Focusing on Process and Learning from Our Mistakes

A short while back, Kate linked to an awesome video about learning from mistakes. Well, following my 7th-graders doing an awesome little project writeup for me this week, I thought I'd wrap up the week reviewing some of their common mistakes from last semester's big exam.

This is how I structured it. First (since it had been a while... we hadn't seen equations since December's big exam), I gave them one problem on the board with an answer written at the bottom of the board. I asked for them to figure out the process for showing how to get that answer, and the first ones to show me the clearest work can put them on the board, and I'll choose another person with the correct work to explain what has been written on the board.

Here was my first problem (not an easy one!):

-4(x – 3) + 1 = 5(3 – 2x) + 70
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x = 12

The kids were instantly into it. (They were engaged by the competition aspect.) After two kids had put up two different ways of solving, I chose a normally very insecure kid to go up and explain their work, and she did great!

Then, we did another problem similarly:
4x – 13x = 2(-x + 8) + 19
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x = -5

This time, a lot more kids were able to successfully complete the problem in a short amount of time. Of them, I picked two kids whose work didn't look exactly the same to put their process up on the board. Another normally unconfident kid agreed to go up and explain the work already put up on the board.

After that, we switched gears and I took the board markers and put up three problems, one at a time. I challenged the kids to quietly put up their hands when they can see where a classic mistake exists, and I waited until over half of the class had their hands up to pick a relatively weak student to tell me the answer.

Here was the first one, which many of them got right away:
-2(3x – 5) = 20
-6x - 10 = 20
-6x = 30
x = -5

They really enjoyed it, so I went ahead and put up:
5x – 3 = 3x + 11
8x = 8
x = 1

This time, a juicy discussion ensued. One of my students thought that the mistake was that 3x + 11 doesn't equal 8 but equals 14. Another student said that the second line should be 8x = 14, because the -3 "should become +3 when it goes across the equal sign." (I put it in quotes because it bothers me when kids say that, but if they've already been taught some basic algebra at home, that tends to be their phrasing.) Finally, some kids correctly identified/explained that the second line should have been 2x = 14.

Then, I put up a third problem, this time with two separate mistakes in it. Again, I challenged the class to find both mistakes.
(1/2)(x – 8) = 50
1/2*x - 8 = 50
1/2*x = 58
x = 29

It was so great! They were very excited that they could find so many mistakes.

It was perfect time to transition into Kate's suggested "My Favorite No" activity. We went through three algebra problems, increasingly more difficult each time, and I had kids submit their solutions on little scraps of paper. I wrote down my favorite incorrect problem on the board, and we started by pointing out all the things that person had done correctly, before discussing where they had gone wrong and why. In doing so, we caught: arithmetic error (some student thought -29 - 27 = 56) because they thought that you apply "the integer rules." We also caught the mistake of subtracting 2x from the same side of the equation twice. (I was so happy when the kids said, "You can't do that, because that would throw the equation off-balance!" They are talking like pros.) We also caught the mistake of going from 2.5x = 10 to x = 2.5/10 = 0.25.

It was brilliant! I think the kids had fun, AND I was able to get them to think hard about some common procedural issues ON A FRIDAY AFTERNOON.

When the class ended, I had just gotten them started on a Row Game involving some more basic algebra. It's their (my) first time doing a Row Game, so the concept of comparing answers even though the problems are not the same was a bit confusing to them. We'll have to continue with this Row Game next week, because it's supposed to address some more common procedural problems that I saw on the December exam. The worksheet I made for that is here if you want it. I am excited to continue it next week! Kids were talking to each other about math and trying to figure it out before turning to me for help (even though they were convinced that they could not have made a mistake and the problems could NOT have the same answers). It was really lovely.

So, yay to Kate, and yay for a day of trying new things and working with our conceptual mistakes instead of pretending that they don't exist.