with students who have English as a first language (L1). What are the discourse features that cause problems for EAL students? An earlier University of Auckland study, involving 80 first-year students, used text, symbolic, and diagrammatic questions to indicate the extent of textual difficulty experienced by EAL students in first-year undergraduate mathematics (Barton & Neville-Barton, 2003). It indicated that, in comparison with native speakers of English, EAL mathematics students have a disadvantage similar to that experienced in arts subjects (about 10 percent). A second study (involving nearly 400 first-year students) found that EAL students self-reported levels of understanding similar to those of English first-language students (Barton & Neville-Barton, 2004). This study focuses on advanced (third-year) undergraduate mathematics. We wish to know whether the disadvantage is as marked at this level, and to understand what features of mathematical English cause difficulty. We notice that the proportion of EAL students taking mathematics drops dramatically at this level. There may be other explanations—for example cultural preferences for other major subjects—however we also suspect that there is a change in the nature of mathematical discourse and its relation to the mathematics presented at this level. There is a considerable body of literature that examines the nature of mathematical discourse. Halliday (1978) is usually credited with focusing researchers’ attention on mathematical language as a special register. Dale and Cuevas (1987) describe the mathematics register in terms of unique vocabulary and syntax (sentence structure), and discourse (whole text features). Also mentioned in the literature above are such features of mathematical discourse as its density, logical complexity, heavy demand on reader’s memory, unpredictability, and the mix of prose, symbols, and diagrams. While these are postulated as potential sources of difficulty for EAL students, whether in fact they do present problems, particularly at higher levels of mathematics, is the issue being considered here. The Study This study involved four third-year undergraduate mathematics courses in the Department of Mathematics at the University of Auckland. Two of the researchers are the lecturers of two of these courses. The study had three phases, two in the first semester and one in the second. The first phase involved one researcher attending lectures of the other two researchers, examining texts and course notes, and trying to identify significant mathematical discourse features at this level of mathematics. In the second phase, these features were transformed into a test/questionnaire that was presented in a tutorial to 12 Chinese-speaking students from one of these courses. The second phase also involved English proficiency testing using the University of Auckland’s Diagnostic English Language Needs Assessment (DELNA).