Responding to students' questions is a critical part of teaching mathematics. A particular response may either stifle a student's inquiry or, ideally, stimulate his or her interest in mathematics. Although formulating responses that have the potential to engage students in developing new mathematical insights is challenging, the authors believe that this skill can be developed. They have found it helpful to view this challenge as a matter of recognizing the mathematics of a student's question--that is, the mathematics the student understands as well as the mathematics he or she may be ready to learn. The two questions analyzed in this article have been used in numerous professional development seminars to engage teachers in seeing "through" the mathematical work presented by students. By making sense of a student's thought processes and anticipating the more advanced mathematical ideas that he or she might be ready to think about, teachers can engage both the student and the class more effectively. In this article, the authors describe the online dialogue from which the questions arose, analyze the two questions in the same way in which they analyze them in their professional development seminars, and conclude with guidelines for responding to students' questions that have emerged from their work with teachers. (Contains 2 figures and 1 table.)