Understanding complex problems such as climate change is difficult for most non‐scientists, with serious implications for decision making and policy support. Scientists generate complex computational models of climate systems to describe and understand those systems and to predict the future states of the systems. Non-scientists generate mental models of climate systems, perhaps with the same aims and perhaps with other aims too. Often, the predictions of computational models and of mental models do not correspond with important implications for human decision making, policy support, and behaviour change. Recent research has suggested non-scientists’ poor appreciation of the simple foundations of system dynamics is at the root of the lack of correspondence between computational and mental models. We report here a study that uses a simple computational model to ‘run’ mental models to assess whether a system will evolve according to our aspirations when considering policy choices. We provide novel evidence of a dual-process model: how we believe the system works today is a function of ideology and worldviews; how we believe the system will look in the future is related to other, more general, expectations about the future. The mismatch between these different aspects of cognition may prevent establishing a coherent link between a mental model’s assumptions and consequences, between the present and the future, thus potentially limiting decision making, policy support, and other behaviour changes.