Mathematics and Martial Arts as Connected Art Forms

The Mathematics Educator
2010, Vol. 20, No. 1, 35-42

Mathematics and Martial Arts as Connected Art Forms
Serkan Hekimoglu


Parallels between martial arts and mathematics are explored. Misguided public perception of both disciplines,
students’ misconceptions, and the similarities between proofs and katas are among the striking commonalities
between martial arts and mathematics. The author also reflects on what he has learned in his martial arts
training, and how this wisdom influences his mathematics teaching. As a result of his martial arts training, his
awareness of how he teaches mathematics has shifted, and his understanding of his students’ struggles has
deepened. Finding the balance between theory and practice enhances the process of learning for both students
and teacher.

At first glance, it may seem that mathematics and
practitioner must become skilled at both the
martial arts are conceptually far apart. However, this is
mechanical side and the creative, humanistic side. It is
not the case. The first thing to understand is that both
possible to perform both mathematics and martial arts
disciplines are difficult, yet creative, human activities.
using strict rules of deductions and a system of axioms
Martial arts are more than just kicks, punches, and
or techniques where all the theorems or moves are then
throws; mathematics is not merely a collection of rules,
obtained and checked mechanically. However, if you
facts, skills, and algorithms. When performed by a
watch a martial arts competition, you will witness a
skilled practitioner, both are art forms that teach us
messy struggle punctuated only occasionally with
ways of learning and a framework of thinking that
something as beautiful as the kata—a synchronized
better enables us to use our bodies and minds by
sequence of combative defensive and offensive
maximizing their efficiency. One cannot achieve a high
techniques in a continuous flow. Achieving the beauty
level of skill in mathematics or martial arts by
and flow of the kata takes more than simply following
following or executing a collection of rules, facts, and
a series of pre-determined steps. This same idea to
techniques. On the contrary, they are arts of
applies to mathematics. The mathematician at work
exploration, discovery, imagination, and creation. The
makes “vague guesses, visualizes broad
practitioner enjoys the excitement of searching for new
generalizations, and jumps to unwarranted conclusions.
results and techniques, the thrill of discovery, the
He arranges and rearranges his ideas, and he becomes
satisfaction of mastering difficulties, and the pride of
convinced of their truth long before he can write down
achieving mastery.
a logical proof” (Halmos, 1968, p.380).
While solidly built on ancient traditions, countless
Mathematics and Martial Arts
practitioners have further developed both disciplines
It may come as a surprise that learning martial arts
by devising and polishing techniques, concepts, and
requires as much use of the brain as the body. The
ideas. With every generation, martial arts and
word dojo means the place of enlightenment. The dojo
mathematics evolve through accumulated knowledge,
is a place for facing one’s weaknesses and for
techniques, concepts, perceptions, and experiences
cultivating a flexible mind and body through hard
built upon by past practitioners. The concepts and
practice. Both martial arts and mathematics are
techniques continue to change over time. Not only are
intersections of art, practical skills, and high ideals that
new concepts and techniques developed, but at the
provide a structure to develop an awareness of life
same time old concepts and techniques are reworked,
through a process of discovery (Devlin, 2000;
modified, and redefined (Bolelli, 2008; Davis & Hersh,
Funakoshi, 1954; Halmos, 1985; Stewart; 2006).
1983; Halmos, 1968). The practitioner, therefore, can
Mathematics and martial arts have a fundamental
only gain a proper understanding of martial arts or
commonality; in order to master either one, the
mathematics through constant practice or study that is
not limited to a technical perspective, but also includes
Serkan Hekimoglu has graduate degrees in several disciplines.
a historical and cultural perspective. Learning
He teaches mathematics, statistics, and economics courses at a
mathematics and martial arts will have a profound
community college. He has practiced wrestling, taekwondo, and
effect on the student since the community plays a
karate. He holds a first degree black belt in Shotakan karate.
powerful role in shaping both the works and lives of
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Mathematics and Martial Arts
their practitioners (Bolelli, 2008; Ernest, 1998;
Misguided Public Perception of Both Disciplines
Gonzalez, 1989).
Martial arts and mathematics both suffer from a
One can perceive martial arts and mathematics as
misunderstanding by the general public. Both
an amazing range of mountains without a single peak.
disciplines have received an unfortunate public image,
We might then describe practicing these arts as
which is quite different from the perspective of the
climbing an endless mountain, with different routes to
practitioner (Funakoshi, 1954; Stewart, 2006). While
the same elevation. As the climber scales the mountain,
the general public usually considers having a PhD in
the view below changes. Able to see more of the
mathematics or a black belt in martial arts a mark of
surroundings, the climber’s sense of where he is and
expertise, the practitioners of both perceive these
what really exists in the world changes. Continuing to
achievements simply as mere demonstrations of a
practice martial arts or mathematics, he will find
committed student (Halmos, 1985; Layton, 1988;
himself able to move his mind in new ways and
Stewart, 2006). Furthermore, in addition to being
gradually discover new strengths that can expand his
confused regarding the goals of martial arts practice or
mental horizons. A mountain climber can always try to
learning mathematics, the general public is equally
reach higher elevations or can choose to be content
clueless as to the true benefits of martial arts and
with reaching a certain plateau, even though there are
mathematics. Many people view mathematics as an
higher peaks in the range. As one climbs higher, the
abstract, non-creative, body of knowledge that is to be
view and the connections between points become more
memorized and applied in a mechanical way (Devlin,
interesting and more intriguing. Martial arts and
2000; Schoenfeld, 1989). On the contrary, mathematics
mathematics also offer many challenges, both external
is a science of patterns which demands creativity.
and internal. The difficulty of certain movements, the
Mathematics requires the use of a vivid imagination, a
complexity of the concepts, exhaustion resulting from
sense of scientific beauty, and the ability to reason in
rigorous practice and study, and the pain of sore
selecting ideas and concepts (Halmos, 1968). In a
muscles or headache can produce a great deal of
similar vein, the true benefit of martial arts does not lie
frustration and discouragement. The journey for each
in its sporting value or as a means of fighting, but in
individual is unique. A master or a teacher can
the opportunity it provides for becoming a stronger,
illuminate principles behind techniques and concepts
more complete individual.
but one must discover the truth for oneself.
Current movies provide a much-distorted picture of
Mathematics and martial arts are pilgrimages of
what mathematics and martial arts really are, as the
self-improvement; driven by human desires to find
philosophy and the subtle beauty of the arts do not
perfection and purity in the human mind and body by
come across well on the screen. Movies about
uncovering the hidden simplicity and complexity that
mathematics (e.g., Pi and Proof) frequently provide a
coexist in the world (Halmos, 1985; Konzak &
negative image of mathematicians by portraying them
Bourdeau, 1984). In both disciplines, the knowledge is
as loner sociopathic savants. At best, movies may
not so much something that one possesses, but rather is
depict a mathematician as an absent-minded nerd
a process of self-discovery. One constructs
engrossed in scribbles and equations, or as a kind of
mathematical ideas or martial arts techniques
human calculator who can perform complicated mental
internally, as a way of dealing with a perceived
calculations with amazing speed and accuracy (Burton,
problem. Therefore, the nature of the objective governs
1989; Furinghetti, 1993; Hekimoglu & Kittrell, 2010;
the selection and the use of tools, whether they are
Lim, 1999; Mendick, 2002; Picker & Berry, 2000).
legs, arms, concepts, algorithms, or techniques. On the
The negative impact of these movies is their unrealistic
journey toward mastery in mathematics or martial arts,
representation of the mathematics problem-solving
the practitioner learns to combine ideas or techniques
process. For instance, the crime drama Numb3rs
through experience, hard work and recognition of what
depicts the main character solving problems in less
is important. Eventually, the practitioner may feel as
than a day. However, in reality, a cadre of
though he is no longer simply using tools and concepts
mathematicians might take months to solve such
as presented to him, instead using their combinations to
problems.
create something new. Depending on the amount of
In striking comparison, the violent martial arts
commitment and energy the practitioner has put into
movies contribute to the corruption of the discipline by
training and studying, there are feelings of hard-won
portraying the stereotypical image of a martial artist as
sense of accomplishment, satisfaction, and self-
a bare-handed, acrobatic, Marlboro Man who screams
improvement.
with flying exotic high kicks (Layton, 1988; Reiter,
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Serkan Hekimoglu
1975). The spectacular and flashy movements that
Glasersfeld, 1995). Through a series of assimilations
require excellent athletic abilities are highly unrealistic
and accommodations, the connections become more
with regard to fighting. Many movies provide a
interesting and more nuanced. The student’s
romantic illusion of fighting, along with a fantasy of
understanding becomes more refined as he or she
what it takes to master in martial arts. The portrayal of
begins to relate to more subtle dimensions of
Daniel Larusso in The Karate Kid provides an example
techniques and algorithms by examining why they
of a martial student who trains with a “master” for a
work and what constitutes the elements of their
short period of time and rapidly becomes proficient in
effectiveness.
karate-d (Weintraub & Avildsen, 1984). Even worse,
Progressive skills and knowledge development are
some recent reality shows present an unattractive
keys to long-term progress in both mathematics and
image of martial artists by portraying them as mindless
martial arts since everything that one learns is merely a
jocks or buffoons, e.g., The Ultimate Fighter Reality
preliminary foundation for the next level. Learning in
Show. Both learning mathematics and training in
martial arts and mathematics is like building a house.
martial arts are vastly complex endeavors that require
A solid foundation is required so that the structural
intense concentration in order to succeed (Barnfield,
integrity of the house remains intact. Similarly,
2003; Brown, 2003; Halmos, 1968; Hardman, 1954).
practitioners need to take the time to build a solid
The transition from uninformed enthusiast to
foundation of basic skills and concepts, and constantly
committed student is a gradual one because it takes
refine and add to this base so that they can expand their
time to develop competence by going through a slow
knowledge. There are neither concepts in mathematics,
and constant contemplative process of change and
nor skills in martial arts, that can exist without a
improvement. Gradually, with practice, reflection, and
foundation. Therefore, failing to develop a proper
experience gained through handling different
understanding of fundamental concepts or skills
opponents or solving problems, one begins to
prevents the student from improving and refining his
understand what mathematics or martial arts are really
skill level. When one learns a new martial arts
about.
technique or a mathematics concept, he or she must
incorporate the elementary principles they already
Learning Mathematics vs. Practicing Martial Arts
know with the new knowledge in order to broaden its
Ideas and visions form the basis for the practice of
scope and applications. When it is difficult to grasp a
both mathematics and martial arts. The process of
new step or concept, a student needs to break it down
learning in these disciplines is a series of realizations
by isolating the appropriate relationships and
or awakenings; the harder one studies, the more
properties, and then practice or study them separately
fascinating the arts become (Bolelli, 2008; Brown,
through continuous self-reflection (Gonzalez, 1989;
2003; Gonzalez, 1989; Halmos, 1985; Stewart 2006).
Hardman, 1954; Skemp, 1971; VonGlasersfeld, 1995).
The practitioners need to make a healthy obsession of
The learning process starts with the introduction of
technical details. It is one thing to understand the
basic concepts or techniques; the instructor then
techniques and concepts, but it is quite another to know
gradually increases the complexity and difficulty of the
them intuitively. In mathematics and martial arts,
material as a student advances. In martial arts, the
practitioners must repeat certain movements,
student starts with learning basic punches, kicks,
techniques, exercises, and algorithms many times so
blocks, and stances. Once comfortable with the basics,
that they can become part of their natural reflexes or
the student learns how to put them together in kata and
thought processes. Practicing a technique or algorithm
fighting practices. Similarly in mathematics, as the
repeatedly not only makes one more proficient, it also
student’s knowledge grows, new ideas and concepts
trains and develops his or her neuromuscular or
are introduced that build upon the previous ones.
cognitive system to act, respond, or think in
To become experts in both disciplines, students
accordance with the technique or concept. The
must not only acquire facts, but also organize their
outcome of a successful learning experience is either
knowledge to facilitate the application to diverse
an assimilation, the integration of new understanding
situations. It is this understanding that makes one a
into the existing neuromuscular or cognitive structures,
mathematical expert or a formidable fighter and
or an accommodation, a reorganization of the existing
enables him to use the knowledge or techniques
neuromuscular or cognitive structures in order to allow
creatively, flexibly, and fluently, in different settings or
one to develop these structures on higher levels of
problems. The learning process requires the ability to
organization (Piaget, 1985; Steffe & Wiegel, 1996; von
shift attention from the objects or techniques to the

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Mathematics and Martial Arts
structure of their properties and relationships. Later,
1994). The execution of katas or proofs will provide a
the student needs to compose parts in such a way that
student with some of the most effective fighting or
they form a coherent whole. For example, one cannot
mathematical techniques ever developed. The
look at proofs and katas as if they were arbitrary
techniques in katas or ideas in proofs can also be a
collections of steps or techniques. There is a need to
springboard for further techniques or concepts not
understand each step or technique and how each is
found in the particular kata or proof under study. They
related to previous and proceeding ones in the proof or
serve as important tools for clarification, validation,
kata. The student should be able to see the proof or
and deeper understanding (Bolelli, 2008; Campell,
kata as a single object by putting the steps back
2005; Fischbein, 1982; Funakoshi, 1954; Gonobolin,
together into one complete object or technique.
1954; Hopkins, 2004; Tall, 1989; Van Asch, 1993; Van
Dormolen, 1977; Volmink, 1990).
Proofs and Katas
Martial arts and mathematics instructors know well
Achieving Mastery
the problems that students have appreciating the need
Mastery in math and martial arts does not just
to practice katas or complete proofs. We often get
happen, one achieves mastery over time. Achieving
frustrated when we hear students saying that practicing
mastery is a slow, gradual, and often frustrating
katas is boring or that practicing katas does not help
process (Brown, 2003; Hobart, 2006; Stewart, 2006).
them learn how to defend themselves. Neither do we
Thus, patience is an essential quality of both martial
like to hear students questioning the importance of
artists and mathematicians. Discipline is crucial since
proving mathematical theorems. What functions do
the improvement is a gradual, day-by-day process. One
katas and proofs have within martial arts and
can only achieve genuine success by making full use of
mathematics and what makes the practice of them a
those valuable experiences sometimes referred to as
meaningful activity? First, katas and proofs provide the
failures. There is no shame in being knocked down by
glue that holds martial arts and mathematics together;
an opponent or being unable to solve a problem. Once
they serve as a means of systematization in both
you have learned how to turn pain and frustration into
disciplines. In mathematics, proofs help us to
self-knowledge and personal growth, the challenges
systematize various known mathematical results into a
focus more on what is being learned and how it can be
deductive system of axioms, definitions, and theorems.
developed more fully. Only those interested in the
In martial arts, a kata unifies techniques by integrating
higher ideal will find martial arts interesting enough to
unrelated kicks, punches, and blocks, leading to an
persevere through the rigors it entails (Halmos, 1985;
aesthetic and efficient presentation of movements.
Bolelli, 2008). Those who do will find that the harder
Another function of proofs and katas are that they are
they train the more fascinating the art becomes. While
forms of discourse. Both serve as a medium for
martial artists pay for their expertise with sweat,
communication and validation of traditions among
bruises, and blood, mathematicians pay the price with
people who share similar backgrounds (Bolelli, 2008;
many sleepless nights and headaches. The more time
Hopkins, 2004; Davis, 1976; Funakoshi, 1954; Gale,
one spends doing mathematics or the harder he trains
1990; Gonobolin, 1954; Hanna, 1989; Tall, 1989).
in martial arts, the more one begins to appreciate the
Katas and proofs also serve as the standard
true depth and beauty of each discipline. This new
measure of the technical basis of competence. A
appreciation does not mean that his previous
student’s understanding of martial arts or mathematics
understanding was wrong; it simply means that he has
can be seen in his performance of the kata or in
moved on to a higher level (Hardman, 1954; Richman
providing proof of a concept. Additionally, proofs and
& Rehberg, 1986).
katas can serve as a challenge. Mathematicians find the
Struggles are also a normal part of both
process of doing mathematical proofs appealing
mathematics and martial arts training processes.
because they test their knowledge and creativity. To
Without perseverance, there is little chance of ever
martial artists, katas provide a physical challenge that
pushing through the hard times. Breakthroughs result
they find as appealing as the mental challenge of a
from sustained effort. In both disciplines, the way to
mathematical proof (Campell, 2005; Manin, 1981;
true understanding must lead through personal
Renz, 1981). Lastly, proofs and katas are teaching and
experience and suffering. Even though there are natural
learning tools. Both help to acculturate students in the
stages in the development of a martial artist or a
discipline since they embody lessons learned by past
mathematician, it takes effort to move from one to the
masters (Campell, 2005; Hopkins, 2004; Wilder,
next. Only those who constantly renew their
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Serkan Hekimoglu
commitment to study and train with interest and
internalize the techniques, concepts, algorithms, or
enthusiasm will attain the highest level of achievement.
movements. In martial arts, the realization that mastery
When you hit a wall in your learning, the key to
can be achieved from endless training has given way to
overcoming the barrier is to immerse yourself
the more popular fantasy of an easily won black belt
completely in the problem or technique. As
status after a few months’ work (Richman & Rehberg,
grandmaster Gichin Funakoshi (1954) expressed, “you
1986). Likewise, in undergraduate mathematics
must be deadly serious in training…I do not mean that
classes, students generally receive a rude awakening
you should be reasonably diligent or moderately in
after the first exam when they realize that they cannot
earnest” (p. 105). Paul Halmos (1985), one of the
begin studying one or two nights before the test and
leading mathematicians of the twentieth century
expect to do well on the examinations. Frequently,
expressed similar ideas by saying that learning
students will become disillusioned with the amount of
mathematics requires complete focus and loyalty: “To
hard work required to excel, and so a large percentage
be a mathematician, you must love mathematics more
of students of both disciplines will drop out or fail
than anything else, more than family, more than
(Brown, 2003; Grady, 2000; Hobart, 2006; Jackson &
religion, more than any other interest” (p. 400).
Leffingwell, 1999).
The truth is that there are no shortcuts or magic
Overcoming Disillusionment and Attrition
formulas for learning mathematics or martial arts. The
The students of typical martial arts dojos or
key to success in both disciplines is simply to become
mathematics classrooms are extremely heterogeneous.
personally accountable for what you learn or do not
Each student brings a unique set of strengths,
learn, and to practice or study as often and as hard as
weaknesses, interests, ambitions, responsibilities,
you can. The skills that look so easy when performed
levels of motivation, and approaches to studying or
by a master martial artist or a mathematician are not
training. Differing physical capacities or mathematical
the result of the martial artist’s unique body or the
knowledge, emotional maturity, and psychological
mathematician’s unique mind; their performances are
factors create varying dynamics for each student. This
the result of long, hard, and dedicated practices.
means that instructors in both disciplines must become
comfortable with the idea of individualizing instruction
Real-World Applications
for their students. Teachers should adjust the vigor and
One should learn the real-world applications of the
degree of difficulty in sparring and the difficulty level
techniques of martial arts and the concepts of
of mathematics problems to the student’s current
mathematics in context. A single movement or concept
developmental level (Piaget, 1985; Vygotsky, 1978).
will have several different applications, and the ideas
The instructor should adjust and enrich the curricula
and techniques can be adapted to achieve various
through differentiations in pace and depth, as well as
goals. Bridging the gap between practice and real-
making changes in their teaching style to match the
world applications will help students to develop a
way students learn. In both disciplines, the instructor’s
proper understanding of what martial arts or
judgment is extremely important in knowing when to
mathematics is and how it relates to the rest of the
press onward with intensive training to stimulate
world. The dilemma is the trade-off between content
learning and when to stop in order to avoid student
and real-world problems in mathematics classes or
injury or discouragement.
forms and fighting in martial arts classes. To learn the
Not surprisingly, attrition remains a significant
fighting lessons of martial arts, a student must
problem in both endeavors. The slow process of
experience a physical encounter through an
growth is often unbearable to many students who have
unchoreographed exchange of techniques (Alsina,
come to expect instant gratification. Students often
2001; Grady, 2000; Kloosterman, 1996; Olson, 2003;
have unrealistic expectations of what they can achieve
Stewart, 2006). Similarly, students need to see the
with martial arts or mathematics and how quickly they
application of mathemathics in different academic
will be finished. Many students want to get their black
disciplines, where extraneous variables complicate
belts or get an A in their mathematics class to gain a
problems or standard algorithms are insufficient.
sense of self-confidence and success (Middleton, 1995;
Teachers must inject realism into a student’s
Layton, 1988; Reid, Wood, Smith, & Petozc, 2005).
training, because actual violence differs greatly from
Students should practice martial arts and mathematics
choreographed training in the dojo, and real problem
for their own sake. One must be willing to spend time
solving processes differ significantly from the polished
outside of regular practice or class time to fully
proofs in mathematical journals. In both martial arts

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Mathematics and Martial Arts
and mathematics training, it is the instructor’s job to
power, but it is my character that earns the respect of
challenge students to seek new levels of excellence. In
my students. I strive to model excellence for my
martial arts, the instructor should help students to avoid
students. This helps to build trust and respect, and will
developing false confidence while working with
hopefully encourage them to raise their level of
smaller training partners. Until a student is certain
performance. As an instructor, it is essential to be
techniques work against a larger person, the student
knowledgeable, challenging, organized, clear, and fair.
has not learned self-defense. In mathematics classes,
But these characteristics matter little without the desire
students sometimes shy away from working with
to encourage students’ learning (Jackson &
complex problems and the instructor must challenge
Leffingwell, 1999; Hekimoglu & Kittrell, 2010; Schon,
students to reach beyond their comfort level and
1987). It is just as important to be committed,
increase their knowledge base. Without knowing the
enthusiastic, and genuinely warm to motivate students
applications of the art, studying mathematics becomes
to give me their best, and to encourage them to strive
merely a mental exercise, and training in martial arts is
for excellence in everything they do. As my martial
no more than exercise for the body.
arts instructors did for me, my job as a mathematics
instructor is to create a stimulating classroom
Teaching Mathematics Based on Martial Arts
environment that inspires effort and achievement.
Principles: A Personal Story
Another lesson from my free sparring sessions that
As a longtime martial arts practitioner, I have
I have integrated into my teaching of mathematics is
discovered that I can apply the principles of martial
the adoption of basic karate principles of ikken hissatsu
arts to the teaching of mathematics. When free
(finish with one blow). In free sparring, one tries to
sparring, the goal is to learn from an opponent and to
finish an opponent with one strike without using fancy
remain deeply attentive. It is imperative to make no
or complicated maneuvers. In my classes, I create
assumptions regarding one’s own actions or those of
lesson plans based on this principle. I try to pare away
the opponent. One must try to develop the correct
anything convoluted and confusing by presenting the
understanding of the opponent’s movements and the
concept and ideas in a clear and logically progressive
proper techniques for responding to them. Translating
manner. Furthermore, free sparring has taught me to
this basic principle to my life as a mathematics
always consider the possibility that I may be unable to
instructor, I strive to be fully present and connected in
conquer my opponent. Likewise, a mathematics
the classroom. As with martial arts, each teaching
instructor should consider the possibility that they may
moment requires constantly adjusting to the needs of
be unable to reach their students with their primary
the student. At the beginning of my teaching years, my
teaching method. When you teach or initiate an attack
focus was on the mathematics, not the students. I used
in free sparring, you should always try to gauge the
to think the students were in the classroom for the
reaction of your students or opponents before you
mathematics, not that the mathematics was there for
proceed. The experienced martial artist or mathematics
the students. If I truly want to motivate my students,
instructor guides his actions by his opponents’ or
then I must find a way to reach their interests. It took
students’ reactions.
me a while to realize that my students do not really
One of the most important things that I have
care how much mathematics I know. Instead, what
learned in martial arts training and have integrated into
they need to see is how much I care about teaching
my mathematics teaching is to make my class a place
them mathematics. A good instructor must act in
where students can confront their anxieties and fears.
harmony with the students, and remember to be the
To become a good fighter, every martial arts student
teacher of the students you actually have, not the
must learn to face fear. If you attack with the fear of
students that you might wish to have.
being injured, your attack will not be fully committed
Through my martial arts training, I have learned
and the probability of being injured increases. For
that it is necessary to develop a sense of self-esteem
students of mathematics, the real enemies are the
and mutual respect between the instructor and students
doubt, confusion, and fear within the students
in the dojo. A good martial arts instructor never tries to
themselves. A student must learn to overcome the
impress students with his own skills and knowledge.
frustration, discouragement, and even depression that
His motto is not let me show you what I can do, but
can result from failure to make satisfactory progress.
rather let’s see what we can do together. When I apply
The presence of fear and anxiety will inhibit the
this philosophy to the mathematics classes I teach, I
progression of learning (Garofalo, 1989; Hackett &
know that my mathematical knowledge may give me
Betz, 1989; Hall & Ponton, 2005; McLeod, 1994).
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Serkan Hekimoglu
The essence of teaching mathematics lies in
Crawford, K., Gordon, S., Nicholas, J., & Prosser, M. (1994).
leading students to believe that they can learn
Conceptions of mathematics and how it is learned: The
perspectives of students entering university. Learning and
mathematics (Crawford, Gordon, Nicholas, and
Instruction, 4, 331–345.
Prosser, 1994; Kloosterman, 1996; McLeod, 1994).
Campell, P. (2005). The five katas of yogi meituku. Journal of
They must be able to visualize success, instead of
Asian Martial Arts. 14(4), 48–61.
focusing on the chance of failure. I always try to create
Davis, P. J. (1976). The nature of proof. In M. Carss (Ed.),
an open and positive environment where setbacks,
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mistakes, errors, and failures are permissible. In this
Mathematical Education. Boston: Birkhauser.
way, students can explore their potential without fear
Davis, P. J. & Hersh, R. (1983). The Mathematical Experience.
of judgment or criticism. I have also learned that
Boston: Hougthon Mifflin.
testing in both martial arts and mathematics is simply
Devlin, K. (2000). The math gene: How mathematical thinking
an opportunity to reflect on the student’s progress and
evolved and why numbers are like gossip. New York: Basic
Books.
allow them to acknowledge their strengths,
Ernest, P. (1998). Social constructivism as a philosophy of
weaknesses, and discover areas for self-
mathematics. Albany, New York: SUNY Press.
improvement.
As an instructor, I must help my
Fischbein, E. (1982). Intuition and proof. For the Learning of
students to realize their own ability to go beyond their
Mathematics, 3(2), 9–18.
self-imposed limitations.
Funakoshi, G. (1954). Karate-d : My way of life. New York:
Training in traditional martial arts is one of the
Kodansha.
most valuable pursuits I know. The more I began
Furinghetti, F. (1993). Images of mathematics outside the
integrating martial arts principles into my teaching, not
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