Currently marking exam scripts.
Do students read the comments on what went wrong for students last year? If so, when they read “many students incorrectly wrote <this>”, perhaps what they remember is <this>.
For example, every year that I ask for the definition, a significant percentage of my students tell me that countable sets must be finite, and moreover they really seem to believe it when they try to prove stuff. [I think it is about 20% of them this year.]
OK, this material is in the last two lectures of the module. But my explicit plea “If it comes up in the exam, please don’t tell me that countable sets have to be finite like many of last year’s students did” may well have the opposite effect to the one I hope for!
Perhaps the problem is because I first tell them what countably infinite means, and then tell them that countable means “finite or countably infinite”. Then in the exam, many of them can define “countably infinite” correctly, but define countable to mean “finite”.
I could define countable to mean “has the same cardinality as some subset of the natural numbers”. This might improve the chances that they give me a correct definition in the exam, if I ask for it, but I doubt that it would improve their understanding.
Still, next time I ask, let’s see if I can beat 20% (in the right way!).