By Allison Henrich, Associate Professor and Chair of the Department of Mathematics, Seattle University
“I am so glad you made that mistake,” I’ve come to realize, is one of the most important things I say to my students.
When I first started using inquiry-based learning (IBL) teaching methods, I had a tough time creating an atmosphere where students felt comfortable getting up in front of class and presenting their work. It is a natural human instinct to not want to expose your weaknesses in front of others. Making a mistake while presenting the solution to a problem at the board is a huge potential source of embarrassment and shame, and hence also anxiety. So how do we—as educators who understand the critical importance in the learning process of making and learning from mistakes—diminish the fear of public failure in our students? For me, the answer involves persistent encouragement. It also relies on setting the right tone on the first day of class.
To prepare my students on Day One of class, I talk about the importance of making and learning from mistakes. I often refer to one of my favorite books on this subject, The Talent Code by Daniel Coyle . Coyle has studied several hotbeds of “genius,” places where an unreasonable number of virtuosos—e.g., world-famous violinists, baseball players, and writers of fiction—emerge. He is interested in discovering just how people like Charlotte Brontë, Pelé, and Michelangelo learn to perform at the top of their fields. The answer involves a simple idea: talented people are those who have made far more mistakes than others and who have deliberately learned from those mistakes. For my students, the takeaway is that the most accomplished people have made many more mistakes than the average person. Consequently, it is of high value for us to make our mistakes public and discover how to correct them together. (As a side note, Francis Su employs the same strategy in his article “The Value of Struggle” .)
After the first day of class, whether I am teaching Quantitative Reasoning, Calculus, or a more advanced course such as Introduction to Knot Theory, nearly every class period begins with presentations of homework problems by student volunteers. Students have homework due each day, and they are required to present problems a certain number of times during the term. The number of problems we do depends on how long the class period is, how complex the problems are, and what I need to teach in the remainder of class. In a course like Introduction to Knot Theory, we might spend 45 minutes or an hour on student presentations, while we will spend 20-30 minutes on calculus homework presentations in an 85-minute class period. This general structure could be modified to fit shorter class periods or weekly recitation sections at universities with larger lecture courses. For instance, we used to teach calculus classes four days a week in 50-minute blocks at Seattle University. Within this structure, I had a weekly “Problem Day” for my calculus classes instead of having daily student presentations of homework. After students volunteer to present problems at the board on a typical class day, all students who are chosen to present simultaneously write up problem solutions while their classmates review the homework or work on another activity. Once all solutions have been written up, we reconvene; one by one, students come to the board to walk us through their solution. This is where supportive facilitation becomes critical.
Encouraging students to make mistakes in the abstract—as I do one Day One—is one thing, but helping students accept their mistakes in front of class is quite another. This is where my new catch phrase comes in. Let’s say, for example, a student is computing the derivative of \(y=x^2\sin x\) at the board and writes \(y’=2x\cos x\). I might say, “I am so glad you made that mistake! You’ve just made one of the most common mistakes I’ve seen on this type of problem, so it’s worth us spending some time talking about. Can anyone point out what the mistake is?” If someone in the class comments that the presenter should have used the Product Rule, I might follow up with, “That’s a good idea. How can we see that this function is a product? Let’s work together to break the problem down into pieces.” Going forward, I facilitate the process of the class coming up with their collective correction of the mistake. Collaboratively working to correct mistakes like this tends to help students observe more subtle differences between different types of problems while building a more sophisticated mental problem-solving framework.
Making and correcting mistakes together can also help address more basic misconceptions. Suppose a student—let’s call them Riley—writes, in the middle of a calculus problem, a line like the following.
\(1/(x+x^2) = 1/x + 1/x^2\)
This mistake will most likely lead to an incorrect final answer. Many of the presenter’s classmates will discover the final answer is wrong, and some will even be able to pinpoint where the computation went awry. How would I address this? Once a classmate has identified the problem, I might say, “Riley, I’m so glad you made that mistake! This is one of the most common algebraic mistakes students make in calculus—I’m willing to bet others in the class made this same exact mistake, so it’ll be really helpful for us to talk about it together. This is a question for anyone in the class: How can we prove that this equality doesn’t hold, in general?” Suppose a student, Dana, in the audience suggests we try plugging in some numbers to see what happens. I’d follow up with, “Riley, could you be a scribe for this part of the discussion? Please write up Dana’s suggestion beside your work. Dana, can you tell Riley exactly what to write?” Once we’ve cleared up the confusion with Riley’s algebra, I might ask them to work through the rest of their problem again at the board, fixing their work accordingly. On the other hand, if Riley appears to be too shaken or confused to fix the rest of the problem or if the actual problem was much more complex than the one that resulted from the algebraic error, I might ask the class to collectively help Riley figure out what to write each step of the way. A third option I frequently use is the “phone a friend” option. I could see if Riley wants to “phone a friend” in the class to dictate a correct answer.
Mistakes can be common in class presentations, but I occasionally have a class that is so risk-averse that very few people offer to present their work unless they know it’s perfect. If I have too many correct solutions presented, but I know some in the class are struggling, I might follow up with a comment like: “That was perfect! Too bad there were no mistakes in your work for us to learn more from. I’d like to hear from someone who tried a method for solving this problem that didn’t work out so well. Would anyone be willing to share something they tried with the class?” At this point, someone may come forward with another (incorrect, or partially correct) way to attempt the problem. If nobody comes forward, I could offer a common wrong way to do the problem and ask my students to identify the misunderstanding revealed by my “solution.” I might even tell a little white lie and say something like, “When I first learned this concept, I had a lot of trouble understanding it. I made the following mistake all the time before I figured out why I was confused.” Alternatively, I could mention, “The last time I taught this class, someone made the following mistake. What’s wrong with this approach to solving this problem?”
Now, let’s say one of my students has just presented a problem at the board. Perhaps they made a mistake, or perhaps they did everything perfectly. What happens next? I will ask the class, “Any questions, comments, or compliments?” The request for compliments is one of the most important parts of this solicitation of feedback. It is so important that, during the first several weeks of class, I make my students give each presenter at least one compliment. Some of the best compliments I’ve heard from students follow some of the worst presentations. For instance, after a disastrous presentation where the presenter appeared clueless and needed their peers to help them complete all parts of a problem, a student of mine once observed, “That took a lot of guts to get up there and make mistakes. I thought you did a great job fixing the solution and taking constructive criticism from us!” If nobody offers up such a supportive compliment after a bad presentation, I might give this feedback myself to publicly recognize the presenter’s courage. What’s more, if a student appears shaken by the experience of messing up so thoroughly, I’ll follow up again after class, reinforcing my appreciation for their bravery. Over time, this strategy helps build a supportive classroom environment.
Looking back on how my classes have evolved, I can see that it is difficult to convince students to be vulnerable in a math class without the three following elements:
(1) setting the stage by sharing my expectations of students making mistakes and being clear about the reasons for these expectations,
(2) encouraging students to help each other come to the right answer while recognizing the benefits of making specific mistakes, and
(3) acknowledging students’ willingness to make mistakes both publicly and privately.
We’ve been primarily focused on how to encourage students to make mistakes, but let’s turn our attention to why it might be important in our math classes. One thing that I found to be particularly striking when I started teaching this way was my students’ exam performance. I typically ask a mixture of conceptual and computational questions on exams. I was surprised to see how much more sophisticated students’ responses were to conceptual questions in courses where I spent a great deal of class time on student presentations. At first, this was surprising to me since we spent quite a lot of time in class working through computational problems. The more I reflected on this phenomenon, though, the more it made sense. The repairing of computational mistakes in class often led to a discussion of the more conceptual mathematics underlying the computations. What’s more, these discussions were sparked by students grappling with problems that they cared about—problems they had spent time outside of class trying to solve—and not simply problems they had just been introduced to in the course of a lecture. Discussion that takes place during a homework presentation session seems to stick with students in a way that a “discussion” (where the instructor is doing much of the talking) during a lecture does not.
There are myriad other benefits I’ve observed, including development of a tight-knit classroom community, increased student self-confidence, and more engaged student participation in all aspects of class. In short, I’m convinced. I’m all in. The benefits of teaching this way far outweigh the costs of redistributing precious class time, making room for students to publicly make and collaboratively fix their delightful mathematical mistakes.
 Coyle, Daniel. The Talent Code: Greatest Isn’t Born, It’s Grown, Here’s how. Bantam, 2009.
 Su, Francis. The Value of Struggle. MAA FOCUS. June/July 2016.