In an attempt to resolve the issues students were having with functional notation, Barry and I worked on a short in-class activity that was meant for students to understand the algebraic and graphical aspects of functional notation. Our hope was to witness a class-wide epiphany! This never actually happened the way we visualized it happening when we were constructing the activity in Barry's office. However, despite the lack of large scale epiphany, a reasonable number of students walked away with at least an inkling of what they had previously been doing wrong with functional notation. A few more students completely understood the point of the activity. While we may have not made our point clear to every student, it is our hope that students will help each other outside of class.
While working on this activity I've learned that just rephrasing something does not make it clear for everyone. Sometimes when we as teacher/tutors/TAs work with a group of students who struggle with the same problem we hope that what we have to say is understandable and clear. However, being so accustomed to concepts in mathematics sometime gets in the way of conveying the math and other means of explanation are sought out.