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.
I also wanted to give a video of a mistake that demonstrates how feeling mathematics in this way can lead to a better understanding. In this video the student starts by walking in the shape of the graph, instead of what the graph represents. Once she understood, she made the adjustment and seemed to have a much stronger understanding of graph. (You might notice her corrected video above.)
The second method for “feeling” mathematics that isn’t demonstrated above and I think is more difficult to represent as it happens inside of a person. It’s similar to a light bulb moment and happens in most disciplines but I don’t think it’s as clear as in mathematics. There have been times in my experience with mathematics when you just know something is true. When it clicks in your mind and it just feels right. Through proof, mathematics can take it one step further. Let me give you an example to demonstrate this is idea.
Suppose I asked you if an odd integer times any odd integer would result in an odd integer. You would probably test a couple in your mind (35=15, 111=11, 7*5=35, etc.) and begin to sense that the conjecture is true. Mathematics can prove that the conjecture is right and the proof in and of itself feels right.(Below is a proof, because I know you’re dying to see it.)
Let p and q be any odd integers. Then there exists integers n and k such that p = 2n+1 and q = 2k+1 (double an integer and add 1). Then when we multiply two different odd numbers we get the following
p*q = (2n+1)(2k+1) = 4nk + 2n + 2k + 1 = 2(2nk + n + k) + 1
So, we get 2 times some integer plus 1, which is, by definition, an odd integer. (Although the thing in blue is kind of ugly, it is an integer). Therefore any two odd integers multiplied results in an odd integer. ∎
For me, the above proof just feels solid. I know that it’s correct. I think if we can help students get to this level of comfort and understanding of mathematics we will end up with better problem solvers. Many students simply do mathematics and wait for the teacher to confirm whether or not they’re correct. There aren’t many that have a good feel for logic and the math with which they’re working. I think if we, as teachers, can get innovative in the tasks we design, then we can develop this feel in our students.
Bernstein, R., & Bernstein, M. (1999). Sparks of genius: The thirteen thinking tools of the world’s most creative people (p. 172). Boston, Mass.: Houghton Mifflin.