Today, in class, I found that I really have gotten the class to begin to discuss things among themselves. After putting up the question of what is a polynomial having $\sqrt{2}+\sqrt{3}$ as a root, the students began to work their way through an answer. They guessed that $x^2-(\sqrt{2}+\sqrt{3})^2$. At one point they started discussing everything together, and I ran to the corner and sat down on a chair watching the students discuss and explore. After Mary finally gave a description of how to get the right polynomial an argument ensued about whether she could do what she was doing (setting the polynomial $x^2-(\sqrt{2}+\sqrt{3})^2=0$, then moving $\sqrt{6}$ to the other side of the equation squaring both sides, and then solving back to $0$ again. At this point, she had a polynomial, and the students pulled out their calculators to check if this polynomial seemed to have the appropriate number as a root.
After the students were convinced that the polynomial was probably correct, I stepped back into the picture to make some remarks. The first was to explain why there was a disagreement, and what it showed. In particular, I was able to discuss the changing role of $x$ in Mary's solution. That is, at first it was a variable in a polynomial that the students had originally guessed to be the correct polynomial for the problem at hand. Then Mary took the polynomial and changed it into an equation and started treating $x$ as an "unknown number" in the problem. Afterwards, she turned it back into a variable. Thus, this problem allowed me to discuss the high school problem of finding a polynomial for a number, in a context associated to the closure of the algebraic numbers, and to then use the students' work as a vehicle to discuss some of the difficulties between variable and unknown number. For me this particular vignette is an example of one of the many ways that advanced mathematical content can be used to allow entry into the high school teachers' world. I only wish I knew better what it is about this class that is allowing me to do this so well.
The rest of the class was spent in showing that the algebraic numbers form
a field.