Computer-based mathematical cognitive tools (MCTs) are a category of external aids intended to support and enhance learning and cognitive processes of learners. MCTs often contain interactive visual mathematical representations (VMRs), where VMRs are graphical representations that encode properties and relationships of mathematical concepts. In these tools, interaction enables learners to perform epistemic actions on VMRs to explore and learn mathematical concepts. Interactivity of VMRs refers to the feel, form, properties, and quality of this interaction. As such, interactivity of VMRs can influence how and what learners learn. A number of factors affect learners' cognitive processes while interacting with VMRs. Researchers from several disciplines have attempted to characterize interactivity and the multiplicity of factors that affect it. However, as many of these characterizations and factors are inapplicable to VMR-based MCTs, understanding of the factors that affect learning and cognitive processes can help in the analysis of interactive VMRs. This article draws on research from various disciplines to identify and describe the applicability of 12 interactivity factors that affect learning and cognitive processes of learners who use VMR-based MCTs. Collectively, the factors can then serve as a descriptive and conceptual framework to help in the design and evaluation of MCTs and to allow designers to discuss and substantiate their design choices of interactive VMRs. (Contains 9 endnotes.)