There is growing concern about misinformation and the role online media plays in social polarization. Analyzing belief dynamics is one way to enhance our understanding of these problems. Existing analytical tools, such as survey research or stance detection, lack the power to correlate contextual factors with population-level changes in belief dynamics. In this exploratory study, I present the Belief Landscape Framework, which uses data about people's professed beliefs in an online setting to measure belief dynamics with high resolution. I provide initial validation of the approach by comparing the method's output to a set of hypotheses drawn from the literature and by inspecting the "belief landscape" generated by the method. My analysis indicates that the method is relatively robust to different parameter settings, and results suggest that 1) there are many stable configurations of belief, or attractors, on the polarizing issue of climate change and 2) that people move in predictable ways around these attractors. The method paves the way for more powerful tools that can be used to understand how the modern digital media ecosystem impacts collective belief dynamics and what role misinformation plays in that process.