In a recent blog, Dan Meyer offered the following example of a table with answers filled in.

The question sheet may have looked something like this:

Marbles Water Level
 0

609

1 626
2 643
3 ?
10 ?
15 ?

The students’ answers, and Dan Meyer says that dozens of students filled it in the way illustrated, show that they got the right and the wrong answer at the same time. Clearly, the point is to understand the difference (17) and apply it to the rest of the table.

Marbles Water Level Student   Correct
0 609      
1 626      
2 643   the students  
3 ? 660 added 17 662
10 ? 677 added 17 779
15 ? 694 added 17 864

The students’ answers are wrong but, Dan Meyer asserts, are not a mistake. They didn’t make a calculating error, or put in the wrong digits by accident / mistake, or reverse the numbers. They actually made a correct calculation and intended to put in the numbers they put in. They meant to put those numbers in and they did. They are not bad mathematicians.

Why might this be important? Dan Meyer says that, it is a mistake, the student must learn from it but the teacher doesn’t have to. “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.”

That’s a pretty dramatic statement! A dictionary definition of mistake is “an action or judgment that is misguided or wrong.” But point of view matters here. From the teacher’s point of view it is a mistake. From the students’ point of view, they did what they intended to do (add 17 each time) and they got the ‘right’ answer. Dan Meyer’s point seems to be that if we label their action, it is more difficult to help them and for them to help themselves.

Psychology Today says that mistakes paralyze us, make us fearful of committing to an action, persuade us to try to avoid mistakes. It also says that mistakes are necessary in order for us to learn. While the first sentence is borne out in everyone’s experience, the second sentence is more difficult. Labeling the answer a mistake makes the interaction between teacher and student transactional and extrinsic – Dan Meyer puts it this way: “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.”

We know that motivation is key to success in all subjects, and maybe even more so in mathematics. If success is possible, I will try harder. Using a student’s answer to understand their thinking and helping them to move forward in their understanding is more motivational than telling they made a mistake and to fix it.

For example, the response to the students might be to ask them to explain what they were thinking as they calculated the answer. Then to applaud their thinking process. Then to ask whether their thinking process aligned with the information given. Eric Mazur shows how peer conversation transforms learning without labels.

The idea that failure is a precursor to success has many problems. Dweck herself said about the growth mind-set that it is “developed over time through learning, mentoring, hard work, good strategies” – not a word about failure there. But failure is for sure a demotivator. Craig Barton quotes one of his students (How I Wish I’d Taught Maths p. 80): “it’s kind of hard to have a growth mind-set when I keep doing shit on tests, sir!”

Approaching our students with an intent to understand them and help them progress on their journey is not helped with labeling. Whatever discipline we teach in, or whatever grade we teach, teaching the child through understanding the child’s own frame of reference is going to benefit all students. Labeling results in winners and losers.

Let’s finish with Dan Meyer: 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?”

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