Overview:
An educator’s enthusiasm and openness to multiple problem-solving paths can transform math from an intimidating subject into a beautiful, curiosity-driven journey of discovery for students.
As my tutoring sessions with Jaylen happily show, how educators view the subject they teach can have a wonderful effect on how their students feel and think about that subject, even one as superficially unwelcoming as math! I’ve volunteered at an elementary/middle school as a tutor for the last eight years, although last school-year was my first with 7th-grader Jaylen. At the end of our second session I mentioned to Jaylen that I thought he had good math skills and asked if he thought so too. “No,” he replied and said he felt that way, “because I think there are other ways to solve the problems and I don’t know what they are.” I praised him for his modesty and curiosity and suggested that much of human knowledge and achievement results from people who ask, I wonder if there’s another way?
At the beginning of our following session I shared another way to solve a problem we completed the previous week and we discussed the merits of one solution versus the other. That’s one of the beauties of math, there is more than one path to the right answer. A few weeks later, Jaylen gave me a thank you note in which he wrote: “Thank you for helping me see the beautiful things in math.”
Math is beautiful, as again revealed to Jaylen and me this year when we were working on problems from a workbook Jaylen’s mother purchased to help him prepare for his high school placement test. Each week Jaylen produced the workbook from his backpack and we’d read a new problem together. Sounds simple. However it’s one thing for me to work through solutions to math problems before a tutoring session, no pressure there, and quite another to solve them spontaneously under the trusting gaze of my student.
The problem confronting us appeared innocent enough: For 30 minutes Nora drove 50 mph, then for 20 minutes she drove 60 mph. How far did she drive in total? Jaylen could see no way forward and he looked to me “in sure and certain hope” that I would guide him to the solution. Oh, ye trusting soul!
I began. Almost immediately I felt like a man wandering through a corn maze. With each line I wrote to simplify my initial equation, I felt like someone taking another step in the wrong direction. My doubts were fully justified. When we checked the answer key my answer was dead wrong. “Well Jaylen,” said I, “that’s one of the beauties of math – there are right and wrong answers. Sadly, mine was the latter. Let’s both think about how to solve this problem and share what we come-up with next week!”
My solution, which quickly came into view after a good night’s sleep, involved two separate calculations and happily produced the correct answer: 45 miles. However, a greater joy followed because I had asked my brother, who is one of my math mavens, how he would solve the problem. His engineering background brings a different and welcome perspective. While the opening paragraph of his email revealed a solution similar to mine, the next one shared his epiphany: “…I noticed that the times of travel made lovely fractions of an hour.” So: ½ * 50 + ⅓ * 60 = 45 miles. Brilliant! And as an aside, what a charming way to describe those fractions. They were indeed “lovely”!
Sharing all this with Jaylen, we once again marveled at the beauty of math. Math challenges us to find the right answer, to navigate the corn maze and discover a way forward, to see what others do not see and, if possible, to express our solution in a way that is wonderful to behold.
Jaylen’s curiosity to discover multiple paths that lead to the right answer is a key element in both math and more broadly, in scientific inquiry. Einstein, not a student of mine, took an untraveled path that revealed, in 1905, the theory of relativity. He began by imagining someone observing a train passing by. When the midpoint of the train coincided with where the observer was standing, lightning simultaneously struck the front and rear of the train. The observer rightly concluded that the strikes happened at the same time. However, a passenger seated at the train’s midpoint concluded that the strike at the front of the train happened first, which seemed correct from the passenger’s point of view but which is, in reality, incorrect. Hence the theory of relativity. The math demonstrating the correctness of Einstein’s theory is relatively straightforward, which is partly why it is beautiful, at least to those who prize simplicity over complexity.
Author Walter Isaacson saw a broader beauty in Einstein’s intellect when he wrote: “He could construct complex equations but, more importantly, he knew that math is the language nature uses to describe its wonders.” It’s a sentiment that Jaylen fully embraces. He sees math as more than rules and computations. It’s a voyage of discovery in which, if you maintain the right point of view, you’ll experience wonder and beauty. And from an educator’s perspective, sharing the wonder and beauty of your subject is fundamental to creating an uplifting, encouraging and rewarding learning environment.
