Students sometimes (often) do not think about their thinking. Most have not had experiences in school that support this. This is easy to think about. If all you do is follow rules and procedures to compute algorithms you don't have to think about your own thinking. All you need to do is follow the thinking of someone else. My intuition about why passive learning fails for most people is that unless you are predisposed to independent thinking, there is little in traditional education that can transform one towards independent, critical thinking. Monitoring one's own thinking is part of the sophisticated set of thinking processes that distinguish experts and novices. And it can be trained!
Where this comes into play in IBL math classes boils down to this: students in IBL math classes must explain their reasoning on a regular basis. I claim that the process of justifying answers engages the metacognitive process. It goes something like this:
Student: "I believe the statement is false."The support of metacognition in IBL classes is much richer than what is presented in the vignette above. Students are stuck on problems. They are proving theorems from first principles, and are asked to write proofs outside of class. Moreoever, students are required to present their proofs to the entire class for peer review. Students cannot get through class without having to think about their thinking.
Instructor: "Can you tell us why?
Student: "Well... I looked at these examples and then I thought that this one here doesn't satisfy the second condition."
Instructor: "Very interesting. What do the rest of you think?..." <discussion ensues>
Some questions to get you to think about your students' thinking about their thinking:
- Do your students think about their thinking and how do you know?
- If you are not sure students are thinking deeply about their thinking, what can you do in class and via assignments to encourage this?
- What mathematical tasks and class setup could you use to support thinking about thinking?