In this paper, we consider the construct of algorithmic thinking in mathematics education. We are convinced that algorithmic thinking is an invaluable way of thinking for students to develop, and so we are motivated to promote theoretical discussions in the field about the nature and utility of algorithmic thinking. We present three examples of algorithmic thinkinga mathematical example, an example from a mathematician interview, and an example from an undergraduate student interview. We then briefly review some relevant literature related to algorithmic thinking, and we conclude with some avenues for future research. Our goal is to further conversations about and refinements of characterizations of this important topic.