Researchers have developed cognitive systems capable of human-level performance at complex tasks, but constructing these systems required substantial time and expertise. To address this challenge, a new line of research has begun to coalesce around the concept of cognitive systems that users can teach rather than program. A key goal of this research is to develop natural approaches for end users to directly train these systems to perform new tasks. However, there does not currently exist a language for describing the key components of cognitive system training interactions and how these components relate to the concept of naturalness for end users. This paper begins to explore this gap. To lay the foundation for this exploration, we review relevant prior machine learning and interaction frameworks as well as the human-computer interaction literature to identify characteristics of systems that have historically been natural for end users to interact with. Based on this review, we propose the Natural Training Interactions (NTI) framework, which decomposes cognitive system training interaction into patterns, types, and modalities, all of which support the acquisition of different kinds of knowledge. Finally, we discuss how this framework characterizes existing research within this space and how it can guide future research.