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Making Mathematics Move

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

Picture of two kids writing

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 pur-ple (2 syllables) to indicate the length the notes is held.
Brown rods are whole notes. The students clap once and say brown-n-n-n (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.

Picture of kids playing patterns on chairs

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 warm-up 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.

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