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
enter the room silently, quietly flexing our fingers - readying
ourselves to create rhythms with our hands upon classroom chairs.
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.
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
do we attain this? Very easily and simply - we kneel upon the
floor in front of plastic chairs. We, ourselves, are the expensive
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
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.
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.
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
Results of the Class
and several other New Mexican schools introduce rhythmic, athletic,
and high energy movement and music to 8 to 12 year olds. While the
children and their families obviously enjoy both the performance and
the work that goes into it, the program has even deeper intentions.
The teachers know that few if any of their students will become professional
musicians or dancers. They do hope that all of them will become life-long
learners who search for excellence in all their activities. The drumming
program nourishes thoughtful habits of mind and helps the children
build contacts between the rhythms they produce and basic knowledge
and understanding they need in everyday life.
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
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
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
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)
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
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 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."
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.
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.
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.
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
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.
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
Attitutdes toward Musical and Kinesthetic Abilities
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.
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.
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.
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
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.
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.
Benefits for Students
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.
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.
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.