I am on Robert Kaplinsky's email list. Recently, I received a lovely email from him on how he gets his students to explain their thinking and he does it without using the words "Explain your thinking."
I sometimes believe when we use those three words cause our students to freeze. It is like the time I was attempting to learn Inupiaq, the language they speak in parts of Northern Alaska, I was bad at it. At one point she asked me something. I had no idea what she said so I answered "yes" complete with a stark look of fright on my face.
He chooses problems like XX + XX and use the digits 1 through 9 only once to find the largest number. He discovered when he asked them how they got their answer or explain their thinking, he'd usually get something like I added 27 + 98 to get 125. They did not explain their thought process, only what they came up with.
One day he asked them to convince him as to why their answer is right or wrong. This directive requires students to provide more explanation than just what they got. They might say their neighbor added a couple of numbers which was lower than theirs. This is one step more towards a good explanation.
I wonder if we ever take time to teach our students what is expected when they explain their thinking. Do they know what makes up an explanation. Its possible the reason we get such a simple answer is because we have not taken the time to help students develop a method for explaining their answers.
I like the idea of asking a student to convince the teacher their answer is right. It means they have to provide a reason to support their answer. Even comparing their answer to another is a good start because comparing items is a tactic used in convincing people.
We might want to take a few minutes to discuss what is involved in convincing people. After all, politicians are always trying to convince people to vote for them, movie trailers are used to convince people to pay money to see the movie, ads are used to convince people to buy a product. We see this happening all around us. So how is it applied to mathematics?
That we can discuss with our students to help them learn the art of persuasion.