Two subject areas that might appear to be impervious to integration are mathematics - viewed as abstract and cold by many - and dance and music - usually perceived as emotional. The work of several New Mexico schools shows that these two areas have much to offer each other. Mathematics and music share a concern with numbers and patterns of change. In music and dance these patterns are called rhythm.

"We enter the room silently, quietly flexing our fingers - readying ourselves to create rhythms with our hands upon classroom chairs.

"The music created from the patterning of our hands tapping the chairs in sync takes us to a mathematic realm as we fit our notes and time into an artistic form. We are lifted to a place and time with a oneness of music and math.

"One-eighth time takes us to a fast movement, a flurry of fingers, a creation of a rhythm above all we have done. We slow down to one-half time, easing our fingers to a slower time frame, artistically drumming fractions.

"How do we attain this? Very easily and simply - we kneel upon the floor in front of plastic chairs. We, ourselves, are the expensive instruments."

Cris Marie Alton, teacher of ten, eleven, and twelve year olds at Alvord Elementary School, a collaboration of the Santa Fe Public Schools and the College of Santa Fe, New Mexico

Each year the students at the Alvord Elementary School repeat this scene. Before their families and friends they drum and leap and twirl. The hall is packed since from year to year community members look forward to receiving their invitations to the Alvord performance. The students have also been looking forward to this day with the combination of excitement and fear that all performers feel. They know that, however they do, their efforts will be acknowledged, rewarded, and remembered by the community in which they live.

The Performance

The performance begins with two drum captains beating their hands on chairs. The simple rhythm they produce guides the performers into the room. Each child, dressed like his neighbors, takes his place behind a chair. Soon all of the children have joined in the rhythm set by the captains as they methodically strike the chairs in front of them. This sound does not last long; soon it moves to ever more complex rhythms. By the end of the hour, the intricate beating has brought the audience to its feet as the whole room joins in stomping and clapping to the performers' sounds.

All semester the students have worked toward this night. Finally, they have put themselves and their abilities on public display and their peers and parents have shown them that their performance and understanding is valued. After the drumming, performers and parents gather to talk over the stimulating evening. In the excitement, only a few remember how much more the students have learned from their percussive work.

The Results of the Class

Through the drumming program, the students realize that performance is a way to share learning. They learn to define and work toward goals consistent with their individual abilities and to cooperate with others in reaching shared goals. As a result of such shared work, they come to see how the work of each person depends on the work of others.

Students who have been through the drumming experience have also learned to link movement, music, and rhythm to basic mathematical concepts. As part of the drumming program, Valdez Abeyta y Valdez, a music teacher at several New Mexico schools, and faculty from Alvord work together to help all children comprehend the mathematics that underlies the world we experience every day. In this program teachers are often as much learners as students. Even those teachers who have been through a drumming exercise many times learn new things each year.

While dance and music motivate most students through exciting action, the excitement these disciplines generate can also help them understand more abstract concepts. They learn to work out mathematical meaning in new and concrete ways. From a natural and intuitive understanding of how his or her own body works, a student can develop an awareness of the working of mathematics in the physical world. (See art standards)

For example, clapping two half beats in the place of one whole beat can help children begin to understand the meaning of fractions. Learning to beat half time, quarter time, and eighth time, children can feel fractions in their own bones as they also begin to work with the larger mathematical theme of patterns and their changes. (See math standards)

The idea of patterns will surface again as the students put together steps, sounds, and movements to create their performance. This early attempt at choreography can also move them into more mathematics, since an interested teacher can help students compare the shapes they make with their bodies and space during dance to similar geometric shapes.

In using rhythm to teach mathematics Alvord is part of a long tradition. Western culture has recognized the connection between music and mathematics since the time of the ancient Greeks. The Pythagoreans (of the famous theorem regarding the square of the hypotenuse of a right triangle) used harmony and rhythm as a basis for their mathematical ideas. Music teachers have long expressed the notion that learning music improves mathematical abilities and scientists recently established experimentally that the link exists.

The Mozart Effect

During the 1990s researchers at University of California at Irvine led by Gordon Shaw, a physicist, and Frances Rauscher, a cellist and a psychologist, studied the relation between music and intelligence. In one study they divided three and four year olds into three groups. One group received piano lessons, another private computer lessons, and a third either studied singing or had no special lessons at all. After six months, the group studying piano was the only one to show a significant increase in spatial-temporal reasoning; in fact, these children scored 34 percent higher than did the next group. (Spatial-temporal reasoning is required for certain higher brain functions and is employed in chess, mathematics, engineering, and composing music. It enables the thinker to put mental images into many different forms without having to use a concrete model of any of the forms.) This increase in spatial-temporal reasoning has been dubbed the "Mozart Effect."

Musical intelligence follows the same sequences as spatial-temporal reasoning, so learning music is like a warm-up exercise for these other reasoning abilities. Researchers believe that music calls on abilities that increase student capacity to learn in other areas. Musical learning, for example, helps students develop such mental skills as concentration, symbol recognition, and memory. Some researchers go so far as to say that musical activity repatterns neurons to improve cortical functioning. Staging an actual performance also teaches students the value of cooperation and collaboration.

Musical training appears to function on several levels. First, musical activities call on the entire body: the muscles of the arms and hands, those that control breathing and the voice, the coordination of movements. At the same time, music can lead students through a series of victories that give them a sense of the necessary sequences of learning and pacing. They find out how it feels to accomplish their own learning goals and develop sought-after self-esteem through actually learning difficult material. Their pictures of themselves as learners become more realistic and each student develops a better idea of how she or he learns.

Multiple Intelligences

The work of Howard Gardner, of Project Zero at Harvard University, has shown that each of us has a mixture of different ways of learning. In his first book, Frames of Mind (1985), Gardner identified seven "intelligences"; recently, he has added an eighth intelligence. These intelligences include the musical and the bodily-kinesthetic, as well as the logical-mathematical. Gardner points out that people are born with all intelligences but usually only one or two are fully developed in any individual.

While he has identified individual "intelligences," Gardner emphasizes that actual intelligence is inseparable since each intelligence involves the others. Once a learner has identified the way she learns best, it is important that she not try to learn only in that one way. If a teacher helps a student identify her most natural way to learn and, as a result, that student begins to say, for example, "I am a visual learner and that is the only way I learn," the student and the teacher will have missed one of the important meanings of Gardner's work: Each of us needs to work on learning in ways that are not our most immediate and natural way in order to become more complete human beings. The visual or the logical learner cannot rely on her most comfortable intelligence. If she is to become a strong learner, she must turn to other ways of learning. The ideal school gives all students experiences of learning in many different intelligences.

In the Alvord music classroom two intelligences - musical and bodily-kinesthetic - are used to open understanding of other domains. Students who are more comfortable in these two worlds can use their natural understandings and abilities to access other areas of knowledge. In addition, students whose strengths are in other intelligences can learn more about the physical and rhythmic aspects of their own lives.

Bodily-kinesthetic intelligence is awakened by movement. (See dance standards) Those with well-developed kinesthetic intelligences typically can control their body motions skillfully to reach certain goals and can do both fine and gross motor work with finesse. Dancers and athletes have developed this intelligence, and so have artisans, surgeons, mechanics, and instrumentalists. Many musicians and others have long noted the close relation between movement and music. Young people, especially, frequently find it impossible to listen to music without moving. These two intelligences seem made to go together in education.

Cultural Attitutdes toward Musical and Kinesthetic Abilities

U.S. culture tends to treat musical and kinesthetic abilities as innate, much as we have long assumed that mathematical ability is innate. Working from the assumption that musical ability cannot be taught, schools often suppose that most students have no musical interests or abilities. In contrast to some other intelligences, such as the linguistic and the logical-mathematical, musical intelligence is not highly valued by our education system. It is usually assumed that only those with special interest or aptitudes in the discipline should study music intensively and for the long term. The bodily-kinesthetic intelligence is also channeled onto the athletic field in the U. S. public school system.

Except for those children with special interests in chorus, band, or the school orchestra, music education essentially ends with the elementary grades in the United States. While being in the band or chorus may be a competitive goal in many schools, its popularity does not translate into broad music training for all students - just as most students do not receive the kinesthetic training that student athletes receive. Since they are assumed to have natural talent, even musically "gifted" students often receive no training in formal processing and understanding of music; instead they only practice and perform. As school budgets shrink, music programs (along with art and other "peripheral" subjects) decrease and what money is available for music will be concentrated on perfecting the performances of the few who are considered expert. Even some elementary schools have cut back so severely on music that young children only have an hour or two a month on the subject, taught by a roving teacher.

Other cultures have different assumptions about musical and kinesthetic abilities and how they should be nurtured. Many cultures see their musical heritage as something each child should understand and be able to perform at some level. Dancing, drumming, and other performances are part of the life of the entire group rather than the domain of a few talented experts. In a few U.S. schools, including Alvord, this approach is also the norm. Alvord faculty also uses music to introduce students to the wide range of cultures in the world and in their own community.

Cultural Knowledge

Movements and music can be used as ancillaries to words and to help make meaning clearer for those who do not speak or do not understand English. Since music and movement do not have to be presented verbally, students with language differences can participate in class work with fewer frustrations.

In addition, music and dance can be part of a cultural learning experience. Alvord students can express interest in their own culture and in other cultures through the drumming they learn and the performances they present. They can weave together sounds and rhythms from the Middle Eastern, African, Native American, Japanese, Flamenco, and other musical traditions.

The teacher can show students that fundamental concepts of mathematics remain the same no matter how they are expressed. The relations between half notes, quarter notes, and whole notes are the same in different musical traditions, no matter how different the notes may sound from each other. In exploring these similarities and differences, children can become aware of the constants of mathematics, the sameness of human cultures and of their differences.

Other Benefits for Students

Cooperating on expressing the emotions of dance and practicing together to get movement and rhythm correct can teach children to work together just as well as working in cooperative learning groups can. Intrapersonal and interpersonal skills can both be improved in a performance. Many dance and music teachers emphasize team building as one of the major benefits of studying their disciplines and one that is as realizable as the team spirit of a school team.

Tying music and dance to another, more traditional, subject helps students see that their abilities have relevance to academic subjects. Children who are unmotivated in other classes can often blossom into hard workers once they have found their niches in music.

Since all students, including special education and gifted and talented students, are in the program, Alvord has offered the students an experience that knits the school together. The sense of unity fostered by the drumming is not artificial or fleeting. It comes out of real learning in an authentic setting. The unity extends throughout the community and between different generations since drumming experiences live on in the students' future learnings and in the memories of their parents and other community members.

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