I’ve seen this particular incorrect answer from dozens of students over the last several weeks.
The work for 10 and 15 marbles is incorrect, but it isn’t a mistake. If I label it a mistake, even if I attach a growth mindset message to that label, I damage the student, myself, mathematics, and the relationships between us.
Mistakes are the difference between what I did and what I meant to Do.
For example, I know that words in the middle of a sentence generally aren’t capitalized. I meant to type “do” but I typed “Do.” That was a mistake.
What we’re seeing in the table above, by contrast, is students doing the thing they meant to do!
When I call that table a mistake, what I’m actually saying is that there’s a difference between what the student did and what I meant for the student to do. Instead of seeing the student’s work as a window into her developing ideas about tables and linear patterns, I see it as a mirror of my own thinking.
And it’s a bad mirror of my own thinking. It doesn’t reflect my thinking well at all!
It’s a bad mirror, so I call it a mistake. “Mistakes grow your brain,” I say. “We expect them, respect them, inspect them, and correct them here,” I say. And if we have to label student ideas “mistakes,” maybe those are good messages to attach to that label.
But the vast majority of the work we label “mistakes” is students doing exactly what they meant to do.
We just don’t understand what they meant to do.
Teaching effectively means I need to know what a student knows and what to ask or say to help her develop that knowledge. Calling her ideas a mistake transforms them from a window into her knowledge into a mirror of my own, and I am instantly less effective.
Our students offer us windows and we exchange them for mirrors.
The next time you see an answer that is incorrect, don’t remind yourself about the right way to talk about a mistake. It probably isn’t a mistake.
Ask yourself instead, “What question did this student answer correctly? What aspects of her thinking can I see through this window? Why would I want a mirror when this window is so much more interesting?”