Function Composition

I was in the process of creating a list of things that my students of calculus need to know but don't seem to and one of the ideas that came to my mind was function composition. It's tricky to say exactly what it is that they don't know. They have no problem composing two functions and they can usually correctly identify the functions that lead to an already composed function. But when they start using the chain rule to take a derivative, the whole idea of the outer function and inner function are totally lost.

Then, I read a posting from Keith Devlin's article, "How Do We Learn Math?" and he had this to say about functions:

Functions, as defined and used all the time in mathematics, don't do anything to anything. They are not processes. They relate things. The "doubling function" relates the number 14 to the number 7, but it doesn't do anything to 7. Functions are not processes but objects in the mathematical realm.

Suddenly, I knew where my students' problems were: they see functions as processes not objects. This is a common misconception, even up through undergraduate mathematics courses. It isn't until Abstract Algebra that students are taught to think of functions like they are objects and find properties about them such as their additive inverses. Before, talking about properties only happened with numbers and how functions look when graphed: even, odd, or neither. Many of my teaching peers looked at further abstraction and ran away in tears or forgot these ideas entirely.

To encourage students from the beginning to think about functions as objects, using the idea of "function machines" may help. Nowhere in our school's curriculum do we explain functions through this lens, although I have often contemplated using this model in my classroom. I have yet to do this because I was unsure of how beneficial it would be and I didn't want to waste any of my precious time with students. I am now rethinking that view.

I know that function machines changed my personal view of functions in high school. I did start to recognize functions as machine (object) that related one number to another, although there still was a process of "doing" happening inside of the machine. This idea did help me to see that to solve a trigonometric equation, I didn't have to divide both sides by sine (as many of my students seem to believe) but instead I had to use the "opposite" function machine (the inverse function) to undo the sine machine.

Another way to have students see this is to work with functions in a much more generic way, giving more examples of functions without giving a specific rule that is associated with it. This happens occasionally in precalculus in terms of looking at piecewise graph, but even at that point, the forward thinking student could create the equation for the function if they so wanted. In calculus, that option happens less frequently because we create functions such as f(x) = x g(x) and ask them to take the derivative. Here they have no option of creating an equation for g(x) but must accept it as an unknown object.

I have even gotten to the point with my calculus students that before we start talking about inverses of functions through function composition, we start by looking at group properties of real numbers under addition and multiplication. Then I lead them to figure out what the identity element of function composition is. From this building, they start to think of functions as objects that work together that have properties as greater structure; they see them as a group of objects and aren't trying to dissect the individual equations. It is a struggle for many students but some start to see the larger picture.

As I think down to the heart of why my students can't see functions as objects and why they struggle when I try to make them think this way, I start to question how my colleagues see functions. If I asked my colleagues to define the word function, what would I get for an answer?