Analogical reasoning has played an important role in the development of modern mathematics. However, there has been critique of analogies for the purpose of learning new mathematics. In this article, I counter that students can productively reason by analogy to learn new mathematics and even develop new mathematics themselves. I display examples of an undergraduate student exploring topics in abstract algebra and describe how they progressively developed productive ways of thinking associated with the mental act of analogical reasoning as they continued to explore. The documented activity provides a blueprint for how analogies might be productively leveraged pedagogically.