Creative individuals often take something that is concrete or complex and abstract it in some way that makes it more meaningful or provides a more useful perspective. When charged with the task of creating an abstract representation of my concept (functions) I was first at a loss. How do I create an abstract representation of something that is fundamentally abstract? The descriptions and media below describe the two methods I chose. For a summary of how these abstractions impact my teaching and understanding of functions scroll to the end of the media.
As I thought about this, I realized that when we write a function, we often only talk about a specific function. Yes, occasionally we will discuss a “family” of functions. But what does that mean and students really understand it? When we discuss a family of functions we are really talking about an infinite number of functions. What does that mean? How can we visualize such an abstract concept? Yes, sliders can help, but it can be visualized more effectively. Below are my ideas for abstracting various families of functions.
This also made me think…Could an abstraction be (at the surface level) more complex than what you’re abstracting? The GIFs show a more complex picture than just a curve with a slider (y=ax^2 with “a” as a slider). A family of functions is an infinite number of functions, right? How do we abstract an idea that is so complex? The GIFs attempt to abstract infinity (or an infinite number of curves) by suggesting what the escalation to an infinite number of curves would look like. (Click a GIF to look at a slideshow of each one individually.)
The task set before me was to abstract my concept in two different ways. The second method I chose for abstracting the concept of functions was to pare downs functions down to their simplest…well…abstraction. Then, I sketched these abstractions and below are the images of my sketches. I decided that at their fundamental level functions have three things: an input, an operation (something that modifies the input), and an output. The images all contain these fundamental pieces.
I think both abstractions can impact my teaching. The GIFs would be something that I could either show my students to give them a visual for the concept of a family of functions. A good project would be to have students create similar GIFs for different functions and their transformations. This solidifies the concept in a visual way and gives us something we can anchor back to throughout the rest of each unit.
The second abstraction might be useful in a different context. I like that the images provide visuals for the simplicity of functions. Often students don’t realize the vastness of functions and their stretch throughout the world. Most things with an input, operation, and output is a function. When you ask a student to give you a function they are likely to say something like y=2x+3 or f(x)=sin(x). They are unlikely to say “The number of cars in the school parking lot is a function of the time of day.” If students have the abstract concept of a function as their fundamental understanding then they might start to see function behavior in their worlds more frequently.