| | Ivars Peterson's MathTrek |
May 28, 1996
He writes: "Before setting out to make my way in the music business, I was in training to become a 'pure' mathematician. Such esoteric subjects as algebraic topology, measure theory, and nonstandard analysis were my preoccupations. I would stay up nights trying to solve knotty mathematical problems, playing with abstract phrases and structures.
"But at the same time, I would be lured away from these constructions by another activity. With an enthusiasm that could come only when critical faculties are in happy slumber, I would listen to or play a musical composition again and again, imprinting my ear and mind and hands with its logic and sense. Music and math together satisfied a sort of abstract 'appetite,' a desire that was partly intellectual, partly aesthetic, partly emotional, partly, even, physical."
Rothstein goes on to say that such an experience is by no means unique to him. He notes that music and math have been associated throughout history.
Pythagoras and his followers saw numbers as models of everything in the physical world, and they identified music with number, noting its scales, tempos, and other regularities. Johannes Kepler envisioned planetary motions as the "music of the spheres." Galileo Galilei speculated on the numerical reasons why some combinations of tones are more pleasing than others. Leonhard Euler considered the same problem in a treatise on consonance and whole numbers.
Johann Sebastian Bach sometimes treated the composition of canons and other types of music as exercises akin to solving mathematical puzzles. Frederic Chopin described the fugue as "pure logic in music." And twentieth century composers have applied sophisticated mathematical theory in their works.
In his 1993 book The Music of the Spheres, music critic Jamie James examines and ponders the history of the concept of a musical universe -- a cosmos envisioned as a stately, ordered mechanism both mathematical and musical. Music and science were once intimately intertwined and united by a grand vision, he points out. But music, like much of the most fundamental art and literature of our culture, has now been relegated to the obscure margins of the curriculum.
"All art, including music, was a much more serious matter before the self-conscious aestheticism of the late nineteenth century took root," James argues. "It is a recent notion that music is a divertissement to be enjoyed in comfortable surroundings at the end of the day, far removed from the hurly-burly of life's business."
I was reminded of these ideas when I read a report in the latest issue of Nature about a study suggesting that a weekly, structured music program can boost reading and math skills in early elementary school. The study was done by Martin F. Gardiner and Alan Fox of The Music School in Providence, R.I., Faith Knowles of the Kodaly Center of America, and Donna Jeffrey of the Start with Arts Program.
The study involved 96 students, between 5 and 7 years old, in eight first-grade, public-school classrooms. Forty-eight of them were exposed to a weekly singing program that emphasized the sequenced development of pitch and rhythm, often through musical games. The second group attended music appreciation classes -- musical training for that age typical of the U.S. public school curriculum.
As it happened, many of the pupils enrolled in the singing program had performed poorly in kindergarten compared to those in the second, "control" group. After seven months of the new program, they had significantly improved their attitude and behavior, caught up in reading ability, and outstripped the control group in mathematics.
The researchers found comparable results when the study continued into the second grade with the addition of a few new students to the structured singing program. Interestingly, those students who had received two years of extra music showed the highest level of achievement in mathematics versus those not in the program or those in it for only one year.
In their Nature report, Gardiner and his colleagues suggest that the pupils responded to the "pleasurable" aspects of the weekly music program, which motivated them to acquire the necessary skills to progress. Such training forced mental development that was useful in other areas of learning, particularly mathematics. "The maths learning advantage in our data could, for example, reflect the development of mental skills such as ordering and other elements of thinking on which mathematical learning at this age also depends," the researchers conclude. That's not entirely implausible. Earlier studies by other groups had suggested links between improved math scores and learning to play a musical instrument.
To my mind, however, the study reported in Nature has too many loose ends to provide satisfactory answers to the issues that it was trying to address. There really isn't enough information to explain how this particular musical program appeared to succeed for this particular group of pupils. Perhaps, the pupils simply benefitted from the special attention they received from dedicated teachers who truly loved what they were doing and conveyed this enthusiasm to their students. Maybe it wasn't the content of the lessons but the spirit that mattered.
Yet, the intriguing interplay between mathematics and music, which goes back to antiquity and possibly much earlier in human history, hints that there may be something deeper and more basic here.
In his conclusion, Rothstein comments that music and mathematics share not only the clarity of their expression but also their beauty and mystery.
He writes: "Our attempt to comprehend music and mathematics, to understand their workings and their purposes, is ... a model for our coming to know at all -- a model for our education, for the ways we make distinctions and connections.
"We begin with objects that look dissimilar. We compare, find patterns, analogies with what we already know. We step back and create abstractions, laws, systems, using transformations, mappings, and metaphors. This is how mathematics grows increasingly abstract and powerful; it is how music obtains much of its power, with grand structures growing out of small details."
In music and mathematics, it may be just a modest but mind-opening step from the kindergarten to the cosmos.
Copyright © 1996 by Ivars Peterson.
Gardiner, Martin F., Alan Fox, Faith Knowles, and Donna Jeffrey. 1996. Learning improved by arts training. Nature 381(May 23): 284.
James, Jamie. 1993. The Music of the Spheres: Music, Science, and the Natural Order. New York: Copernicus.
Rothstein, Edward. 1995. Emblems of Mind: The Inner Life of Music and Mathematics. New York: Times Books.
Weiss, Rick. 1996. Pedagogics: Arts program pays off in math. Washington Post (May 27).
Wertheim, Margaret. 1995. Pythagoras' Trousers: God, Physics, and the Gender Wars. New York: Times Books.
Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.