Imagery: A Key to Understanding Math
How can teachers help students find the beauty in math? There may be roadblocks already set up in math education — students’ disposition toward math anxiety, and pressure to cover material quickly. Or maybe it has something to do with the curse of knowledge — the gap between what experts know and non-experts don’t.
It’s easy for math professors to see the beauty in math, said New York University neuroscientist Pascal Wallisch, because they already have an obvious connection with it. “They perhaps had the luck to enjoy a positive math experience in school,” he said. “Or, frankly, from a cognitive neuroscience perspective, there is little doubt in my mind that they have quite a bit of a different brain than the average person who is trying to unlock the wonders of math, or just learn some math in order to get by.”
Wallisch said that as both a brain scientist and someone for whom math did not come easily, the key to mathematical beauty (and understanding) is visuals. “As primates, we are mostly visual creatures. A good amount of the cortex in primates (upwards of 30%) is dedicated to visual processing in one way or the other. Put differently, things that look interesting or appealing are bound to attract curiosity,” he said. Looking at a picture (or a movie or video) is the same thing as looking at an equation. But while it represents the same information, one method is inherently more appealing to our brains than the other.
According to Wallisch, mathematical imagery is what students are missing, and what causes confusion. He used the example of reading the words “Statue of Liberty,” and how it evokes an immediate image in the mind. But if a person couldn’t read, or had never heard of the Statue of Liberty, they would visualize only letters and words, not Lady Liberty holding her torch — and the same goes for math novices. Since they have no experience, the mix of mathematical symbols on the page don’t mean much. “Mathematicians see equations by imagery built by long-term practice manipulating them,” he said. “The trick is to use software to visualize the equations so that those who don’t have the practice (or the unusual brain) can see the same.”
Wallisch began creating moving mathematical images for himself using technical computing environment Matlab, and said that, although he uses it for high-level research computations, high school students can just as easily build visual mathematical models with some guidance. By creating images of equations and playing with the variables, Wallisch now sees what all the fuss is about. He wrote in a blog post: “Personally, I’m betting on aesthetics, with Kant: ‘Beautiful is that which is appealing without interest.’ As we can’t presume interest, aesthetics can serve as an important bridge(head).”
California College of the Arts math professor Michael S. Schneider agreed that imagery is the best way to show students the beauty of math. He has been helping students connect mathematics to visual imagery for nearly forty years, and wrote the book A Beginner’s Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art and Science to show humans are surrounded with mathematical imagery — including right outside, in nature.
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“As a youngster I wasn’t particularly outstanding in math,” he said. “But at 16 I became earnestly interested in the variety of shapes nature produced and wanted to understand why regular shapes keep recurring in nature. I remember pondering the same hexagonal shape found in the beehive, quartz crystal and metal hexnuts. I could understand how a crystal grew mechanically in this precise geometry by accumulating atoms, but how did bees know how to produce the pattern which holds more weight of honey than, say, a checkerboard pattern? I wasn’t comfortable with the ‘trial and error’ explanation, and even if it was in their DNA, how did that knowledge of superior design get there?” Then, he said, he wanted to know more about logarithmic spirals – “in the bathtub, swirling leaves, tornadoes, hurricanes, solar systems, galaxies.” Schneider said he had good math teachers, but these topics were never covered in school; books he looked up on the subjects covered them one at a time, but never altogether in the same place.
Schneider said he became obsessed with understanding the language and shapes of “nature’s geometric alphabet”: circles, spheres, triangles, squares, and more — the shapes that surround us every day if we simply take the time to notice them. “A circle represents the number one,” he said. “Most people can feel why a circle represents unity, its wholeness, completeness. A circle holds more inside it than any other shape having the same perimeter. So it’s practical to know that round pizzas hold more toppings than squares or rectangles having the same length of crust.”
Armed with this set of nature’s images and symbols, Schneider found that numbers and shapes have personalities, each playing different roles in the cosmos. “The universe becomes a book and then a great play with great actors in great parts telling great stories,” he said. “Mathematics is a way to read the world of nature and technology around us. If a teacher can convey this, the entire world becomes an exciting textbook.”
Schneider admits that today’s math teachers are strapped for time and resources to really explore the beautiful part of math, partly due to the way textbooks are constructed, and the pressure to cover material quickly. He believes that for students to see the beauty in math, teachers need more time and freedom. “I think that math education gets too abstract too quickly without first providing a sense-based foundation,” he said.
But appreciating the beauty in mathematics could start by just having students look around them. “The universe may be a mystery, but it’s not a secret,” he said.