THE MATH INSIDE DANCES, SONGS,
STORIES, POETRY AND GAMES
Alleli C. Domingo
Institute of Mathematical Sciences and Physics
College of Arts and Sciences
University of the Philippines Los Baños
+639174010594
allelidomingo@yahoo.com
ABSTRACT
For mathematics education to be relevant in the 21st century, it
must offer an integrated, comprehensive and extensive view of
the world. As an alternative to the fragmented systems so
prevalent today, quantitative reasoning should enable learners to
think broadly and across disciplines. Viewing math from
multiple perspectives could go a long way to helping pupils
overcome their fear of what they may have wrongly perceived
to be a difficult, unnatural pursuit.
Illustrative examples are presented to highlight the presence of
mathematical concepts involved in dances, regional songs,
stories, poetry and indigenous games. Possible activities using
movement and rhythm are suggested to make learning much
more fun.
Engaging the children in their mother tongue naturally creates
spontaneity and reduces math anxiety. This experimental step
taken towards an inter-disciplinal approach to the teaching of
mathematics is a recognition of the role played by multiple
intelligences in today’s classrooms and a response to the need of
making mathematics accessible for all students with varied
cultural backgrounds and learning styles.
Keywords
Mathematics, reasoning, patterns, integration, baybayin numerals
1. INTRODUCTION
From the mighty pen of Francisco Baltazar flowed the following
lines:
“Natarok ko ang lalim ng pilosopiya
Aking natutunan ang astrolohiya
Natantong malinis ang kataka-taka
At mayamang dunong ng matematika.”
- Talata 216, Florante at Laura
What is it in mathematics that appealed to the sense of wonder of
a brilliant poet? In his book “The Math Gene” mathematician Dr.
Keith Devlin, argues that “mathematics is not about numbers,
but about life. It is about the world in which we live. It is about
ideas, and far from being dull and sterile, as it is often portrayed,
is full of creativity. It is the science of patterns.”
Consider the patterns explored in some branches of mathematics:
in logic we study patterns of reasoning; in arithmetic, we work
with patterns of number operations; in algebra, we examine
patterns of finding unknown quantities; in geometry, we
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investigate shapes and sizes; in trigonometry, we examine patterns
of triangles; in calculus, we deal with patterns of changes in
quantities; and in statistics we are concerned with patterns of data
analysis.
Dr. Devlin expounded that “doing math” involves all kinds of
mental capacities such as numerical reasoning, quantitative
reasoning, linguistic reasoning, symbolic reasoning, spatial
reasoning, logical reasoning, diagrammatic reasoning, reasoning
about causality and the ability to handle abstractions. He observed
that such mental abilities, developed by our ancestors thousands
of years ago to survive in a sometimes hostile world, are basic
attributes important to our daily lives.
When mathematics is defined as the science of patterns, and
doing math is viewed as reasoning about patterns, the presence of
mathematics is keenly perceived here, there and everywhere. As
gleaned from the Greek root word mathematikos (which means
“inclined to learn”), math is a “learnable knowledge” that can
be made accessible to all, whatever language that is in use.
2. SHALL WE DO THE MATH DANCE?
In 1978, C.R. Fowler, in Dance as Education, pointed out that
“the basic components of dance – patterns, lines, form, shape,
time, rhythm, and energy – are pivotal concepts in may other
curricular areas and can therefore be integrated with and enhance
mathematics and the social sciences as well as the language arts.”
Dance is geometry in motion. A unique integration of math
concepts and dance skills that maintains the integrity of both
disciplines is modeled by “Math In Your Feet” (MIYF) – a
standards-based, weeklong artist residency program in the USA
for 4th and 5th graders. MIYF physically and mentally challenges
students to experiment with traditional percussive dance while
focusing on an inquiry into patterns, algebra, geometry, and
problem solving. MIYF puts discrete math concepts like angles,
degrees, center, zero, origin, symmetry, sequence, and directions
directly into students' feet and bodies. In general, MIYF takes
what students know and puts it in a new context. This process
connects previously unrelated ideas together, directing students to
the crossroads where disciplines meet.
An episode of Discoveries and Breakthroughs Inside Science (a
syndicated science and engineering news service for local
television newscasts all over the United States) in May 2008
featured the math dance program of mathematician-dancereducator Karl Schaffer, Ph.D., and actor-dancer-composer-teacher
Erik Stern, who described how this approach can help students
who have trouble relating to mathematics. Mr. Stern observed that
“many math-phobic adults and children are put off by math
because they are given symbols before they have a real solid
experience on which to base these on.” To which Dr. Schaffer
added “For many people, having a kinesthetic experience of an
abstract idea is extremely helpful in understanding what that
abstract idea is.” In this alternative way of knowing, mathematical
symbols and patterns are translated into choreography and
conversely dance patterns are translated into math.
In a paper presented at the April 3-4, 2008 Science Summit held
in the University of the Philippines Baguio, Dr. Christopher C.
Bernido of the Research Center for Theoretical Physics, Central
Visayan Institute Foundation, referred to the circle as a rich,
unifying concept. As a synthetical level example to illustrate a
feature of the Dynamic Learning Program that he and his wife Dr.
Marivic Carpio-Bernido are administering in Jagna, Bohol, he
enumerated the topics that can be discussed based on the circle
such as Venn diagrams, circumference as a linear function,
circular area as a quadratic function, coordinates of points on the
unit circle, circular functions, polygon of infinite sides, nontrivial
topology, wave functions and simple harmonic motion.
The simple closed curve that is the circle is fundamental to
dancing. It is the dance formation that must been used for the
longest time. There is a wealth of resources on circle dances,
which speaks of the universality of the very ancient tradition for
the celebration of special occasions, community-building and
bonding. Up to this time, cultural circle dancing is alive and well
in the Philippines and in many parts of the world
The wholly intuitive performance of the Balinese candle dancers
continue to impress international audiences. The candle dance
trick involves rotating the hand, palm-side up by 360 degrees,
resulting in an arm twist. To undo this arm twist, a second 360degree rotation in the same direction is needed. That a 720degree turn is the identity instead of 360 degrees is also
instinctively demonstrated by Filipino dancers in their Binasuan
dance routine. Children can experiment with the movement and
prove that they can extricate themselves from orientation
entanglement. A glass filled with colored liquid is held over the
right hand straight out in front. Then it is brought to the left, and
in a counterclockwise direction under the underarm, it is brought
around front in a circular, 360 degree rotation with the elbow
straight up in the air. To get out of the awkward twisted arm
position, the counterclockwise movement must be continued, but
this time the arm must be swung around over the head. At 720
degrees, the hand is back to its original straight arm, palm up
position, with no drop of the colored liquid spilled.
All the dance steps of the delightful traditional Filipino bamboo
dance Tinikling are combinations of only three basic 4/4 steps,
called singles, doubles, and hops. The choreography can be
altered by simply changing the combinations. A new formation
can be attained by rotating the poles through varying degrees.
Listed in Table 1 are culturally-rich illustrative examples of mathladen creative movement activities that can be undertaken inside
and outside the classroom.
Ed
Table 1. Math concepts involved in movement
ACTIVITY
MATH CONCEPTS
*Handshake Dances : Kumusta
ka, halina’t magsaya
*Clapping combinations to the
tune of “Bahay Kubo”
*Tinikling bamboo dance
Counting, combinations, sets
*Ifugao Uya-uy wedding ritual
*Kanyao festival dance
*Khmer Rambong
*Hooky Pooky
Circle, coordinates of points on
the unit circle; linear function,
quadratic function, polygon of
infinite sides
Moving with giant tangrams
*Ilocano/Visayan kumintang
*Shoulder roll variations
*Full range hip rolls
*Filipino Binasuan
*Honeybee Waggle Dance
Shape, angle, area, spatial
relations
Rotation, direction, range
Angles, measurements,
direction
Cultural enrichment is one of the benefits derived from math
dance. The students get to see a world beyond the walls of the
classroom as they try different dances. Such exposure paves the
way to cross-cultural understanding and peace education
2. HEARTFELT MELODIES
Filipinos are musically-gifted. The numerous prestigious awards
reaped by individual artists and choral groups from international
competitions are there to prove it. The entire nation sings and
dances. In any gathering, the fun is doubled if there is karaoke
singing. Everyone aspires to win in a singing contest. It is
abundantly clear: our hearts are brimming with melodies, which
can fill our classroom walls with laughter and songs! The regional
rhymes and rhythms can be explored to introduce learners to math
wonderland. Table 2 shows a sample list of some of the songs
that can brighten a math classroom.
Table 2. Math concepts inside songs
SONGS
Maysa dua, baduya/Tallo uppa,t
patupat/Lima innem, kankanen
Pito walo. ginao/Siyam pullo…
Enero, Pebrero Marso, Abril
Mayo, Hunyo, Hulyo. Agosto…
Uppat a pato ti nakitak
dua’t nalukmeg, dua’t nakuttong,
agkukuyog da/Ngem diay
kabassitan/Atiddog ti ipus na
Bassit a lawwalawwa/Immuli
diay sanga/Immay ti tudo, natnag
diay baba…
Ulo abaga hawak dapi-dapi
Tuhod, tiil, tuhod, tiil
Bahay Kubo
Manang Biday
MATH CONCEPTS
Counting numbers/
positive integers
Precedence relation, ordinal
numbers:1st, 2nd, 3rd,…
Counting numbers,
Measurement, size
Length, order relations
Magnitude, direction
Order relation, direction
Set , cardinality
Shapes, area, perimeter,
direction, height,
symmetry, sets, cardioid
3. THE POWER OF STORYTELLING
Children are fond of stories. They love to hear about “the birds
and the bees and the flowers and the trees.” What about
honeybees and honeycombs? What is it in the hexagonal structure
of the mass of wax cells built by honeybees to contain their
stores of honey and pollen that makes these winged creatures
such remarkable natural optimizers? A fascinating introduction to
the language of sizes and geometric shapes is the six-sided
honeycomb that allows the maximum space using the least
amount of wax. And that is not all there is to the incredible
behavior of the honeybees. Aside from their economic instinct,
they are also into dance! The waggle dance takes the form of the
figure eight and involves angles, lines, distances and cycle
lengths. It is an elaborate mode of communication used by
honeybees to signal to hivemates the location of food and
potential new nesting sites. A role-playing game that calls for the
active participation of elementary-age students was designed by
Daniel Herms [10], a resident entomologist at a botanical garden.
The simulation of the waggle movement affirms that math dance
can also raise awareness of the benefits of the foraging foray of
bees.
Narratives should be harnessed for maximum learning. In the
process of telling and re-telling, math concepts can be brought
into sharp focus. Take the case of how the Philippine flag came
to be. Our national symbol is peppered with geometric shapesequilateral triangle, rectangle, congruent quadrilaterals, three stars
equidistant from each other, a circle, eight rays of the sun - and
these are replete with significant historical meanings.
The account of how our ancestors came up with an efficient
system of numeration is riveting. The development of a
sophisticated Filipino number sense would captivate kids who
grew up playing computer games. Table 3a shows the ancient
syllabary equivalent of the numerals (the ten single-digit whole
numbers-zero and the first nine counting numbers) as gleaned
form Vocabulario de Lengua Tagala: San Buena Ventura 1606
and UST Archivos Libros . Samples of baybayin numerals can be
found in the Cave Petroglyphs of Angono, Rizal.
Table 3a. Numerals in baybayin form
They engaged in barter trade and had a way with ten per cent
and large numbers. Table 3b shows the ancient syllabary for ten
as a base with negative one as an exponent, and ten raised to the
first eight counting numbers.
Table 3b. Powers of ten in baybayin
SYMBOL
0.1 = 10-1
Daan
1 000 =103
Libo
10 000 = 104
Laksa
100 000 = 105
Yuta
1 000 000 = 106
Angao
10 000 000 = 107
Kati
100 000 000 = 108
Bahala
The concept of “powers of 10” lends itself naturally to place
values. In Ilocano, what comes after sangapulo (10) is sangapulo
ket maysa (10 and 1, or 11). Intuitively, the child knows that 11
consists of a “ten” and a “one”. In sangapulo ket dua or 12,
there is a “ten” and two “ones.” Twenty five in Ilocano is
duapulo ket lima: 25 is two tens and five ones. Eighty seven is
walo pulo ket pito: 87 is eight tens and seven. (This system of
counting is similar to how the Chinese and the Laotian languages
tackle these numbers.).
Primary grade pupils can easily recognize that when all of the ten
numerals (corresponding to their ten fingers) have already been
exhausted, these can be used again by creating a second column to
the left, giving rise to two-digit numbers. The first number that
needs the second column is ten, hence such column is called the
tens column. The first number that needs the third column is 100,
hence that column is referred to as the hundreds column. When
the counting gets to the level of 99 (siyam nga pulo ket siyam),
that would logically lead to one hundred (sangagasut). The
additional columns that are subsequently created correspond to
the powers of ten, as reflected it the last three rows of the
following matrix of place values.
Powers of ten and place values
=10 000
Ten thousands
1
Ed
Powo
100 = 102
104
Long before foreigners reached the Philippine archipelago, our
forefathers had a counting scheme that involved powers of ten
Saikapowo
10 = 101
Table 3c.
(© 2010 Comandante)
BAYBAYIN
103 =1000
Thousands
102 =100
Hundreds
101 = 10
Tens
1
100 =1
Ones
1
2
5
8
7
1
0
0
1
0
0
0
0
0
0
0
Once the concept of place value for the decimal system (base 10)
is firmly fixed, the young learner is ready for a gentle
introduction to binary arithmetic (base 2) that makes use of only
two numerals, 0 and 1. It never ceases to amaze technology-savvy
students that binary arithmetic governs the operation of computers
and calculators that have lots of circuits through essentially
on/off switches, represented by 1 and 0, respectively.
The weaving patterns, music and kinship
system of the
Kankanaey of Mt. Province (and other regions) can also be tapped
as sources of interesting stories that can lead to a deeper
understanding of math concepts. Ethnomathematics, the study of
the relationship between mathematics and culture. is
contextualized learning at its best.
4. GAMES WE USED TO PLAY
Psychologists tell us that one way to develop the positive selfworth of children is to encourage them to play. In the words of
practicing clinical psychologist and retired professor Dr. Ma.
Lordes A. Carandang, “Play helps the child make sense of what
is going on in the world. When a child plays, he is able to have a
sense of power over his environment and to impose his wishes on
his surroundings.”
The learners entrusted to our care belong to a generation that has
been profoundly shaped by television, computer technology, the
Internet, cell phones, iPods, and the phenomenon of OFW
parents. They grew up with senses addicted to speed in a
distraction-filled environment: blaring frenzied music, video
games, Facebook, fast-paced cartoon TV shows, animated movies,
reality TV; programs dominated by sex and violence, and
accessible unlimited texting – all of which have resulted to
shorter attention span.
A viable alternative to the sedentary lifestyle spent in front of a
TV or computer screen is revisiting indigenous Filipino games
that entail muscle memory and develop team spirit, sportsmanship
and goodwill.
The “Stone Scissors and Paper” game, believed to have originated
from the Far East, seems to be well known in various parts of the
world. It is commonly called “Jack n Poy” in the Philippines.
Two players simultaneously present a hand in one of three
positions: an open hand (paper), a closed fist (stone) or two open
fingers (scissors). A stone “breaks” scissors, and is paid 2 coins.
Paper “covers” stones an is paid 3 coins. Scissors “cut” paper and
is paid 4 coins. If both players present the same form, there is a
tie, and hence, there is no payoff. As the players try to outguess
and outsmart each other, they are initiated into the process of
strategic thinking and developing winning strategies.
One uniquely Ilocano game is kukudisi. A stick (the an-anak ) is
placed on a baseline scratched into the ground. One player makes
the stick jump in the air; the other player tries to catch it before it
hits the ground. If the latter cannot do so, a second, longer stick
(the in-ina ) is laid across the baseline; the player then tries to hit
it with the an-anak. The next two phases of the game involve
competing to see who can hit the an-anak (which has been tossed
in the air and stuck into the baseline, respectively) with the in- ina
the farthest. (ww.everyculture.com/wc/Norway-to.../Ilocanos)
The mechanics of the game are of course best explained in the
mother tongue, to make sure everyone knows the rules for fair
Ed
play. Table 4 is a list of sample games that can be incorporated
in the curriculum.
Table 4. Math concepts inside games
GAMES
Kudisi
Kumbato
Sungka
Luksong tinik
Jack n Poy
Pitik-bulag
“Seven Up” and its
variations
Patintero/Patalunton
Tug of War
Jackstone
Langit at Lupa
MATH CONCEPTS
Measurement, length,
relative size
Points, lines, shapes, areas
Whole numbers
Addition/subtraction
Combinatorics
Measurement, height
Maximization of gains
Minimization of losses
Equivalence
Multiples of a number
Odd/even numbers, Prime
numbers
Lines, perimeter,area
Equality
Numbers
Number sets
Complementation
Traditional games were meant to be enjoyed. But these can also
contribute to the development of creative and critical thinking.
Game theory is a relatively new branch of mathematics designed
to help people who are in conflict situations to determine the best
course of action. The theory does not only govern parlor games,
but it has been applied successfully to decision-making problems
in economics, business, psychology, sociology warfare and
political science. Educationally-rich games that require the
animated participation of the entire class will make each child
realize that indeed, math is not a spectator sports.
People skills, team-player skills and group problem solving skills
are some of the highest-level skills that can be sharpened in our
children when they are thrown into the arena of games.
5. NATURE AND NURTURE
According to National Scientist Dr. Dolores R. Ramirez, an expert
in genetics, the inherited aptitude for mathematics is only twelve
per cent. (12%). This implies crucial influence educators have in
making a difference in math literacy. As W.B. Yeats put it, we
need to “light the fire” and provide an enabling environment
where the passion for learning can thrive. Children can see a lot
more when they survey the world around us with a mathematical
eye. To be mathematical means to be curious, open-minded and
interested in always learning more.
Let us build partnerships and stand in solidarity with our allies in
applying the principles of multi-lingual education to help awaken
students to the joys of mathematics as the science of patterns.
With the heart language, we can entice them into encounters
with what they do not yet know about mathematics while
honoring and celebrating what they already know. Our task is
to equip young people with skills they need to become
contributing citizens of the country and to be successful in the
world.
6. ACKNOWLEDGEMENT
Gratitude is expressed to Engr. Bonifacio F. Comandante, Jr. for
his invaluable insight into the origins of the baybayin numerals.
7. REFERENCES
[1] Carandang, Ma. Lourdes A. and Lee-Chua, Queena N. The
Filipino Family Surviving the World, Anvil Publishing, Manila ©
2008
[2] Devlin, Keith,
The Math Gene - How Mathematical
Thinking Evolved and Why Numbers Are Like Gossip, Basic
Books, ©2000
[3] Domingo, Alleli C. “Harnessing the Power of Math Dance to
Broaden Cultural Horizons” Paper presented at the 12th
UNESCO-APEID Conference on Quality Innovations
for
Teaching and Learning, Bangkok, Thailand, 24-26 March 2009
[4} Domingo, Alleli C. “Philippine Agenda on Education and
Transformative Learning”, I Belong Philippines’ 1st National
Youth Summit: Values for Success, Marikina Convention Center,
16 May 2009
[5] Domingo, Alleli C. “Determining the Appropriateness of
Materials in the TX: Readability and Suitability of Topics,
Activities and Exercise” Orientation Seminar in Developing TXs
and TMs for Public Schools, Department of Education
Instructional Materials Council Secretariat, Manila Manor Hotel,
6 May 2009
[6] Domingo, Alleli C. “Armed for the Revenge of the Right
Brain A discourse on what the flat world requires of UP
GURO” , Math and Science Teaching Society Seminar, ACCI
Auditorium, 25 February 2009
Ed
[7] Domingo, Alleli C. “Kwentahan, Kwentuhan, KatotohananDramaturgical Notes on Math as a Theatrical Tool” , Playbill
for Bagong Cristo, Makiling Performance Garden, UPLB, 12-16
March 2008
[8] Domingo, Alleli C. “Math of Dance and Music in
Agriculture”, Crop Science Cluster Seminar, Rm.104, Agronomy
Building, 1 September 2008
[9] Domingo, Alleli C. “The Classroom as Ballroom - Teaching
G. E. Mathematics Using Movement and Rhythm”, UP System
G.E. Professorial Chair, 30 June 2008
[10] Herms, Daniel A. “The Honeybee Waggle Dance: An Active
Participation, Role Playing Game”, Entomology Note#22
copyright Michigan Entomological Society, December 1990
[11] Rapanut, Teofina A. et. al. Algebra of the Weaving Patterns,
Music and Kinship System of the Kankanaey of Mt. Province,
Department of Education, Culture and Sports (DECS) and Center
for Integrative and Development Studies (CIDS), ©1996
[12] Schaffer, Karl, Stern, Erik and Kim, Scott. “Math Dance
with Dr. Schaffer and Mr. Stern”, Move SpeakSpin, Sta Cruz,
CA, USA © 2001
[13] Stein, Sherwin K. Strength in Numbers. Discovering the Joy
and Power of Numbers in Everyday Life , John Wiley and Sons,
©1996