I recently taught complex analysis to our second year students. One particular problem jumped out whilst marking the exam. One question was “Define the length of a contour.” This was only worth about 2 marks and the bulk of the students got it mostly right. My point is that students’ responses can tell us something about how they see mathematics and perhaps how they do mathematics. The main mistakes were 1. Not giving enough information. 2. Not being mathematical. 3. Giving the procedure. 4. Trying to memorize without understanding. Let’s deal with these in turn (the most important is number 4!). Number 1. Not giving enough information. A good answer to “Define the length of a contour” is “Suppose that is a contour. The length of the contour is .” Instead many students slapped down . They lost a mark because they didn’t tell me what , and were. This happens a lot, students focus on the equation and forget about the surrounding information. If I did not know the definition of a contour, then the equation doesn’t tell me enough. I wouldn’t know where the , and were coming from and their relevance. 2. Not being mathematical. Another problem with definitions in general, not just this one, is that students give a hand waving definition, e.g. “It’s the actual distance that the curve moves.” This is not very mathematical and would not help anyone understand length except in an intuitive way. (In this case you could probably guess from the name that length is to do with distance!) 3. Giving the procedure. Another very common mistake with definitions is confusing the definition with a procedure used to calculate the object defined. For example, “Define the order of a pole” is often incorrectly answered...