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.