[Spoiler alert: This post is, in part, a reflection on the show “In & Of Itself“, written and performed by Derek DelGaudio. If you are near New York City, I strongly encourage you to see this show before it ends on 8/19/18 and before reading below.]
This summer, I have helped lead a professional development workshop for mathematics educators on student-centered pedagogy. One session of the workshop  is organized around a paper by María Trigueros and Sally Jacobs entitled “On Developing a Rich Conception of Variable”. Trigueros and Jacobs argue that the concept of “variable”, which seems unified from an expert perspective, is multifaceted. Moreover, they point out the ways that this multiplicity is challenging for students and that there are structural issues with our curricula that fail to support the development of a rich conception of variables in most students. Faculty can easily take a deficit perspective on students whose conceptions are unlike our expert perspectives, and this concept in particular is at the root of a lot of the blame we lay on students when they think differently than we expected.
At the workshop, we use this image of an old parable about scholars and an elephant. The elephant represents the concept of variable, and the scholars each describe an important facet of the concept: as unknowns, general numbers, parameters, co-varying quantities, or something else. The point is that student conceptions of variables are reasonable attempts to make sense of the contexts in which we have placed them and judging them for not having integrated those contexts into a unifying concept feeds into some of the structural issues with our educational system.