Translation between external representation systems in mathematics: All-or-none or skill conglomerate?

The study described herein represents an initial, exploratory attempt to understand what it means to translate between external representation systems. Researchers have traditionally considered translation as an all-or-none activity. We hypothesize that translation is comprised of both knowledge and skill components, and accordingly construe translation as an activity that our framework allows us to define as partial or complete. We examined the effects of part–whole knowledge in change unknown subtraction situations and the structure of three different external representations on first grade students’ ability to translate in a partitioning task. Based on three interviews conducted with each student, results from our mixed methods analysis showed that translation skill was affected by what students knew about subtraction and the structure of the ER. Further, whereas high knowledge students demonstrated greater difficulty with translating to the ERs, low knowledge students had difficulty translating from the problem statement as well as translating to the ERs.