Karen Wootton, Director of Curriculum & Assessment, karenwootton@cpm.org
Last fall, at the NCTM Regional Conference in Seattle, I attended Steve Leinwand’s session on quality lessons. During his presentation, he made a comment that, due to my bad memory, I cannot remember, but prompted me to record this in my notes: “Are bad conversations worth as much as good conversations? Mistakes!” After the conference when I was reviewing my notes, I realized I needed to know more about what Dr. Leinwand had said. In particular, I had a pressing need to know since I was (and still am) in the process of reading feedback on the pilot course developed to support students who might be struggling in the Core Connections, Course 3 class. Piloting teachers had expressed concern over students not always getting the right answers. A few teachers expressed their frustration over students working for 45 minutes on a problem and not being “successful.”
Personally, I would be celebrating students who would work 45 minutes on math, particularly when they are not arriving at the “right” answer! Talk about perseverance! But while I might be thrilled with this, how valuable is it for the students? I wrote Dr. Leinwand, about this. Specifically, I asked if it was worthwhile for students to spend a year in a support course, if they never mastered anything.
Dr. Leinwand replied to my email, saying he thought it worthwhile to be working on math whether or not the students were getting the right answers. He said “I base my approach on two views: 1. That good math begins with answers – often a wrong answer – and the key is justifying these answers, and 2. That mistakes are to be expected and honored because often, more is learned from mistakes than from correct answers.”
I was still somewhat hesitant to embrace this completely, and worried that I might be considered the math equivalent of Harold Hill, the con man from The Music Man. If you are unaware of the plot of this musical, Professor Hill claims that by using the Think method, anyone can learn to play a musical instrument. He sells the instruments to all the families of the children in the small Iowa town, claiming they will learn to play by thinking about the music. When I expressed this concern to Dr. Leinwand, he replied “Yes, Harold Hill was a con man and usually just telling someone to think doesn’t work, but getting students to actually think productively is 3/4 of the battle. And yes,… the struggle [is] essential.”
He went on to say “I often think about writing as a way to think about math. There is never an on-off switch where suddenly you can write, but with the right experiences your writing always gets better. Same with math.“
I do not think this means we should be content when students have incorrect answers, but rather we have to help them understand WHY the answers are incorrect, and then guide them to form correct ideas. These “bad” conversations are the rough draft talk that leads to correct thinking. And by going through the process, having “bad” conversations, or conversations that contain misunderstandings and incorrect pieces, students make sense of the mathematics.