Have you ever experienced that magical moment in the classroom when everything seemed to be just perfect? The class session when you might not have even planned for the activities but the students took you down a path where one thing led to the next and before you knew it…there was magic?
I’ve been very fortunate over my 23-year history teaching mathematics and have experienced several of those moments. It was those moments of magic when I knew the students were engaged, learning from each other, and I was the proud teacher on the side. The moments when the students were working together in groups and found they didn’t agree with the answers from another group. This is what I always referred to as a “controversy”, followed by telling students that “Controversy is good for the soul and this is the time to listen and learn from each other”.
It happened quite by accident. I was teaching College Algebra (MAT151) and I had a wide range of learning levels in this class. Actually, this was a regular occurrence in my College Algebra classes. Some students would enroll in this class after having taken the mathematics placement test, which usually meant they had a very high level of mathematical knowledge and ability, while other students enrolled because they had earned a C or better in the previous class, Intermediate Algebra (MAT121). This latter group usually had a mixture of mathematical knowledge and ability. This is the group that Michael came from.
Michael was a very large and imposing young man, probably in his early 20s, and an admitted felon. Michael was a nice young man with a quick smile, big laugh, and a limited understanding of algebraic rules. He tried hard, came to class everyday (probably because his probation officer made him), participated in all of the group activities, but wasn’t passing the class.
We came to the part of the class where we were solving radical equations…sorry to all the non-math folks for the technical part of this story, but it is very important. All semester, I had been working with the students and helping them to understand that there was more than one way to solve an algebraic equation…there was the traditional algebraic method, but there was also a numerical method and a graphical method. We regularly used a graphing calculator, TI-84, for just about everything in the class but particularly to see the numerical and graphical answers.
So, we were knee deep into learning about radical equations and they were familiar with the possibility of having an Extraneous Solution…that pesky problem where you might be able to algebraically solve the equation but the solution doesn’t check out when you substitute the answer back into the equation…blah, blah, blah.
My students were working in groups and five of the six groups all came up with the same answer… all except Michael’s group. This is where the magic happened. In Michael’s loud booming voice, he said, “You all are wrong, there’s No Solution!” To which I said, “Michael, can you come to the front and show everyone what you mean?” We regularly used the Pad Camera for the class so Michael makes his way to the front and using the graphing calculator, shows the class that the two sides of the equation, each representing a function, did not intersect. The class erupted in cheers and Michael had his moment of validation in the sun.
I don’t know what happened to Michael after the semester, I can only hope and pray that he stayed on the straight and narrow path and is living a happy life. I know that he made a huge difference in my life and serves as a reminder of the potential that lives within all of our students.