Reasoning is central to what Cuoco, Goldenberg, and Mark refer to as mathematical "habits of mind" (1996). "Reasoning is an integral part of doing mathematics. Students should enter the middle grades with the view that mathematics involves examining patterns and noting regularities, making conjectures about possible generalizations, and evaluating conjectures" (NCTM 2000, p. 262). Mathematical habits of mind are the practices that mathematicians use in their work, such as searching for patterns, experimenting, tinkering, describing, inventing, visualizing, and conjecturing. Teachers who wish to promote these habits in their students, however, are faced with the daunting challenge of finding tasks that can develop and further such habits. A paper-folding task can be an effective vehicle for such mathematical reasoning. When one thinks of origami and its connections to mathematics, images of intricate folded geometric models come to mind. In this capacity, origami has great potential for enriching mathematics classrooms (Cipoletti and Wilson 2004; DeYoung 2009; Higginson and Colgan 2001). When building an origami model, however, the person who is folding the paper is typically following a set procedure. The important mathematical thinking resides in the reasoning about the model, not the folding process. In this article, the author demonstrates a novel pedagogical approach to origami and how habits of mind can serve as a foundation for mathematical exploration.