Thursday, July 25, 2013

A Study Report on Ausubel’s Theory of Meaningful Learning

1. Introduction
          Understanding of theories about how people learn and the ability to apply these theories in teaching mathematics are important prerequisites for effective mathematics teaching. Many people have approached the study of intellectual development and the nature of learning in different ways. This has resulted in several theories of learning. Although there is still some disagreement among psychologist, learning theorists, and educators about how people learn and the most effective methods for promoting learning, there are many areas of agreement. The different theories of learning should not be viewed as a set of completing theories, one of which is true and the other false. Each theory can be regarded as a method of organizing and studying some of the many variables in learning and intellectual development and teachers can select and apply elements of each theory in their own classes. You may find that some theories are more applicable to you and your students because they seem to be appropriate models for the learning environment and the students with whom you interact.
          In the past many mathematics teachers, educators neglected the application of theories about the nature of learning and centered their teaching methods on knowledge of the subject. Recent findings in learning theory, better understanding of mental development and new applications of theory to classroom teaching now enable teachers to choose teaching strategies according to information about the nature of learning. The purpose of this report is to present the Ausubel view on Learning Mathematics.

Rationale of the study:
          Different types of learning theories are adopted to teach in the mathematics class on the basis of size of the class, class environment as well as about the prescribed curriculum course book and topic. We have the knowledge those learning theories which are used in the class room by studying learning theories, we can find that the history of theory, rationalization of the invented theory, relation to the other theories, implications in the teaching room and limitations.


Limitations of the study:
          In this section, it is tried to explain about learning theory developed by David P. Ausubel. Although this elaboration may not be the fulfillment due to the lack of preparations duration as well as the reference books and mathematical reports. This study is limited on the basis of our course of study prescribed in the level of Master's Degree of education. So all of the required knowledge about this theory may not be provided by this short note or group report.

Purpose of this paper:   
          This group report on David P. Ausubel's Theory of Meaningful learning mathematics has been prepared for the practical fulfillment of the requirement of the completion practical part of math Ed.519: Foundation of Mathematics Education. This report attempts to cite and critically appraisal of Ausubel's Theory of meaningful learning in varied prospective specially  this report contains introduction, meaning of mathematics and its philosophy, steps in learning, factors affecting learning  theories, different learning theories, details of Ausubel's theory of meaningful learning, implication in mathematics teaching and conclusion and suggestions.
          Learning and teaching of mathematics is not an easy job as it is thought to be. Mathematics teachers come across various discipline and pedagogical problem while teaching mathematics in classroom. Many mathematicians and mathematics educators have investigated and found out various method of solving those teaching learning problem in terms of mathematics learning theories. But none of them have arrived at a universal theory that can be applied for all learning situations. So the choices of learning theories and teaching strategies are left open for the individual teacher. So that he could make a rational decision in formulating his own strategy that can best suit his students. So a teacher must understand all the theories of learning and method of teaching.
          The general objective of this study is to know about the learning theory developed by David P. Ausubel but the behavioral (measurable) objectives are as follows:
1) To introduce Ausubel's views on learning mathematics.
2) To describe process of learning mathematics.
3) To elaborate stages of learning mathematical concepts.
4) To express principles of meaningful verbal learning.
5) To write problems in Ausubel's view.
6) To apply Ausubel theory in the classroom

2. Factors affecting learning process: 
          Learning is a mental process due to the development so it has not equality for all persons. From classroom achievement it is proved that all the students on the same class cannot solve any problem on the same level because of some affecting factors of learning. To be a good teacher we should be familiar with those affecting factors. They are as follows:
i.                   Environment
ii.                 Maturity
iii.              Role of teacher
iv.              Learning method
v.                 Motivation
vi.              Intelligence
vii.            Physical and mental health
viii.         Emotional factors
ix.              Heredity
x.                 Nature of material
xi.              Interest and aptitude.

3.  Steps in Learning:
   We couldn't earn the learning knowledge and skill automatically. In any subject we could gain the knowledge and skills by study, research and discovery and with the interactions of environment. According to psychology we learn anything in different steps such steps are as follows:
i.                   Goal
ii.                 Motivation
iii.              Recreation
iv.              Obstacles
v.                 Response
vi.              Reinforcement
vii.            Generalization

  4. Learning Mathematics Learning Theories:
         The word 'learn' is derived from American English language which means to receive knowledge and comprehension through experience of the study. Learning is the basic behavior of human beings. The main fact of learning is to change the behavior of learner. It plays vital role in development of innate power of the learner.
         Especially, how does the teacher teach? Before we can answer this question, we must answer first another question. How does pupil learn? The art of teaching, if is to be effective; we must be based on adequate theory of learning. Since, the year 1900, the psychologist has analyzed the working of learning in a very board manner even including the learning by animals. A learning theory is a theory about the mental growth of the child and about interaction of mental growth and structure of a provided subject matter.

4.1 Condition of Learning Mathematics:
       Learning is facilitated when the learner has the opportunity to see relationship of new behavior or knows how the new behavior is part of some larger plan.
       The guidelines consisting of  assumptions that have been found to be useful in 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.     Learning 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 outcomes of the task to be learned is known to the learner.
6.     Learning is facilitated when the learner is actively participates in the performances of the task to be learned rather than passively receives information.
7.     Learning is facilitated when erroneous behavior is eliminated and when the learner can distinguish between behaviors which may be confused with the 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 of spaced continuous development of the behavior is provided.
10.                         Learning is facilitated when the learner has the opportunity to practice the task with a wide variety of problems in many contexts different from that in which behavior was learned.

4.2 Different Mathematical Learning Theories
       Mathematical concepts are very difficult problem to understand and to know. It can make the learners nervous sometimes. So the most of the number contributed to the development of mathematical learning theories. These learning theories play vital role in teaching mathematics and to solve the problem in mathematics. Among of them some important learning theories that are very useful in teaching mathematics are following:
a.     David P. Ausubel's theory of meaningful learning.
b.     Zoltan Paul Diene views on learning mathematics.
c.      R.M. Gagne's multiple learning theories.
d.     Jean Piaget theory of cognitive development.
e.      J.S.Bruner's theory of learning mathematics.
f.       R.Skemp's psychological learning of mathematics.
4.3 Needs of studying Learning Theory:
       The teacher of mathematics should understand the most recent developments in learning theory. Undoubtedly competence in school administration beings with competence in the essential of the learning process. The social goals of learning, the  psychological factors of learning, the method or procedure of learning, the subject matter content of learning, the aids to learning such as laboratories and library, the physical development of youth and the physical and mental health of the students. In spite of the fact that on single theory of learning has been developed which supplies a total explanation of human learning. There are six major agreements among the educational psychologist which should be understood by the teachers of mathematics in school.
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 term of goals of the learners.
3.     That mere repetition or exercise or drill is not necessarily conductive 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 change while learning might be variable. It is usually related to goals or purpose of learners rather than to purpose of teachers.
5.     That response are modified by their consequences; plans of action which seem to propel learners toward their goals are more likely to learned; those which seem to divert learners from their goals are less likely to be learned.
6.     That learning is in pair a process of discriminating one situation or one plan of action from another in meaningful patterns which are related to learner's goal.



4.4 Misconceptions about Learning:
         In 1944, Burton outlined some basic misconceptions about learning.
1.     The process of learning is often regarded as mere memorizing.
2.     Learning is thought as purely intellectual.
3.     Facts and verbalisms are looked upon as satisfactory outcomes.
4.     Transfer of training is automatic.
5.     Attitudes and emotional reactions are secondary.

5. Ausubel’s Theory of Meaningful Learning
5.1 Reception and Discovery Learning

David P. Ausubel raised the issued of Reception learning fiercely in the field of mathematics learning about 1960 A.D. His main concern was when we go to inter in the learning process we should provide the clear concept to the students, what is gong to learn? Which is the main way of learning? In the reception learning students do not discover the new matter they get the content which is meaningfully presented. The learning environment should be appropriate, first of all try to make the positive attitude of student to the subject matter which in going to learn. Learning should be meaningful and related with prerequisite knowledge. The learning is generally known of Reception learning.

In general discovery means attaining knowledge by the use of one’s own intellectual or physical resources. In a narrow sense, discovery is learning, which occurs as the result of learner manipulating structuring and transferring the information, so that the pupils field the new information. In the discovery learning the learners may make a conjecture, formulate and hypothesis, or find a mathematics truth by using inductive or deductive processes, observation and extrapolation. There are two types of discover learning, one is pure discovery and another is directed discover method. Discovery methods of learning make the students able to decide any infrastructure and develop the felling of independency also the habit of inquiry. Discoveries can be made by students working together on the small group or by students working individually on laboratory exercises. There are mainly two types of discovery lessons which are inductive disorder lesion and deductive discovery lessons.   Inductive discovery lessons in lower school mathematics are most appropriate for the children. Who are well into Piaget’s stages of concrete operations? Deductive discovery lessons are most appropriate into the class of higher school mathematics in order to the order students who are well into Piaget’s stage of formal operation because these students are better equipped intellectually.

Meaningful and Rote Learning:
 Meaningful leaning refer primarily to a distinctive kind of learning process and only secondarily to a meaningful learning outcome attainment of meaning that the completion of such a process. Meaningful learning is as a process presupposes, in turn, both that the learner employs a meaningful learning set and the material, he/she learn is potentially meaningful to him/he. Conversely no matter how manful the learner is set may be, neither the process nor outcome of learning can possibly by meaningful nor learning task itself is devoid of potential meaning. There are three hampering factors of meaningful learning. They are related with mental development of the student, the motivational factor & the memorization.
The rote learning mainly concern with the memorization learning. It is ancient types of learning system. It is related to the knowledge to the students. It provides the single subject matter on which the student knows that subject matter by memorization Rote learning ignore the meaningful learning. In the presently rote learning is the burning problem in the field of mathematics. Most of Under Developed Countries practice rote learning in the class of mathematics. So mathematics became so hard subject. And we call “no mind no mathematics”. Mathematics is gaining through memorization like wise Ramayana Veda etc.

 5.2 Preconditions for Meaningful Reception Learning:
As Ausubel has given opinion about preconditions for meaningful reception learning (MRL). There are mainly two preconditions for meaningful reception learning. Firstly, MRL can be taken place in such a student who has a good meaningful learning set. Such kind of students connects new learning to his previous knowledge. There are some students who do not understand the underlying meaning of subject matter rather what then do is that then just memorize the new information and do not grasp the real concept. Such student forgets and become confuse with previously memorized mathematical structures.
And the second precondition for MRL is that the learner’s cognitive structure should be related to the learning task. The previous meaningful learning provides an anchor for new learning so that new learning and retention do not require rote learning. This second precondition needs the well organization of learners existing knowledge.

5.3 Factors Hampering Meaningful Learning
There are so many factors affecting or hampering meaningful learning. Some of the factors have been mentioned and described precisely.
1.     Level of mental development: To learn the mathematical concepts meaningfully there should be mature enough and the level of mental development should match learning concept.
2.     Lack of sufficient motivation: The learners must be motivated before teaching/learning activities so that they learn meaningfully.
And, thirdly, some teachers think that the learners must follow only my methods of teaching and learning. Such teachers expect definitions to be reproduced, steps of rules. Problem solving rules to be done as he wants. As a result, the learners can’t learn in their own way.
5.4 Strategies for Meaningful Verbal Learning:
       Since, the Ausubel believes, the major job of education is to teach the discipline, tow conditions must be satisfied .First the discipline must be presented to student so that the structure of the discipline is stabilized within each discipline is extremely powerful and that the learner can make maximal use this ideas by them with his old ideas and making new kind of structures.
              The second condition in teaching a discipline is to make the materials meaningful to the learner. To ensure meaningful learning, the teacher must help students build linkage between their own cognitive structure and the structure if the discipline being taught. Each new concepts and principles within the disciple must be related to relevant, previously learned concepts and principles, which are in the learner's cognitive structure.
5.5 Implications of Ausubel's Theory in mathematics teaching: 
             The Ausubel's theory of meaningful learning is very useful as well as effective in mathematics teaching when they are connected in different situations of mathematics teaching.
i.                   Before teaching new mathematical concepts, teacher should motivate the students and relate the new concepts with pre-knowledge of the students so that the students will be positive in the concepts and ready to learn. Teaching should be done according to the need and interest of students in meaningful way.
ii.                 Teaching materials should be meaningful to the learner. To ensure meaningful learning, the teacher must help students to build linkage between their own cognitive structures and structure of the discipline being taught. For example; to teach fundamental concepts of geometry the expository method is better than discovery method.
iii.              Before teaching a mathematical concept, the teacher should determine the steps that are to be in sequential structure, where teacher should describe each step meaningfully and appropriately, otherwise students can not learn in meaningful way. Description of each step of learning is possible only in good expository method as that it is applicable in meaningful learning and weak students can learn easily. The problem solving or discovery method helps to the learner to memorize the solution of the problem that leads to the learner to follow the rote learning which is meaningless.
iv.              Using teaching materials related to subject matter to make teaching effective, permanent and meaningful must do mathematics teaching. For example, the teaching concepts of circle will be very effective if circular objects are used as teaching materials and the concept of circle can be taught effectively relating it with radius, diameter, circumference, centre etc. it is possible only by expository method and not by discovery method.
v.                 Expository method should be used in teaching new concepts in mathematics. If other methods, such as discovery and problem solving method are used, they create confusion to the students as well as consume much time in concept development. For example; in teaching Pythagorean relation, pure discovery method becomes time consuming and less effective while expository method is an effective method and problem is saved easily by this method. Expository method helps the students to understand the solution of the problem in meaningful way.
vi.              The students having good mathematical skills and verbal capacity should be taught by verbal expository method. This method increase debating, reasoning, imaginary, problem solving powers and conjectures making ability to the students. For example, to prove sum of interior angle of polygon is 180(n-2), we can use expository method more than by problem solving and discovery methods.  
5.     Conclusion:
The teaching strategy that Ausubel suggests in order to promote meaningful verbal learning through the use of progressive initiation and interrogative reconciliation is to apply advanced organizer. An advance organizer is a preliminary statement, discussion or other activities which introduce new materials at a higher level of generality, Inclusiveness and abstraction in the actual new learning task. The organizer is selected for its appropriateness in explaining and integrating the new materials and its purpose to provide the learner with conceptual structure into which they will integrate the new materials.
   

Study on Gerem Bruner's Theory of Learning in Mathematics

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 designed and introduce the faculty of education with aiming to produce qualified manpower in the field of education. There are various subject mailers and content in this field. But it is directly related to the teaching learning process. Many learning theories developed by famous scholar arc included in various courses. They are the main content of each course but everybody may not be known their importance and implications. Similarly everybody is talking about Bruner but not many people are reading him. His views are widely misinterpreted and misunderstood. So it is necessary first to know meaning, implications of learning theories, Bruner learning theory with including his biography, his theory implications some experiments. Therefore I think this study is rationale. So I prepared this study for the fulfillment of practical paper studies in mathematics education (Maths Ed. 591)


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 where he majored in psychology, earning the AB degree in 1937. Bruner then pursued graduate study at Harvard University, receiving the MA in 1939 and the Ph.D. in 1941. During World War II, he served under General Eisenhower in the Psychological Warfare Division of Supreme Headquarters Allied Expeditionary Force Europe. After the war he joined the faculty at Harvard University in 1945.
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's schools. Although numerous work area reports were issued, to Bruner fell the task of writing a chairman's report. The end result was The Process of Education, which became an immediate best-seller and was eventually translated into 19 languages. Bruner centered on three major considerations: a concept of mind as method applied to tasks - e.g., one does not think about physics, one thinks physics, the influence of Jean Piaget, particularly that the child's understanding of any idea will be contingent upon the level of intellectual operations he has achieved, and the notion of the structure of knowledge - the important thing to learn is how an idea or discipline is put together. Perhaps the element that is most remembered is Bruner's statement that "any subject can be taught effectively in some intellectually honest form to any child at any stage of development."
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. His research now came to focus on cognitive development in early infancy. In 1980 he returned to the 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 and director of the New York Institute for the Humanities.
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 on the implications of affirmative action within the context of the university. Bruner also contributed to an educational videocassette, Baby Talk (1986), which provides excellent insight to the processes by which children acquire language skills.
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. Their discussions centered on the importance of teaching the structure of disciplines, readiness for learning, intuitive and analytic thinking, and motives for learning. General principles such as those stated in the following list emerged from the Wood Hole conference:
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
:         
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).
          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. So It necessary to learn and know his theory. Here I have made a summary of his learning theory listing in following important points.
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.