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It has been demonstrated that most people have a limited understanding of atmospheric CO2 accumulation. Labeled stock-flow (SF) failure, this phenomenon has even been suggested as an explanation for weak climate policy support. Drawing on a typology of knowledge, we set out to nuance previous research by distinguishing between different types of knowledge of CO2 accumulation among the public and by exploring ways of reasoning underlying SF failure. A mixed methods approach was used and participants (N=214) were enrolled in an open online course. We find that ostensibly similar SF tasks show seemingly contradictory results in terms of people's understanding of CO2 accumulation. Participants performed significantly better on stock stabilization tasks that explicitly ask about the relationship between stocks and flows, compared with a typical SF task that does not direct the participants' attention to what knowledge they should use. This suggests that people possess declarative and procedural knowledge of accumulation (knowing about the principles of mass balance, i.e., what and how to use them) but lack conditional knowledge of accumulation (knowing when to use these principles). Additionally, through a thematic analysis of answers to an open-ended question, we identified three overarching ways of reasoning when dealing with SF tasks: system, pattern, and phenomenological reasoning, providing additional theoretical insights to explain the large difference in performance between the different SF tasks. These more nuanced perspectives on SF failure can help inform interventions aimed at increasing climate science literacy and point to the need for more detailed explorations of public knowledge needed to leverage climate policy support.