11313244024
International Mathematics Education 2011
UNDERSTANDING OF SUPPORTING
ELEMENTS ETHNOMATHEMATICS
Ethnomathematics has many supporting element. Following exposures of a short answer test with Mr. Marsigit in international class mathematics education, along with the explanation that I got from several sources.
Mathematical communication is a form of potential correlational
vitality, which has the properties of the appointment or determine, namely
the characterization of the properties
that designated based on the properties of the bookmark.
Dimensions of communication is determined by the nature whether the nature of
the subject or the object has such a
way in or a
direction parallel to the direction of the
outside, dimensions of communication are also determined by the number of potential units of math
involved and a variety
of vitality resulting . Literally, the
crystallization of the dimensions of mathematics communication gives meaning to the mathematical
material communication, formal communication of
mathematics, and mathematics
normative communication.
Mathematics
communication
material, dominate by horizontal properties from
the direction of its vitality. In
terms of engagement, the potential number of units involved is to be
minimal when compared with the other dimensions of
communication. So some people can gain the awareness that mathematics communication material is the
communication with the lowest dimension.
Correlational nature of the parallel, to mean equality
between the subject or object of
communication. The implications of the equality of subject and object are that they have a very
low position in the nature of
the appointment.
Formal
communication
of mathematics is dominated by correlational properties
out or inside, of
the vitality of its potential. Correlation
to the outside or to the inside has meaning difference between the properties of the outer and the properties inside. The correlation between the different properties that determines the nature from the subject or
object communication. Implications of the differences
in the properties of the subject
or properties of the object provide reinforcement of differences in
the nature of the appointment. The vitality of the
subject of mathematics with
greater potential, will to establish himself to survive as the subject, while the vitality of the subjects with a smaller potential shifts
the role of the subject himself, became the
object of a role for the subject. Intuition two-Oneness
will help the subjects
of mathematics to understand mathematical objects.
Communication
normative
mathematics,
marked
by decreased
the
appointment of a correlational
nature
appointment
on
the
subject and
the
object. However,
normative
communication
are
said to have a higher
dimension because the
involvement
of more potential
units,
larger
and more complex.
Decreased
nature
of the appointment
correlational
horizontal
not
due to lack of
potency
and
vitality
of
communication, but simply
because
of the wide range and
because
of the involvement
of units of
potential
and vitality,
both on themselves and
on the subject of
self-object.
Thus
the
normative
communication
can
be described the properties
on
the subject and object
as
the subject that has the
potential and vitality
of
mathematics are high,
but
have a low
correlation
horizontal.
It
is understandable that
the
normative
mathematical
communication,
correlational
properties
inside
and out
is
getting
stronger. They are
more
powerful than
the
formal
communication
or
the materials communications.
The situation can be described As
with a "cease fire" between the potential and vitality of mathematics
in and out. The structure of such
communication,
it is a communication
structure that are better
able to accommodate
the
characteristics of the subject
or
object of mathematical
communication.
Communication
normative
mathematics,
characterized
by the properties
of an ideal,
an
abstract of the
potential
and vitality of the
subject
and object of mathematics,
such
as good or
bad
math,
appropriate
or
not
appropriate
other math,
math
should
or
should
not,
useful
or
not a mathematical
concept,
and so on.
Correlational
properties
out
of mathematical concepts
shows
the state of more clearly and
specifically
whether
in
the form of out
upward
or
outward
down.
Correlational potential and vitality
of mathematics
to
the top will
transform
communications
to
the dimensions
above
that is spiritual communication
of
mathematics, whereas the
correlation
potential
and vitality
to
the bottom will transform
mathematical
forms
of communication to the
dimensions
under
the
formal
mathematical
communication,
or
communication
materials
mathematics. So, spiritual communication of mathematics is accommodate existing and all
communications that may
exist. While communications into, will give the absolute
nature of the appointment for subject and object of mathematics. While communication to the outside to the top, to break down
all of the properties of the subject and object of mathematics, so that in
reaching a state subject and object of communication, by the properties without
the properties. The state of
the
subject with properties
without
the
properties is a state
in
which the subject and
object of communication
also decays
into
a state in which the
subject
and object of mathematics
cannot
be distinguished anymore.
That
is no subject and
object of communication
of mathematics at the level of
metaphysical.
Then spiritual communication
can
be identified uses
correlational
relationship
potential
and vitality of
the
subject and the object.
Correlational relationships
in,
and
then transform
all
the potential and vitality of
mathematics
to
the absolute
subject.
Absolute
subject are subject
with
highest
dimension,
which
overcome
all subjects and
objects of communication
as
well as overcoming all
types
of communication that exists and
may
exist.
If we
start
with a pure
mind
is
the
mind that is still
clean
and
free
from any kind of
burden
of
consciousness then we
are
faced with the original
question about the
nature
of mathematics,
but if we start
with an impure
mind
then
we
are directly
involved with the
awareness that
other
than awareness
about
the reality
of
mathematics. From
this
contradiction, it is felt
there
is a need for
a
solution. On the one hand,
the
pure mind
will
generate awareness
about
the reality of
mathematics,
namely
the
facts
that
are "a
priori" but on
the other hand we
need
"eviden"
derived
from
human
experience that produces
mathematical
reality
as
reality
"synthetic".
So
the
mathematical
reality
in
our
minds, we cannot
disconnect
from
the
eviden
from
our
experience. It is true
that
according to Immanueal Kant,
mathematical
reality
is
"synthetic
a
priori".
Two kinds of
way
of
knowledge,
it is up
and down.
Up
is the
experience
that continues to grow as
time passes and the more
mature
person.
Down
is the logic
of
human thought,
which
continued
to fall.
The three
pillars
of learning,
ontology,
epistemology,
and
axiology.
Ontological
approach
is
a reflection, in order to capture
the reality of
mathematics
as
fact
has
been discovered.
In
this self-consciousness,
then
the people who think
mathematics
is the
closest
to the reality
of
mathematics, and from here,
then
he can begin
to
discover the reality of
the
whole of mathematics, and his
relationship with mathematics.
Mathematical
reality
can
be understood by
the
entire contents, density,
autonomy
and
communication
potential,
both
in material,
formal, normative and transcendent.
Ontological
awareness
tried
to
reflect and
interpret
the mathematical
reality
then
implicitly
present
it as a
useful
knowledge in association
with
others, as well as
explicitly
can
be formulated in formal forms
to
obtain
consistent
themes.
Mathematical
fact
implicitly,
has
been included along with the
existence
of actor’s
mathematics.
The
next issue
is
how to formulate
a
formal mathematical
reality
that
is implicit
that?
Later it was realized that
the existence of the
self is the
last
background
which
includes
all mathematical
reality
into
a single overall
vision
of the reality
of
mathematics. Thus
the ontological approach of tried think back deepest understanding of the
reality of mathematics that has been contained in reality itself and concrete
experience. Examining the most common foundation of mathematics is a way of
thinking of philosophy as the beginning and end of the reflection of the
reality of mathematics. Ontological approach moves between two poles, the
experience will be the fact that concrete mathematics and existence of
mathematical reality in which each pole interpret each other. Based
on the experience of the reality
of mathematics,
it can be aware of
the nature of existence of
mathematical
reality:
but existence of mathematical
reality
will
give
concrete
experiences for
themselves
about
the
nature of mathematical
reality.
Therefore,
an
ontological approach
to
understanding
the mathematical
reality
is
hermeneutic
circle
between
experience and presence
without
being able to say
which
comes first. Ontological
coverage
of cannot be
provided in advance
but
will
look through
ontological
description,
it means the study
of
mathematics ontologically
cannot begin by determining
definitions
or
theorems
about
the true
basic
mathematical since
it
will
thus narrowing
the
boundaries of thought and thus
will
close the way of
thinking to another. Thus, the
ontological description of the reality of mathematics
can only be revealed while running the ontology of mathematics as a branch of philosophy
of mathematics.
Epistemological questions can be
submitted, or example can we define mathematics? Defining
the phrase means revealing something else is more understandable. So when we
try to define, we will see "infinite Regress", the endless
explanations of the terms in question. Surely this is impossible to do. If we
want to get knowledge about the "nature of math" then such knowledge
is the simplest and most basic (sui generis). Such mathematical knowledge
cannot be simplified anymore and cannot be explained using other expression.
Therefore epistemological approaches need to be developed so that we can know
the position of mathematics in the context of science. One way is to use the
language of "analog". With this approach, we have thought that the "there is" mathematics are "analog", with "there is" other objects in the study of
philosophy. If knowledge of the others we call "idea"
and is
in our minds, then mathematics can
also be seen as "ideas"
that are within our minds. If we
think of knowledge as a form of "linguistic" then we can also think that mathematics is a form of
"language". So we are thinking about the general philosophy is "isomorphism" with our thinking about the philosophy of mathematics and science philosophies other. In other words, the position of mathematics is "isomorphism"
with others knowledge in the study of
philosophy. The next
question is how far the role of consideration subject, in his attempt to explain mathematical concepts, and how can we know that such considerations are true or not? Is a consideration so require "evidence" or not. If "yes" then what is actually called evidence or evidence math? From these questions, it is clear we have found the distance
between the consideration and
the evidence. Immanuel Kant explains that our knowledge in general and knowledge of
mathematics is the meeting between the knowledge that is "super serve" and knowledge that are "sub serve". Mathematical knowledge that is sub serve, derived from evidence, while mathematical
knowledge that is super serve,
comes from the imagination in
our minds. According to Kant, judgment is the final stage of the process of thinking; latter stages of producing knowledge. So Kant would say that mathematics is
the science of reasoning itself.
Axiological
approach
studied
the philosophical
essence
of the value
or
the value
of
mathematics. Do the math as the reality of value or a given value? Is the value of the mathematical reality is intrinsic, extrinsic or systemic? Is math a pragmatic or semantic? Is math scores are subjective or objective? Is mathematics is essential or temporary? Is math scores are free or dependent? Is math singular or plural? Is there an
element of beauty in mathematical reality, and how the
relationship between mathematical reality with the art? Is there a responsibility to
the mathematical reality? The investigation of the values contained
in
the reality
of
mathematics, has been
doing
since contemporary
philosophy.
There is
3-dimensional
mathematical
value that is
intrinsic, extrinsic, and systemic.
If
someone is master
mathematics just for
her
then
the knowledge of mathematics intrinsic.
If
he can apply
mathematics in daily life,
the knowledge of mathematics
is
extrinsic. If he
can
develop mathematics
in
the
social
arena
of mathematics,
the
mathematics
knowledge
is
systemic.
We
can describe the
hierarchy
math
scores someone with
a
simple diagram
as
follows:
If S is a mathematical value that is systemic then it will load the value of mathematics that is extrinsic (E) then S contains E or can be written mathematically S ⊃ E.
Every math extrinsic value, it must be supported by its intrinsic value (I), so the value of extrinsic load intrinsic value and can be written mathematically as E ⊃ I. Finally, the relationship between the three values can be described as: S ⊃ E ⊃ I, meaning that S contains E contains I.
If S is a mathematical value that is systemic then it will load the value of mathematics that is extrinsic (E) then S contains E or can be written mathematically S ⊃ E.
Every math extrinsic value, it must be supported by its intrinsic value (I), so the value of extrinsic load intrinsic value and can be written mathematically as E ⊃ I. Finally, the relationship between the three values can be described as: S ⊃ E ⊃ I, meaning that S contains E contains I.
Type
of political dimension of education is industrial
trainer (traditional), conservative trainer, humanistic trainer, progressive,
and public educators. The indicator of public educators is a democratic
system of government which should be consistent from head to foot. Reality in
Indonesia said that Indonesia's education system is inconsistent, tend to
industrial trainer.
The dimensions
of students'
abilities according to
Piaget are
the
sensorimotor stage, the
preoperational stage, the concrete
operational stage, and the formal operational stage. Piaget
concluded that human development involves a series of stages. Each stage
prepares the child for the succeeding levels.
The
sensorimotor Stage is the first stage Piaget uses to define cognitive
development. During this period, infants are busy discovering relationships
between their bodies and the environment. Researchers have discovered that
infants have relatively well developed sensory abilities. The child relies on
seeing, touching, sucking, feeling, and using their senses to learn things
about themselves and the environment. Piaget calls this the sensorimotor stage
because the early manifestations of intelligence appear from sensory perceptions
and motor activities. Countless informal experiments during the sensorimotor
stage led to one of the important achievements. They enable the infant to
develop the concept of separate selves, that is, the infant realizes that the
external world is not an extension of them. The sensorimotor stage is also
marked by the child's increasing ability to coordinate separate activities. An
example of the fundamental importance of this is coordination between looking
and reaching, without this an action as simple as picking up an object is not
possible. Infants realise that an object can be moved by a hand (concept of
causality), and develop notions of displacement and events. An important
discovery during the latter part of the sensorimotor stage is the concept of
"object permanence". After a child has mastered the concept of object
permanence, the emergence of directed groping" begins to take place.
In the
preoperational stage a child will react to all similar objects as though they
are identical (Lefrancois, 1995). At this time all women are 'Mummy' and all
men 'Daddy'. While at this level a child's thought is transductive. This means
the child will make inferences from one specific to another (Carlson &
Buskist, 1997). This leads to a child looking at the moon and reasoning; 'My
ball is round, that thing there is round; therefore that thing is a ball'. From
the age of about 4 years until 7 most children go through the Intuitive period.
This is characterized by egocentric, perception-dominated and intuitive thought
which is prone to errors in classification (Lefrancois, 1995). Most
preoperational thinking is self-centred, or egocentric. According to Piaget, a
preoperational child has difficulty understanding life from any other
perspective than his own. In this time, the child is very me, myself, and I
oriented. Egocentrism is very apparent in the relationship between two
preschool children. Imagine two children are playing right next to each other,
one playing with a colouring book and the other with a doll. They are talking
to each other in sequence, but each child is completely oblivious to what the
other is saying.
The Formal
Operational stage is the final stage in Piaget's theory. It begins at
approximately 11 to 12 years of age, and continues throughout adulthood;
although Piaget does point out that some people may never reach this stage of
cognitive development. The formal operational stage is characterized by the
ability to formulate hypotheses and systematically test them to arrive at an
answer to a problem. The individual in the formal stage is also able to think
abstractly and to understand the form or structure of a mathematical problem.
Another characteristic of the individual is their ability to reason contrary to
fact. That is, if they are given a statement and asked to use it as the basis
of an argument they are capable of accomplishing the task. For example, they
can deal with the statement "what would happen if snow were black".
During this
stage, children begin to reason logically, and organize thoughts coherently.
However, they can only think about actual physical objects, and cannot handle
abstract reasoning. They have difficulty understanding abstract or hypothetical
concepts. This stage is also characterized by a loss of egocentric thinking. During
this stage, the child has the ability to master most types of conservation
experiments, and begins to understand reversibility. Conservation is the
realization that quantity or amount does not change when nothing has been added
or taken away from an object or a collection of objects, despite changes in
form or spatial arrangement. The concrete operational stage is also
characterized by the child’s ability to coordinate two dimensions of an object
simultaneously, arrange structures in sequence, and transpose differences
between items in a series. The child is capable of concrete problem-solving.
Categorical labels such as "number" or "animal" are now
available to the child.
Zone of Proximal Development
(ZPD) theory
is presented by Vigotsky.
Vygotsky (1978) sees the Zone of Proximal
Development as the area where the most sensitive instruction or guidance should
be given - allowing the child to develop skills they will then use on their own
- developing higher mental functions. Vygotsky also views interaction with
peers as an effective way of developing skills and strategies. He suggests that teachers use cooperative
learning exercises where less competent children develop with help from more
skillful peers - within the zone of proximal development.
Assimilation
is a part of Piaget's theory that is evident in the way children
perceive the outside world. Assimilation refers to a process by which something becomes more and more
similar to something else until it becomes totally absorbed and loses its own
identity. In psychology, the term Assimilation is used in two contexts. First,
in the context of cultural assimilation in which someone from one culture
assimilates into another, so that they can no longer be tell apart from the new
culture. Assimilation is also a process described by the famous psychologist
Jean Piaget who identified two cognitive processes (Assimilation and
Accommodation) at work in the normal learning process of children. According to
Piaget, when a child becomes aware of something new that it has never seen
before it has two choices for making sense out of that thing. It can interpret
that thing in terms of what it already knows (Assimilation), or it can learn a
new way for making sense of that thing (Accommodation). Taken together, these
two processes make up adaptation, or the child's ability to adapt to his or her
environment.
Equilibrium
theory
is presented by Piaget.
This is the force, which moves
development along. Piaget believed that cognitive development did not progress
at a steady rate, but rather in leaps and bounds. Equilibrium is occurs when a child's schemas
can deal with most new information through assimilation. However, an unpleasant
state of disequilibrium occurs when new information cannot be fitted into
existing schemas (assimilation). Equilibration is the force which drives the
learning process as we do not like to be frustrated and will seek to restore
balance by mastering the new challenge (accommodation). Once the new
information is acquired the process of assimilation with the new schema will
continue until the next time we need to make an adjustment to it.
Architectural theory presented by Immanuel Kant. 2 kinds of
intuition
are space and time.
According
to Immanuel
Kant,
the beginning of the
knowledge
of
mathematics is
"the
awareness of the meaning of
mathematics".
Consciousness
is
thus considered as a container
of
mathematical
reality.
Mathematical
awareness
is
always bi-polar
is
aware
of the meaning of
mathematics.
Awareness
was
on our minds
"reason".
So
when
the reality
of
mathematics is in
consciousness,
then
knowledge of mathematics
has
been in
my
mind. So
there
is a distance
between
the
reality
of
mathematics content
and
container is
my
mind. Inside
that
distance,
there
is intuition
"space"
and "time". So
my
knowledge of mathematics
is
in the
intuition
of
space and time.
In
modern
times, appears
Widget Stain who
sparked mathematics
as
a language and
science
underlies
all over the world. Mathematics
was
always related to
history.
Work
of civilization
which
is a monument
in
mathematics in physical form
is
artifacts.
Mathematics
is
written on the
palm
leaves on
papyrus.
The
first king who produces
letters
of
Java is
Prabu Ajisaka.
Java
cases
characters as many as 20.
In
Java, there is a
market
day, calculated indirectly
using
mathematics.
The market day are Legi, Pahing, Pon, Wage, Kliwon.
Element of
the
concept of mathematics is
a form or container and contents.
The
content is the
reality
of
mathematics, and
the
container is
human
reason. Ethnomathematics
methods
using
grounded theory to
construct
theory of ethnomathematics. Ethnomathematics
supported
by anthropology. Science
from
the point of anthropology
is
Ethnography.
In
the class, there are three parts of various interactions that are whole class interaction, small group
interaction, and individual interaction. Whole class interaction is an
interaction between someone, for example teacher with all students in the
class. Teacher explain the material in front of class which is intended to all
students other example is teacher give a chance for students to share their
idea in front of class. Students’ reflection is included in whole class
interaction. From the whole class, teacher can divide class into small group
then distribute some problems and let students to discuss and solve it. In
small group discussion, there is an interaction between students with others.
They can share their idea and express their opinion then discuss together to
solve the problem that have been distributed by teacher. In the small group
discussion, teacher offer something that is needed by students and identify
difficulty one so an interaction between students in group and teacher is
occurred. In individual interaction, the interaction can be a communication
between teacher with a student when ask question or interaction between a
student with one student. The advantages when teacher use various interactions
in the learning are make learning become an interesting learning, increase students’
spirit in the class, and help students to appreciate others opinion.
Socially,
there
are two kinds of mathematical
knowledge that is
subjective
and objective.
Implementation
of character
education in mathematics
education can be achieved
on
the basis of an understanding
of mathematical
knowledge that is
objective
and mathematical
offender
is
subjective in
its
efforts to obtain
justification
about
the
truth of mathematics
through
the creation,
formulation,
representation,
publication
and
interaction. Explicitly
implementing
character
education in mathematics
education based
on:
(1)
knowledge
of mathematics
in
a
variety of dimensions,
including the nature,
justification
and
its occurrence, (2)
mathematical
objects in
many
dimensions which include
the
nature and
origin,
(3)
the
use of formal
mathematical
which
include effectiveness
in
science, technology and
other sciences, and (4)
mathematical
practices
in
different
dimensions
more
generally, including the
activities of the mathematician
or
mathematical
activity
of
students
in elementary school.
The beginning of
knowledge
is
awareness.
According
to Immanuel
Kant,
the beginning of the
knowledge
of
mathematics is
"the
awareness of the meaning of
mathematics".
Consciousness
is
thus considered as a container
of
mathematical
reality.
Mathematical
awareness
is
always bi-polar
is
aware
of the meaning of
mathematics.
Awareness
was
on our minds
"reason".
So
when
the reality
of
mathematics is in
consciousness,
then
knowledge of mathematics
has
been in
my
mind. 2 kinds of
knowledge
that
cannot be defined
with
mathematical
knowledge
is
primitive
/
base
/
start
knowledge and Intuition
knowledge.
SOURCE:
Marsigit. 2011. Pengembangan
Nilai-nilai Matematika dan Pendidikan Matematika sebagai Pilar Pembangunan
Karakter Bangsa. Presented
at
the University of Semarang.
