Algebra students may often demonstrate a certain degree of proficiency when manipulating algebraic expressions and verbalizing their behaviors. Do these abilities imply conceptual understanding? What is a reliable indicator that would provide educators with a relatively trustworthy and consistent measure to identify whether students learn algebraic concepts beyond procedures? How might teachers know when the transition from "operational" or "process conception" to "structural" or "object conception" takes place? Assessing mathematics students' conceptual understanding is critical for educators to make informed decisions when selecting curriculum, planning instruction and developing an assessment program. These decisions should be supported by strong educational foundations and be rooted in theory that provides sustained basis for understanding how mathematics students construct knowledge and how mathematics can be taught. The multiyear research study described in this article attempts to answer these questions. The article introduces a framework for assessing middle school algebra students' levels of understanding of linear equations with one unknown. The framework is rooted in Orit Hazzan and Rina Zazkis's ideas of reducing level of abstraction, Uri Wilensky's and Victor Cifarelli's ideas of the degree of abstraction, Anna Sfard's theory of operational and structural conception, Ed Dubinsky and Michael McDonald's APOS theory, and the concept of representation in mathematics and mathematics education. (Contains 3 figures.)