*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 [1]. 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” [2].)

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.

**References**

[1] Coyle, Daniel. *The Talent Code: Greatest Isn’t Born, It’s Grown, Here’s how*. Bantam, 2009.

[2] Su, Francis. The Value of Struggle. *MAA FOCUS.* June/July 2016.

I am so happy you made this post. I fundamentally agree with it on every level. As a former IB student I was thankful to learn this necessary skill in high school rather in college where I have seen a lot of my peers learn it from my example. For the uninitiated the IB is the International Baccalaureate program. It is a world wide recognized high school education system which when completed and if you earn the IB diploma you can go to college anywhere in the world and you get automatic Sophomore standing at such prestigious universities such as Harvard. In my IB education I was fortunate enough to take HL or higher level mathematics. A course that’s rigor and intensity I have yet to see an equal of even as I am graduating from university this semester as a mathematics undergraduate. I was by far not the smartest or the most mathematically talented student in the class but in it I learned an extremely valuable less. My teacher for that class is still one of my favorite professors I’ve ever had and by far has been the one who impacted me most profoundly. I would struggle and profusely make mistakes in that class not only because of the material, which as a junior and senior in high school doing calculus 1 and 2, linear algebra, imaginary algebra, statistics, as well as learning about a plethora of other fields of mathematics was difficult for me. But it wasn’t only the difficulty of the course but the speed at which we went through the material I struggled with. But my teacher never once made me feel stupid or inadequate in the class. I would have to frequently go and ask questions about the material I was having trouble grasping and for help with the mistakes I had made and didn’t understand what I had done wrong. But because of that one professor I have succeeded in university. He made me realize one critical thing that has helped me to this day. Not everyone is perfect. Not everyone is Pascal, or Newton, or Einstein. Struggle is a necessary part of life as well as education. Struggle breeds hard work, which breeds understanding, which in turn breeds success. Not everyone is going to get the material right away. But as long as students put the time and effort into understanding the material and get the help they need they can and will succeed. It is from this that I found my motto. Victory from Defeat. From every mistake I have made in my academic career I have learned from it. I have succeeded because I have failed. I learned that making mistakes is critical to the learning process and it’s through those mistakes that you learn how to succeed. It’s is because of this lesson that I have been able to breeze through my higher level mathematics courses with minimal difficulty. I learn from my mistakes and I think it is a critical lesson that all students must learn to succeed. It is because of this that when my peers in my classes are afraid to answer the professor’s questions in fear of failure I raise my hand and give the best answer I have, even if it’s not the exact correct answer, without a moment’s hesitation. Because what’s the worst thing that could happen? I could be wrong? So? Being wrong isn’t the worst thing in the world to happen. As long as you learn from the mistake and are able to understand why you were wrong you learn. And I believe that is extremely important to the learning process. Everyone makes mistakes. No one is perfect. And more importantly some of the most important mathematical and scientific discoveries where found because of mistakes. It’s because of this motto that my favorite mathematics professors in my university, who is the professor for the class I am doing this post for, and has helped me out and made things easy for me in dire times, has said before in class when no one answers his questions, “Jacob you must have an idea you aren’t afraid of mistakes. The rest of you should be more like Jacob. There is no shame in being wrong and making a mistake. Even the greatest of mathematicians have made mistakes. It’s because they learned from their mistakes that they succeeded. Not because of some innate mathematical ability. But because of drive and perseverance.” The will to succeed despite failure is something that has been ingrained in me at an early age. I have always been lucky enough to have at least a few teachers who genuinely care. And it’s because of the push those teachers have given me and the drive they unlocked I had buried inside of me that I have succeed despite the many hardships I have faced in my life. In my opinion the best teachers/professors aren’t ones who pass the most students. But are ones who teach their students it’s okay to make mistakes and help them learn from them. And I will always be eternally grateful for those professors as I imagine your students are grateful they had you as a professor. My point in this long ramble is that learning from one’s mistakes is a critical ability not only in mathematics but in life itself and I commend you for promoting this idea to your students.

Good write-up. I certainly love this website. Thanks!