CHAPER - I
1.1 Introduction
Now a day, mathematics is considered
as an important subject called the queen of all science. It is directly related
to the cultural, political and social conditions of our society. Some people
define mathematics as the subject of basic computational skills and they
believe that mathematics deals with only the number and some arithmetic
operation. Of course the numbers and arithmetic operation are very useful in
our life but it is unfair to define mathematics as the subject of only
arithmetic operation. In broad sense, mathematics is a body of knowledge
on the area of science with its won symbol, terminology, content, theories and
techniques. It is a way of thinking, a way of analyzing, organizing and
synthesizing body of data. Now a day it is accepted as a common language of the
world and considered as an art as well as science.
It is said that learning began from
the origin of the mankind. In the beginning, people started learning from
elders and even from nature. Gradually people in the ancient age started
counting those things which could help them to sustain their lives. In due
course of time, learning of mathematics was felt necessary for daily
life and also fur religious work. Mathematics became also necessary for
astrologist. Therefore, philosophers showed strong interest in mathematics.
There are so many learning theories started coming in use to make learning of
mathematics easier.
To understand the facts for the
fulfillment of immediate needs, the sense of learning then changed. Then the
learning was developed in formal form to avoid the difficulties. Many
educationalists put more efforts to make learning effective by joining the term
strategies and education of a child was meant as overall development of a
child. Different types of learning theories have been developed to
eliminate the difficulties of the teaching learning process and to fulfill
the needs of the existing society. As the needs of society arc dynamic and
increasing day by day. The learning theories and strategies must be changed or
upgraded.
Learning theories play vital role in
teaching learning process. Educationists and psychologists are still trying
their level best to make teaching more and more effective by using more and
more materials. Any way learning can't be imposed to them because it depends on needy, interest and
innate power of learners.
There have been many different
theories of learning each having different implications for the practice of
teaching. These theories may be classified according to their views of relation
between the child and his environment. One type of theory, which saw the basis
of the traditional subject centered school. A second type of theory, which saw
the child as the primary active factor led to the child centered school. A
third type of theory, which recognizes that both the child and his environment
play an active role in the learning process. It is the foundation for
contemporary teaching. This theory is of particular importance to mathematics
teachers. Now, everybody is talking about Bruner but not many readers are
theory of Bruner in teaching mathematics.
1.2. Objectives
of the study
i)
To define the term learning and learning theory.
ii)
To define Mathematics learning theories.
iii)
To define conditions of learning Mathematics.
iv)
To define different Mathematical learning theories.
v)
To introduce Bruner's life span.
vi)
To explain the Basic concept of Bruner's Theory of instruction.
vii)
To explain the phase of intellectual development.
viii)
To name application of Burner's learning theory to the instruction on
mathematics.
1.3. Limitation
of the study
i)
This report is based on meaning and implication of learning theories.
ii)It
is bound to explain different learning theories in mathematics in detail except
the constructivist theory of Bruner.
iii)
This report des not go more than basic concept of Bruner's theory of
Instruction, phase of intellectual development according to Bruner, theorems on
learning mathematics and application of Bruner's learning theory in maths
teaching.
1.4.
Rationale of the study
Tribhuvan
University of Nepal and the government of Nepal
CHAPTER – II
2.1. Learning and Learning Theories
There
are many theories about how children learn a subject matter, e.g.
mathematics was developed. Especially since the year 1900 the psychologist has analyzed the act of learning' in a very
broad manner. Different educationists define the term "Learning
Theories" in their own words.
The word 'learn' has been taken from American language which means to
gain knowledge, comprehension and mastery through experiences. Learning is the
modification of behavior through training or experiences. So learning is the
acquisition of habits and knowledge. The main aim of learning is to
change the behavior of the learner. It is a dynamic process which helps the
learner to develop psychological and mental status. It changes the thinking and
feeling of the learner's- It helps to store all knowledge gained in the
mind of learners. The following are the definition of learning by some
educationists.
1.
"Learning is the modification of the behavior through the experiences and
training' - Grades & Friends.
2. "Learning is the acquisition of
habits knowledge and attitudes" -
Crow and Crow
3. "Learning is the process of
progressive behavior adaptation"- B.F. Skinner.
4. "Learning is the change in
behavior resulting from existing behavior"
- Gillford.
5. "Learning is shown by change in
behavior as the result of experiences" -
Crown Back.
6. "Learning is a relatively
permanent change in behavioral tendency and is the result of reinforced
practices" - Kimble and Garmazy.
In
the same way the word theory means a formal set of ideas that is intended to
explain why something happens. But in the field of education, theory is known
as the principles on which a particular subject is based. Thus we have
defined that the term learning theory is the principles which is used in the
field of teaching learning process. These theories are directly related to the
mental growth of the child and about the interaction of mental growth and the
structure of a given subject matter (Mathematics)
There
are many theories about how children learn a subject matter, e.g. mathematics
was developed. Especially since the year 1900
the psychologist has analyzed the act of learning' in a very broad manner.
Different educationists define the term "Learning Theories" in their
own words.
These
theories are classified according to their view in child and environment.
Generally, these theories can be classified in three types. First type of these
theories is known as traditional theories which emphasize the teacher on
the children. Second type of theory is known as child centered theories and
third type of theory is more reasonable from above two types and it emphasized
both are child and environment.
In
the first type of learning theories, teacher literally gives the child an
educations and the child is in the role of some one who passively receives it
various scholar used to describe the theories of these type as "the
child's mind is Tabula-Rasa on which the teacher can write it we or the child's
mind is like clay, to be molded into a socially desirable form". Theories
of these types have been rejected in favor of a different views that analysis
the theories of the second and the third type, and had pictured the environment
as active4 and the child as relatively passive. The first critics of theories,
reacting against them went to the oilier extreme and formulated theories of me
second type which pictured the child as active and the environment as
relatively passive. Since these theories have influenced contemporary teaching
in many ways.
A
composite model of some typical theories of second type would include
proposition like these: before we can educated the child, we must study the
child to find out his nature and needs. The child is growing and his growth is
to be understood as an unfolding form within. The functions of the school are
to supply growth-promoting nutrients, as soil does to growing plants. The
child learns only through his own activity. These activities should be directed
by the child's needs at the moment and not by any adult propose. The child
learns anything's from words that are divorced from meaning, so verbal learning
should be avoided. The growth of the child takes him through, the several
qualitatively distinct stages of development. Instruction during any particular
stage should be tailored to what the child needs and is ready for at (he stage.
If he is not yet ready for a particular type of learning experiences, it is
necessary to wait until he/she becomes ready through spontaneous maturation.
It
is true that we should study the child; we must leach and recognize that he/she
is growing. [I is not true that his growth is a process of unfolding from
within. An important aspect of his growth is his assimilation of parts of the
cultural heritage of his generation. Their growth is not spontaneous growth, it
is directed growth and the teacher plays an important part in giving its
direction. It is true that words divorced from meaning are useless but it is
not true that all verbal learning is useless. It is true that we should not teach the child what he is not
ready for. Instead of waiting passively, we should actively help him to become
ready.
The
principle errors of the theories of the second type are avoided in the recent
theories of the third type which recognize the active role of both the
pupil and his environment in the leaning process. One of these newer theories
is the developmental psychology of Bruner.
2.2. Mathematics Learning Theories
Especially since the year 1900, the psychologists
have analyzed the act of learning in a very broad manner, even including the
learning by animals. Many theories about how children learn a subject matter
e.g. mathematics, were developed. Many of these theories were contradictories
and even nowadays their exist may different theories of learning. In fact, a
learning theory is a theory about the mental growth of child and about the
interaction of mental growth and the structure of a given subject matter, e.g.
mathematics. he studying of learning theory enables the teacher a better
understanding of how a child develops, why at a certain age reactions of
children in front of a given mathematical topic are the same. This
understanding may prove of practical utility in classroom. It may help the
teacher to master a difficult teaching situation.
We know more today about the way
children learn mathematics and the general nature of mathematics they are
capable of learning at various stages of cognitive development that we have
ever know before. Ironically, we still do not know precisely how children
learn, but the effects of the researchers are continuously providing new
evidence to support and often refute various learning theories. Since learning
is an individual matter and invariably dependent upon the numerous variables,
some of which are quite elusive, it is highly unlikely that a comprehensive
learning theory in which is “all things to all people” will ever evolve. We are
continuously adding to the existing bank of knowledge in this very complex
area.
The objective of this chapter is not
to explore specific theories in detail, but rather to provide a forum for a
variety of learning theories that seem to have different implications for
mathematics teachers. Hence, several schools of thoughts on how children learn
mathematics are represented in this chapter. The intent is not to provide a
psychological or learning theory that meets all the needs of teaching
mathematics. The most important things to realize is that choice are available
in selection of models to guide one’s teaching. Further, the selection of a
model should be a function of the instructional objectives and classroom
situations with which one is dealing. No one can prescribe general rules for
the teacher to follow. However, if one is unaware of the important role of
learning theory has to play in guiding instruction, or if he is unfamiliar with
the essential theoretical features that control the outcome of a learning, then
he lacks a necessary component for teaching. Just as know teacher can divorce
himself from some sort of methodology, no teacher can function in a classroom
without employing some elements of learning theories. There are multitude of
learning theories from which each teacher selects “bits and pieces” to
formulate his own unique theory of leaning. Hopefully, the different learning
theories provided in this section will provide the students with a basis for
structuring, or perhaps restructuring, a viable theory of how children learn
mathematics.
2.3 Conditions of Learning
Mathematics
The following are the guidelines consisting of
assumptions that have been found to be useful and certain empirical statements
that have been found practice to mathematics learning outlined by Bassler, Otto
C. and Kolb, John R. in their book Learning to Teach Secondary Mathematics.
1. An individual learns all new behavior because of his
interaction with his total environment.
2. Learning is facilitated if the behavior to be
learned is consistent with the learner’s view of the world and his view of
himself and his role in the world.
3. Leaning is facilitated when the learner has
previously attained and can recall all the prerequisite knowledge necessary for
learning a specified behavior.
4. Learning is facilitated if the learner perceives the
task being meaningful and applicable.
5. Learning is facilitated when the goal or outcome of
the task to be learned is known to the learner.
6. Learning is facilitated when the learners is
actively participates in the performance of the task to be learned rather than
passively receives information.
7. Learning is facilitated when erroneous behaviors are
eliminated and when learner can distinguish between behaviors which may be
confused with behavior to be learned.
8. Learning is facilitated when the learner is informed
of his progress in relation to the goal to be attained.
9. Learning is facilitated when a planned program if
spaced continuous development of the behavior is provided.
10.
Learning is
facilitated when the learners has the opportunity to practice the task with a
wide variety of problems in many contexts different from that in which behavior
was learned.
Learning is
facilitated when the learner has the opportunity to see relationships of the
new behavior or knows how the new behavior is part of some larger plan of
recurring techniques.
2.4 Different Mathematical Learning Theories
Learning theory play a vital role in teaching
learning process. Different types of learning theories have been developed
solve the problems of teaching learning process to meet the needs of
contemporary society and to that of learners. The need of the society is
dynamic and changeable day-by-day. So learning strategies are developed
according to the needs of learners and type of subjects matter to be taught
that make the teaching learning process effective and meaningful.
Many
psychologists, mathematicians and educators have contributed to the development
of the mathematical learning theory. Among them important learning theories
that are useful in teaching and learning of mathematics are (a) Ausubel’s
theory of meaningful learning (b) Dienes’ views on learning mathematics (c)
Gagne’s multiple learning theory (d) Skemp’s psychological learning theory (e)
Bruner’s theory of learning mathematics (f) piaget’s theory of cognitive
development. The common propose of learning theories are:
·
To make learning
effective
·
To transfer of
learning
·
To make
meaningful learning
·
To emphasize on
long term memory
·
To teach orderly
and systematically.
2.5 Needs of Studying Learning
Theory
The teacher of mathematics should understand the
most recent development in learning theory. Undoubtedly competence in school
administration begins with competence in the essentials of the learning
process: the social goal of learning, the psychological factor of learning, the
method of procedure of learning, the subject matter content of learning, the
aids to learning such as laboratories and libraries, the physical development
of youth, and the physical and mental health of the student. In spite of the
fact that no single theory of learning has been developed which supplies a
total ex0lanation of human learning, there are six major agreements among the
educational psychologists which should be understood by the teachers of
mathematics in schools.
1. The motivation is essential; that learners should
desire to learn or learning is not likely to occur.
2. The transfer of training is not likely to happen
automatically; transfer of training is more likely to occur if experiences are
meaningful in terms of goals of the learners.
3. That mere repetition, or exercise, or drill is not
necessarily conducive to learning, but they are likely to be when repetition or
drill is experienced because learners see that these activities are related to
their goals.
4. That learning is not merely a matter of chance;
while learning might be variable, it is usually related to goals or purpose of
learners rather than to purpose of teachers.
5. That response is modified by their consequences;
plans of action which seem to propel learners toward their goals are more
likely to be learned; those which seem to divert learners from their goals are
less likely to be learned.
6. That learning is, in part, a process of
discriminating one situation or one plan of action from another in meaningful
patterns which are related to learners’ goals.
2.6.
Implication of the Learning theories
Instead
of going through the traditional teaching method, we can apply the different
strategies highlighted by different learning theories. The implications of such
theories not only focus on concepts learning but also direct the effective
teaching method. It guides how to use it in proper way- No doubt the
implication of learning theories is beneficial for students, teacher and
teaching learning environment. The implication of learning theories help in the
following ways:
v Learning
theories helps to know students' cognitive level and shows the way to improve
it.
v It
helps the teacher to determine behavioral objectives.
v It
helps to present learning matter in easy and interesting way so that learning
becomes more effective and long lasting.
v It also helps (he teacher to build professional
skills,
v It
maintains the psychological link between the teachers and students.
v It
also helps to promote educational standard of the society with the use of
different learning theories.
v As
different learning theories put stress on different aspect of Learning,
implication of such theories may be included in curriculum pattern,
Many
educationist and psychologists have developed different learning theories to
make mathematics learning is easier and effective. It is the major concern of
the educationist to present the new concepts in simple manner. Realizing the
needs of such theory the learning theories of above three types is included in
mathematics education. These theories have tried to make mathematical concepts
simpler, easier and interesting. It has helped to construct and implement the
mathematics curriculum.
CHAPTER –
III
3.1. Biography
of Jerome Bruner
Jerome Seymour Bruner was born on
October 1, 1915, to polish immigrant parents, Herman and Rose (Gluckmann)
Bruner. He was born blind and did not achieve sight until after two cataract operations while he was still an infant. He attended public
schools, graduating from high school in 1933, and entered Duke University Harvard
University Harvard University
When Bruner entered the field of
psychology, it was roughly divided between the study of perception and the
analysis of learning. The first was mentalistic and subjective, while the
second was behaviorist and objective. At Harvard the psychology department was
dominated by behaviorists who followed a research program called psychophysics, the view that psychology is the
study of the senses and how they react to the world of physical energies or stimuli. Bruner revolted against behaviorism and psychophysics and,
together with Leo Postman, set out on a series of experiments that would result
in the "New Look," a new theory of perception. The New Look held that
perception is not something that occurs immediately, as had been assumed in
older theories. Rather, perception is a form of information processing that
involves interpretation and selection. It was a view that psychology must
concern itself with how people view and interpret the world, as well as how
they respond to stimuli.
Bruner's interest moved from perception
to cognition - how people think. This new direction was stimulated by Bruner's discussions in the early 1950s with Robert
Oppenheimer, the nuclear physicist, around whether the idea in the scientist's
mind determined the natural phenomenon being observed. A major publication to
come out of this period was A Study of thinking (1956), written with Jacqueline Goodnow and George
Austin. It explored how people think about and group things into classes and
categories. Bruner found that the choice to group things almost invariably
involves notions of procedures and criteria for grouping. It may also involve
focusing on a single indicator as a "home base" and grouping things
according to the presence of that indicator. Furthermore, people will group
things according to their own attention and memory capacity; they will choose
positive over negative information; and they will seek repeated confirmation of
hypotheses when it is often not needed. A Study of Thinking has been called one of the initiators of the
cognitive sciences.
Center for Cognitive Studies
Soon Bruner began collaborating with
George Miller on how people develop conceptual models and how they code
information about those models. In 1960 the two opened the Center for Cognitive
Studies at Harvard. Both shared a conviction that psychology should be
concerned with the cognitive processes - the distinct human forms of gaining, storing,
and working over knowledge. Bruner was drawn toward new developments in
philosophy and anthropology: linguistic philosophy for insight into human
language capacities and how thoughts are organized into logical syntax and
cultural anthropology for insight into how thinking is culturally conditioned.
To the center came some of the leading figures in psychology, philosophy,
anthropology, and related disciplines who made contributions to the study of
cognitive processes. In retrospect, Bruner said of those years that what he and
his colleagues most sought was to show "a higher order principle"
that human thought included language capacities and cultural conditions and not
only a mere response to a stimulus.
In spite of his many contributions
to academic psychology, Bruner is perhaps best known for his work in education,
most of which he undertook during his years with the Center for Cognitive
Studies. He held the position that the human species had taken charge of its
own evolution by technologically shaping the environment. The passing on of
this technology and cultural heritage involved the very survival of the
species. Hence, education was of supreme importance. As Bruner admitted, he was
not fully appreciative of this importance until he was drawn into the
educational debate gripping the United States
following the launching of Sputnik, the first satellite, in 1957 by the former Soviet Union .
In 1959 Bruner was asked to head a
National Academy of Sciences curriculum reform group that met at Woods Hole on Cape Cod . Some 34 prominent scientists, scholars, and
educators met to hammer out the outlines of a new science curriculum for America
A Controversial Curriculum
Bruner's educational work led to an
appointment on the Education Panel of the President's Science Advisory
Committee. He also worked on a new social studies curriculum for Educational
Services, Incorporated. Called "Man: A Course of Study," the
controversial, federally funded project drew the ire of various conservative and
rightwing pressure groups because it did not push values and traditions they
felt were important. The controversy led some school districts to drop the
program, and federal funds were withdrawn from any additional development. The
program was continued in some American school districts, and it was also
adopted by many schools in Britain
and Australia
In 1972 the Center for Cognitive Studies was closed, and
Bruner moved to England upon
being appointed Watts Professor of Psychology and Fellow of Wolfson College at Oxford University United
States and for a short time served again at Harvard
until, in 1981, he was appointed to the position of the George Herbert Mead
professorship at the New School for Social Research in New York
Bruner never tried, in his own
words, to construct "a 'grand' or overarching system of thought." His main interest was on
"psychology of the mind," particularly perception and cognition, as
well as education, during a long and productive career.
Later Works and Publications
Bruner published a series of
lectures in 1990, Acts of Meaning, wherein he refutes the "digital processing"
approach to studies of the human mind. He reemphasizes the fundamental cultural
and environmental aspects to human cognitive response. In 1986 he had put his
own professional slant on varied topics such as literature
and anthropology in his book Actual Minds, Possible Worlds. During that
same year he participated in a symposium at Yale
University
3.2. Jerome Bruner on Learning and
Instruction
The
well-known psychologist, Jerome Burner, has written extensively on learning
theory, the instructional process and educational philosophy. Since he has
modified his position on the nature of instruction and his philosophy of
education between 1960 and 1970, any comprehensive consideration of Bruner’s
work must include the comparison of his changing attitudes. In the late
nineteen fifties Bruner and many other educators, notably those people who were
beginning to develop the new curricula in mathematics and science, appeared to
regard the structure of the disciplines as a very important factor (maybe even
the most important factor) in education. At last, it would not be incorrect to
say that the content issue was of major concern to many of the developers of
the several variations of a modern mathematics curriculum. Burner’s highly
acclaimed book, the process of Education,
which was written in 1959-60, reflects the then current thinking of the
scholarly community with regard to primary and secondary education. This book
is a synthesis of the discussions and perceptions of 34 mathematicians,
scientists, psychologists and educators who met for ten days at Woods Hole on Cape Cope to
discuss ways to improve education in schools in the United States
1. Proper learning under optimum conditions leads
students to “learn how to learn”
2. Any topic from any subject can be taught to any
student in some intellectually honest from ant any stage in the student’s
intellectual development.
3. Intellectual activity is the same anywhere, whether
the person is third grader or a research scientist.
4. The best form of motivation is interest in the
subject.
Studying the structure of each subject
was thought to be so important that four reasons for teaching structure were
formulated. First, it was thought that an examination of the fundamental
structure of a subject makes the subject more comprehensible to students.
Second, in order to remember details of a subject, the detail must be placed in
a structured pattern. Third, the optimum way to promote transfer of specific
learning to general applications of learning is through understanding of
concepts, principles and the structure of each subject. Fourth, if the
fundamental structures of subjects are studied early in school, the lag between
current research findings and what is taught in school will be reduced.
These
general principles of instruction and the more specific arguments for teaching
structure were thought to constitute the basic rational for the curriculum
changes which were under way in 1960. However, in his article “ The Process of
Education Revisited,” which appeared in 1971 in the Phi Delta Kappan journal,
Bruner assessed the major notions about education which were prevalent ten
years previous and them to be quite inadequate. In reference to the educational
thinking of 1959, Bruner, in comments critical of that type of thought, stated
in 1971 that:
The prevailing notion was that if you
understood the structure of knowledge that understanding would then permit you
to go ahead on your won; you did not need to encounter everything in nature in
order to know nature, but by understanding some deep principles you could
extrapolate to the particulars as needed. Knowing was canny strategy whereby
you could know a great deal about a lot of thing while keeping very little in
mind.
The movement of which The Process of Education was a part was
based on a formula of faith; that learning was what students wanted to do, that
they wanted to achieve and expertise in some particular subject matter. Their
motivation was taken for granted. It also accepted the tacit assumption that
everybody who comes to these curricula in the school already had been the
beneficiary of the middle-class hidden curricula that taught them analytic
skills and launched them in the traditionally intellectual use of mind.
Failure to question these assumptions
has, of course, caused much grief to all of us.In this same phi Delta Kappan article, Bruner states
his more recent viewpoint of the school curriculum as follows:
If I had my choice now, in terms of a
curriculum project for the seventies, it would be to find a means whereby we
could society back to its sense of values and priorities in life. I believe I
would be quite satisfied to declare, if not moratorium, then something of a
de-emphasis on matters that have to do with the structure of history, the structure
of physics, the nature of mathematical consistency, and deal with it rather in
the context of the problems that face us. we might better concern ourselves
with how those problems can be solved, not just by practical action, but by
putting knowledge, where we find it and in whatever form we find it, to work in
these massive tasks.
3.3. Bruner’s Theory of Instruction
Bruner was one of the founding
fathers of constructivist theory. Constructivism is a broad conceptual framework with numerous perspectives,
and Bruner's is only one. Bruner's theoretical framework is based on the theme
that learners construct new ideas or concepts based upon existing knowledge.
Learning is an active process. Facets of the process include selection and
transformation of information, decision making, generating hypotheses, and
making meaning from information and experiences.
Bruner's theories emphasize the
significance of categorization in learning. "To perceive is to categorize,
to conceptualize is to categorize, to learn is to form categories, to make
decisions is to categorize." Interpreting information and experiences by
similarities and differences is a key concept.
Bruner was influenced by Piaget's ideas about cognitive development in children. During the
1940's his early work focused on the impact of needs, motivations, &
expectations (“mental sets”) and their influence on perception. He also looked
at the role of strategies in the process of human categorization, and
development of human cognition. He presented the point of view that children
are active problem-solvers and capable of exploring “difficult subjects”. This
was widely divergent from the dominant views in education at the time, but
found an audience.
Bruner
presents his viewpoint of the nature of intellectual growth and discusses six
characteristics of growth. He also gives two general characteristics which he
believes should form the basis of a general theory of instruction and discusses
four specific major features which he thinks should be present in any theory of
instruction
3.4. Four Key themes emerged in
Bruner's early work:
Bruner emphasized the role of
structure in learning and how it may be made central in teaching. Structure
refers to relationships among factual elements and techniques. See the section
on categorization, below.
He introduced the ideas of
"readiness for learning" and spiral curriculum. Bruner believed that any subject could be taught at any
stage of development in a way that fit the child's cognitive abilities. Spiral
curriculum refers to the idea of revisiting basic ideas over and over, building
upon them and elaborating to the level of full understanding and mastery.
Bruner believed that intuitive and
analytical thinking should both be encouraged and rewarded. He believed the
intuitive skills were under-emphasized and he reflected on the ability of
experts in every field to make intuitive leaps.
He investigated motivation for
learning. He felt that ideally, interest in the subject matter is the best
stimulus for learning. Bruner did not like external competitive goals such as
grades or class ranking.
Eventually Bruner was strongly
influenced by Vygotsky's writings and began to turn away from the intrapersonal
focus he had had for learning, and began to adopt a social and political view
of learning. Bruner argued that aspects of cognitive performance are
facilitated by language. He stressed the importance of the social setting in
the acquisition of language. His views are similar to those of Piaget, but he places more emphasis on the social influences on
development. The earliest social setting is the mother-child dyad, where
children work out the meanings of utterances to which they are repeatedly
exposed. Bruner identified several important social devices including joint
attention, mutual gaze, and turn-taking.
Bruner also incorporated Darwinian
thinking into his basic assumptions about learning. He believed it was
necessary to refer to human culture and primate evolution in order to
understand growth and development. He did, however, believe there were
individual differences and that no standard sequence could be found for all
learners. He considered instruction as an effort to assist or shape growth. In
1996 he published The Culture of Education.. This book reflected his changes in
viewpoints since the 1960's. He adopted the point of view that culture shapes
the mind and provides the raw material with which we constrict our world and
our self-conception.
3.5 Phases of intellectual
development
Bruner’s
approach was characterised by three stages which he calls enactive, iconic and
symbolic and are solidly based on the developmental psychology of Jean
Piaget. The first, the enactive level,
is where the child manipulates materials directly. Then he proceeds to the iconic level, where
he deals with mental images of objects but does not manipulate them
directly. At last he moves to the symbolic
level, where he is strictly manipulating symbols and no longer mental images or
objects. The optimum learning process
should according to Bruner go through these stages.
1. Enactive mode
When dealing
with the enactive mode, one is using some known aspects of reality without
using words or imagination. Therefore,
it involves representing the past events through making motor responses. It involves manly in knowing how to do
something; it involves series of actions that are right for achieving some
result e.g. driving a car, skiing, tying a knot.
2. Iconic Mode
This mode deals with the internal imagery,
were the knowledge is characterised by a set of images that stand for the
concept. The iconic representation
depends on visual or other sensory association and is principally defined by
perceptual organisation and techniques for economically transforming
perceptions into meaning for the individual.
3. Symbolic mode
Through life one
is always adding to the resources to the symbolic mode of representation of
thought. This representation is based
upon an abstract, discretionary and flexible thought. It allows one to deal with what might be and
what might not, and is a major tool in reflective thinking. This mode is illustrative of a person’s
competence to consider propositions rather than objects, to give ideas a
hierarchical structure and to consider alternative possibilities in a
combinatorial fashion.
3.6.
Characteristics of Intellectual Growth
According
to Bruner, intellectual growth is characterized by a person’s increasing
ability to separate his or her responses from immediate and specific stimuli.
As people develop intellectually, they learn to delay, restructure and control
their responses to particular sets of stimuli. One might understand, and even
expert, a seventh grader’s uncontrolled, angry response in form of harsh,
vulgar words and unacceptable physical action to criticisms from his or her
teacher. However, one would neither expect no tolerate a teacher’s swearing at
or striking a student in response to the student’s criticisms of the teacher.
One of the general objectives of education is to assist students in learning to
control their responses and to make socially acceptable responses to a variety
of stimuli.
A
second characteristic of growth is development of the ability to internalize
external events into a mental structure which corresponds to the learner’s
environment and which aids the learner in generalizing from specific instances.
People learn to make predictions and to extrapolate information by structuring
sets of events and data. In one sense, the totality of a person’s capabilities
to extend and apply his or her previous learning is greater than the sum of
that person’s specific learning activities. Mathematical theorem-proving and
problems-solving require this somewhat intuitive and creative ability to
generalize specific learning.
A
third characteristic of mental development is the increasing ability to use
words and symbols to represent things which have been done or will be done in
the future. The use of words and mathematical symbols permits people to go
beyond intuition and empirical adaptation and to use logical and analytical
modes of thought. The importance to mathematics of appropriate symbol systems
has already been illustrated. Without symbolic notion, mathematics would
develop very slowly and would have limited applications for modeling physical
and conceptual situations.
The
fourth characteristic is that mental development depends upon systematic and
structured interactions between the learner and teachers; or anyone who chooses
to instruct the learner. According to both Bruner and Piaget, intellectual
development will be severely retarded if children do not have variety of
contacts with other people. One thing that may school teachers tend not to do
is to exploit the unique abilities which student have for teaching each other.
On many occasions, students are better able to learn concepts by discussing
them with each other and explaining them to each other than through exclusive
instruction form the teacher.
Bruner’s fifth characteristic of growth is
that teaching and learning are vastly facilitated through the use of language.
Not only is language used by teachers to communicate information to students,
language is necessary for the complete the primary ways for students to
demonstrate knowledge and understanding of mathematical ideas is through the
use of language to express their conceptions of the ideas.
The sixth characteristic is that intellectual
growth is demonstrated by the increasing ability to handle several variables
simultaneously. People who are intellectually mature can consider several
alternatives simultaneously and can give attention to multiple, and even
conflicting, demands at the same time. The influence of Piaget’s work upon
Bruner’s thinking is apparent in Bruner’s formulation of this characteristic of
intellectual growth. You will recall that Piaget’s research has shown that
small children who are still intellectually immature are able to deal with only
a single characteristic of an object at one time.
3.7 Basic Concept of Bruner's Theory of Instruction:
3.7 Basic Concept of Bruner's Theory of Instruction:
According to Bruner a theory of instruction should
be prescriptive and normative. A theory of instruction is prescriptive if it contains principles for the most effective
procedures for teaching and learning facts, skills, concepts, and principles.
That is within the theory there are prescribed processes and methods for
attaining the learning objectives of instruction. In addition, the theory
should contain processes for evaluating and modifying teaching and learning
strategies. A theory of instruction is normative
if it contains general criteria of learning and states the conditions for
meeting the criteria. That is, the theory should contain general learning
objectives or goals and should specify how these objectives can be met.
Bruner distinguishes between a theory of
learning, or a theory of intellectual development and a theory of instruction.
Learning theories are descriptive, not prescriptive. A theory of learning is a
description of what has happened and what can be expected to happen. For
example, Piaget’s theory of intellectual development describes the stages
through which mental growth progress and even identifies mental activities
which people are of are not able to carry out in each stage. However, Piaget’s
learning theory does not prescribe teaching procedures. A theory of instruction
is prescriptive and does have learning objectives. A theory of learning will describe
those mental activities which children are able to carry out at certain ages,
and a theory of instruction will prescribe how to teach students certain
capabilities when they intellectually ready to learn them. For example,
Piaget’s learning theory describes the fact that young children cannot
understand one-to-one correspondence; however, an instructional theory might
prescribe methods for teaching one-to-one correspondence to students who are
intellectually ready to master this concept.
Theories of learning and theories of
instruction are important in education and are, in fact, inseparable. While
Piaget’s major research efforts are designed to describe the nature of
learning, he is not unconcerned with theories of instruction. Much of Bruner’s
work has been developing theories of instruction, but his theories of
instruction are related and compatible with elements of certain learning
theories.
Bruner’s believes that any theory of
instruction should have four major features which prescribe the nature of the
instructional process.
The first feature is that a theory of
instruction should specify the experiences which predispose or motivates
various types of students to learn; that is, to learn in general and to learn a
specific subject such as mathematics. The theory should specify how the
student’s environment. Social status, early childhood, self image, and other
factor influence his or her attitudes about learning. Predisposition of
learning is an important aspect of any theory of instruction.
Second, the theory should specify the manner
in which general knowledge and particular discipline must be organized and
structured so that they can be most readily learned by different types of
students. Before it is presented to students, knowledge should be organized so
that it relates to the characteristics of learners and embodies the specific
structure of the subject. Burner believes that the structure of any body of
knowledge can be described in three ways: its mode of representation, its
economy, and its power, each of which varies according to learner
characteristics and disciplines.
The mode of representation of a body of
knowledge can be either sets of examples or images of the concepts and
principles contained in the body of knowledge, or sets of symbolic and logical
propositions together with rule for transforming them. For seventh graders, the
concept of a function could be represented quite appropriately by sets of
action such as adding 2 to specified set of numbers, halving each measurement in
a set of measurements, or converting a set of Fahrenheit measurements to the
Centigrade scale. High School sophomores could be given examples of functions
such as sets of ordered pairs of objects, or could be shown linear relations
such as y = 2x, y = 
, and y
= -x, all of which are appropriate examples of function for students in
high school. high school students in advanced mathematics classes could be
given a symbolic representation of the function concept in the form: y = f(x) is function of x of for every element a belonging
to a set X there exists a unique
element b belonging to a set y such that a is mapped into b
according to b = f(a).
, and y
= -x, all of which are appropriate examples of function for students in
high school. high school students in advanced mathematics classes could be
given a symbolic representation of the function concept in the form: y = f(x) is function of x of for every element a belonging
to a set X there exists a unique
element b belonging to a set y such that a is mapped into b
according to b = f(a).
Economy
in representing the structure of a discipline is the quantity of information
which must be stored in memory in order to understand elements of the
discipline. The less information one must remember in order to understand a
concept, principle, or process in mathematics, the more economical is the
representation of that particular idea or procedure. It is more economical to
remember the formula for converting a Fahrenheit scale measurement to a
Centigrade scale measurement than it is to remember a table of specific
conversions. Economy of representation depends upon the way in which
information is organized and sequenced, the manner in which it is presented to
students, and the unique learning style of each students.
The power of the structure of a body of
knowledge for each learner is related to the mental structure which he or she
forms in learning the information and is the learner’s capacity to organize,
connect, and apply information which has been learned. A learner, who has
structured his or her learning of the mathematical concepts group, ring, and
field in such a way that he or she sees no relationship among these three
mathematical ideas, has mentally structured the concepts in a manner which is
not very powerful.
The third feature of the theory of
instruction that the theory should specify the most effective ways of
sequencing material and presenting it to students in order to facilitate
learning. Dienes believes that material in mathematics should be sequenced so
that students manipulate concern representations of the concepts in the form of
games before they proceed to more abstract representations. Gagne’s
hierarchical sequencing of mathematics topics suggests that some material
should be sequenced using a bottom-to-top approach with prerequisite and simple
material being presented first. In contrast to Gagne’s sequencing of material,
Ausubel suggests a top-down approach which begins with an advance organizer to
subsume subordinate material and provide an anchoring mental structure. The
problem of sequencing material in mathematics is very complex and is closely
related to each student’s individual learning characteristic.
Bruner’s fourth feature of a theory of
instruction is that theory should specify the nature, selection, and sequencing
of appropriate rewards and punishments in teaching and learning discipline.
Certain students, especially younger children, may require immediate
teacher-centered rewards such as praise and grades on a frequent basis; whereas
many older students may learn more effectively when the rewards are intrinsic,
such as self-satisfaction and the joy of learning a new skill. Some high school
students regard grades and school awards as artificial and not very meaningful;
however other students are motivated to a large extent through their desire to
obtain high grades and teacher approval.
These four features of a theory of
instruction (developing a predisposition to lean, structuring knowledge, and
sequencing the presentation of materials, and providing rewards and
reinforcement) suggest corresponding activities which mathematics teachers
should engage in when preparing to teach courses, units, topics and lessons in
mathematics. Motivating students to learn mathematics, while no within the
exclusive control of the teacher, usually is the responsibility of the teacher.
Structuring of knowledge and sequencing of topics in mathematics has been done
in part for teachers, by the writers of mathematics textbooks. However, many
perceptive teachers find that student learning can be improved by some
judicious resequencing of textbook topics, by selecting supplementary topics,
and even by changing textbooks. The primary extrinsic reward system in schools
is the grading system; although many good teachers encourage students to learn
mathematics by developing learning activities which provide internal reward
such as satisfaction in work well done and appreciation of the nature and
structure of mathematics as an interesting intellectual activity.
3.8 Theorems on
Learning Mathematic
In
order to identify factors involved in teaching and learning mathematics, Bruner
and his associates have observed a large number of mathematics classes and have
conducted experiments on teaching and learning mathematics. As a consequence of
these observations and experiments, Bruner and Kenney (April, 1963) formulated
four general “theorems” about learning mathematics which they have named the
construction theorem, the notion theorem, the theorem of contrast and
variation, and the theorem of connectivity.
(a)
Construction Theorem
The construction theorem says that the
best way for a student to begin to learn a mathematical concept, principle or
rule is by constructing a representation of it. Older students may be able to
grasp a mathematical idea by analyzing a representation which is presented by
the teacher; however Bruner believes that most students, especially younger
children, should construct their own representations of ideas. He also thinks
that it is better for students’ to begin with concrete representations which
they have a hand in formulating. If students are permitted to help in formulating
and constructing rules in mathematics, they will be more inclined to remember
rules and apply them correctly in appropriate situations. Bruner has found that
giving student’s finished mathematical rules tends to decrease motivation for
learning and causes many students to become confused. In the early stages of
concept learning, understanding appears to depend upon the concrete activities
which students carry out as they construct representations of each concept.
(b) Notation Theorem
The
notation theorem states that early constructions or representations can be made
cognitively simpler and can be better understood by students if they contain
notation which is appropriate for the students’ levels of mental development.
Efficient notational systems in mathematics make possible the extension of
principles and the creation of new principles. Until efficient notational
systems for representing equations were formulated, the development of general
methods for solving polynomial equations and systems of linear equations
progressed very slowly. Students should have say in creating and selecting
notational representations for mathematical ideas; simpler and more transparent
notions should be used when concepts are being learned by younger students.
Since seventh and eighth graders have just learned to use parentheses as
symbols of grouping in arithmetic representations such as (2+3) + (5-7) =
(7-4), they are not yet ready to use the notation y = f(x) to represent the concept of a mathematical function. For
students in these grades, a better way of representing functions is to use a
notation such as ž = 2D+3; where ž
and D denote natural numbers. Students in a beginning
algebra class will be able to understand and apply representations such as y = 2x + 3 for functions, and students
in advanced algebra courses will use y =
f(x) to represent functions. This sequential approach to building a
notational system in mathematics is representative of the spiral approach to
learning. Spiral teaching and learning is an approach whereby each mathematical
idea is introduced in an intuitive manner and is represented using familiar and
concrete notational forms. Then, month-by-month or year-by-year, as students
mature intellectually, the same concepts are presented at higher levels of
abstraction using less familiar notational representations which are more
powerful for mathematical development.
(c) Contrast and
Variation Theorem
Bruner’s
theorem of contrast and variation states
that the procedure of going from concrete representations of concepts to more
abstract representations in values the operations of contrast and variation
most mathematical concepts have little meaning for students until they are
contrasted to other concepts. In geometry, arcs, radii, diameters and chords of
circles all become more meaningful to students when they are contrasted to each
other. In fact, many mathematical concepts are defined according to their
contrasting properties. Prime numbers are defined as numbers which are neither
units nor composite numbers, and irrational numbers are defined as numbers
which are not rational. In order for any new concept or principle to be fully
understood, it is necessary that is contrasting ideas be presented and
considered. Contrast is one of the most useful ways to help students establish
an intuitive understanding of a new mathematical topic.
If
students are to learn general concepts in mathematics, each new concept must be
represented by a variety of examples of that concept. If not, a general concept
may be learned in close association with specific representations of itself.
There have been cases in elementary school where children learned the concept
of a set through examples of sets, all of which were represented in the
textbook and by the teacher as being enclosed in braces, i.e. { }.
Consequently, students who were shown set of objects such asD
ž O would not identify the collection as a set because
the objects were not enclosed in braces. When teaching mathematics, it is
necessary to provide many and varied examples of each concept so that students
will learn that each general, abstract mathematical structure is quite
different from more specific and more concrete representations of that
structure.
(d) Connectivity
Theorem
The
connectivity theorem can be stated as follows; each concept, principle and
skill in mathematics is connected to other concepts, principles, or skills. The
structured connections among the elements in each branch of mathematics permit
analytic and synthetic mathematical reasoning, as well as intuitive jumps in
mathematical thought. The result is mathematical progress. One of the most
important activities of mathematician is the search for connections and
relationship among mathematical structures. In teaching mathematics is not only
necessary for teachers to help students observe the contrasts and variations
among mathematical structures, but students also need to become aware of
connections between various mathematical structures. Gagne’s development of
learning hierarchies for structuring the teaching of mathematical content
involves searching for connections in mathematics. The structure of mathematics
is condensed and simplified and learning mathematics is made easier by
identifying connections such as one-to-one correspondence and isomorphism.
Infant the many of the modern mathematics curriculum projects have attempted to
illustrate the connection within each branch of mathematics and connections
among various branches such as algebra, geometry, and analysis. Not only are
connections important for the progress of mathematics, but awareness of
connections is also important in learning mathematics. Since very few
mathematics topics exist in isolation from all other mathematics topics,
connection among topics must be illustrated and understood if progressive,
meaningful learning is to be accomplished by students.
Chapter – Iv
4.1. Applications of Bruner’s Work
Bruner’s earlier works, as well as his
recent writings, are relevant and useful for teachers and students of
mathematics. Hi viewpoint regarding the importance of institution and discovery
learning for meaningful learning provides mathematics teachers with a balanced
contrast to the structured, expository approach to teaching and learning which
Ausubel has promoted. In closing or discussion of Bruner’s contributions to
mathematics education, we will consider an illustration showing how his four
theorems for teaching and learning mathematics can be applied to a topic in
mathematics – the topic of limits.
Calculus is a difficult subject for
many students, and some people who complete high school or college calculus
course do so by memorizing rules and problem types and have little
understanding of the conceptual nature of the subject. The creation of calculus
was motivated by the need for mathematical techniques to handle continuous
processes in nature, such as the moment of bodies in our universe. While
algebraic concepts and skills are quite satisfactory for dealing with discrete,
finite processes, the concept of a limit is needed to attack those continuous
and infinite processes in natures which are now commonly studies in calculus
and related subjects. The fundamental concept of calculus of limits is also the
fundamental sources of many difficulties in learning and applying the subject.
Throughout school, until calculus comes along, little is said about limits in
spite of the fact that the limit concept is indispensable to any serious
consideration of continuous nature processes.
For each year in school from seventh
grade on, Bruner’s theorems of construction, notion, contrast and variation and
connectivity suggest spiral procedure of teaching and learning the very
important mathematical concept of limit. It should be noted that Bruner’s four
theorems are not many to be a chronological sequence of steps in the teaching
/learning process. In teaching different mathematics topics, it may be
appropriate to apply several of his theorems simultaneously or not use them in
various sequences depending upon the characteristics of your students and the
nature of the mathematics topic being studied.
In
teaching mathematics other implication of Bruner’s Learning theory are as
follows;
Ø The process of education is more important than
product. Further, knowing is a process not products.
Ø To teach new concept for learner first of all uses
the symbol of concrete, semi concrete, and then abstract.
Ø Emphasis should be given in individual teaching then
group teaching.
Ø Teaching should relate to interest of the child for
this interview, observation and questionnaires can be use.
Ø According to Bruner problem solving is the first
stage of teaching.
Ø In teaching mathematics emphasis should give in
learning by discovery.
Ø New experiences should relate to learners previous
knowledge.
Ø In teaching mathematics the inactive, the iconic and
the symbolic mode must be used.
4.2. Conclusions
There have been different theories of
learning is having different implications for the art of teaching. One type of
theory said the environment was the most important factor in learning. The
other type said the child himself was the primary factor. The first type is
more traditional and implies that the child is like a piece of clay, the
teacher can form at his own will. The second type would imply that the child
can develop completely by himself, isolated from any environment that he can
discover all by him. A third type of theory now a day seems more reasonable:
both the child and his environment play an active role in the learning process.
One of these newer theories is the developmental psychology of Bruner. Since it
is based in part on studies of how the child develops his intellectual this
theory is of particular important to mathematics teachers. There are very
implications of Bruner's learning theories in teaching learning process, which
is applicable in the context of Nepal
Þ Bruner
propounded three stages of intellectual development. They are enactive, iconic
& symbolic.
Þ Predisposition
to learn, structure of knowledge effective sequencing & reinforcement are
the basic concept of Burner’s theory of instruction.
Þ Bruner’s
theory emphasize the significance of categorization in learning to practice is
to categorize, to conceptualize is to categorize, to learn is to form
categorizes, to make decision is to
categorize.
Þ Interpreting
information and experiences by similarities and differences is a key concept.
Þ To
generate knowledge which is transferable to other contexts fundamental
principles or patterns are best suited.
Þ Burner’s
theory introduced the idea that people interpret the world largely is terms of
similarities and differences.
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