Johns Hopkins Magazine -- February 1998
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In an era when school budgets are being slashed, music is one of the first things to go. That's a shame, say many educators, who've found that music can unlock the doors for kids to learn math and other core subjects.

H U M A N I T I E S    A N D    T H E    A R T S

Arithmetic of the Soul
By Joanne P. Cavanaugh
Illustration by Kim Barnes

"Music is a secret exercise in arithmetic of the soul, unaware of its act of counting."
--Gottfried Leibniz, philosopher and mathematician.

IN THE WISDOM OF ANCIENT GREEK philosophers, most notably Plato and Aristotle, music was not solely a matter of artistic, divine, or romantic expression. Music was math. Following that theory, later philosophers named the "quadrivium" of mathematical arts: Geometry, Arithmetic, Astronomy, and Music.

"Music was not emotional," says Susan Weiss, a musicologist and faculty member at Hopkins's Peabody Conservatory of Music. "It was based on solid evidence of how the moon, the sun, and the stars worked, more than on the localized, anecdotal experience of hearing sound.

"When you see the Greek treatises on music, you see elaborate drawings that looked like something out of a geometry textbook," Weiss says. "They weren't musical notes, but they described notes."

Over the centuries, math and music remained intertwined. Before the Renaissance, much of Western-based music relied on mathematics. The baroque era's Johann Sebastian Bach, who used a highly calculated approach to composition, was famous for his submerged numerical symbolism. Take the sacred choral work St. John Passion, in which Bach composed five notes to mark the German toten (to kill) at the narrative point when Pilate hands over Jesus. The number five symbolized the Fifth Commandment, according to German musicologists. Bach repeats the theme 10 times, to musically evoke the Ten Commandments.

Clearly, Bach's listeners won't necessarily note such "cabalistic" symbolism. Any mathematically derived code instead may land in the subconscious. In recent centuries, the connection between math and music has become more complex as Western composition pushed away from its origins. Among other influences, the Hindu raga, Japanese pentatonic scale, and other non-Western melodic patterns were introduced to the artistic mix. Today, despite the early links between math and music, the bridge between calculus and concertos is being debated and tested, applied consciously or ignored.

Susan Weiss: "We try not to think that we deal with numbers in music. It's a little off- putting."
Photo by Jay Van Rensselaer
"We try not to think that we deal with numbers in music. It's a little off-putting," says Weiss. "And math has developed to such an extent that its connection to music is very vague and probably blurred."

In popular culture and academia alike, there's now a theoretical No Man's Land between the scientific and the artistic. Higher math is considered coldly abstract, while music is often warm, unpredictable and passionate; one is perceived as left brain, the other right.

"Today we think in terms of math/science people or verbal/artistic people. There's that division," says Wayne Parker, senior researcher at Hopkins's Institute for the Academic Advancement of Youth (IAAY). "In the past, math, music, and reading held the liberal arts together. The idea that educated people say today, `I don't do numbers,' would be unheard of. It would be just as bizarre as saying `I don't do letters.' Of course you do, you're in college."

Across Hopkins, math and music are again merging. And it's likely no coincidence the ancient Greeks considered the two studies part of the same intellectual journey. Apparently, music can play an integral role in the cognitive development of children.

A Hopkins study, now being conducted by Parker, will explore links between music training and math test scores among elementary school children. At Hopkins's Kennedy Krieger Institute in East Baltimore, therapists use music to help autistic and other developmentally disabled children learn math and other subjects. Meanwhile, infants being studied by Hopkins psychology researchers may reveal just how early in life music is recorded as patterns in the brain.

And the math-music connection may continue well into adulthood, reaching into the classrooms of academia. Like other universities in the country, Hopkins's math and mathematical sciences departments include several serious musicians, and the Peabody Conservatory is infiltrated with math-types. At Peabody, in fact, math has become an integral element of composition through advances in computer music.

"To create music using computers, you need to understand how sound works," says Ichiro Fujinaga, Peabody faculty and associate director of information technology. "To understand that, you need calculus. Not very high level calculus, but it helps."

Fujinaga, who came to Peabody with degrees in both math and music, says computers bring new sounds into the tonal universe by tapping elements of science: "We introduce sounds and figure them into equations. We can calculate every pitch, and since sound can be a combination of sine waves, we can engineer the sine waves to get what sounds we want.

"And now that we have automatic jazz accompaniment, you can jam with a computer and it does exactly what a musician might do," Fujinaga says. "If you are jamming, how do you know there's not a little algorithm already running up there in your head. I think we do that, and one day we won't be able to tell the difference."

At Tench Tilghman, "Music theory sparks a level of understanding," says principal Betty Turner.
Photo by Mike Ciesielski
IN THE BRIGHT BLUE AND YELLOW recreation hall next to Tench Tilghman Elementary School in East Baltimore, a group of fifth graders gathers on the floor, sitting cross-legged, chins resting on upturned palms.

"In music, you have a measure that has four beats," Lynn Davidoff, a Peabody Preparatory music teacher, tells them. "Which is the most important beat? Beat number one, number two, number three, or number four?"

"Number four?" answers a child in a flannel shirt and Air Jordan high-tops.

"No, beat number one!" calls out half the class.

The children cluster around xylophones resting on the linoleum. They break into groups to play an "ostinato," a repeated rhythmic pattern that helps them learn to keep a steady beat and work as an ensemble. Using small mallets, two children tap the C & F notes on the first beat, while others hit C on the first beat and F on the third, and the last group--the leaders--play the same alternating notes in a 1,2,3,4 succession. As the children listen to each other, the leaders speed up. Spaces between the notes grow smaller, a fraction of the original tempo.

For some students, the four-beat pattern is a hidden lesson in math. "We talked about four beats in a measure, but they don't necessarily realize it's math. There's lots of repetition," Davidoff says afterward. "They listen to each other. Although [the leaders] pushed the tempo, the guys on the outside stuck with them."

The Peabody Outreach Partnership Program, created 11 years ago, is geared toward students in prekindergarten through fifth grade. Taught in four different Baltimore City schools in the past decade, the program is currently being used at Tilghman and is set for two other city schools next year. Funding has often been a struggle, and is usually a mix of private foundation, Peabody Prep, and limited public school money.

Through classroom games that incorporate physical movement, or instruments such as bells, triangles, and drums, the children learn musical concepts such as pitch or beat. The hope is that the lessons students learn in music class will transfer to other academic skills--including speech development, math, and problem-solving.

Davidoff is also working with some fifth graders to help them compose their own works. "It's what I call the creative composition process," she says. "By using games emphasizing patterns, the goal is to have a well-arranged composition by the end of the year. It has to be a step-by-step, sequential thing."

Tench Tilghman also receives music-oriented classroom studies from the Baltimore Symphony Orchestra through the Arts Excel program, a nationally known music curriculum that has fueled improvements in math and reading scores among students. And, as part of Peabody Prep's program, a chorus of fourth and fifth graders this year is learning to read music, including complicated pieces that use dissonance.

Vicki Downer, who directs the Outreach program, practices during lunchtime one day for a Peabody Conservatory recital she is performing with some of the choral students. A handful of young girls gather around a school-issued upright piano, working on their pronunciation and pitch. Downer plays a section they're having trouble with, and stops. "Don't sing `anxious'... Listen to how that sounds--aankshus. Sing it ahnkshus. Come closer so I can hear you all."

Afterward she reflects on her students: "A lot of them may never have this kind of experience again. A child is born with an innate musical aptitude that is nurtured in their environment. Whether or not they achieve depends on their experience."

Parker wants to document music's influence on learning. Using standardized testing--done before and after a school-year span of music lessons--the IAAY researcher hopes to see whether Tench Tilghman students who receive music training improve their math and reading skills.

Parker: "In the past, math, music and reading held the liberal arts together."
Photo by Jay Van Rensselaer
The idea was ignited after Parker, leafing through a study on mathematically talented fifth grade students from IAAY's 1992 talent search, noticed that a high percentage of the math-savvy students had formal instrumental music training. In fact, nearly 93 percent of the females and 83 percent of the males had studied musical instruments, according to a survey of the 825 students whose overall school performance is being followed from fifth grade through college.

"It seems like it's one of two things or a combination," Parker says. "First, you have a family that is very achievement-oriented and has an intellectually stimulating environment. Or, maybe the activities the kids are involved in increased their intellectual functions. Probably it's both."

Parker decided to study the children at Tench Tilghman, most of whom have no outside musical training, to better gauge music's influence on math. Located just a few blocks from Johns Hopkins Hospital, Tench Tilghman sits squarely in a high-crime section troubled by gun violence. About 80 percent of the school's children come from single-parent, young-parent households, and family incomes range from $0 to $25,000, according to music program directors.

For many of these kids, family support is lacking. Says Davidoff, "One of our students got really locked into the recorder. She came in here to practice on her lunchtime. But she couldn't take it home, so she dropped out of the music club. Her mother was sure the kid would lose the $7 recorder or break it, so she wouldn't sign a contract to take the instrument home."

Parker also plans to test a control group of students at a school with no classroom music training but an otherwise similar environment. "There's more than academic interest in this," he says. "In an era when school budgets are being slashed, one of the first things to go is school music programs. But if rigorous music programs increase achievement in other subjects, the school board needs to know that."

In Baltimore's 121 elementary schools, there are 54 music teachers, with only 15 offering instrumental instruction. Some city schools budget for one or two music teachers, while many have none: "There have been cutbacks over the past 20 years. We are quite thin," says Augustus Brathwaite, curriculum specialist for Baltimore city schools. "The problem has been funding. When there is a need to cut positions, it's easier to cut a fine arts position than an English teacher."

Tench Tilghman, which funds one teacher's salary for the Peabody program, has struggled to provide even the 45 minutes of music the 750 students now get about once a week. Class times have been cut short and class sizes increased because of conflicts over space at the city-owned recreation center next door. Such limits may make Parker's study more difficult to complete as well.

One Tuesday last fall, Principal Betty Turner contemplated the effects of music training on students as her school lunch cooled inside a generic brown box marked Pizza.

"Music theory sparks a level of thinking. There is a relationship to tempo, and what students later see in fractions and in math. It makes it tangible. They can work it out," Turner says. "It's a piece of the puzzle that will be missing for youngsters who have to put the puzzle together later. If you take it away, you take away their chance to be a literate adult."

Anita Cooper, a faculty member at Peabody Prep who developed the Outreach program, says music enhances children's cognitive ability. Among other reasons, success at the keyboard or on the violin can build self-confidence that spills over into other academic subjects.

Opening up neural pathways for learning: Peabody Prep now offers music classes from birth.
"Music is a wonderful platform to learn everything else from. It's a hard thing to get parents and school officials to understand: that games are children's work," Cooper says.

"Composing music is like problem solving--you have to put text and rhythm together," she continues. "Musical rhythms are mathematical equations, like 4/4 or 2/4 time. You use addition and subtraction skills: How many more beats do you need? How many more ways can we divide it? Music also helps children develop language. A musical phrase is like a verbal sentence. It has a beginning, a middle, and an end. And the inflection changes in different parts of the sentence."

AT HOPKINS'S KENNEDY KRIEGER School in East Baltimore, a similar approach is being used. There, music therapist Alicia Barksdale works with elementary- and middle-school-age children who are developmentally or learning disabled, including autistic youths. Among other methods, she uses rhythm to teach arithmetic.

"Somehow a child who is not able to understand the process of a math problem can understand four beats per measure and set rhythm within a measure," Barksdale says. "They understand that the beats have to add up, even if they can't do math. Autistic children also have a hard time remembering things so I use music as a way to get it into their heads. I put it in a song so they can get the answers to the times table."

In one exercise, Barksdale teaches the children to count by using drum beats and a song. "It goes, `Listen, listen, here I come. Someone special gets the drum.' Then that person taps the drum so many times, like four or 10. I think something about the rhythm allows the information to be imprinted," she says.

Rebecca Landa, director of the Center for Autism and Related Disorders at the Kennedy Krieger Institute, also uses music to draw responses from autistic children--teaching them to recite lengthy excerpts from songs when they may rarely speak otherwise: "Music and simple repetition is predictable, so for autistic children it reduces the amount of novelty they have to process," she says. "The predictability allows them to devote more cognitive energy."

Several studies in recent years draw a link between music and cognitive development. Researchers at the University of California at Irvine and the University of Wisconsin found that three- and four-year-olds who studied piano for six months scored 34 percent higher on spatialtemporal reasoning tests than did preschoolers who received private computer training, singing lessons, or no extra training at all.

In the study, published in Neurological Research in February 1997, the authors referred to earlier work that indicated exposure to music might "excite and enhance the cortical firing patterns" used in spatial-temporal reasoning. Such reasoning transforms mental images--without a concrete, physical model--and is required for higher brain functions such as chess, mathematics, and engineering. Or composing music.

Based on such research, Peabody Preparatory now offers music classes to children from birth. There are separate classes for two-year-olds, three-year-olds, and other age groups. Fran Zarubick, dean of Peabody Preparatory, says that babies aren't exactly soaking up what they need to compose sonatas: "The real thing that's going on with a child is putting music in a positive environment, creating a bonding experience with parents. They are not really learning something. But they are having a good time and the neural pathways they use to listen to music can open up other neural pathways for learning."

In a Hopkins study begun this fall, Peter Jusczyk, professor of psychology and cognitive science, is exploring how early in life these musical patterns are recorded in the brain. The research parallels a study Jusczyk published in Science magazine in September that showed children apparently file away words at eight months old, and perhaps earlier (see The Origins of Babble). "We found that they learn sound patterns, such as language, and store them," Jusczyk says. "But would the same thing happen with music? My guess is that it will, and if it does, then it's really a general ability to retain highly structured material once a kind of pattern is repeated."

In the new study, Hopkins researchers are playing 32-minute piano concertos to babies for 10 days over a two-week period. At least 15 babies are then brought into the lab for testing: in a cubicle they will again hear the music played, but only as long as they stare at a blinking red light near the speaker. The idea: the longer they stare, the higher their interest. One projection: they'll stare longer at the light when the music they recognize plays, versus unfamiliar music.

A related study, which Jusczyk conducted several years ago while at the University of Oregon, showed that babies as young as four-and-a-half-months prefer music that's sequential--not broken up by odd, misplaced pauses. Babies stared longer at the blinking light when Mozart minuets were played correctly. So, even babies' brains may understand the concept of order.

Then, the question remains: Could early music training--including playing Beethoven to babies in the womb or giving piano lessons to two-year-olds--really help create math whizzes, or vice versa? As ongoing studies evaluate possible connections, adding to the wisdom of the ancient Greeks, modern anecdotal evidence may support the idea. For high-level mathematicians, for instance, the music/math connection is almost cliché.

Dan Naiman, Hopkins professor of mathematical sciences and a specialist in statistics, is also among a fraternity of math instructors who are musicians. "Go to any math department in the country and you'll find some serious musicians," he says. There are several musicians in Hopkins math and math sciences genres. They include Lenore Cowen, assistant professor of math sciences, who plays violin; Leslie Hall, associate professor of math sciences, who plays piano; and Steven Zucker, Hopkins professor of mathematics, who also plays classical piano.

FROM PERSONAL EXPERIENCE, Naiman can draw some connections between learning music and math. He started playing clarinet in the third grade and later switched to saxophone, performing for years in various jazz quartets and bands. In the fourth grade, he learned probability from a book his father gave him about poker. By fifth grade, he was studying algebra.

"In jazz, especially, you look for patterns that have a mathematical structure--the repetition, the way you put it together, the exercises are very mathematical," he says. "But I don't want to say that the product itself is mathematical. There are big differences. Music is passionate and very sensual. Mathematicians may see math that way, but try to explain that to anyone else and you sound like an idiot."

Joanne P. Cavanaugh is a senior writer for Johns Hopkins Magazine.