Reality and perception - by Kathy Jaqua

 Have you ever looked at a cloud and seen an old man’s face?  Did you squint your eyes to blur the distracting details so that you could see the image more clearly?  Did you really think there was an old man in that cloud? I’m guessing that you have seen a figure in a cloud, you did ignore the troublesome details that detracted from that image, and that you didn’t really believe what you saw was really there.  You moved between perception and reality seamlessly.

In theoretical mathematics, we need to be able to easily transition between what we know to be mathematically true, and how we can depict that knowledge.  Even drawing a graph of a simple linear function requires us to suspend the reality of a line being one-dimensional and infinitely long because we have to provide some width and some finite length for it to be visible and to fit on the page.  We generally aren’t picky about how thick the line is, and we add arrows to the end to indicate the infinite length; but in reality, we are only approximating the linear function. What we strive for is to capture the essence of the mathematical ideal through a “perfect enough” example that demonstrates that ideal.  This movement between mathematical reality and mathematical perception is inherent in discussions of mathematical knowledge and is a trait of math selfies. 

When students talk about their selfies, I often hear comments like “I know it’s not really like this, but it looks like it” to refer to some relationship being depicted.  Before reading my thoughts, look at the picture on the right and think about your mathematical perception of the trashcans and the reality of them. 

I think this selfie illustrates the distinction between reality and perception perfectly.  In this photo, the student noted the similarity of the trashcans, but went on to say that really they were all the same, but it looked like they were similar.  You might think this is a problem with selfies, but it actually revealed the student’s understanding of the difference between congruent and similar.  The student knew that all of the trashcans were congruent based on side lengths and angle congruence from the manufacturing process, while simultaneously noting that in the photo, perspective allowed the image to shrink uniformly creating an appearance of similarity.  The interweaving of the reality of congruence and the appearance of similarity was not a problem for this student or for me to understand his point. When showing this photo to others, the most common interpretation of what is being depicted is similarity.  What did you see?  Does the reality of congruent trashcans negate the perception of similarity shown in this photo?  I don’t think so.

Here’s another kind of distinction between reality and perception.  In the photo on the left, the student in the selfie was interested in the shapes of the giraffe’s spots, which she saw as polygonal.  It wasn’t critical to her that the sides be perfectly linear.  It was as if someone had drawn the polygons free-handed to show a variety of straight sided shapes. She saw the reality of a mathematical object in the appearance of a giraffe’s spots. The fact that she could look beyond the amazing experience of touching a giraffe in Kenya to see the shapes of the spots on the coat demonstrates the kind of mathematical awareness that I want to encourage.