I am a compulsive problem poser. When presented with a mathematical context, I have an incredible amount of difficulty leaving it alone. My teacher instinct is to push, prod, tinker, and/or alter it in someway. As I do so, I wonder what students might do and what new actions might be sponsored by any changes.
Problems are sandboxes and playing in them is a sort of imagination fuel for teaching.
Fractions, factors, and functions.
A large portion of my career to date has been spent musing over how to engineer classroom environments that infuse meaning into these three mathematical structures. When it comes to polynomial factoring, the area model has provided the most success. After connecting 2-digit by 2-digit multiplication, the area model becomes a beautiful visual to make sense of the “adds to ___; multiplies to ___” phrase that echoes around the room.
But we don’t keep the area model around forever. Once we’ve used the model to build meaning, we mobilize that understanding in more symbolic situations in a careful, deliberate march toward mathematical abstraction.
Every year, my students study the general characteristics of polynomial functions. We investigate the various shapes of various functions and slowly shift parameters to watch changes in the graphs. Eventually, we deduce the roles of the constant term, leading coefficient, and degree.
It should be noted that Desmos makes this process much easier than years previous. Just set up the generic polynomial, add sliders, set specific ones to play (depending on what you want to investigate), and have students discuss in groups.
I have only been teaching for 2 years, but am already beginning to encounter the recursive nature of the profession. I have had several repeat classes in my 4 semesters of teaching, and they require the achievement of the same outcomes. This does not bother me, in general, because I am excited to see the improvement in my teaching. There is one unit, however, that has already frustrated me. Its ability to sabotage creative exploits is unrivalled throughout the mathematics curriculum; I am speaking of the unit on polynomial factoring.