It is almost a year later and I think another piece of the puzzle about abstraction and functions has finally fit into place. I was speaking with the co-coach of my school's math team and the resident physics teacher, when he brought up the fact that students may learn things in math and then can't apply them in physics. He related it to the math team topic Arithmetic with * Operations to explain when things are tweaked a little, students no longer understand. As he spoke, I got a faraway look in my eye as things started connecting.
In calculus, it's easy to use function notation to manipulate functions. You can even use as a function to have students differentiate and then interpret at different points. In precalculus, you can give them a graph of an abstract, piecewise function and ask them to transform it. But how do you make functions more abstract in Algebra I and Algebra II? One way would be to give them lots of abstract notation to interpret.
In teaching my SAT Math Prep class, I have noticed students who are in precalculus that can't solve a problem like . In fact, they don't even know where to begin. They have never seen an asterisk in a problem before but to them it looks a lot like a multiplication symbol. Once I finally get them to see it is just a symbol that signifies a rule that explains how to manipulate a pair of numbers to associate them with a third number, they become confused if I try to use an asterisk again in a different problem and assign it to a different set of points.
Just this week in my calculus class, we reviewed functions. I gave them a set of three points and asked them if it was a function. They looked at me quizzically and then as if I had a second head. "No," a student said. When pushed to explain his answer, he said, "That is only three points. That can't be a function." I brought them back to our definition of a function: A set of points where every x-value is associated with at most one y-value. Our three points made a set. Each value of x that was defined had a unique y-value. All other x-values had no associated y-value, therefore, it fit our definition. My students were blown away. Never had they seen a function with such a limited domain and range.
As I move on through my work on my masters and I think more deeply about the material I teach, I wonder about the other math teachers in my department. What would be their answer if I gave them three points and asked if it was a function? How often do they expose their students to abstract ideas and different notation? I think I am ready to put together a small talk for my department about functions and abstraction. I'll be sure to post my results and further thoughts.