- Upper Grades Program
We often think about mathematicians as highly talented individuals who, with a stroke of genius, come up with a formula or a theorem. Their work usually resembles a cacophony of symbols and numbers to the untrained eye. But there are a number of examples in which years of hard work, years of collaboration or multiple failures are the major driving forces behind the discovery of groundbreaking theorems or proofs. Or, in some cases, individuals have achieved beautiful results that require only the knowledge of high school mathematics.
G. H. Hardy, an English mathematician, provides us with one of the finest examples of the importance of collaboration in the world of mathematics. Although he was a highly talented and accomplished mathematician, he is better known for his very successful collaboration with another mathematician, John Edensor Littlewood.
Like many other mathematicians throughout history, Hardy loved mathematics not just because he found a deep desire to dedicate his life into it, but also because he felt mathematics offered some of the purest forms of beauty.
When comparing mathematicians to painters or poets as “makers of patterns,” Hardy claims that mathematicians’ patterns are permanent than theirs, “because they are made with ideas.”
Last week, having almost completed all high school math curriculum, a group of 12th graders set foot to find beauty outside the classroom. This time, they looked for patterns not in a sequence of math problems, but ones hidden in the works of artists. As a prelude to their yearlong math-art project, students visited galleries in Chelsea. One of the stops was home to works by artist Ruth Asawa.
When you enter the gallery to see Asawa’s timeless pieces and let the wire sculptures surround you, their sheer resemblance to fractals, sequences and series, volume, multivariable functions, rotations, cones, spheres and various solids that morph into each other transport you to different ages in the history of mathematics, as if you are in a time machine. Each piece charmingly, albeit quietly, illuminates one brick of the wall of mathematics built painstakingly over the centuries.
As you get closer to each sculpture and observe the details and the beauty hidden in such intricate work, and feel the excitement, it is hard not to think about similar works that lead to amazingly simple but ingenious mathematical results. One of the most elegant of those is known as Euler’s equation, eiπ + 1 = 0, that our Calculus students will soon get the honor of demystifying.
According to Hardy “beauty is the first test” in mathematics, and he argues, “there is no place in the world for ugly mathematics.” But in order to understand and appreciate such beauty, one needs to learn to be patient and hardworking, and one needs to demonstrate perseverance—for trial and error, investigations, explorations and failure are all winding roads to success both in art and in math.
Our job as math teachers at Avenues, first and foremost, is to instill the love of producing sound but also aesthetically pleasing mathematical solutions in the hearts and minds of our students. The source of satisfaction that comes from solving math problems can be found in the process of creating meaningful solutions—not in the final answer, or in how fast one can solve a problem. We invite our students and the members of our community to see mathematics not as a list of procedures to follow, but as a pursuit of beauty and enjoyment.
- 12th Grade
- Upper Division
- Upper Grades Program