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Mathematics and Dance: A Mutual Friendship

Brittany McCarthy

MATH 416-X02

Spring 2012

MATHEMATICS AND DANCE: A MUTUAL FRIENDSHIP

Table of Contents

Abstract 2

Mathematics and Dance: A Mutual Friendship 2-3

Ways in which Math is Affiliated with Dance 3

Why Dance Could Affect Math Concept Comprehension 3-4

Teaching Math through Dance 4

Algebra and Dance 5-6

Kids get Excited for Math 6

Multiple intelligences 7-8

Viewing the Dance floor as a Cartesian Plane 8

Dance and Math Through Time and in Space 9

Calculations Involved with Dance Turns 10-11

The Musical Aspects Contributing to Mathematical Understanding 11

Choreography and Figures 12

Studies/Charts/Statistics 12-14

Different Styles for Different Concepts 14-15

Personal Experience 15

Analysis 16

References 17

Page 1


Abstract

This essay examines the dynamics of both mathematics and dance, each of which has an

influence on the other. From counting beats and doing turns, to maintaining formations on the

dance floor, and using different shapes, angles and levels to put together a combination, reading

this paper will prove that the relationship between math and dance is closer than one may think.

Nowadays, many teachers use dance to teach math classes and keep the children engaged, while

learning kinesthetically. The rhythms and patterns of dance are mathematical in nature, and thus

mathematics can be part of how dance is taught. This works from the opposite perspective, also.

Students can learn concepts of geometry and mathematical sequences by acting them out through

dance (Burg, 2005). Charts and stories will be introduced that show just how effective this

method is for teaching children no matter their learning style. The mathematical aspects of

dance will spark interest for those open to the idea of dance and math coexisting.

Mathematics and Dance:

A Mutual Friendship

What do dance and mathematics have in common? More than one might realize. Dance deals

with spacing across what one may think of as a Cartesian plane. The dancer needs to travel from

point A to point B and make it look effortless while using the music to hear the many beats.

Different styles have different counts and of course different shapes and formations to put

together entertaining choreography. When it comes to teaching, Westreich says that dance is

often taught by having students repeat the organizational structure (the steps) until they can

perform the entire sequence without any thought. Similarly, mathematics is often taught by

having students repeat formulas or calculations (e.g., multiplication tables) until they have

memorized them- again, without any thought or analysis (2002). Teaching math to dancers will

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help their understanding as the reader will learn, and teaching dance to mathematicians will open

their eyes to how fun and abstract math can be when applied to movement and one’s own body.

Ways in which Math is Affiliated with Dance

Mathematics is made up of many elements and uses terminology that is also used when

choreographers teach routines. A choreographer must keep in mind that the dancers must paint a

picture for the audience by creating symmetry and levels. While one dancer may be leaping

across the stage, another may be on the floor. As said by Jennifer Burg, dance movements and

choreography can be described in mathematical terms, e.g., symmetry, correlations, patterns,

geometrical shapes, tessellations, and even chaos theory (Burg, 2005). Without these

mathematical patterns, dance would be boring and still; In fact, it would not be dance at all.

The simplicity, precision, purity and elegance of dance are also inherent in the discipline of

math. Dance often attracts those who use their abstract reasoning capabilities in science and math

fields (Hackney, 2006). It is easy to see why many dancers are intrigued by mathematics.

Why Dance Could Affect Math Concept Comprehension

As discussed before, there are many concepts that relate both to math and dance. When students

act out mathematical concepts with steps, movements, and gestures, the ideas become real to

them. Weaving dance into mathematics instruction is viewed as a way of integrating the arts into

the mainstream curriculum and, at the same time, making mathematics more interesting and

extending the attention span of students (Burg, 2005). When a person dances he or she is using

their body to understand the concepts of patterns, shapes, levels etc. which is more than just

hearing the terms and learning them. Hackney says, dance instruction may enhance students’

mathematical skills, whether they are kinesthetic, visual/spatial, auditory/musical learners or

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strong in the traditional intelligences, linguistic and logical (Hackney, 2006). This brings up the

question: If dance and math are so closely related, why is dance not used in math classrooms to

further the understanding of students? The answer to that question is that it is already being used

in the classroom, in several schools!

Teaching Math through Dance

Aleta Margolis tries to give teachers ideas for stepping away from textbooks and worksheets. A

former teacher, she once asked sixth graders to choreograph a dance using parallel lines,

perpendicular angles, right angles, obtuse angles, and other elements of geometry. She then

added a written component to the assignment, telling the students to write out the instructions

and see if other kids could recreate the dance (Zuckerbrod, 2011). This kind of lesson would

teach many of the students in a manner that will allow them to remember the concepts, because

whether they are the ones actually dancing, or watching the routines, they will, feel and most of

all understand the concepts in a way that applies to real life. In one study, there were math

classroom set up, one of which was taught through dance and one that was taught as a

discussion. The results after several weeks were considerable. Overall, there was a significant

difference between the dance/math students’ and the non-dance/math students’ attitudes toward

math. This is something one would expect, especially if one group of kids is getting up and

getting involved in the classroom, while the other group of kids has to remain seated and only

watch and listen. On the motivation inventory post-test, dance/math students score significantly

higher than their non-dance/math counterparts. Where the non-dance/math students became

more negative or stayed the same, the dance/math students became more positive or stayed the

same (Werner, 2001). This method of teaching involves the students in a way that gets them

excited and anticipative about learning math.

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Algebra and Dance

Algebra is used in many aspects of dance. A teacher, Susan Hartley, did an activity with her 7th

and 8th graders to experiment with algebra as it pertains to dance. She first had the students use

aspects of geometry by asking them to find personal space on the floor, sitting with their legs in a

diamond shaped position so that their body is perpendicular to the floor, creating right angles.

Then by using an upbeat 4/4 musical selection, she had them each "arc" their torso forward 8

counts, then sideward 8 counts, then backward 8 counts, and then arc to the other side 8 counts.

She then divided the counts by 2, which means they arched 4 counts to each side; Then again she

divided by 2 = 2 counts to each side; And then by 2 again = 1 count to each direction. She then

switched their legs to a parallel shape straight out in front of the body and repeated the

combination of directional arcs and division of counts. She then changed the leg position once

more to create a right angle with the legs on the floor and repeat the directional and division

combination (Hartley, Algebra and Dance).

So what does this teach? Susan Hartley explains that it teaches “X” possibilities. "X" has the

potential in this lesson of being any number "1" through "5". Whether it is teacher generated or

student generated, the goal is to teach or create five movements. For example, the five

movements could be wrap, spiral turn, reach, squat, and jump. These are your five possibilities of

"X". The teacher or students could first demonstrate the equation, 2X + 3 = 13. "X" equals 5, so

you perform all five of the movements in any order, two times, and then create 3 completely new

movements. Students observing will identify all of the "X" movements performed twice (2X),

and will observe three new movements (+3) to total a combination of 13 counts. Therefore, "X"

= 5, and the equation as demonstrated is 2X + 3 = 13. Another similar example is if the students

were to demonstrate the equation, 4X + 1 = 9. "X" equals 2, so you perform only two of

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movements in any order, four times, and then create one new movement. Students observing will

identify only two of the "X" movements performed four times (4X), and will observe one new

movement (+1), to total a combination of 9 counts. Therefore, "X" equals 2, and the equation as

demonstrated is 4X + 1 = 9. So as one can see, algebra is used in many features of dance and

can be used by teachers to aid classes and make them more fun and interesting (Hartley, Algebra

and Dance).

Kids get Excited for Math

There are programs in the United States that use dance as motivation. One such case is that of

the Math Academy program. Participating students get three hours of math tutoring each day for

eight weeks during the summer, then blow off steam in a one-hour freestyle hip-hop dance class,

in which math principles like counting, numbering, segments and patterns are disguised behind

cool moves. "When you dance, you have to say what count each move is on, how long you do it

and stuff like that," explains TDC member Bret Easterling (Jarrett, 2003). In the study done by

Werner with the dance/math students vs. non-dance/math students, it was clear that the

dance/math students were much more excited about learning new concepts. On the fall pre-test

survey, students from the dance/math classrooms scored the same as their non-dance/math

counterparts on their attitudes toward math. However, after having one year of dance/math, the

dance/math students scored much higher than the non-dance/math students. In general, the

dance/math students either stayed the same or increased their scores on the survey, whereas the

non-dance/math students stayed the same or decreased their scores (Werner, 2001). A student

being excited about learning is a significant step in the right direction for teachers. It is

extremely important for students to want to learn, or they will not retain the information in years

to come.

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Multiple intelligences

Some might argue that not every student wants to get up in front of a class and dance around in

order to learn math. It is true that every student is different, and respectively learns differently.

There are visual learners, auditory learners and kinesthetic/tactile learners.

Visual learners like to keep an eye on the teacher and sit in the front of the class. Often, visual

learners will find that information "clicks" when it is explained with the aid of a chart or picture.

Auditory learners are those who learn best through hearing things. They may struggle to

understand a chapter they have read, but then experience a full understanding as they listen to the

class lecture. Kinesthetic learners are those who learn through experiencing/doing things

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(Zuckerbrod, 2011). The argument to this is that students need to be opened to all three types of

learning. Dance encourages social interaction so interpersonal intelligence is also tapped. By

using dance, one tries to reach students’ multiple intelligences and to encourage them to try new

problem solving techniques (Hackney, 2006). When students learn in all three ways, they are

much more likely to be successful in understanding the concepts.

Viewing the Dance floor as a Cartesian Plane

In math, many teachers draw a Cartesian plane on the board in two-dimensional space. Once

they begin teaching three-space they often refer to a wall and floor in the room, but they do not

often describe the floor as a Cartesian plane. It would make sense to think of the floor in this

manner to better understand how objects and points transpose, rotate etc. There are many

geometric and abstract concepts that can be taught using the floor, or in this case the dance floor

as a Cartesian plane. Hackney says, dance provides a tangible material to learn geometric ideas,

but it also necessitates the use of abstract concepts that must be generated within the cognitive

portion of our brains. This is wonderful preparation for geometry and higher math when one

must routinely conceive imaginary lines and planes in space, which can be difficult for some

students to conceptualize (Hackney, 2006). In partner dancing, the boys, who lead, must also

navigate and be aware of where they are in the room relative to their own partner, and to the

group, respectful of their coordinate system as they stay within the perimeter of the circle. In

another style called the basic Foxtrot step, the children must draw clear right angles in space, by

stepping directly forward, back and side, moving gracefully and specifically. Clearly delineated

borders and patterns must lie within their minds because the teacher does not draw patterns for

them on the floor (Hackney, 2006). The concepts are subconsciously learned through this style of

dance.

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Dance and Math Through Time and in Space

When a person dances, they experience the proportional relationship of time and space

(University of Arkansas, 2011). The movement in dance can be combinations of circles and

lines that one’s body moves through in space in circles and arcs. There is a physical experience

or visual experience of geometry through movement. One can create shapes while moving in

patterns across the floor. Dance can be very abstract, and can also be symmetrical. There are

consistent patterns that are reflected when dancers repeat routines.

More importantly, says the Professors from the University of Arkansas, dancers can physically

experience amounts of time, and those can be added, divided and multiplied. One example they

gave is as follows: You can do a movement that takes 3 pulses, then add a movement that takes 2

pulses; you can repeat a 3 pulse pattern 3 times, to make 9; two dancers could dance a 12 count

phrase, with one dancer repeating a 3 count movement 4 times, and one repeating a 4 count

movement 3 times. Dancers learn by rhythm; without it, it would be impossible for them to

remember their steps or synchronize their movements. Certain dances like tap or a waltz requires

participates to link the world of numbers and counting on beat to the movement it takes to master

their dance.

The ways that dance can divide space are as follows: Imagine two dancers moving across a room

in a straight line: one takes 7 very large steps, and one takes 14 steps, each half as long. Then one

wild character might cover the same space in 56 teeny-tiny steps, followed by a leaper eating up

the space in 3 and a half. One person can dance all of those relationships alone, or they could

decide not to make any patterns at all; but as soon as you add another dancer, there is a

relationship created that could be described by numerical, or geometrical, concepts (University

of Arkansas, 2011).

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Calculations Involved with Dance Turns

There are calculations involved in the physics of a spinning dancer. These calculations can be

approximated if that dancer is looked at as a spinning top. The difference between a dancer and

a “top” are that a dancer’s toe on the ground is not frictionless. The movement through the air is

not frictionless, since there is air drag and the dancer's body and clothes are not completely rigid

during the turn. Due to the friction, the dancer's body may not be turning exactly around its

center of mass. However, there is friction involved in a spinning top as well. A top with no

friction will spin forever. Looking at a dancer through a physics perspective, an important thing

that a dancer can do is change the moment of inertia by bringing their hands/arms closer to their

body during the spin. As this happens, the moment of inertia decreases, and the angular velocity

has to increase to keep the energy the same (again, ignoring friction) (Ask Dr. Math, 1996).

There is an analogy between the equations of linear motion and the equations of rotary motion

that could help better describe the physics involved with a spinning dancer. If you take almost

any equation of linear motion, and substitute angular velocity for velocity, angular momentum

for momentum, moment of inertia for mass, angular acceleration for acceleration, torque for

moment of inertia, and so on, the exact same equations hold.

So F = ma [force = (mass) (acceleration)]

becomes:

T = Iα [torque = (moment of inertia) (angular acceleration)]

Or E = ½ × mv2 [kinetic energy = (half the mass) (velocity squared)]

becomes:

E = ½ × Iw2 [k. energy = (half mom. of inertia) (ang. velocity squared)]

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This last equation explains the increased speed when the dancer pulls their arms in. Dancers

learn at a young age that it is important to “spot” so that they will not get dizzy. This involves

locking eyes on one thing, turning the neck to keep spotting, and then whipping it around to lock

again. The mass of the head is not huge, but it is not insignificant, either, and a spotting dancer

will therefore have an irregular spinning rate (Ask Dr. Math, 1996). Physics is a form of

mathematics that is extremely present in many elements of dance, and should not be looked over.

The Musical Aspects Contributing to Mathematical Understanding

Most dances are not only centered on the movement of the dancers. Music brings dance to life,

providing beats, timing and of course mathematics. Music training has been shown to correlate

with better math performance. Both music and math involve ratios, fractions, proportions and

thinking in space and time. When dancing, one must make music with one’s body by moving on

the beat. Dance and Music are so integrally linked that it is hard to separate the two into a mere

discussion of only one (Hackney, 2006). As any dancer or musician would say, music is not

always counted in eight-counts. This is common in many styles of dance, but in ballroom styles

such as the Waltz or the Foxtrot, the counts are much shorter. The steps in Foxtrot have different

timing. The Foxtrot’s rhythm is composed of half, half, quarter, quarter, or Slow, Slow, Quick,

Quick in ballroom terms. This adds up to one and a half measures, in musical terms. The dance

steps “ride over” one four-beat measure and extend into the next; therefore, its fractional

organization does not correspond to that of the music. One dances six beats to music having four

beats per measure. Often dance timing blows the minds of trained musicians, but for students, it

provides an example of how composite ideas can be broken down into elements and rearranged

(Hackney, 2006). Dance timing is also a main component that choreographers use when making

up routines.

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Choreography and Figures

Math is convenient when choreography routines in order to determine how many dance figures

are needed. For instance, choreographer Phil Seyer says that if one wants to determine the beats

per minute (BPM) of a song, he or she would need to first count the number of beats that happen

in 5 seconds, then multiply by 12 to get the beats per minute. If 9 beats happen in 5 seconds then

the BPM would be 108. If a song style has two steps, for example, the song is usually

approximately 108 beats per minute. If the song is about 2 minutes long, there would be 216

beats for the entire song. Each figure is usually 8 beats of music, so one would need 216/8 or

about 27 dance figures (Seyer, Private Dance Lessons). This math helps figure out how often

and how many formations would be adequate for the routine, so the dancers do not remain in one

formation for the entire dance.

Studies/Charts/Statistics (Werner, 2001)

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Table 1 shows the classrooms A-G that incorporate dance in the classroom and compares them to

classrooms AA, EE, FF, GG and HH that teach traditional math lessons without using dance.

These classes range from second to fifth grade. Each class was given a fall survey to test their

knowledge with the mathematics the students were learning and their interest levels.


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Table 2 shows the comparison before the

classes began of similar classrooms on the

fall survey: A with AA, E with EE and G

with GG. It is obvious that the scores are

very similar with both classrooms. This

will help to show the improvement in the

different classrooms by the spring.

Table 3 shows the difference in scores

from the fall to the spring semester. There

is a significant difference in scores.

Table 4 shows a comparison in grades two,

three and five with classrooms that were

dance/math and non-dance/math. The stars

indicate that there is a significant difference

it test scores with the fall and spring survey

tests.

Table 5 shows that within the

comparison, the scores indicate that

scores either increased significantly or


Different Styles for Different Concepts

As mentioned previously, diverse styles provide dancers with different time signatures to follow.

One main reason for this is the various origins of the music. Madeleine Hackney says, on the

tango dance floor, I have personally met many mathematicians and physicists who are fascinated

by the geometry and possibilities of this improvisational dance (2006). The tango uses many

turns and lines with the arms and legs. The couple must match up perfectly for the geometry of

the two bodies to have the correct effect on the audience. The Merengue poses a simple

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Table 6 shows the effect sizes for the total group and classroom comparisons. In addition to

tests of statistical significance, effect sizes, which indicate the strength of the “treatment,” were

also calculated. The result shows that the dance/math project had a large effect on student

attitudes toward math.

Table 5 shows that within the

comparison, the scores indicate that

scores either increased significantly or


rhythmic task because all beats have equal value. This mirrors the integer natural numbers of

most children’s math classes until the fourth or fifth grade (Hackney, 2006). The idea of parts

within a whole is beautifully illustrated by the Cuban Rumba. The rhythm is composed of

quarter, quarter, half, (Quick, Quick, Slow), which makes up half of a basic box step, composed

of six alternating steps. Two half-basics make a whole. These box steps are put together in a

well-choreographed routine, with other steps that elaborate upon the basic idea. Students’

progress onto the rectangle step composed of three boxes. The students must now count up to

twelve to keep track of where they are in the music and dance (Hackney, 2006).

Personal Experience

Dance and Math are, and have always been a significant part of my life. I began dancing when I

was only four years old and have been enjoying it for eighteen years now. I was always the

dancer who needed everyone to dance on the correct counts or beats, because if not, the dance

would be completely off. I started teaching and choreographing at my studio at the age of

twelve, where I started to learn how important symmetry, levels, and formations were when it

came to organizing a routine. When I was sixteen I realized that I wanted to go to college to

become an elementary school teacher. In high school I did a program called Big Brother Big

Sister, which allowed me to go to a classroom every day and witness the teacher singing and

dancing with her students to promote recognition of terms. I always enjoyed math, but the

thought of being able to teach math through dance really sparked my interest. If teachers make

math fun, using dance as a means of involving them in the classroom, then more children in the

future will have an interest in learning more about mathematics.

Analysis

Page 15


To conclude, dance and mathematics are correlated in ways that many would not normally

consider. The concepts are similar, using elements of geometry, physics and number systems.

There is mathematics found in music, movements and levels on the dance floor, and for that

reason dance is now being used as a means of teaching math to young students. Children are

becoming more confident with their math skills and enjoying learning new concepts which was

shown statistically. Whether students learn through hearing, seeing or using kinesthetics, dance

will push the students to step out of their comfort zone and learn these concepts not only by

hearing and seeing them but also by means of physically moving through the material. Physics

is also a factor that influences dance, when it comes to leaps and turns across the floor. There are

rhythms, patterns and sequences that dancers and mathematicians live and breathe every day.

Luckily for some people, they get to experience both, and not only write down patterns, but feel

them and use them as an escape.

References

Page 16


Ask Dr. Math (1996, May 31). In The Math Forum. Retrieved March 22, 2012, from Google.

Burg, J., & Luttringhaus, K. (2005, November). Entertaining with Science, Educating with

Dance. In http://www.cs.wfu.edu. Retrieved February 19, 2012, from Googlescholar.

Hackney, M. (2006). Dancing Classrooms Enhance Math Skills. Connect Magazine, 19(4), 23-

25.

Hartley, S. (n.d.). Algebra and Dance. In NC Standard Course of Study. Retrieved April 28, 2012

Jarrett, S. (2003). Giving Back. Dance Spirit, 7(7), 98-165

Professors at the University of Arkansas. (2011, March 16). Dance and Math. In Mathematical

Thought math 2033 at the University of Arkansas. Retrieved April 28, 2012

Seyer, P. (n.d.). Private Dance Lessons: Using Math in Dance Choreography. In Love Music

Love Dance. Retrieved April 28, 2012, from www.google.com.

Werner, L. (2001). Arts for Academic Achievement. Changing Student Attitudes toward Math:

Using Dance To Teach Math.. Retrieved February 19, 2012, from Education Resources

Information Center.

Westreich, G. (2002, August). Dance, Mathematics, and Rote Memorization. JOPERD: The

Journal of Physical Education, Recreation & Dance. p. 12.

Zuckerbrod, N. (2011, April). From readers theater to math dances: bright ideas to make

differentiation happen [Electronic version]. Instructor, 120(5), 31.

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