While trying to find patterns in the context of functions, I struggled with the massive number of possibilities. I settled on the consistent application of functions being their use in the context of data sets. When trying desperately to give real world contexts of functions we often give students data and then fit functions to that data.
Below are a few examples of where we might use a function to model real data. The first is a model I came up with for iPhone sales since 2007 by quarter. It’s not a friendly function but it’s pretty close considering how erratic the data is. (If you think you can do better, play around with my sliders here.) The second is the temperature (in degrees Fahrenheit) of freshly brewed coffee after being poured into a coffee mug.
There’s nothing wrong with this application of functions, but it’s the typical application. As I started to think about how all functions (a rather large area of mathematics) could be applied in a different context, I realized that we often use them for data but rarely apply them to things we see in our everyday life. If we begin to look for patterns in our everyday life, what kind of functions will we find? I began to ask myself, “If I start to pay more attention, can I find functions providing structure in my world that generally seems to lack structure? Or possibly, is the structure we assume to be there, the only type of structure?”
Following my search, I quickly noticed that functions in the Cartesian system are limiting. Many interesting patterns can only be described by different types of equations (parametric, polar, and possibly others that I’m not aware of). See the image “lamp 2” for an example of parametric equations. You might also find it interesting to look at the equations or functions that yielded each curve.
Robert and Michele Bernstein, authors of Sparks of Genius, point out, “Our ability to recognize patterns is the basis for our ability to make predictions and form expectations.” One problem with math education and also my most profound takeaway from this process was that we often give students the patterns. We say “Here is concept x and it is applied in this way.” The problem with this is two fold. First, students immediately assume that the given pattern is the only pattern that exists. Second, it discourages students from looking for patterns on their own. If we want our students to be creative then we must provide situations in which they look for patterns and extrapolate the importance of them. I understand that this can be difficult because it takes more time, but when students find patterns on their own they are more likely to remember them.
Finding functions in their daily experience could be a worthwhile task for my upper level math students. However, I think it’s more beneficial to view this process in the context of all mathematics. The important idea for math teachers is that we encourage students to see math in their everyday life. I’m not talking about arithmetic or fractions. I’m talking about structure where their seems to be none. This is the essence of mathematical thinking. If we teach students to find patterns in their daily lives, then we will end up with more creative problem solvers.
Bernstein, R., & Bernstein, M. (1999). Sparks of genius: The thirteen thinking tools of the world’s most creative people (p. 30). Boston, Mass.: Houghton Mifflin.