Four-component decomposition of sense making in algebra

This article presents a case in which a pair of middle-school students attempts to make sense of a previously obtained by them position formula for a particular numerical sequence. The exploration of the sequence occurred in the context of two-month-long student research project. The data were collected from the students' drafts, audiotaped meetings of the students with the teacher and a follow-up interview. The data analysis was aimed at identification and characterization of events and algebraic activities in which the students were engaged while making sense of the formula. We found that the students' conviction, by the end of the project, that the formula "makes sense" emerged when they justified the formula, checked its generality, discovered a geometry mechanism behind it, and found that it came to cohere with additional formulas. The findings are summarized as a suggestion for a four-component decomposition of algebraic sense making.