# The Intersection of Two Unlikely Worlds: Ratios and Drums.

When mathematics lessons are linked with personal experiences, typically, the result is that the student gains a stronger understanding of the content than if the lessons are isolated and unconnected. This premise was recently supported in a local fifth-grade classroom. The students learned to play three mathematically disparate rhythms on conga drums as an introduction to an exploration of ratio. Ratios connect naturally with African and Afro-Cuban drumming because the drummer's combination of many rhythms, each with a pattern repetition of different length, results in a polyrhythmic song. The pattern repetitions are comprised of a given quantity of one type of beat mixed with a specified quantity of another type of beat, or a ratio of one beat to the other one. Although we completed this lesson with a group of children of whom half were African American, we believe this lesson can be powerful and meaningful for children of all ethnic backgrounds.

Ratios

Mathematically, a ratio is a comparison of two quantities. Figure 1 is a visual representation of the ratio of circles to squares, which in this instance is 6:8. Ratios exist in nearly every aspect of children's lives, such as in comparing the number of hits with the number of times at bat, the number of red marbles with that of blue marbles, or the calories with tablespoons of sugar. In fact, children probably use ratios without realizing that their activities are mathematical. For example, counting, an activity learned in early childhood, is an example of a 1:1 ratio. When children create a bar graph to represent data and compare the lengths of the bars, they are, in effect, dealing with ratios.

Kieren (1988) created a model for describing a student's developing knowledge about rational numbers. In his model, the first level of knowledge is referred to as "ethnomathematic." In general, ethnomathematics is a field of study, founded by D'Ambrosio (1985), that emphasizes the sociocultural environment in which a person "does" mathematics. (See "In My Opinion: What Is Mathematics, and How Can It Help Children in Schools?" by Ubiratan D'Ambrosio in this issue.) According to Kieren, this ethnomathematic knowledge is uniquely constructed because it is based on a person's life in a particular environment. In addition, the person using this knowledge may not even see it as "mathematical." Thus, a teacher can use a student's unique real-world knowledge to help the child learn about ratios. The teacher can also bring out-of-school contexts into the classroom to create a common collection of real-world experiences for all the children in the class. The teacher can then facilitate the children's understanding of the mathematics of these experiences, such as in this example, the study of ratio through drumming.

Teaching in accordance with Kieren's (1988) theory, the teacher then plans lessons to move the students toward and through more advanced levels of knowledge about ratio. The next two levels are most relevant to elementary school children. At level 2, children develop their intuitive knowledge by combining thought, informal language, and image. After identifying the understanding of ratio embedded in the ethnomathematic knowledge of the children, the teacher can plan lessons that will help them think more formally about those experiences. At level 3, children come to understand and use symbols through conventional language, notation, and algorithms. This growth beyond the ethnomathematic level is most likely to develop through social interactions during instruction (Kieren 1988).

Rich mathematical content is inherent in African rhythms and embedded in drumming a given song. Because of its mathematical complexities, African and Afro-Cuban drumming can be used to explore the notion of ratio through class discussion. Several other mathematical connections can also be emphasized. Children could measure and compare the drums using a variety of units, such as pounds, cubic centimeters, or inches. They could build drums, exploring the notion of mathematical similarity, which is an application of ratios. We focused on the concept of ratio because it appeared to offer the richest cultural connection. Cultural etiquette dictates that a person must know some history about African drumming before learning to play African drum rhythms. In addition, some rudimentary knowledge about drumming is essential for critical examination of the mathematics lesson described in this article.

Drumming

The oral traditions of African villages indicate that ritual drummers have existed in parts of Africa since the fourteenth century and probably much earlier (Grund 1985). From the perspective of professional and community drumming circles, the human body possesses and operates by many rhythms, the most central of which is the heart (Anderson 1994; Cohen and McFall 1994). As a direct consequence of this knowledge, professional drummers are trained to find the "pulse," or "heartbeat," of a crowd and then to use that natural rhythm to sway the crowd (Steed 1984). This effect is possible because the drummer's rhythm and the crowd's corporate rhythm are, in essence, a mathematical ratio. When the drummer locks into the crowd's rhythm, the ratio is 1:1. When the drummer wants to move the crowd, he or she moves away from the 1:1 ratio. For centuries, the Griots (pronounced GREE-ohs) and other ritual drummers of Africa have used this principle to sway the dancers who move to their spirited drumming (Charreaux 1992; L ocke 1987). In application, a drummer's success in swaying the dancers is directly proportional to his or her ability to handle complex mathematical patterns in the form of rhythms.

The ritual drummer is separated from the ordinary drummer by the extensive training that he or she undergoes and the level of expertise that results. For example, the Batimbo, or master drummers, of the Batutsi family of Burundi could spend a month or more in one drumming ceremony alone. They and the drums they played were reserved for the listening pleasure of the king, Mwami, and his ancestors (Grund 1985). The role of this intense ceremony was simply to honor the king, who was viewed as godlike in the cultural traditions of the Burundi people. The master drummers' special position and relationship with the king afforded them the rank of nobility in their home villages. This rank came at great cost, however, because drummers sometimes died from exhaustion during the month-long drumming ceremonies.

Not only did the drummers undergo extensive training, but their drums also underwent a yearlong preparation process that culminated in the Mwami ceremony. After the close of the ceremony, the Batimbo were allowed to play the drums for only a few days longer before the drums were returned to their home until the following year (Grund 1985). Their home was a special hut made exclusively for drum storage.

Rhythms

Children's school knowledge of patterning is rigorously developed because much of mathematics is based on patterns. Beginning in kindergarten, children copy, extend, and create patterns (NCTM 1989). A musical rhythm is a special kind of pattern, so a natural place to blend music and mathematics is through analyzing patterns in various rhythms.

In the African music previously described, several rhythms are often played at the same time. Just as polygon is a generalization for all geometric shapes that are many sided, so is polyrhythmic a description of music that results from the simultaneous playing of many rhythms. The polyrhythmic complexities of ancient African music, such as that in the Mwami ceremony, are sometimes heard in modern Afro-Cuban and Afro-Caribbean music. Such musical styles as salsa and rumba have elements of the original music brought to the Latin American region by Africans who were sold as slaves to the owners of sugar and tobacco plantations (Boissiere 1992). Although many rhythms contribute to the total Afro-Cuban repertoire, this article describes only three: (1) clave, (2) bembe ostinato, and (3) tumbao. The clave is discussed first. See figure 2.

The word clave, or "la clave," as it might appear in writings of Afro-Cuban origin, is the rhythmic foundation for Afro-Cuban music and gives that music its distinctive feel (Caruba 1995; Boissiere 1992; Gerard and Sheller 1989). In Western understanding, the two-measure clave rhythm extends across twelve beats, including rests. Figure 2 shows a visual image of two "measures" of the rhythm; the X is a hit, and the space is a rest. Although the clave rhythm is not always played, it is ever present in the drummers' minds.

The second rhythm discussed is the bembe ostinato. An ostinato is a short melody or phrase that is repeated in the context of a song (Bozzio and Olson 1993). The bembe ostinato was chosen for use in this activity because it contains a particular ostinato (see fig. 2).

Finally, the tumbao (see fig. 2) is an Afro-Cuban rhythm typically played on the tumbadora drum, that is, a large conga drum (Cohen and Cohen 1992; Boissiere 1992). Like all Afro-Cuban rhythms, the tumbao is based on the clave, but it is also the most popular rhythm on which to base other conga-drum rhythms (Boissiere 1992). Typically, beginning conga drummers learn the tumbao rhythm immediately after grasping the importance of the clave (Cohen and McFall 1994). Figure 2 contains a visual image of the tumbao rhythm.

Playing techniques

A distinguishing feature of the tumbao is the variety of ways in which the conga player's hands strike the drum. These varying techniques produce different sounds from the drum and engrave the rhythms with musical textures. In playing the tumbao, the unique rhythm emerges through the particular tones resulting from a specific eight-beat combination of four different styles of the hand's contact with the drum. The strokes are (a) open tone (using the top of the palm, just below the fingers), (b) slap (with the fingers), (c) heel (of the palm), and, (d) toe (fingertips). See figure 3 for a symbolic representation of the strokes (Dworsky and Sanshy 1994).

Finding ratios in rhythms

Ratios can be found in music by analyzing the music and comparing its rhythms. The bembe ostinato has six beats, and the tumbao has eight beats. When these rhythms are played at the same time, an ethnomathematical understanding of the ratio 6:8 can be demonstrated. Because children typically have out-of-school experiences with music, the teacher's use of a musical idea in teaching about ratios addresses the necessary first step suggested by Kieren (1988), that is, constructing ethnomathematical knowledge. The teacher then transforms the children's ethnomathematical knowledge of music into a school mathematics-of-music experience. At that point, the teacher draws on the students' familiarity with, and interest in, the musical rhythms to build the lesson about ratios.

In addition, by analyzing the result of two rhythms played simultaneously, we can begin to understand how the drummer's awareness and use of ratios can cause dancers to interact with the songs. This analysis further exemplifies the ethnomathematics of the ratios of African and AfroCuban drumming. The two-rhythm song created by the tumbao and the bembe ostinato matches the dancers' heartbeats at the simultaneous beginning points and falls away from their heartbeats during the course of the song. This constant fluctuation creates in each dancer a kind of rhythmic dissonance that must be resolved by the body. When the agitation is resolved, both rhythms are at a simultaneous count of "one," the beginning of a measure, and the dancer or listener is brought back into the music. When the agitation is at its height, both rhythms are away from the simultaneous count of one. In this manner, the drummer uses the ratio of the rhythms to influence each dancer's personal rhythm or heartbeat, which compels the dancers' bo dies to sway.

This rhythmic dissonance is the ethnomathematical basis of this lesson on ratio. The listener or learner who feels the agitation experiences a deeply personal interaction with ratio. This experience is also an ethnomathematical introduction to least common multiple (LCM). When the number of beats is the LCM of the two rhythms, the rhythms are at a simultaneous count of one, which is the point at which the dancer is unknowingly brought into the music.

The following lesson describes a fifth-grade teacher's attempt to give students a feel for ratio and her efforts to facilitate some discussion about ratios.

The lesson

In this four-day lesson, fifth-grade students learned some drumming history so as to play the tumbao and the bembe ostinato. The students played and quantified the rhythms by counting the beats and comparing those values. The primary goal was for the students to appreciate the ethnomathematical connections in the rhythms while they came to understand and manipulate ratios. The lesson culminated with the children using ratios that were based on their own created rhythms.

We introduced students to the background of drumming. Without this contextual knowledge, drumming can become an isolated artifact of a disjointed lesson. With this knowledge, however, students can learn about the number of beats in the different rhythms that they hear and about how African and Afro-Cuban music is played with those different rhythms.

The drummer opened the lesson by playing the tumbao against the ostinato. Students were asked to consider how this polyrhythmic sensation felt. One child said that the experience felt as if the two rhythms "were at war with each other." After discussing other students' similar feelings, the teacher introduced the conventional mathematics word ratio to help students describe the situation.

Next, the students collectively repeated the ostinato rhythm until they could hold it against the eight-count tumbao rhythm played by the musician. A different sense of ratio emerges as students themselves must maintain the six-beat rhythm against the eight-beat rhythm. "It was hard to keep the beat against Mr. S.," said one student. The students were able to develop a beginning understanding of ratio as a definitive part of African and Afro-Cuban music, agreeing that the rhythms could be described in a 6:8 ratio. The children also recorded the rhythms (see fig. 4). Once they were able to hold the bembe ostinato's six-count rhythm against the drummer's eight-count tumbao, the drummer taught the students the various hand strokes that he used in drumming the tumbao.

Students were then urged to consider how many times the ostinato and the tumbao would be repeated before simultaneously coming back to a beginning, to the count of one. Figure 5 shows the hand strokes noted by the teacher as the students instructed her to record the two repeating patterns simultaneously. As shown, she recorded the information in a string, using the symbolic representations for the hand strokes. This visual presentation of the rhythmic patterns helped the students determine that the ostinato would be played four times and the tumbao would be played three times. When we asked the students to explain why they thought it was four times and three times, one student said, "It took twenty-four beats for both rhythms to go all the way around." When pressed further, a different student said, "It might be because 6 and 8 are each factors of 24." This student added LCM to the lesson, which is another natural connection with African and Afro Cuban drumming. This class discussion related to the last ques tion on the record sheet.

Finally, the students invented two different rhythms. First, they created and recorded counted rhythms. Then they created rhythms that required the various hand strokes, as in the tumbao, and recorded those rhythms according to the symbols. Figure 6 shows Kaitlyn's creation of a counted rhythm. Figure 7 shows Kavaughna's work in creating a hand-stroke rhythm.

After finishing the creative work, each child played his or her rhythm against another rhythm, usually the tumbao or an ostinato, with the conga drummer. Each child decided whether the conga drummer would play the child's new rhythm while the child played the ostinato or whether the child would play his or her rhythm while the conga drummer played the ostinato or tumbao.

Some students had difficulty maintaining the beat because of the disparate ratio of the two rhythms. The students were then asked two questions: (1) "What is the ratio of your rhythm to the conga drummer's rhythm?" and (2) "When will the two rhythms come together again?" For the counted rhythm, students answered these questions out loud in class. Kavaughna wrote the answers on her worksheet and appears to have gained an understanding of ratios. In class, the students described the number of times that their rhythms would need to be played against the clave for both rhythms to land on a count of one. These ideas laid the foundation for an exploration of LCM.

By the end of the fourth day, several of the students demonstrated an application of LCM. They had learned to feel the point at which the various rhythms had come full circle to a common beginning. This understanding was demonstrated when both the drummer and the child stopped playing at the same time without any prompting; they both felt the rhythms come back to a count of one. Students explained the number of repetitions of their created rhythms that would be needed to come full circle if the rhythms were played simultaneously with the tumbao. Some students answered with a common multiple, but more often, they gave the LCM. One child explained, "The bembe ostinato and the tumbao would meet back up after twenty-four beats because 6 and 8 both go into 24." He further explained that the two rhythms would also come together at forty-eight beats! This explanation grew from his work with the drums, which clearly set the conceptual foundation for his future study of common multiples.

Conclusions

Most children like drums and have had experiences with playing, or at least hearing, rhythms. Drumming represents one possible avenue for a mathematical foray into children's ethnomathematical knowledge. More specifically, the ways in which conga drummers play rhythms against one another make a natural setting for the study of ratio. "I had fun learning about bembe and conga ... and I can even explain ratio," said one student. We believe this statement represents the thoughts of all the students in our group and shows strong support for the notion that students can learn mathematics through related personal connections with culture.

When the students were encouraged to invent their own rhythms, a few of the creations were incredibly difficult. At times, the students forgot to count rests in the number of beats in their rhythms. One student created a rhythm that impressed the expert drummer. When it was played against the ostinato, one female student in the class could hardly keep from dancing. This delightful opportunity reminded students about what a ratio "feels like" while helping them understand the way that African dancers feel when the rhythms are "off," as one child put it. Experiences like this one embed the ratio concept in the memory in a way that traditional classroom settings often fail to do.

This mathematical experience also helped the children actually feel the ratio. Given that mathematical ideas exist in the mind and cannot be touched, the opportunity to feel a ratio gives students an additional tool to reflect on their developing knowledge of ratio. "I think it was cool that ratio has a lot to do with the drums," said one student. She continued, "Now I know what ratios ....."

When asked about the ratios of their rhythms to the ostinato, all children were able to use correct technical symbols to record the ratio n:6, in which n is the number of beats in the student's rhythm. "I had a fourteen-count beat and he had a six-count one; that makes 14:6," said one student. This student learned about ratio in the context of creating and analyzing her own rhythm.

All the children came to appreciate the historical background of African and Afro-Cuban drumming. The children of African or Afro-Cuban descent learned more about themselves and their cultures. Similar lessons could be constructed using drums from other parts of the world to help children of other ethnic backgrounds assimilate information about their cultures while learning mathematics. We found ethnomathematics at its best in the intersection of drums and ratios.

Anthony Stevens, acstevens@iastate.edu, and Janet Sharp, pipurr@iastate.edu, are colleagues at Iowa State University, Ames, IA 50011. Stevens is a quantitative research specialist who infuses his lifelong percussion expertise with mathematics and statistics for his work with children. Sharp teaches elementary and secondary methods courses to both undergraduate and graduate students and is interested in developing innovative, effective mathematics pedagogy. Becky Nelson is a teacher and assistant principal at Perkins Elementary School in Des Moines, IA 50311. She enthusiastically embraces new opportunities to share the wonder of learning with her students.

References

Anderson, J. R. The Big Bang: In the Beginning Was the Drum. Ellipsis Arts, 1994. CD and book.

Boissiere, Guy, producer, and Karen Panica, director. Conga Drumming and Afro-Caribbean Rhythms. Parkside Queens, N.Y.: Alchemy Pictures, 1992. Film.

Bozzio, Terry, producer, and Rex Olson. Melodic Drumming and the Ostinato. Brea, Calif.: Paiste America, 1993. Film.

Caruba, Glen. Afro-Cuban Drumming: A Comprehensive Guide to Traditional and Contemporary Styles. Fullerton, Calif.: Centerstream Publishing, 1995. CD and book.

Charreaux, Jacques. Percussions Mandinques, by Adama Drame. Playa Sound Collection, 1992. CD and book.

Cohen, Martin, and Wayne Cohen, producers, and J. Fedele, director. LP Adventures in Rhythm: Close-Up on Congas, Vol. 1 with Richie Gajate-Garcia. Garfield, N.J.: LP Music Group, 1992. Film.

Cohen, Wayne, producer, and Michael McFall, director. Community Drumming for Health, Happiness with Jim Greiner. Garfield, NJ.: LP Music Group, 1994. Film.

D'Ambrosio, Ubiratan. "Ethnomathematics and Its Place in the History and Pedagogy of Mathematics." For the Learning of Mathematics 5(1) (1985): 44--48.

-----. "In My Opinion: What Is Mathematics, and How Can It Help Children in Schools?" Teaching Children Mathematics 7 (February 2001): 308-10.

Dworsky, Alan, and Betsy Sanshy. Conga Drumming: A Beginner's Guide to Playing with Time. Minneapolis, Minn.: Dancing Hands Music, 1994.

Gerard, Charley, and Marty Sheller. Salsa! The Rhythm of Latin Music. Crown Point, Ind.: White Cliffs Media, 1989.

Grund, Francoise. Les Maitres-Tambours du Burundi. Paris: Arion, 1985. CD and book.

Kieren, Thomas E. "Personal Knowledge of Rational Numbers: Its Intuitive and Formal Development." In Number concepts and Operations in the Middle Grades, edited by James Hiebert and Merlyn J. Behr, 162-181. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1988.

Locke, David. Drum Gahu: The Rhythms of West African Drumming. Crown Point, Ind.: White Cliffs Media, 1987.

National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989.

Steed, Rick. Personal communication, 1984.

Where's the Math?

The least common multiple (LCM) of two numbers is the smallest positive number that is a multiple of both. One way to find the LCM is to list the multiples of both numbers until you find one that is common. See the chart below for an example using 3 and 4.
```X 1 2 3  4  5
3 3 6 9  12 15
4 4 8 12 16 20
```

Picture two drums. One drum is hit every three counts, and one is hit every four counts. They would both hit the twelfth count. Challenge the students to tell you when the drums would again be hit simultaneously.
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