Unknown, placeholder, or variable: what is x?

• DOI: 10.1007/s13394-011-0029-9
• Publication Date: 2012-02-21T02:02:22Z
• Authors: Robert Ely; Anne E. Adams
• Keywords: Confusion

One of the most significant steps in learning algebra is understanding the change in the role of letters in mathematical expressions from unknowns to variables. We describe the historical development of this change in usage, starting with the ancient use of mathematical unknowns, detailing several important changes in practice that allowed for the idea of the placeholder, the birth of symbolic algebra, and the development of the variable. Focusing on these changes in practice, we interpret some classroom examples of 8th-grade students who interpret letters in terms of their experience with unknowns, rather than in terms of variables, to the confusion and dismay of the teachers. We also discuss how particular curricular and pedagogical treatments can support student learning by deliberately focusing on these changes in practice in the transition from unknowns to variables in the middle grades.