R. Skemp's Psychological Learning & its Implications in Teaching Mathematics

Background
i.Introduction to the study
Evaluation on Foundation of Mathematics in master's level consists of two parts in its curriculums. First is theoretical formal examination which is taken by Tribhuwan University and the second is practical part. On that sequence according to the instruction of syllabus we should be given to prepare a report on one of the learning theories learnt of be in Math. Ed. 501 for master's degree. So we were given to prepare a brief report on "R Skemp's Psychological Learning and its implication on Teaching Mathematics" as the partial fulfillment of the practical work, which made us search about that and present as a report and we have tried to show it well.
iiObjectives of the study
The objectives of the preparing this report are as follows:
· To present the different phases and types of R Skemp's Psychological Learning
· To present the implications of R Skemp's Psychological Learning and its implication on Teaching Mathematics
iii. Limitations of the study
As it is well known fact for the related level, there are certain limitations on any type of research. Similarly there were also some limitations on our study and for preparation of this report. We were given the time period of one month. In that period we consulted with different books and T.U. Photocopy papers and some websites. The instruction of teacher was also very useful for to prepare this report.















Methodology
i. Sources of Materials
To prepare this report main source is the part of notes given by our instructor. Furthermore we have taken the help of different photocopy papers from TU and books written by different psychologists such as Hillgard, Deway and others. We have also taken the information from other books which are mentioned in the part of bibliography.

ii. Work Division
After getting the instruction of way of preparing report we four members of our group divided the work in to two parts and started to collect the information from different sources of materials.

iii. Discussion
After collecting material we and information we chose one place for further discussion and discussing with each other we drew conclusions. There were some problems too but we consulted with subject experts for the correct solutions of the problems appeared.

iv. Draft Preparation
Completing the group discussion and drawing conclusions prepared the draft of the report and tried to present in the proper and simplest way.




















About Learning
Background of Learning
We will agree with the fact that we have schools in order to prepare children for different roles in the society. The main objectives of school as an institution is to bring certain desirable changes in the behavior of children through the process of learning. The process of learning begins from the birth of the child and continues till his death. All living creatures learn. When the child is born, his mind is just like clean slate. As soon as he comes in the contact with his environment, he starts reacting and in its process of interaction of the individual and his environment, the foundation of learning are laid down.
History of Learning
Learning is a continuous process which has started from ancient time. At that time elders used to give instruction to the smaller and instruct them, who had knowledge about something they have used to explain for them, similarly in a family father, mother used to teach their son and daughter and instruct the things what is wrong and what is right. Gradually learning has got the system of religious way and with their behavior. In western side, Christianity in eastern Hinduism and in Buddhism was speared who would spray knowledge according to their religion. They use to give education in the cottage.
In the middle age, it was slightly changed in the education system prevailed in the past. In this age learning process was started in the formal system. It was accepted to solve the problems of the society. In this age learning was taken the adjustment in the changing society and able to adjust. In short, this stage learning was used to avoid religious miss concepts and making healthy society.
In the modern age, educational psychology was developed and with the help of this psychological education was started. In fact, learning was achieved from different ways such as formal, informal, non-formal way. In this age of 21st century we should not take learning as a process of teaching, reading, writing and understanding only. Learning can be done anywhere house, society, school, film hall, or on the way in any way formal, inform or nonformula. Learning is not the system of rote learning. it is the process of self realization and experience learning. It is also not the process of getting only certificate, owner of degree. It is wrong thinking to be proud with getting certificate, but it is the system of changing behavior in activities or in practical life.
Meaning and Definition of Learning
Learning means to bring changes in the behavior of an organism. It is very difficult to give a universally acceptable definition of learning because various theories developed by psychologists attempt to define the term from different angles. Learning in psychology has the status of construction. Construct mean an idea of image that cannot be directly observed like elections or genes but which interfere the behavior of the organism. Learning is a relatively enduring change in behavior which is the function of prior behavior( usually called practice.).
Ø What is theory:
ü Theory is a systematic interpretation of knowledge, ideas and facts.
ü Theory is a set of properties on which a system of ideas are intended to explain the facts or events.
ü Theory is the principle on which subject of study is based.

Ø What is learning:
ü Learning is a process of change on human behavior through experiences and training.
ü Learning is acquisition of knowledge, skill and attitude.

· Learning process is different according to different fields of psychology. For example
Behaviourism
Stimulus response experiences learning behaviour change
Cognitivism
Problem perception central meditation new relationship insight learning behaviour change
Learning is a behavior. learning is a process of behavior ways of human action.
Human behavior
ü way of thinking
ü way of feeling
ü way of acting / doing

v Why learning:
Learning is necessary for:
· learning for development (Qualitative + Quantitative)
· learning for growth (Physical, mental, social , emotional)
· learning for experience (Concrete or abstract)
· learning for adjustment (Internal or External)
· learning for behavior change (way of thinking, way of feeling, and way of doing)
· learning for knowledge (Facts, principles, events, trends, definition, theory, etc.), skills (Manipulation, handling, constructing, dancing, writing, reading, singing, etc.) and attitude (values, belief, feeling, etc.)


Learning theories
Formalized learning theory in late thirties hen proponents of various approaches attempted to build their own theory to explain the problems of learning. we can quote the definition of a theory as "a provisional explanatory proposition or set of propositions, concerning some natural phenomena and consisting of symbolic representations of 1) the observed relationships among independent and dependent variables, 2) the mechanisms or structures presumed to underlie such relationships, or 3) inferred relationships and underlying mechanisms intended to account for observed data in the absence of any direct empirical manifestations of the relationships." (Learning Theories edited by Melvin H Marx)
Implications of learning theories
The implications of learning theories are as follows:
a. learning theories helps to identify their physical, mental condition and helps to go forward further step giving guidelines.
b. it increases the psychological relationship between students and teachers.
c. it helps them to develop occupational capacity.
d. it helps the teachers to measure the behavioural objectives.
e. to make interest over subject matter of teaching and learning subject and to remember the subject matter for long term in the memory learning theory is used.
From above mentioned things of points we can notice that learning theory is helping directly or indirectly in all round of own society or individuals, such theories are implemented in different subject teaching activities. some mathematical matters area complex and abstract but are very important, we can use different learning theories in practice. many mathematicians have implemented learning theories in mathematical problems. among them Jean Piaget, J.P. Dines, Gagne, R. Skemp are main mathematicians who implemented the learning theories and helped to make learning mathematics easier, faster and in good environment. mathematics learning theory has helped to develop new curriculum and to implement it. among them the mathematicians who help to improve the state and show the path way for the younger generations of mathematics.
Areas of learning mathematics
A. Programmed Learning
laboratory approach of learning is more effective way of learning. in this method teachers should have to prepare a model. a package of learning strategies for certain contents. there should be enough teaching materials and reference materials to expose the learner into content. content should be presented in simple to complex, particular to general, and bottom to top of learning.
B. Contents programme
topic for the programme can be designed to be unfamiliar to the students but that should be included related to previous knowledge in each and every case.
C. Measuring students' perfomance
after completion of teaching learning activities teachers have to evaluate the students to forward their relation. those types of evaluation may be test of performance.
- test of transfer
- test of subordinate knowledge
D. Variables in learning programme
E. There should be variety in teaching activities so teacher can compose different activities during his learning programme.
Skemp’s Psychological Learning Mathematics
Richard Skemp was born at Bristol of London in 1919. He had done post graduate in mathematics from Hartford Collage in 1939, and worked in Royal Signal from 1939 to 1945 as a mathematics teacher. During that period he was interested to solve psychological problems such as
i. How can students learn knowledge that is more mathematical?
ii. How can teacher teach to the students more effectively?
Skemp is mathematician cum psychologist. According to him, the principle of mathematics learning is the interaction between mathematics and psychology. The psychology of students plays a vital role in teaching mathematics. He was increasingly concerned with the problems of those pupils who, though intelligent and hard-working, could not do well in mathematics. Surely the main ability required for mathematics was the ability to form and to manipulate abstract ideas.
Habit Learning and Intelligent Learning
According to Skemp, there are two types of learning. They are :
a. Habit learning
b. Intelligent learning
These two types of learning differ from each other in qualitative point of view. Habit learning refers memorization of mathematical concepts by rote learning and intelligence refers understanding learning. Main principles of Skemp’s learning are based on the following three questions:
i. What is intelligence?
ii. What is understanding?
iii. How mathematics learning can be improved?
Understanding
The term understanding is used to mean the ways of interpreting, organizing and analyzing the experiences. This is done by means of concepts, conceptual structures and organizing principles. The concepts could be handled only through symbols. The interpretation and analysis have to be judged for their rightness, truth, and adequacy.
Factors Affecting Learning Mathematics
According to Skemp, six factors affect learning mathematics. If we properly use these factors in mathematics teaching, then students can easily understand the mathematical concepts and they can develop understanding capacity. The six factors are as follows:
Ø Formation of mathematical concepts
Ø The idea of schema
Ø Intuitive and reflexive intelligence
Ø Symbols
Ø Different kinds of imagery
Ø Interpersonal and emotional factors
1. The formation of mathematical concepts:
There are certain steps to the formation of mathematical concepts which are as follows:
a. Abstracting and classifying
b. Naming
Communication of Concepts
According to Skemp, there are two types of concepts. Primary concept is formed on the basis of sensory and motor experiences of outside world such as red, motorcar, heavy, hot, sweet are called primary concepts. Secondary concepts are the collection of primary concepts or those concepts which are abstracted from other concepts. Secondary concepts are of higher order than primary concepts.
Concept as a cultural heritage
A concept is a way of processing data that enables the users to bring past experiences in use.


The Learning of Mathematical Concepts:
i. The concepts of higher order than those that a person already has, cannot be communicated to him/her by a definition but only by arranging for him/her to encounter a suitable collection of examples.
ii. Mathematical concept is invariably ….than other concepts. It must first be ensured that these are already formed in the learners’ mind.
Learning and Teaching
Learning and teaching are interrelated to each other if both of them have a good combination then entire learning and teaching will be effective. For this purpose, teacher has to teach effectively by using different methods with the help of different teaching materials. And students have to bear proper attention to learn.
2. The idea of schema:
Schema is a mental structure which deals about how the concepts fit together to form conceptual structure. The formation of simple concepts based on formation of mathematical concept is the process of schema.
Functions of idea of schema:
- The integrative function of schema
- Schema as a mental tool for further learning
Steps of schema
- Understanding
- Implication for the learning of mathematics
3. Intuitive and reflexive intelligence:
4. Symbols:
The functions of symbols are as follows:
a. Communication
b. Recording of knowledge
c. Communication of new concept
d. Making multiple classification straightforward
e. Explanation
f. Making possible reflective activity
g. Helping to show structure
h. Making routine manipulation automatic
i. Recovering information and understanding
j. Creative mental activity
5. Different kinds of imagery:
- Visual and verbal symbols
- Visual symbols in geometry
6. Interpersonal and emotional factors:
Since each individual is different, they have to be treated accordingly. They have their own peculiar interpersonal and emotional characters which are to be addressed while teaching mathematics.
7. Implications of Skemp's Learning Theory in Mathematical Instruction
1. The concept of higher order then those which people already have cannot be communicated to them to encounter a suitable collection of examples. This means plenty of carefully selected examples should be used in the text books and definitions should be used only to consolidate a concept when that is already formed.
2. In a mathematical curriculum the logical priority of concepts should be clearly worked out and order of learning of concepts, where implied by a careful conceptual analysis, should be followed.
3.Since mathematical concepts are abstract there is a danger of their becoming mere symbols, there fore, in the early stage a lot of relevant experience should be provided for in the curriculum itself. At the same time, they should be formed in such a way that they can become free from reliance on the apparatus, then only shall they become genuinely mathematical in nature.
4. Intuitive intelligence may or may not be true because sometimes it may be wrong in the situation teacher should be responsible to correct the problems or errors done by students. As a result student know their mistake and they get a right chance to correct and from onwards and they do not repeat that mistake. In a way, they can solve the problems without any help of teacher.
5. A variety of examples should be chosen so that the concepts learnt can become free from the incident features of the examples.
6. To develop the intuitive intelligence of learners , teachers should applied the many distinct method. As it is impossible to find their intuitive intelligence for even thought it is even necessary to encourage towards true intuitive intelligence. Not only that but also the teacher should be capable to develop the intuitive intelligence by deep study . If the teacher does not have the knowledge of intuitive intelligence, he will not be success to give clear knowledge to the students.
7. The mathematical teacher should focus on the theory that "process is more important then the product" during teaching learning activities that helps the students to develop strong concepts, to prepare mentally ready for learning and to help further learning concepts by means of previous one. For example
12 = 4 + 4+ 4 adding 4 three times(previous learning)
12 = 4 x 3 multiplicative form with the help of additional process (new learning)
From the combination of these two, students understand that multiplication comes after addition operation.
8.Teracher must used multi techniques to solve problems of mathematics that helps students to increase the mental power so that learners able to search themselves different methods and choose appropriate process. At the result, students can make the scheme for further solution. For example, to find the HCF of x2-4x+4 and x3-8, we can use factorization method, Venn diagram , division method, etc.
9. Skemp learning theory focuses the leaner's imagery and motive the learners. If the learners are motivated through the teachers, they search the problems themselves, them thinks themselves, they solve themselves and finalize, and they derive the solution. In this way learner can learn.
10. Teacher should careful about individual differences of students in intelligence, logical, imagery, problem solving abilities, learning capacities etc. Therefore teacher should focus the individual learning based on interpersonal and emotional factors.
11. Teacher should create the environment of thinking, speculating and processing of many concepts so that they are able to make imagery themselves and use the imagery which helps to make mathematical concepts prolong and applicable in behavior.
12. Intuitive intelligence cannot always true. Teacher should correct the results carried out by students and if students are good in way then teacher should praise the students.
Bibliography
Pandit, Ramjee Prasad.2006. Foundation of Mathematics Education.
Budhathoki, Tara Bahadur & Ram, Surendra. 2007. Ganit Shikshako Aadhar.
Upadhyaya, Hari Prasad. 2005. New Trends in Mathematics Education.