A great debate: Polynomials

My senior year of college I ran into my former Linear Algebra professor leaving the bathroom, who was at the time dusted head to toe with chalk. Middlebury's math professors were amongst those who loved the "classic" feel of chalk, so most things in the math building had a fine layer of grit if you stuck around long enough, but she was COVERED. When she saw me she acknowledged how ridiculous she looked, then explained her appearance with an exuberant "I was teaching the Fundamental Theorem of Calculus!"

That memory sprang to mind vividly today during my junior Algebra 2 advanced class, when I was getting excited about the discussion my students were having to develop rules for the end behavior of polynomials. After discovering and agreeing on how the degree and leading coefficient can help you figure out the very basic shape of a graph, they moved on to how we could use factored form to graph polynomials more specifically. How we could work backwards from a solution followed, and as we went along I could feel my voice growing louder with excitement. I prefer modern technologies, so in lieu of chalk dust by the end of class I found my hands covered in whiteboard marker flecks, but in that moment I was very much channeling Professor Proctor. I at one point exclaimed "...and all of that is so important that we call the idea the Fundamental Theorem of Algebra!" so red-faced and loudly that my students asked if I was alright, while one kid (lovingly) pointed and shouted "Nerd!"

In one poignant moment of our discussion a student, M, raised her hand and said "So to solve a polynomial you just need to know its x-intercepts?" We talked about this for awhile and my excitement ballooned as everybody realized that the x-intercepts also tell us the solutions of a polynomial and that now we can put our previously learned quadratics skills around graphing and factoring and solving to work to get results from the standard form of a polynomial. M raised her hand again and asked in a somewhat annoyed tone "Why didn't you tell us all this months ago? It would have made things so much easier!" Smugly, I responded with "Because all that work we did months ago led to all of you coming to the conclusions we just reached. You guys just discovered everything on your own." And as she slowly released her breath in a damn-it-you're-right-but-I-still-want-to-be-mad-at-you-for-keeping-this-from-me way, I considered dropping the metaphorical mic and walking out of my classroom, my work complete.

The more I teach and read about teaching, the more I question the practice of assigning nightly homework. My students and I have agreed that all of our polynomials work will happen in the classroom, with all class minutes focused on productive work and discussion. If a student feels the need to work independently at home (or needs to catch up after absences), they may, but I'm not assigning homework. We'll see how it goes as the weeks progress, but since these moments of great conversation happened 3 days into the pilot, I'm feeling pretty good about the decision so far!