This blog post was produced by our talented postdoctoral fellow Dr. Christina Barbieri, who is an expert in the teaching and learning of mathematics. Kathy Hirsh-Pasek is a co-author on the post.
Walk into any middle school and you'll see hallways plastered with motivational posters that read, "Mistakes are proof that you are trying!" or "Life is all about making mistakes & learning from them." Now take a step outside of schools and into social media. You see a meme of a disheveled Albert Einstein, with a quote, "A person who never made a mistake never tried anything new." Or consider what Thomas Edison said to a reporter who challenged him, "I have not failed. I've just found 10,000 ways that won't work." But are we practicing what we preach about learning from our mistakes?
In many US math classrooms, reflecting upon an error is often discouraged. Why? Teachers are concerned that having students think about a mistake could accidentally encourage students to make that mistake again. However, in Japanese and Chinese classrooms, errors are frequently discussed and even seen as an important part of the learning process. In these countries with a history of high math achievement, errors are embraced! Understanding why something is incorrect and how to fix it is much more common than here in our own classrooms.
Let's go to Germany next. Researchers from the University of Augsburg find that if middle school students think that their teachers and classmates will judge them harshly for making errors, they are more likely to back off when working on difficult math problems. Avoiding errors, instead of using them to learn, makes students timid. Why lose face among my teachers and peers? Why would I keep trying? And if I'm not willing to try, why would I bother thinking about why something was incorrect in the first place? It's just wrong, and that's all that matters...right?
But can students actually learn from errors? Several studies conducted in real-life math classrooms around the US show that learning from errors happens all the time. Having students explain common mistakes and compare these mistakes to the correct solution helps students learn the material more deeply. Many studies even show that error reflection is much more beneficial than simply practicing math problems. When students are given the opportunity to consider why something is incorrect, they may be more likely to persevere and make sense of confusing material.
Reflecting on errors may be especially important when starting to learn more complicated subjects such as algebra. Without recognizing that making mistakes is inevitable in learning something new - and even a good thing - struggling algebra students might think, "I'm just not good at math" and give up! Unfortunately, some educators say things like this out loud in classrooms. Students are then okay to think, "Oh, even my teacher thinks I'm just not good at math." In countries with many math whizzes like China and Japan, success in math is seen as a product of hard work. American students need to understand the importance of hard work when learning challenging material. They know this in the physical realm: Who has ever ridden off into the sunset the first time on a bike? Now we just have to get our students to think this way about math.
But should we even care about whether students are learning more advanced mathematics? After all, how many of them will actually go on to become mathematicians? Consider that Algebra I is called a gatekeeper course for later math. Students who don't succeed in algebra often can't take higher level math and science courses. This reduces their chances of being admitted into college, regardless of their career goals. Do we really want to snuff out children's opportunities before they even leave high school and know what they want to do in life?
But move beyond math and science for a moment. Do we want to raise children to become adults who avoid challenging situations for fear of making mistakes? Did you give up on doing the laundry after you shrunk your favorite wool sweater? Did you stop offering ideas at the office when your last one didn't pan out? If learning from errors is not only possible but also a particularly effective learning tool, why not use this tool more often in schools? Many of our children could come to see the world through Edison's eyes, "Many of life's failures are people who did not realize how close they were to success when they gave up."