Fear
keeps many teachers from trying to teach mathematics through music
or rhythm. Teachers often think they don't know enough about the
relationships between these subjects. The relationship is not that
mysterious and help does exist (for a start, see the resources
section). Moreover, showing students how an adult goes about learning
a new subject may well be one of the most important lessons a teacher
can pass along. Admitting to students that we don't know something
can be a daunting task for teachers, but the lessons learned from
this experience can stay with students for a lifetime.
People
can come up with other objections to tackling a project like this:
Supplies are bound to be expensive. The project will take up too
much valuable class time. Who can help me with this complicated
stuff? None of these problems is insurmountable.
Overcoming
Initial Difficulties
Expensive
equipment for music and dance is a luxury not a necessity. Students
can explore rhythm and movement with objects lying around the classroom
and at home. The class at Alvord uses the seats and backs of the
plastic chairs that fill classrooms throughout the United States.
Pots, discarded plastic bottles, and odd pieces of metal also make
resonating sounds, as do the children's own arms, legs, and chests.
Mention the popular dance group Stomp  they use all kinds of found
objects in their performances  and the students will probably be
able to name and find some of the objects the group uses. If a school
feels that it must have percussion instruments, simple and inexpensive
hand drums are adequate.
A
teacher who feels "rhythmically challenged" might want
to invest in one more piece of equipment: a simple metronome or,
even, a clock that ticks. A metronome makes any beat steadier as
the class grows in its understanding of rhythm.
Time
for this project is flexible, it can take a few days or a semester.
If the class is truly interdisciplinary, this investigation can
be a wiser use of time than having separate music, mathematics,
and science classes. A school could integrate the topic vertically
so students increase their learning in each year, rather than having
one teacher devote a large block of time to the exploration during
one year.
Students
may be the greatest resource the teacher has for this project. Many
students probably already have some kind of musical training. They
can explain concepts to the class or show the others how a rhythm
works on the instruments they play.
Other
teachers and people from the community can also be helpful in integrating
this material into the classroom. The teacher might consider inviting
the band instructor, cast from local dance productions, drummers
in local bands, choir masters from area churches, and similar experts
to help with these classes.
Exploring
Rhythm
Most
students have an intuitive understanding of music and rhythm. The
teacher's goal is to help them use that understanding to form a
bridge to unfamiliar material in mathematics. Start with something
they already understand  clapping their own hands. Young children
are used to clapping in the classroom. Often teachers clap for attention;
some classes regularly applaud those who have reached some milestone
in life or done well on an assignment.
Begin
by clapping with and for the children in a very simple pattern like
four equal and fast beats. Ask the students if they can repeat this
pattern. Then vary the beat (two fast and two normal or an easily
recognized rhythmic pattern from popular music). Ask the students
to complete the pattern of a rhythm they all know.
The
goal at this stage is to enjoy and explore the rhythm for its own
sake. Eventually, the students will need to consider certain basic
questions, even though they may not know the answer when the questions
are first posed:

What is the difference between clapping like this (fast beats
the equivalent of half notes) and clapping like this (steady
beats the equivalent of whole notes)?

Can you see the relationship of the fast beats to the slower
beats?
The
students will see that a whole note includes two half notes or four
quarter notes, but understanding of that concept should not be rushed.
At this stage the ideas of patterns, change, and repetition are
the important concepts to consider.
Building
Patterns
Now
the students can begin writing their own version of musical notation.
Like any system of notation, this version helps make music concrete
and preserves it for future use. The students will use this notation
system to present their own rhythmic ideas to each other and to
people from outside their classroom.
As
an introduction, use objects to stand in for specific beats. Something
the students are already familiar with is best, for example, Cuisenaire
rods. Colored strips of paper, beans of different colors and shapes,
or buttons are other possibilities. (For simplicity's sake the rest
of this activity will be written as if Cuisenaire rods are being
used.)
Assign
values to the colors. (For example, red Cuisenaire rods could be
designated as quarter notes, white rods as eighth notes, purples
as half notes, and browns as whole notes; these values will be followed
in the rest of this activity.) Since the rods of one color indicate
a specific beat length, the color helps the students control the
frequency and speed of beats.
The
students lay out the rods on a piece of chart paper or on a large
sheet of butcher paper with grids marked on it. The rods are arranged
to make a pattern  say, two reds, one purple and a brown  and
this pattern is repeated several times. The children then "read"
the patterns by clapping or beating their instruments  in the above
example, two quarter beats, followed by a half beat and a whole.
It helps to keep the rhythm steady if the students say the color
word as they clap or beat. When tapping out quarter notes, for example,
they say "red" with each beat. Point out that using this
system will make it possible to write out any rhythm the students
can think of.
On
their own some students will notice that clapping two reds (quarter
notes) equals the time for clapping one purple (half note). See
if these students can discuss their observations with the class.
Help all the students realize that the reds must be clapped twice
as fast as the purple. Show the students that two purple rods make
one brown rod. Clap the beat for them. Have them clap the beats
with partners and talk about how many of any one beat it takes to
make a whole beat. Have confident students demonstrate to the rest
of the class that four beats of the red rods equal one beat of the
brown and that changes can be made in the patterns. (See mathematics
standards)
Testing
Patterns
Now
the students develop their own rhythms, write them in the new notation
system, and test to see if they can move rhythms out of their own
minds to the understanding of others. Give them an assignment: For
example, each small group or pair of students is to develop a rhythm
using four red rods and four whites (whites should be used in pairs
at this stage). The working groups draw their rhythmic pattern on
graph paper and then clap it for themselves.
If
you have Macromedia
Flash installed, you can create your own rhythms with the interactive
rhythm builder.



Red
Cuisenaire rods are quarter notes 



White
rods are eighth notes and are clapped twice in the same
frame as one red quarter note 



Purple
rods are half notes. The students clap once and say purple
(2 syllables) to indicate the length the notes is held. 



Brown
rods are whole notes. The students clap once and say brownnnn
(as 4 syllables) to indicate the length the note is held. 
















red 
red 
red 
red 
red 







red 
red 
red 
white 
white 
red 








pur 
ple 










brown 
n 
n 
n 



When
the members of the small groups have agreed that they understand
their own pattern, they ask others in the class to clap it also.
In this way, each group tests to see if the notation they have used
is understandable to others. (They can tape their clapping to see
if the other students match it when they reproduce the pattern.)
Now
the students rearrange their rhythms  without increasing or decreasing
the numbers of counters. After they have rearranged their patterns
they clap the new rhythm. How many ways can the eight rods be rearranged?
Can they clap out each rearrangement? Can they write it out so others
can clap it?
The
teacher can begin to compare the relationship between whole, eighth,
and quarter notes and fractions. Help them to see how quarter notes
and half notes, for example, make up whole notes. How many changes
can you make to get a whole note? What would happen if you had an
extra half note?
The
class and the teacher need to work at a pace that is comfortable
to them. If the teacher thinks it is possible, the class might work
on this project over an entire semester. Eventually, the students
will work out their understandings of the relations among the notes,
of how to indicate these relations on paper, and of transferring
the notations from paper to practice.
Eventually,
the teacher will have to introduce the concept of rests: Explain
that sometimes in a piece of music no noise or sound is needed but
the beat has to go on. This place in the music where there is no
sound is called a rest. Rests are useful for varying the beat. (This
concept may be conveyed best by playing a few selections and asking
the students to indicate where the rests are.) As a group the class
needs to work out a way to indicate rests. For example, they may
want to use a nonsense sound to indicate a rest and to put it above
the line rather than on the line with the other rods. (At Alvord
they use "phtt" as their sound indicator and put the appropriate
rods above the line.) To understand each other's notations, the
whole class will have to agree on the same system.
Preparing
for Performance
The
students are now ready to put their learning into practice by preparing
for a public performance. Performance will be both a celebration
and an assessment of their learning. They will need to continue
their rod exercises as warmup exercises for their performance and
as a method for explaining the rhythms they have imagined to each
other. They will also begin experimenting with different materials
and rhythms from many sources. What kinds of sounds can they make
from abandoned tires or blocks of wood? How can these differences
be incorporated into their performance? What effect would they have
on the finished product?
The
teacher could introduce the students to rhythms from other sources.
Other cultures emphasize different rhythmical structures and patterns.
Do the students find these more difficult to beat out than those
they have written on their own? Students can discuss folk dances
with their parents and others in the community, and guests might
visit the class to discuss their musical traditions. Recorded music
and videos might also be useful in bringing rhythms of other cultures
to the attention of the students. The teacher can use these discussions
to show that the mathematics behind the music remains the same across
cultures.
The
teacher can also present rhythms that occur in nature: crickets'
chirps, frog calls, raindrops. How will the children interpret these
sounds with their instruments? Can they see any relations between
these sounds and the rhythms that form human music? Is the mathematical
structure behind these rhythms the same as the structure in others
they have studied?
The
students now narrow their search to a rhythm or series of rhythms
they feel comfortable with and begin describing it in their notation
system and beating it out in the classroom. Soon they add movements
to the sounds and put them together into a beginning choreography.
In
private conferences with students and work groups the teacher can
ensure that the mathematical concepts are clear in their minds.
Ask them to explain the patterns they see in their rhythms. Probing
the use of terms like "whole note" and "quarter beat"
will show the level of mathematical understanding each child has
reached. These individual assessments can then feed into polishing
the public assessment, the final performance.
The
performance can be a powerful assessment piece. The teacher is not
alone in telling children "how you did." Even the reactions
of audience members will not be the ultimate assessment. The performers
can judge their own work as they present it. If the relations between
rhythm and mathematics have been made clear to the students, the
performance itself will further embed their memories with knowledge
of mathematics as well as the joy of performance.
