This study investigates how students in England and Shanghai understand linear function. Understanding is defined theoretically in terms of five hierarchical levels: Dependent Relationship; Connecting Representations; Property Noticing; Object Analysis; and Inventising. A pilot study instrument presented a set of problems to both cohorts, showing that the English cohort generally operated at the levels of Property Noticing and Object Analysis, whereas the Shanghai cohort reached the higher level of Inventising. The main study explored understanding levels and students' errors within each cohort in detail, in order to gain insights into reasons for apparent differences. The instrument used in the main study included two overlapping items, which were the same for both cohorts, while others were pitched at levels of understanding revealed in the pilot. Analysis of students' solutions revealed that the English students' errors were manifested in a lack of basic skills including dealing with negative numbers, while the Shanghai students showed weaknesses in their ability to use graphs. The discussion highlights different views of understanding as a possible background reason for the contrasts observed. Errors and apparent difficulties suggest implications for teaching linear function in each context.