How to facilitate students’ understanding of mathematics is a major concern for the mathematics education community as well as education authorities, especially in England, UK and Shanghai, China. However, research into such understanding in these two regions is still in its infancy. The aim of this thesis is to contribute to this research area by investigating how well students understand a particular mathematical concept, linear function, and describe how their understanding has been shaped. A model of understanding function is defined in terms of six levels: Variable Perspective, Dependent Relationship, Connecting Representations, Property Noticing, Object Analysis, and Inventising. These six levels are developed by examining the most prominent theories from existing Western and Eastern literature on understanding function. Using this model, three perspectives around understanding linear function are investigated: what the official documents expect; what students actually achieve; and teachers’ views of how students’ understanding of linear function develops. Mixed methods are adopted to portray a holistic view of understanding function in the two regions. The quantitative data analysis includes three curricula and seven selected textbooks to identify their characteristics and requirements. The main study also analyses student tests from 403 Year 10 Higher Level English students and 907 Grade 8 Shanghai students. Findings demonstrate that the Shanghai students have more abstract understanding than the English Higher Level students, and are more comfortable with algebraic expression, which is emphasised heavily in the Shanghai curriculum and textbook. The graphic representation dominates the Higher Level English students’ solution approaches, which is again emphasised in their textbooks. This study recommends that the more emphasis should be on algebraic expression for understanding linear function in England and graphic representation in Shanghai.