When Errors Lead To Understanding

I gave a test on equations and inequalities recently. From earlier posts, you may remember I am not a huge fan of tests. Certainly I understand that this is a controversial opinion, but testing seems arbitrary to me. We decide that by a certain date students should master material we teach them. The truth is that some students will master concepts by a specific date and some might not be there yet. Regardless, instead of simply offering support and troubleshooting without judgment, grades speak the words teachers do not wish to say. Well, at least teachers like me don’t wish to say because I know they are often statements riddled with falsehoods.

On a recent test, my students did not get the “right” answers at a percentage that assured me that they mastered the material. Due to a curriculum calendar, I knew I should move on with the content, but I just couldn’t. I scored each question as either right or wrong, took copious notes to the errors that abounded and quietly reflected. The truth was, most students clearly understood the process and concepts of equations and inequalities. They dropped a negative sign, they accidentally performed the same operation in one step, but correctly used the inverse operation in others. They made sense of the story problems and set up tables to organize their information, they translated the English to Algebra, but then, forgot to distribute a factor in their solving. In other words, what I saw was clear understanding of concepts marred by precision errors.

How do you get students motivated to improve precision?  How do you even make them aware that they are making the errors? There is an excellent reason precision is one of the mathematical practice standards we emphasize.  I remembered reading a post by Andrew Stadel about how much students love to find other people’s mistakes and how valuable it is in their learning. In addition, there was a wonderful Teaching Channel clip my entire math department watched (and then I re-watched multiple times) that highlighted a math teacher showcasing her favorite mistakes to her students so that they could all learn from them. I adopted this practice long ago, but after these test results thought, I need to do more.

Since I had corrected the tests and left my error notes on my desk, I decided to get to my classroom at around 6 a.m. the following day. I recreated similar problems completed with the common errors. I told students about the precision errors on their tests and that I knew were not a reflection of their total understanding. Their task for the day was to prove me right. After distributing the common error sheets, they were challenged to work together to discover what was wrong on each question. I circulated around the room and students pulled me over to say, “Oh my gosh, I do this too. I think I did this on my test.” I also heard, “I feel so sorry for this student, look at all the great understanding all the way through this problem and to get it wrong at the last step, ugh, heartbreaking.” Exactly my sentiments, I thought.

After this exercise, I saw my class again during a rotation period and gave them back their tests. There were no notes on their tests, just circles of wrong answers. Students were challenged to now find their errors.

The results:  Amazing. Every single student was able to find and correct mistakes without any specific feedback on the individual tests.

The next day, I asked the students what they thought about the exercise. The response was overwhelming. Every single student raised a hand when asked if they found it helpful. Students were speaking over each to explain why it had been such a productive exercise.  One in particular mentioned how much easier it is to find someone else’s mistakes than their own. Another talked about how when you can find an error, that means you truly understand the math enough and it strengthens understanding (seriously, those were her words). It was student comment after comment that made me say, gee, I should do this type of exercise more often.

The question remaining is, how often?


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