This paper reports on a two-part investigation into how students think about and use summation (sigma) notation. During an instructional design experiment, two participating students struggled with this notation, but also reasoned about it in creative ways. This motivated a follow-up study in which we administered a free-response three-item survey to 285 students enrolled in a variety of undergraduate mathematics courses. Our findings suggest that it is not uncommon for students to produce incorrect responses when encoding and decoding sums using summation notation. More significantly, our findings reveal that students are capable of reasoning about this notation in a variety of viable, but unconventional, ways. We argue that these findings have implications for the teaching and learning of both summation notation specifically and mathematical conventions more broadly.