By NCTM President Linda M. Gojak
NCTM Summing Up, January 8, 2013

 “It has been a long trip,” said Milo, climbing onto the couch where the princesses sat; “but we would have been here much sooner if I hadn’t made so many mistakes. I’m afraid it’s all my fault.”

“You must never feel badly about making mistakes,” explained Reason quietly, “as long as you take the trouble to learn from them. For you often learn more by being wrong for the right reasons than you do by being right for the wrong reasons.”  
―  Norton Juster,  The Phantom Tollbooth

How often do our students consider their mistakes to be signs of failure? How many students, as well as parents, believe that the goal of learning mathematics is solely to get the correct answer? How often, on arriving at an answer, do students believe their thinking about the problem is finished? In The Phantom Tollbooth, author Norton Juster offers a valuable contrasting perspective:

Just as Reason explains to Milo, students can use their mistakes as learning opportunities to develop a deeper understanding of the mathematics that they are doing—although we may need to help them along the way.

Can you recall an experience when an important mistake helped you to understand an idea or a skill that was not clear? For me, it was learning to sail. I took a sailing class in the adult education program at our local high school. No matter how many times I read the explanation, I could not understand the difference between a “tack” and a “jibe”—two methods of turning a boat—one much safer than the other. Later, while sailing on Teachers’ Pet, a 22-foot sloop, a heavy wind came up. We needed to head back to the dock. I called out the warning “hard-a lee” as I prepared to tack. Before I realized my mistake, the wind hurled the boom and the mainsail across the cockpit. It was too late. I had pushed the tiller in the wrong direction. Luckily, no one was hurt. Only after this hands-on experience did I clearly understand the sailing phrase “turning on the wind,” the description of tacking that I had read so many times in the textbook. I never confused a jibe with a tack again! I often share this story with my students after telling them they have made an important mistake. I know they don’t understand all of the nuances of sailing, but I suspect they know I had learned something important by making that mistake many years ago.

Helping students to learn from their mathematical mistakes can give us insight into their misconceptions and, depending on our instructional reactions, can enable them to develop deeper understanding of the mathematics they are learning. Meaningful assistance encompasses more than fixing a careless error related to poor study habits, such as misreading directions, miscopying a numeral, or forgetting a sign. Although we can easily and explicitly address careless errors in our instruction, it can be more challenging—but potentially much more rewarding—to address errors that fall into the special category of productive mistakes. These are the mistakes that have the potential to promote rich learning.

How we respond to productive errors can encourage or discourage student thinking and learning. As you reflect on your own instructional practice, which of the following describes your response to productive errors?

Do you—

• assign a smaller number of carefully selected problems, with the expectation of having students devote more time to reasoning and making sense, rather than requiring them to go through the full litany of exercises found in the text?
• ask students to explain their reasoning about whether an answer is correct or incorrect?
• take the time to analyze students’ errors, both written and oral, to determine misconceptions that students might have and how you can address those misconceptions?
• follow productive errors with probing questions that offer students different approaches for reflecting on their thinking, or do you simply mark errors as wrong and demonstrate the “right way” to do the tasks?
• provide students with mathematical tasks that—
• allocate sufficient class time to discuss students’ strategies and thinking?
• explicitly work to persuade students and parents that mathematics is more than getting the correct answers?

As you strive to implement standards designed to ensure that students master mathematical practices and mathematical content—whether they are the Common Core or your own state or provincial standards—remember that how you respond to“good” mistakes has the potential to discourage students or to help them become more confident in their ability to do mathematics. This new confidence can transform student attitudes toward learning mathematics.

Along the way, you can share some  examples of mistakes that have changed the world. Some of the more notable “mistakes” include Coca-Cola, sticky notes, rubber tires, chocolate chip cookies, and penicillin. You never know what the next great mistake may be!