Getting students to really understand...

From Research In Practice, a great thought for all teachers:

Take-home lesson: never underestimate your ability to fool yourself into believing your students understand something when really what they are doing is watching you. To force them to engage the material it is often necessary to restrict their access to you or systematically confound the signals they get from you.

I think this is a central issue for modern math teachers. We need to explicitly develop ways of question-posing and interacting with our classes and individual students that hide or disguise our intentions for how they are supposed to respond. This needs to be part of the core training of math teachers, much more than it already is.

This is central to developing actual thinking in our students. Students have much practice in picking up clues from the teacher in order to arrive at right answers. This does not mean that they understand. Give them less in order to work and understand more.