observed during psychophysical experiments. Formally, our approach consists of a Hidden Markov Model (HMM) with states corresponding to different decision-making strategies, each parameterized by a distinct Bernoulli generalized linear model (GLM). We fit the resulting model (GLM-HMM) to choice data from two large cohorts of mice in different perceptual decision-making tasks. For both datasets, we found that mouse decision-making was far better described by a GLM-HMM with 3 or 4 states than by a traditional psychophysical model with lapses. The identified states were highly consistent across animals, consisting of a single "engaged" state, in which the strategy relied heavily on the sensory stimulus, and multiple biased or disengaged states in which accuracy was low. These states persisted for many trials, suggesting that lapses were not independent, but reflected state dynamics in which animals were relatively engaged or disengaged for extended periods of time. We found that for most animals, response times and violation rates were positively correlated with disengagement, providing independent correlates of the identified changes in strategy. The GLM-HMM framework thus provides a powerful lens for the analysis of decision-making, and suggests that standard measures of psychophysical performance mask the presence of slow but dramatic alternations in strategy across trials.