If the problem was done incorrectly, it involved multiplying 12 by 4. For a number of my freshmen this is NOT a number fact they own. And it is something that they would most likely be wanting a calculator to do....or at least use pencil and paper.
The problem asked them to not only find where the student went wrong, but to correct it as well. We had a multitude of answers (which freaked me out considering how many of these we have done, but it was an excellent learning opportunity). After we talked about it for a while and decided which answer was the correct one, I asked if there were some people who wanted to share where they got derailed. I talked about how mistakes are when our brains grow. We get the chance to go "OH!!! I see what I did, but this is what I was thinking..." and how we ALL learn from that.
There was one young woman who was secure enough to put it out there. In the process of explaining what she did, she mentioned that she multiplied 12 by 4 and got 48. She is not one with real strong number skills so I wondered how she had done that so quickly. After we were done talking about the errors and how we were thinking, I asked her to share how she multiplied those numbers.
It involved halving!! I nearly fell over with excitement (but contained myself very well, I think).
Hers is strategy #1. "When there are even numbers, I like to cut them in half until they get to where I can multiply in my head. I cut 4 in half to get 2, and I know that 2 times 12 is 24. I did that twice because I broke the 4 into 2 pieces. Then I just added them together."
When I asked for other strategies, hands went up! One young man gave us strategy #2:
"I just did 4 times 11 because that is so easy, and then added one more 4."
"Why did you add one more 4?"
"To make 12 fours. And so 4 times 11 is 44 and 4 more is 48."
Then another young lady shared strategy #3.
"I multiplied 4 by 1 and got 4, then 4 by 2 and got 8."
When I wrote it the same way I wrote the first one, I said "and that gives us.....?" She could "see" that the way I wrote it was going to make 12, so she asked if she could think a minute. Then she and her partner came up with the strategy #4. "Oh, the 4 is in the tens place, so it must be 4 times 10 to make 40, then 4 times 2 to make 8 for 48"
Honest: this REALLY happened today....and I didn't plan it. I told them I loved them.