When I was in high school, there was nothing I hated more than math. And in math class, there was nothing I hated more than showing my work. It felt like a tedious chore with for no real purpose—a hoop somebody wanted me to jump through. Math was not open to poetic interpretation; the answer was either right or wrong. And no one ever offered me partial credit for getting an answer partially correct, or for demonstrating an interesting, if flawed, process. As far as I could tell, nobody even looked at my work. So why did I have to show it?
Today, if my high-school aged son is to be believed, the situation doesn’t seem much improved. And that’s a shame. The Common Core State Standards in mathematics--and the new math standards in many non-CCSS states—make clear that reasoning, arguing, and critiquing are vital skills within mathematics, and that teachers should be instilling in students a devotion to precision and perseverance. None of those mathematical habits of mind reveal themselves in the calculation sitting to the right of an equal sign and a question mark. They reveal themselves in the process—in the work a student does along the way to an answer. There are real and strong reasons to show your work. But if a teacher doesn’t understand those reasons and make use of them, the practice can easily remain an exercise in hoop-jumping.
As Elizabeth Green explains in her excellent book, Building a Better Teacher, the right answer is often the least interesting piece of data available in a classroom. It is the wrong answers that give teachers real information about how students are thinking and where they may be going astray. And while a seasoned veteran may be able to intuit a problem from nothing more than a final answer, most of us will have an easier time understanding what’s going on by making the student’s thought process visible. In written assignments, that’s precisely where have students show their work is important.
What about in the heat of a class discussion? Students can show their work here, as well, by talking about their process. Asking “what did you do?” is far more valuable (and interesting) than asking “what did you get?” It allows the class to explore multiple problem-solving strategies and assess what works and what doesn’t. It helps students become comfortable and fluent in “talking math.” And it can make error an interesting topic of conversation rather than something to be embarrassed about and try to hide from.
There’s a lot of talk about Growth Mindset in the education world. Well, a vital component of Growth Mindset is the willingness to fail, and try, and fail again—the understanding that failure is a natural part of the process of learning, a necessary step on the way to success. Athletes understand this perfectly well. A baseball player with a batting average of .300 is considered pretty good; a player who bats .400 is considered outstanding. Outstanding—with a 40% success rate! In basketball, perhaps the greatest player of my lifetime, Michael Jordan, said this: “I’ve missed more than 9,000 shots in my career. I’ve lost almost 300 games. Twenty-six times, I’ve been trusted to take the game-winning shot, and missed. I’ve failed over and over and over again in my life. And that is why I succeed.” He doesn’t say that he succeeded in spite of his failures; he says he succeeded because of them. Because he was willing to put himself out there, and make a mistake, and learn from the mistake, and press on.
We need to find ways to model this more positive view of failure for our students. Instead of focusing solely on end-results, we need to show students the long road it took to reach that goal. How many drafts did it take for an author to reach the final manuscript? How many publishers rejected the book before it was accepted? How many experiments did the inventor go through before hitting on the right combination of ingredients? What challenges and setbacks did a historical figure endure before doing the great work for which they are remembered?
And what about us? Are we willing to model failure personally in front of our students? Can we admit to them when we’re wrong, or when we don’t know the answer, or when an activity we planned didn’t go as well as we had hoped? Can we show them that there’s no shame in falling down, as long as we get back up again, brush the dust off ourselves, and push on?