How many squares can be found on a typical 5 × 5 geoboard? Teachers who are unfamiliar with this question may wish to stop here and reflect a bit. The question can lead to wonderful student discourse: How can someone tell if something is a square? Should squares be counted that are the same size (but in a different location) or just different sizes? How is it known if squares are different sizes? How many squares are on an N × N geoboard? Teachers who are familiar with this question, which is often used to promote open-ended investigations, may think of what is called "usual squares" because they all have vertices on the pegs. Note that only single, isolated geoboards are being considered throughout this article; the authors are not using "connected geoboards," which can be created by combining separate geoboards. Teachers know that one of the many joys of the profession is encouraging students to explore mathematics, often leading to new and unexpected questions. All three authors have spent time exploring what they are calling Off-the-Peg (OTP) squares with classes of middle school students as well as with preservice middle school teachers. In this article, the authors will showcase what students meant by OTP squares and how they redefined what it meant to ask, "How many squares of differing sizes really can be found on a typical 5 × 5 geoboard?" Along the way, they will profile some insights they gleaned from students about what it really means to "do mathematics" as they progressed in investigating their redefined question.