Response to "Learning from learners: robust counterarguments in fifth graders’ talk about reasoning and proving”

In Vicki Zack’s article titled, “Learning from learners: robust counterarguments in fifth graders’ talk about reasoning and proving” a group of fifth graders are given a problem that has been incorrectly solved by someone their age.  The students were asked to formulate counterarguments to the incorrect solution first by themselves by writing in a math log, then with a partner, then eventually with 12 other classmates (there were 26 students in the class, so the teacher split the class in half and within each half of 13 students, each student had a partner to share ideas with before their shared with all 13 students in their half).  The problem they were given was:

Task 1: Find all the squares [in a four by four grid given as a figure]. Can you prove you have found them all?

Task 2: What if ... this was a 5 by 5 square? How many squares would you have? Extensions were subsequently posed.

Task 3: What if this was a 10 by 10 square? What if this was a 60 by 60 square? How many squares would there be?

The incorrect solution was:

Imagine that two of your classmates, Ted and Ross, came up with the following solution for the 60by60: The answer for the 10by10 grid is 385 squares. So take the answer for the 10by 10 square (385) and 10 x 6 = 60 so multiply 385 x 6 = 2310 and you have the answer for the 60by60. What would you say?

There were about six different counterarguments the students came up with, which I will not try to describe here.  What I will do is list a couple of ideas that I thought were really interesting in reading this article.

The first idea was:

A look at the counterargument helps me as a teacher to come to a better understanding both of the mathematics, and of the children's understanding of the mathematics.

This is something that I think we often times can forget as educators.  Obviously we try to meet students where they are at with their mathematical knowledge, and help them grow from there, but it is so easy for us to impose our way of thinking onto our students.  I really like the idea of using counterarguments as a way of discovering how students understand different mathematical concepts.

The second idea I thought was interesting was:

When Adele keeps saying that the Ted-Ross strategy sounds good, stating: "I think it's right but I don't know how it works," Maggie (who has just finished presenting CA #1 to her group of five) insists: "Then you have no reason to think it's right."

It is amazing how often we just teach concepts to students and tell them that it is correct and the one way to do it, but we don’t let them necessarily figure that out for themselves.  We tend to make “Adele’s” who just say things like “I don’t know why this works, but my teacher said it works this way.  So this is what I am going to do” instead of making “Maggie’s” who ask questions like “Why does it work that way?  How can we be sure?”