Over the past 100 years the world has changed fundamentally. Information is no longer scarce. Technology has advanced exponentially and will continue to do so. Science fiction is, in many cases, becoming reality. If a person is to be successful in this world the way in which we navigate it has to fundamentally change. Also, and maybe more importantly, in the coming decades we will face the biggest challenges humanity has ever faced. From climate change to overpopulation to cyber-security, now, more than at any time in human history, we need a generation of creative problem solvers and divergent thinkers.
Over the course of the last few months I’ve had the opportunity to learn about the thinking tools used by creative individuals. I want to make the case that all stakeholders in education, from students to teachers to administrators, need to intentionally integrate creativity into their work. You can read my white paper that outlines the importance here. The above elevator speech is a synopsis of the paper. Thanks for reading and I’d love to hear your thoughts and ideas in the comments!
Edit: In my original post there was a small mistake in the proof. I have now corrected it. Thanks to Carleton for pointing it out!
For this assignment in my Creativity in Teaching and Learning class I had to come up with a way to “feel” my concept (functions). “Embodied thinking”, or the idea that “feeling” a concept can help us understand it in a useful and deeper way is at the heart of this assignment.
Getting a feel for mathematics can happen in a number of ways. I want to discuss a couple of the methods, starting with how real world situations are translated into mathematics. This assignment actually spurred an idea for a project in class in which I give a group of students a position versus time graph and have them walk out the graph as if they were the particle being described by the graph. I recorded it, put the videos on Youtube, and then had the other groups sketch graphs based on the videos. We then held a competition to see who could get the most accurate graph and also which group did the best at walking out their function. Position v. time graphs, which come up frequently in calculus, became much more real. It truly gave the functions a feel. My hope is that when students look at these graphs in the future they might imagine how a particle would feel while tracing the graph and that might help them get glimpses into the velocity and acceleration of the particle. This would be similar to how Robert and Michele Bernstein, authors of Sparks of Genius, point out that Stanislaw Ulam, a mathematician who worked on the atomic bomb, apparently “imagined the movements of atomic particles visually and proprioceptively” (1999). Below are a couple of the videos along with the graphs that went with them.
Update: As I glanced at the forum I noticed that I evidently did this assignment incorrectly. I apologize for misreading the instructions and will be more careful in the future.
The focus of this assignment was to take a song and change the words to reflect an understanding of my concept area (functions). My song is below. I also want to explain how creativity often stems from creativity.
I used to think that the best way to be creative was to give yourself a blank slate, time, and the tools and “create”. This is actually not an environment that is conducive to creativity. If we view a particular concept as a device with knobs that can be adjusted, then Henrikson, Mishra, and the Deep Play Research Group in their article “Twisting knobs and connecting things: Rethinking Technology and Creativity in the 21st Century” point out that “A creative person… works with those knobs to figure out possible variations on the original concept, and to incorporate anything that will produce something novel, effective and/or aesthetically pleasing (2014).” I’ve found this to be true repeatedly in my career. If I sit down with the thought “okay, create an activity for concept ____” then I really struggle to come up with something good. My most creative and useful activities come when I spend some time looking at activities that were created previously. These almost instantly spark ideas for new activities and usually what gets implemented is a variation on or combination of several other activities. Continue reading →
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.
I spent a lot of time trying to think of people I knew that would fit the description of creative. I realized that I don’t know many artists or musicians or other people who are stereotypically deemed creative. However, I do know Steve Kelly, and he is by far the most creative person I know.
Steve is a national board certified math teacher at St. Louis High School in mid-Michigan. He was also my math teacher, cross-country coach, college instructor, and currently my collaborative partner. We’ve worked on many projects together in the short time I’ve been an educator and the majority of the really creative ideas we develop come from him. (Check out our first project and a more recent project if you don’t believe me.) Although we have created together for some time, I never asked him many of the questions I came up with for the interview. Below is a synopsis of our discussion.