Knowledge about the structure of arithmetic expressions enables people to reason effectively about such expressions, including an ability to judge equivalence under transformations. This paper reports an empirical study of six middle-school children who judged the equivalence of three sets of three-term arithmetic expressions with an addition and a subtraction operator. Analyses of thinking-aloud protocols on this task reveal that the students (a) use several different methods to parse and judge the equivalence of such expressions, (b) sometimes use a different parsing or judging method with the same expression, depending on which expression it is compared against, and (c) are able to work with different conceptual interpretations of expressions. Additional results are provided about specific errors that were made and trends in the students' application of these methods. The results are discussed briefly along with three comments on their educational implications. (Author)