Recently, a question and answer from a math test made the Internet rounds. The question read, “Come up with an equation that is true when x = 7. (Be creative; you can make the equation as simple or as complex as you want.)” The student’s answer was simply “x = 7”; the comment from the teacher, in bright red marker, was “Really?” Someone somewhere in Internetland commented that this lesson had turned out to be a lesson for the teacher.

I laughed out loud when I saw the item because I have been that teacher many times: writing an assessment, trying to think outside the box, and hitting on an idea for a good, nonstandard question—only to be schooled by my students. I do a fair amount of on-the-job learning as a teacher, and much of it comes the hard way.

In contrast, mathematics itself has always come fairly easily to me. This is more of a liability than an asset in my teaching. When my students struggle with concepts, I have to work hard to turn off my intuition and try to see through their eyes. My colleagues would talk about the importance of building students’ confidence, but only when I experienced it myself did I appreciate the difference that confidence building can make.

A couple of years ago, I happened on the website Brilliant.org. Every week, the mathematicians running the site would issue two problem sets containing some rather challenging mathematics problems. I typically found the first problem or two to be fairly straightforward, but the rest genuinely intimidated me. I would scan them and then skip them.

 After a day or two, however, I would return to one of those intimidating problems. More often than not, a little time thinking about it produced a small revelation: “Wait a minute . . . This problem is not nearly as difficult as I thought!” And some number of minutes (or hours) later, out popped the answer.

These revelations did not occur with every problem, mind you—plenty were still beyond my reach—but every time I solved a problem that I initially suspected was too difficult for me, my confidence grew. And as my confidence grew, so did my patience and my perseverance. And as all these grew and reinforced one another, I was able to solve even more problems. The validation was incredibly addictive, and only then did I understand that this was the kind of experience that my students needed.

Unfortunately, building confidence is somewhat of a chicken-and-egg problem. How can we help our students solve challenging problems? By building their confidence! How can they build their confidence? By solving challenging problems! In my next post, I’ll talk about how I’ve tried to break into that cycle and help my students change their beliefs about what they are capable of accomplishing.

Matt Enlow, matt.enlow@danahall.org, preaches the gospel of mathematics at Dana Hall School in Wellesley, Massachusetts. He is a regular contributor of (mostly) original math problems to Brilliant.org and tweets (mostly) mathematical musings at @CmonMattTHINK.