Metacognition seems to be all the rage in education these days and I'm thinking it is not just a passing fad but rather a real key to making a positive difference in the lives of our students. My guess is that metacognition has always been part of teaching but at present I am interested in getting a handle on it and seeing if I can use it in a more systematic fashion for the benefit of my students. The definition I've gotten for it is that metacognition is thinking about thinking. (That wording, "thinking about thinking" appears in Really Raising Standards by Philip Ady and Michael Shayer.) Knowing the definition surely isn't enough to go on as far as figuring out how to implement it (or bump up the level of implementation if it is already being implemented) so I figure I've got some reading to do (hopefully, some of it right here on this list) but I've actually started my study by, rather than reading about it, trying to engage in it in a conscious fashion. Here's an example:
When John Denker posted the first version of his web page <http://www.av8n.com/physics/understanding-parallel-r.htm> I sent him an off-list message with among other things a "correction" to what are, at the time of this writing, equations 28 and 29, namely the shorthand expressions for DeMorgan's theorem and corresponding relations among the operators discussed in the document. (Spoiler alert: I have the word "correction" in quotes because I was wrong, the expression was correct the way he wrote it in the first place.) I'll make my point here in terms of the first shorthand expression:
−(·) maps to + [28a]
which is given as a shorthand expression corresponding to
− (A · B) = −A + −B [26a]
John made some edits to the document but he didn't make the "correction" in question (I thought he needed the operator on the right, in the shorthand expression, to address the fact that it wasn't the operands that get ORed on the right-hand side of the longhand expression, but rather it is the negation of one operand that is ORed with the negation of the other operand). So I sent him another message, more specific in that it addressed only that point, and more strongly worded--I remember using the word "wrong", and including what I thought was a strong argument). He said nope, the way he had it said what he meant for it to say and what he meant for it to say was what it was supposed to say but that he had added some more text to the document to make it clearer that it was correct.
I still wasn't convinced. I sent him a completely different argument that I thought was just as strong as the first one. He said the way he had it was still correct but that he had added further words of clarification.
I finally got it. Then I explained it back to him in my own words and applied it to a couple of different cases.
Then it was time to metacognate on it. Here's what I arrived at. My problem was that I was making an assumption, an underlying unstated assumption that was wrong. The assumption was that the compound operator −(•) on the left of 26a was:
AND two variables and NOT the result.
This is an incorrect assumption.
So the puzzle-solving strategy that I am reminded of by thinking about my thinking in this case (where the puzzle was that a thoughtful person is telling me something is right when my own reasoning is telling me it is wrong) is:
1) Look for underlying unstated assumptions and if you find them, test them.
2) If looking in one place is not getting you anywhere, look someplace else. (I was looking at the right side of the expression, I needed to look at the left.)
So the question is, is this an example of what metacognition is all about? Also, if you can, please provide more explanation of metacognition as well as some examples.