The Joy and Purpose of Challenge

In my first decade at Dana Hall, I had the pleasure of coaching our squash teams. I fondly remember a conversation with one scholar-athlete who was lamenting the prospect of playing a formidable opponent: “They are so good. We will never beat them, and then we won’t be undefeated anymore.” While I found the words to assure her that there is more to the game than an unblemished record, the thought running through my head was, the day we are undefeated is the day we are national champions. While that would be a tall order for our young program, I knew that to improve as squash players, we needed to play tough teams. We learned the most when we faced opponents who were just a little bit better than us. We could earn some victories but remain hungry to learn the next set of skills. Conversely, the matches I grew to dislike were those against much weaker teams, where we were rewarded with wins despite playing without focus, direction, or determination. Over time, the Dana Hall squash players began to recognize that, especially when playing opponents whom they knew they could easily beat, they struggled to make good choices and stay focused. This realization fostered a group of athletes who, for many years, longed to play as many tough matches as possible.

This same philosophy applies to learning just about anything: make it too easy, and apathy and boredom take over the experience; make it too difficult, and the result is anxiety and lost opportunities to learn. As I plan my math classes each week, I eagerly search for just the right balance of “spicy” and “basic” problems. This process looks very different depending on the class and what else is going on in my students’ lives.

In my Middle School Geometry class, we start early in the year by presenting more challenging problems as whole-class exercises. We provide ample time for students to sit—however uncomfortably—with the feeling of not knowing, in the hopes that it will transform into wonder and curiosity. As the year progresses, we introduce “spicy” problems to our assessments. They often begin as bonus questions but eventually make their way into the tests with small point values. All the while, we explicitly discuss the joy and wonder that challenging problems can spark.

Early in Trimester II, I introduced a complex problem set. We started by establishing known definitions and theorems, forming a framework upon which students could build their understanding. Each night for homework, students would productively struggle for 30 minutes. Then, each day in class, they collaborated with one or two peers. This solo work, followed by group discussion, allowed students to spend time wondering and searching for a pathway through the problem before articulating their ideas and brainstorming alternate solutions. The result was an experience of deep thinking, discovery, resolution, and understanding. The students’ smiles showed genuine pride when they saw the complexity of their own work on the summary test at the end of the unit.

I recently asked the class what they think of “spicy” problems in comparison to “cookie-cutter” ones. One student wrote:

“When spicy problems are presented, I normally feel a bit nervous at the beginning if it is something that we have not covered in class. Ultimately, though, I feel excited at the prospect of succeeding by solving the problem. Throughout the year, as we have been asked to solve many difficult problems, I have realized more that the process of getting the answer is more important than the answer itself. Working through the problem correctly and creatively is what helps us the most, rather than finding the answer through an answer key or from a friend.”

We often talk about the “aha” moment—that instant when you have been grappling to find a pathway forward, only to suddenly remember a kernel of knowledge that unlocks the entire solution. It is a gratifying experience shared by almost everyone.

Another student noted:

“One of my favorite parts of class is our conversations and debates, where we talk through spicy or very difficult problems. I think our class would lack this element if all the problems were ‘cookie-cutter.’ Though sometimes I think we all wish that the problems were easier, being presented with challenges makes us stronger and better students and geometers.”

One of my greatest challenges as a teacher is helping students reach the point in the year where they truly buy into the philosophy of tackling difficult problems. But it is worth the effort, every single time. Once we are there, we can take on increasingly complex concepts, finishing the year with the confidence that these students are ready for whatever challenges come next.

This practice is just as exciting (albeit quite different) in my Upper School Precalculus Accelerated class. Thanks to my colleague Melissa Palmer, I was introduced to a wonderful way of finishing each trimester. Students are given a culminating problem set. While they may use their textbook, past tests, and notes, they may not use the internet or outside resources. In fact, they may only work on the problem set during class time. They have four class periods to work on it, but during the third class, they are given 45 minutes to collaborate with their classmates. Because these problems are significantly more difficult than what we’ve covered, and because the students have spent two days pondering them, this 45 minutes of collaboration is the most lively and chaotic scene I’ve had the pleasure of calling "productive learning." Students must explain their thinking, which means there isn’t enough time to simply copy what a peer wrote. They have to understand the solution and make notes of their thoughts so that during the final day of independent work, they can write out their own unique solution and explanation.

Three times a year, I ask my students to answer a set of questions as a form of personal reflection. Without fail, the vast majority of end-of-the-year responses highlight the positive journey of solving a problem, rather than just getting the right answer. I find this easily relatable for anyone who enjoys the daily Crossword, Wordle, Connections, or other puzzle games: it isn’t just the final, completed grid that people admire and appreciate—it is the joy of the journey.