## Wednesday, 25 July 2012

### Unit - 8 Methods and Teaching aids

METHODS AND TECHNIQUES OF TEACHING MATHEMATICS

For teaching of every subject a method is very important. It is nothing but a scientific way of the presentation of the subject, keeping in view the psychological and physical requirements of the students. It is through method only that it is possible to make a subject interesting and useful. Without a method, teaching would be pretty hard and difficult. The selection of a suitable method depends upon the objectives of the lesson, needs of the learner and nature of the content.

The following are the methods which are suitable for classroom teaching of mathematics.
•  Lecture method
• Inductive and Deductive method.
•  Analytic and Synthetic method
•  Problem Solving method
•  Project method
•  Heuristic method
•  Laboratory method

Lecture method
The ‘Lecture method’ is otherwise known as ‘the chalk and talk method’ where in verbal explanation is given for facts. This is a teacher centered method, teacher is an active participant and child is a passive learner. It is a one way traffic or flow of ideas. This is the method of imparting information through a speech. This is a one man’s show because the children remain passive through out the process. This method is most commonly used in colleges and not a very suitable method for teaching mathematics at high school classes. In this method the students are provided with readymade information by the teacher. The teacher goes ahead with the subject matter at his own speed.

Advantages of   Lecture method
•  It is an easy, concise and attractive method. Using this method the teacher feels safe and secure.
•  Using this method, a large amount of subject matter can be presented within a short time and   the prescribed syllabus can be covered easily.
•   It can be used for a large number of students.
•  This method gives the students as well as the teacher a sense of satisfaction and achievement.
•  Lecture method trains good listeners who are able to concentrate on subject matter for a long duration.
•  This method suits the teacher. It is convenient for him because he has not to give individual help to the students.
•  It is economical with respect to time and money.

Limitations
• This is a teacher- centered method. So this is against psychological principles.
•  It does not provide opportunities to develop various mental abilities like reasoning, logical thinking etc.
•  In this method students become passive recipients of information as their involvement in classroom interaction is negligible.
•  Receiving information is not mathematics learning and hence it does not enhance  mathematical ability of the students.
•  In this method there is no way to ensure the students concentration and understanding of the subject matter presented to them.
•  It does not provide for individual differences and individual needs.
•  This method does not help in developing problem-solving skills.
•  It does not provide for corrective feed back and remedial help to slow learners.
•  It does not call for the development of mental faculties such as power of observation, reasoning, critical thinking, independent thinking and so on.

Uses
It can be used
• while introducing a new concept.
•  for explaining abstract concepts.
• for giving demonstration.
• for summing up and reviewing certain concepts.
• while initiating a discussion.
Inductive method
Inductive method is based on induction. Induction is the process of providing a universal truth or a theorem by showing that it is true of any particular case, it is true for the next case in the same serial order and hence true for any such cases. So the technique of making transition from particular facts to generalizations about these facts is known as the process of induction. Thus it is a method of arriving at a formula or a rule by observing a sufficient number of particular instances. If one rule applies to a particular case and is equally applicable to different similar cases, it is accepted as a general rule or formula. Therefore, in this method we proceed from particular to general, from concrete instances to abstract rules and from simple examples to complex formula. A formula or a generalization is arrived at through inductive reasoning.
This method has been found to be very suitable for teaching of mathematics because many mathematical formulae and generalizations are the results of induction.

Example
Angle sum property in a triangle
When a student measures the angles of several triangles and finds that in each case the sum of the angles approximates to 1800, he has the background to generalize that sum of the interior angles in a triangle is 1800.
Establishing the formula (a + b)2 = a2 + 2ab + b2
Students may be asked to find out the square value in each of the cases like (x + y)2, (m +n)2 , (p +q)2  etc by the simple method of multiplication.  After doing these different multiplications, they may be helped in generalizing that ( 1st term + 2 nd term )2 = (1 st term)2 + (2 nd term)2 + 2 (1 st term) (2 nd term).

Steps in Inductive method
• Selection of a number of cases.
• Observation of the cases under given conditions.
• Finding common relations.
• Arriving at generalization.
• Verification or application.
•   It is a logical method and develops critical thinking.
•   It encourages active participation of the students in learning.
• It provides ample opportunities for exploration and observation.
•   It sustains the interest of students as  they proceed from known  to unknown.
•   It enhances self-confidence.
•   It discourages cramming.
•   It helps in increasing the pupil- teacher contact.
Limitations
• This method is limited in range and is not suitable for all topics.
• This method is not suitable for higher classes because higher order mathematical principles cannot be generalized through the observation of concrete cases.
• It is time consuming and laborious method.
• Inductive reasoning is not absolutely conclusive because the generalization made with the help of a few specific examples may not hold good in all cases.
• The ability and capacity of problem solving cannot be developed  by the use of this method.

Uses
Inductive method is most suitable where
• Rules are to be formulated
• Definitions are to be formulated
• Formulae are to be derived
• Generalizations of laws are to be arrived at
• It can be used in lower classes where certain simple generalizations are to be made.

Deductive method
Deductive method is the opposite of the inductive method. It is based on deduction. Deductive reasoning is the process of drawing logical inferences from established facts or fundamental assumptions. In this method  we proceed from general to particular and from abstract to concrete. The students are asked to apply these rules to solve more problems. This approach is not suitable for exploration.
In this method, we begin with the formula or rule or generalization and apply it to a particular case.
Example-
(a + b)2 = a2 + 2ab + b2
Find (3 + 2)2
By applying the identity,
(3 + 2)2 = 32 + 2 . 3 . 2 + 22
= 9 + 12 + 4
= 25

Steps in deductive method
Clear recognition of the problem: A clear recognition of the problem statement provides the basis link for thinking process and the solution to the problem.
Search for a tentative hypothesis
Search for a tentative hypothesis, a tentative solution to the problem.
Formulation of a tentative hypothesis
Search for the solution leads to the formulation of a tentative hypothesis that appears to have promise as a possible or probable solution to the problem. The tentative hypothesis has its basis on certain axioms or postulates or propositions or rules and formulae that have been accepted to be true.
Verification
Finally the hypothesis that has been formulated is to be verified as the right solution to the problem at hand.

• It saves time and labour for both the teacher and the student.
• It enhances speed, skill and efficiency in solving problems.
• It is a short and elegant method.
• It helps in fixation of formulae and rules as it provides adequate opportunities for practice and revision.
• This method suits all types of students.
• It helps in increasing the memory power of the students, as the students are required to memorize a large number of laws, formulae etc.
Limitations
• It is not a psychological method.
• It encourages cramming.
• It is not suitable for beginners.
• It does not encourage students involvement in learning.
• It is not suitable for the development of thinking, reasoning and discovery.]
• Since it is based on blind memory, once the formula or rule is forgotten, it is not possible to rediscover them.
• Students get disinterested as they are only passive listeners.
Uses
• This method is suitable for fixation and retention of facts and rules.
• It is suitable for teaching demonstrative geometry.
• This method is suitable for average students.
• This method provides skill, speed and efficiency in solving problems.

Inductive and Deductive Method

 Inductive method Deductive method Based on inductive reasoning Based on deductive reasoning Particular to General General to Particular A psychological method An unpsychological method A method of discovery that stimulates intellectual powers A method of presentation that does not develop originality and creativity Emphasis is on reasoning Emphasis is on memory Encourages meaningful learning Encourages rote learning Most suitable for initial stages of learning Suitable for practice and application Suitable for lower classes Most suitable for higher classes Enhances active participation of the students Makes the student passive recipient of  knowledge Lengthy, time consuming and labourious Short, concise and elegant Facilitates discovery of rules and generalizations Enhanced speed, skill and efficiency in solving problems It gives opportunity for the better pupil teacher relationship In this method the relationship remains nominal

Analytic method
The word ‘analytic’ is derived from the word ‘analysis’ which means ‘breaking up’ or resolving a thing into its constituent elements. This method is based on analysis and therefore, in this method we break up the problem in hand into its constituent parts so that is ultimately gets connected with something obvious or already known. In this process we start with what is to be found out (Unknown) and then think of further steps and possibilities which may connect with the known and find out the desired result. Hence in this method we proceed from unknown to known

Example
If a/b = c/d  P.T ac + 3b2    =   c2 + 3bd
bc                  dc
To prove this using analytic method, begins from the unknown
The unknown is   ac + 3b2    =   c2 + 3bd
bc                  dc
ac + 3b2    =   c2 + 3bd        is true
bc                    dc
If   ac + 3b2    =   c2 + 3bd        is true
b                   d
If d( ac + 3b2)   =   b( c2 + 3bd)

If dac + 3b2d  = bc2 + 3b2d

If dac = bc2

If da = bc

If  a/b = c/d  which is given to be true.

ac + 3b2    =   c2 + 3bd
bc                  dc

•  It helps in developing reasoning power of the students
•  It leaves no doubt in the minds of the students as every step is justified.
•  It facilitates clear understanding of the subject matter as every step is derived by the student himself.
•   It is a psychological method.
•   No cramming is required in this method
•   Students take active role in the learning process resulting in longer retention and easier recall of what they learn.
•  It develops self-confidence in the pupil.

Limitations
• It is a time consuming and lengthy method. So it is uneconomical.
• This method may not be suitable for all topics of mathematics.
• In this method facts are not presented in a neat and systematic order.

Uses
•   This method is suitable for teaching arithmetic, algebra and geometry.
•   This method is suitable for solving complicated problems

Synthetic method
Synthetic method is derived from the word synthesis. Synthesis is the complement of analysis. To synthesis is  to combine the constituent elements to produce something new. In this method we start with something already known and connect it with the unknown part of the statement. Therefore, in this method one proceeds from known to unknown. It is the process of combining known bits of information to reach the point where unknown information becomes obvious and true.
The usual form of statements of proofs found in textbook are examples of synthetic method. Beginning with known definitions, assumptions and axioms, the sequence of steps are deducted and conclusions are arrived at.

Ex
If a/b = c/d  P.T ac + 2b2   =   c2 + 2bd
bc                  cd

a/b = c/d  (given)
Add 2b/c to both sides (Why? It is not explained)
a/b + 2b/c  =  c/d  + 2b/c
ac + 2b2   =   c2 + 2bd     ( It was to be proved)
bc                  cd

•   This method is logical as in this method one proceeds from the known to unknown
•   It saves time and labour
•   It suits majority of students
•   It can be applied to majority of topics in teaching of mathematics
•   It is a neat method in which we present the facts in a systematic way.
•  It is short and elegant

Limitations
•   It leaves many doubts in the minds of the learner and offers no explanation for them
•   It makes the students passive listeners and encourages cramming
•   This is an unpsychological method
•   In this method there is a scope for forgetting
•   There is no scope for discovery and enquiry
•   It does not provide full understanding
•   If the student forgets the sequence of steps, it would be very difficult to reconstruct the proof/solution.

Uses
• This method can be used by an average teacher and also suits for the average student
• This method is suited for the final presentation of proofs of theorems and solutions to problems in a logical and systematic manner

Comparison of Analytic and Synthetic methods
 Analytic method Synthetic method Here one proceeds from unknown to known Here one proceeds from known to unknown Here one starts from what is to be proved and then comes back to what is given Here one starts from what is given and ends with what is to be proved Analysis means breaking up into components Synthesis means combining the elements to get something new Lengthy, laborious and time consuming Short, concise and elegant Encourages meaningful learning Encourages rote learning Easy to discover Once forgotten, not easy to recall Valid reasons to justify every step in the sequence No justification for every step in the sequence Application of inductive reasoning Application of deductive reasoning Active participation of the learner Learner is a passive listener A psychological method A logical method

Problem solving method
The problem solving method is one which involves the use of the process of problem solving or reflective thinking or reasoning. It may be noted that all problem solving doesn’t necessarily include reflective thinking or reasoning because problem may be solved through trial and error or analogy.
Problem solving method, as the name indicates, begins with the statement of a problem that challenges the students to find a solution. In the process of solving the problem the students may be required to gather data, analyse and interpret information to arrive at solution to the problems.
According to James Ross ‘Problem solving is an educational device where by the teacher and the pupils attempt in a conscious, planned, purposeful manner to arrive at an explanation or solution to some educationally significant difficulty

Steps in problem solving
1. Identifying and defining the problem
Teacher has to select problematic situation and motivate children to ask questions. Then the children have put a number of questions. Then the problem is identified. The identified problem should be based on the requirement of children their competency and intellectual level. The students should be able to identify and clearly define the problem.

2. Analysing the problem
The problem should be carefully analysed as to what is given and what is to be found out. Given facts must be identified and expressed, if necessary in symbolic form.

3. Formulating tentative hypothesis
The focus at this stage is hypothesizing searching for a tentative solution to the problem. Analysis of the given data and analysis of interrelationship among the given facts helps the students in formulating hypothesis or educated guesses as the solution to the problem at hand.

4. Testing the hypothesis
Appropriate methods should be selected to test the validity of tentative hypothesis as a solution to the problem. If it is not proved to be the solution, the students are asked to formulate alternate hypothesis and proceed.

5. Checking the result or verification of the result
At this step the students are asked to determine if their results substantiate the expected solution

•  Problem solving provides a real life experience to the children
•  It develops in pupils good habits of planning, thinking, reasoning and independent work.
•  It develops initiative and self-responsibility among the students
•  It takes in to account individual differences
•  It helps the students to develop reflective thinking
•  It helps the students to approach future problems with confidence
•  This method is helpful in the development of harmonious relationship between the teacher and taught
•  It helps the children develop mental traits of open-mindedness and tolerance as the children see many sides to a problem and listen to many points of view.

Limitations
• Not all the students are problem solvers.
• It cannot be suited to large crowded classed
• It is time consuming and consequently it is not possible to cover the syllabus on time
• All the topics or subject areas cannot be taught by this method
• It does not suit the students of lower classes

Uses
This method trains the pupils in problem solving
Pupil learn to sense, analyse, reflect, organize and solve the problems
It helps at every step in the teaching-learning process

Project method
Project method is developed and applied practically by Dr.Kilpatrick. It is the outcome of the pragmatism- ideas of Sir John Dewey. “ What is to taught should have a direct relationship with the actual happenings in life”, this central idea forms the basis of project method.
According to Dr. Kilpatrick ‘A project is a unit of whole hearted purposeful activity carried on preferably in its natural settings. Ballard defined project ‘as a bit of real life that has been imparted in to school’. All definitions stated above emphasis the project should be purposeful activity related to life and it should be carried out in a natural environment. In project method, teaching and learning are considered from the child’s point of view and in this method knowledge and skills are learnt by pupils through practical handling of problem in their natural setting. This method is an ideal way of promoting creativity, arousing curiosity and inculcating the spirit of enquiry among the students

Basic principles of project method
The project method is based on the psychological principles of learning namely
(i) Learning by doing
(ii) Learning by living
(iii) Childern learn better through association, co-operation and activity

Steps involved in  project method
•   Providing a situation
•   Selecting and purposing of the project
•   Planning of the project
•   Executing the project
•   Evaluating the project
•   Recording

Providing a situation
The project should arise out of the felt needs of the students. The teacher should provide such situations to students which may arouse some suitable questions to which the students seek answer. It should look important, must be interesting and purposeful for the students. The teacher can provide a variety of situations through discussion, questioning, library work, field work etc

Selecting and purposing
The selection of the project is done by the students themselves. The teacher  should refrain from purposing any project; otherwise the whole purpose of the method would be defeated. How ever the teacher can guide the students in the selection of a good project, keeping in mind the interest, aptitude and ability of the students. In this step the nature and goal of the project is clearly determined as well as the limits and scope of the project is clearly defined.

Planning
For any successful project planning is essential. Planning involves the selection of the unit appropriate and feasible set of activities to be executed. The students should choose the most practical plan of action. The students themselves should do the planning with the teacher as a guide. While planning the points to be taken in to consideration are (i) The degree of complexity of the project. (ii) time allotted to finish the project and (iii) availability of material resources. Discussion may be held among students before the final draft of the plan is agreed upon.

Execution
In this step the teacher helps the students in assigning work to different students in accordance with their interest, aptitude and capabilities. Each member of the group should be actively involved in the execution of the project. The teacher should carefully supervise and guide the students in the execution of the project as per the proposed action plan.

Evaluation
The students along with the teacher should review the progress of the project at frequent intervals. This is to ensure that the students are progressing towards the realization of the objectives of the project. Without evaluation, the project can’t move out of focus. The evaluation of the project has to be done in the light of
(i) proposed plan (ii) difficulties in the execution and (iii) achievement results.

Recording
The students are required to maintain a complete record of work including the choice of the project, the planning, the discussions held and duties assigned. Also reference and books consulted and readings taken, difficulties faced, guidance sought, details of places visited and surveyed and so on should be carefully recorded.

• It is based on sound psychological principles
• It provides scope for independent work and individual development
• It provides for individual differences as the students can select the activity and work at their own pace.
• It promotes social interaction, inculcates spirit of co-operation and exchanges of experiences among the students
• It provides opportunities for children to acquire a lot of skills- observation, reference and interpretation
• In this method children are active participants in the learning task
• It develops self-confidence and self-discipline among the students
• It upholds the dignity of labour
• It arouses interest of the pupils
• The facts are easily remembered
• It makes the learning more interesting and facilitates better understanding of the subject matter as the learning is related to reality and the world around him.

Limitations
• The project method is un economical in terms of time and not possible to fit in to the regular time table
• It does not provide any training in mathematical thinking and reasoning
• The learning is incomplete and uniform learning or balanced learning is not possible for all students as each student perform a different activity
• For the success of this method the teachers should be exceptionally resourceful and gifted and knowledgeable
• It is an expensive method as it makes use of a lot of resources which are not immediately available in the school
• Syllabus cannot be completed on time using this method
Uses
• Project method can be used to arouse interest and encourage initiative and to provide a sense of achievement to the students
• Project method provides a practical approach to learning

Heuristic method
This method involves finding out by the pupil, instead of merely telling of everything by the teacher. This method demands complete self activity or self-education on the part of the learner
The term ‘heuristic’ is derived from the Greek word ‘Heurisco’ which means ‘I discover’. This method was advocated by Professor H.E Armstrong, who felt that by placing the student in the position of a discoverer, he would learn much more than being merely told about things. In this method the teacher becomes on looker and the child tries to move ahead independently with out any help or guidance. This method makes the student self reliant and independent. The teacher’s job is not to solve problems for the pupil, but to enable the student to solve problems for himself. It aims at developing a scientific or heuristic attitude in the pupil

Ex:-
The population of a city is 25,000. If the percentage increase in population is 2% per year, then find out the population of the city after 2 years.

Solution
Teacher will tell the students to repeat this question many times and will tell them to find the solution with the help of suitable research questions and he will encourage them for finding correct solution. We can understand the solution of the problem with the help of questions asked by the teacher.
The questions asked by the teacher and expected answers give by the students are listed in the following table

 Questions Expected answers What has to be calculating in the given question? Population of the city after 2 years How can we find out this? First find the population after one year What is the rate of increase in population? 2%  per year How can we find population at the end of first year? Students can find that increase in population at the end of first year = 25,000 X 2/100 = 500 In this way what will be the total population at the end of first year Students can tell after solving that population at the end of first year = 25,000 + 500 = 25,500 Population increase for II year can be calculated on which figure? Students tell that population increase for II year can be calculated on 25,500 How can we calculate this increase? Students can solve by their efforts. Population increases at the end of II year = 25,500 X 2/100 = 510 Now how to calculate the population at the end of II year? Students will be able to understand that population of 2 years will be 25,500 + 510 = 26,010
In this way solution to the given problems can be found by Heuristic method.

•   It develops the habit of enquiry and investigation among the students
•   It is a psychological method, knowledge is not thrown upon the child rather he is made to discover things himself.
•   It develops self-confidence, self-reliance and scientific attitude
•   It provides for individual differences as each student can work at his own pace.
•   This method demands quite alertness and presence of mind on the part of the student for responding to the heuristic questions asked by the teacher.
•   Heuristic method gives enough opportunity of developing teacher-pupil intimacy.
•   It helps in real understanding and mastery over the subject matter.
•   It helps in the development of social skills as the students have to co-operate with one another.

Limitations
•  This method is feasible only with a highly resourceful teacher and a small class
•  It is lengthy, time consuming method and hence it is difficult to cover the prescribed syllabus in time.
•  It is not suitable for beginners as the students need enough guidance at early stages to have a proper foundation in the subject.
• Not all students can cope with this type of learning
•  This method requires special preparation on the part of the teacher
•  The results found out by the student may not be correct. In the absence of individual checking, all the istakes may not be rectified.
•  If the teacher is not able to distinguish between false and true heuristic questions, he/she may do more harm than good.

Uses
•   Cultivating a heuristic spirit in students, this method can be used.
•   To develop efficiency in problem solving skills, independent thinking and reasoning, self learning and self-reliance among the students

Laboratory method
Laboratory method of teaching mathematics is that method in which we try to make the students learn mathematics by doing experiments and laboratory work in the mathematics room or laboratory on the same lines as they learn sciences by performing experiments in the science rooms or laboratories. It is based on psychological principles of learning such as ‘learning by doing’, ‘learning by observation’ and so on. Laboratory method is quite competent to relate the theoretical knowledge with the practical base. This approach makes the learning process more interesting, lively and meaningful.
The success of the laboratory method depends on an able skilled mathematics teacher as well as the availability of a well-equipped mathematics laboratory.  According to J.W.A young “ a room specially filled with drawing instruments, suitable tables and desks, good black boards and the apparatus necessary to perform the experiment of the course is really essential for the best success of the laboratory method”.

•  This method is psychologically more sound as in it we proceed from known to unknown
• This method is based on the principle of learning by doing
•  It stimulates the interest of students to work with concrete material
• Knowledge and skills acquired through experiments help in better understanding and longer retention
•  It promotes self-confidence and self-reliance and a sense of achievement among the students
• It provides opportunities for social interaction
• This method suits the average and below average students and to such student who cannot understand things at the abstract level
• The things learnt experimentally can be used by the pupil in many situations

Limitations
• It is an expensive method in terms of time, equipment, laboratory facilities and number of skilled and able teachers
• Only very few topic in mathematics can be taught through this method and hence it has limited applicability
• This method is not useful for large classes
• In higher classes this method is not suitable. In these classes students are expected to grasp abstract ideas and concepts
• Text books in use do not indicate the practical work can be done by the students
Uses
It can be uses in lower classes where we have to make clear certain mathematical concepts and ideas

Seminar method
The English author Francis Bacon rightly said long ago, “Reading makes the full man, writing the exact man and conference the ready man”. That is the skills of reading, writing and speaking are essential for development of the personality. These skills are well combined and well developed through the seminar method. The term ‘seminar’ generally refers to a structured group discussion that may precede or follow a formal lecture. It may be either in the form of an essay or a paper presentation. It may be done by experts or academicians at a seminar or conference organized by an institution or an association or by an individual student in class or group of peers as a part of the source of work. In any case, the audience critically examines the paper and discusses the content or findings of the paper. They make queries for clarification or elucidation on any point and the writer of the paper is required to clarify these queries with further facts and figures. The seminar method is very suitable for higher education or professional discussion when the level of attainment of the group is relatively high and nature of discussion is expected to be analytical and technical

Characteristics
•   The seminar method is stimulating and motivating
•   It is an effective mode of testing the students understanding and knowledge in the subject or the topic
•   It evaluates their skill in arranging, formulating and presenting facts in a systematic manner
•   It develops the sense of responsibility and co-operation as well as the powers of self-reliance and self-confidence
•   It is an advanced means of socialization and instruction involving students participation, preparation and interaction
•   It breaks the monotony of traditional teaching and promotes academic excellence
•   It stimulates and tests the students power of comprehension and understanding
•   Seminar is mostly subject oriented and deals in depth a particular topic
•   Questioning power is developed through participation

Types of seminar
On the basis of levels of organization, the seminars are of 4 types (i) Mini seminar (ii) Main seminar (iii) National seminar and (iv) International seminar

Mini seminar
A seminar organized to discuss a topic in class is known as mini seminar

Major seminar
Such seminars are organized at departmental level or institutional level on a major theme

National seminar
A national seminar is organized by an association or organization at national level. Ex:- NCERT organizes seminar at national level on different themes like Educational Technology, Population Education,  Distance Education, Vocational Education etc.

International seminar
Generally such seminars are organized by UNESCO and other international organization

Limitations
•  A seminar cannot be organized on all the content of a subject matter. Some topics are highly structured
•  This technique cannot be used for all levels of education. It can be used for higher level of education
•  When a seminar is being organized, the persons who speak too much, dominate the discussion of the seminar and do not provide opportunities to take part in the discussion. It means that the discussion confuses only few persons rather than whole group
•  During discussion, groups are formed in two ideas, anti-ideas and provide on the theme. As a result they try to win over the other. The purpose of the seminar is not served
•  The opposition group generally try to oppose even for the contractive or relevant ideas of the other group. This instructional situation of such discussion is not conducive for learning.

Symposium
Symposium technique is also one of the techniques of higher learning. It is also an instructional technique which is used to achieve higher cognitive and affective objectives.

Definitions
“The symposium technique serves as an excellent device for informing an audience, crystallizing opinion and generally preparing the listeners for arriving at a decision, policy, value, judgement or wider understanding”.
The symposium is a type of discussion, in which two or more speakers talk from ten to twenty minutes, develop individual approaches or solutions to a problem or present aspects of a  policy, process or programme. The speeches are followed by questions or comments from the audience. The speeches may be persuasive, argumentative, informative or evocative. Each speech proceeds without interruption. The chairman of the symposium introduces the topic, suggests something of its importance. Sometimes indicates the general approaches. The symposium forum serves an excellent device for informing an audience, crystallizing opinion and in general preparing the listeners for arriving at a decision policy, value, judgement or understanding. All members of the performing group can sit in a straight line behind a table or adjoining chairs, with chair in the middle or to one side of the speakers or if the symposium is to present two conflicting points of view, the seating arrangement can separate the speakers on the platform in order to indicate differences in opinion or in order to preserve peace.

Scope
The symposium  technique is used to realize the higher cognitive and affective objectives
The following are the main topics on which symposium techniques is used:
•  Use of television for education
•  Scope of distance education in our education
•  Semester system in education
•  Use of essay and objective type tests
•  Use of micro teaching in teacher education
•  Use of team teaching in schools

Characteristics
•   It provides the broad understanding of a topic or a problem
•   The opportunity is provided to the listeners to take decision about the problem
•   It is used for higher classes to specific themes and problems
•   It develops the feeling of co-operation and adjustment
•  The objectives as synthesis and evaluation (creativity) are achieved by employing the symposium technique.

Limitations
• The chairman has no control over the speakers as they have full freedom to prepare the theme for discussion
• The discussions and presentation of theme is not summarized at the end. The participants take decision according to their own. Hence mature persons can make use of this technique.
• There is a probability of repetition of the conduct because every speaker prepares theme as a whole. The different aspects of theme are not prepared separately. It creates difficulty of understanding to the listeners.

Work shop
The workshops are organized to develop the psychomotor aspects of the learner regarding practices of new innovations in area of education. Under this technique participants have to do some practical work to produce instructional, teaching and testing material.

Definitions
Workshop is defined as assembled group of ten to twenty five persons who share a common interest or problem. They meet together to improve their individual skill of a subject through intensive study, research, practice and discussion.
Generally workshops are organized for three to ten days duration. The period of workshop may be 40 days, it depends on the nature of task assigned to the workshop. It is organized in three stages

 First stage Presentation of the theme of providing awareness Second stage Practice the approach for its applicability Third stage Evaluate the material prepared by the participants and follow-up programme

First stage:
The resource persons or experts are invited to provide the awareness and understanding of the topic. This stage is like a seminar type. The paper reading is also alone to discuss the different aspects of the theme. The trainee or participants are given opportunities to seek clarification. Stage continues for first two days of schedule.

Second stage:
In the first stage theoretical background is provided to the participants. In the second stage the group is divided in to small groups. Eg: - A workshop for lesson planning or writing objectives in behavioural terms, question bank preparation etc. The group formed on the basis of different subjects such as Hindi, English, Maths, science etc. A resource person or expert provide, guidance and supervision their work of each trainee of his group. Every trainee has to complete his task within the given period. At the end they meet in their groups and discuss and present their task to be completed.

Third stage:
At this stage all groups meet at one place and present their reports of work done at second stage. The participants are given opportunities to comment and given suggestions for further improvement

Follow-up:
The effectiveness of a workshop is ascertained by a follow up programme. After workshop the trainees have to go back to their institutions. They are asked to continue their task and examine the workability and usability in their institution. The participants are invited to meet again and present their experiences regarding applicability of the topic or new practices

Scope:
The workshop technique is used mainly in the following areas of education
1. New format of lesson planning
2. Writing objectives in behavioural terms
3. Preparing instructional material or teaching model
4. Workshop for micro-teaching

• It is a technique which can be effectively used for developing understanding and proficiency for the approaches and practices in education.
• The teaching proficiencies can be developed by using the workshop technique for in-service teachers.
• It provides the opportunities and situations to develop the individual capacities of a teacher.
• It develops the feeling of co-operation and group work.
• The new practices and innovations are introduced to in-service teachers.

Limitations
• The workshops cannot be organized for large group so that large number of persons are not trained.
• The effectiveness of a workshop technique depends on the follow up programme which does not take place in workshop technique

Group discussion
There is no comprehensive definition of group discussion but it is considered as a democratic teaching strategy. The pupils are more active in it. Teacher’s job is to supervise and to provide guidance to pupils activities. It is child centered teaching strategy. It may be of two types
(i) By the teacher and
(ii) By the students

By the teacher: This type of discussion is more autocratic in style.
By the student: In this situation, discussion is more permissive on democratic in style.

Objectives: The affective objectives and higher order cognitive objectives may be achieved.

Structure: The group discussion is organized in two forms: Formally and informally. In formal group discussion, proper schedule is prepared and certain rules are observed. In the absence of a teacher pupils have to select a student to act as a leader of the group. The leader of the group prepares a plan for discussion. In the group discussion, due weightage is given to the answer and questions of the pupils.

Principles: This strategy is based up on the following principles of teaching:
1. Principle of active participation
2. Principle of freedom for work
3.Principle of group work and equal opportunities to as questions and to answer them.

Advantages: The following are the main advantages of this strategy
•   It develops the feeling of group work and group participation
•   It helps in developing the creative ability and thinking among pupils
•   It develops the problem solving attitude and the tolerance to hear one’s own criticism.
•   It helps in developing the feeling of co-operation

Limitations
It has the following limitations
•  It has chances for deviation from the main topic
•  In the discussion, only a few students dominate and to discuss more and large number of students do not participate in the discussion
•  Generally groups are formed in this strategy and they are critical to one another.
•  It involves more criticism rather than relevant discussion.

Suggestions
•  It may be improved and can be effectively employed by observing the following precautions:
•  Every student should be provided an opportunity to participate in the discussion
•  Students should be encouraged and motivated to participate in discussion
•  Irrelevant criticism should not be encouraged.

Task analysis is as the name implies a way of describing how a task is performed. A task analysis involves a careful examination of the tasks and or objectives in an effort to determine such things as to how task will be taught, to whom and when. The task analysis breaks down the task into its constituent parts and involves detailed listing of component elements of a job or task.
Task analysis enables us to bring principles desired from learning theory to bear on the sequencing of instruction. The task analysis assists the teacher to determine the specific task the student has failed to perform. In task analysis the knowledge to be taught must be systematically organized so that salient points are identified.

Techniques of Task analysis
1. Interviews: Individual and group
2. Direct observation

Role of teacher in task analysis
1. He should administer diagnostic pre-task before beginning instruction. This enables the teacher to understand the position of the learner.
2. The teacher should meet the individual differences of the learners.
3. The teacher must give cues to the learner which he needs.

Use of computer for teaching of mathematics
In this modern age of computers through their valuable services and applications represent the most important gift and contribution of the science and technology to the man kind. The use of computer in teaching of mathematics may be supported on the following grounds.

1. Diagnosis of the entry behaviour of learners:
Computer programmes can help him in the early diagnosis of the strength and weaknesses of his students in terms of their entry behaviour for determining their potentiality.

2. Providing a rich treasure of knowledge and information:
Computers may help in a big way for the collection, storage, classification and dissemination of knowledge and information related to the teaching and learning of mathematics. More over internet provides enormous opportunities for the utilization of information and knowledge for the attainment of teaching learning objectives in the subject mathematics.

3. Generating individualized instructional plans:
Depending upon the need, characteristics, nature and individuality of the learners computer softwares are able to generate and organize individualized instructional plans for the benefit of a countless learners at one or the other times at the demand and pace of the individual learner.
4. Providing valuable teaching aids for instruction:
Computer can help in generating and producing valuable teaching aids for the teaching and learning of a topic of a particular unit of concept in the subject mathematics with the use of MS Office, corel draw, graphics etc we can do the same.

5. Computer help in providing sufficient drill and practice:
Computers can help a lot in providing sufficient amount of practice and drill work for the fixation of knowledge and skills already learned. Computer as a machine works quite instantly and provide proper feedback. Moreover all the students can work here with their own speed and will to carry on the practice work.

6. Help in doing practical and project work:
Many of the practical skills like construction work in geometry, surveying and measuring may be well demonstrated through the well prepared programmes.

7. Providing opportunity for improved self-study:
Computers have the ability and capacity to provide individual instruction to a limitless number of students at a single time and there by increase their self learning ability. More over the student doesn’t remain passive as may happen in case of listening to a classroom lecture.

8. Monitoring of progress:
Computer may prove quite effective in monitoring the progress of each and every individual learners in a quite satisfactory way.

9. Providing remedial instruction:
On the basis of the learning difficulties diagnosed and the probable causes detected, the softwares can suggest all the possible remedies helpful in the planning and organization of remedial instruction to the needed learners.

10. Organisation of testing and evaluation programmes:
Computer softwares may provide valuable services in the task of managing and organizing testing and evaluation programmes related to classroom instruction andeducational progress of the students.

11. Generating all types of reports and records related to the teaching learning of mathematics:
The data regarding the available instructional resources, aids and equipments unit wise organization of the curriculum and the progress regarding to the attainment of instructional objectives are readily and to the learners for self learning and to the teachers for imparting appropriate guidance and educational help to the students. The testing, evaluation and progress reports of the individual students and the group as a whole can be very well generated through computers.

Importance of teaching aids
There are so many abstractions in mathematics which cannot be easily followed by the students. To make such abstract ideas or complex ideas less abstract, teacher takes help of various teaching aids. Since most of the learning in children takes place at the sensory level, so the senses are gateways of knowledge. Teaching aids influence the minds of learners through their senses.
There is a famous saying,
"I hear, I forget
I see, I remember
I do, I understand"
• They help in clear understanding of the subject and clarifying abstract ideas.
• The appeal the senses of the pupils and so they satisfy their innate tendencies and interests.
• They stimulate pupil participation. Since they are based on the maxim ‘learning by doing’.
• They make teaching –learning process interesting.
• They help in saving time and energy because it takes a long time to classify an abstract idea verbally.
• Promote self-activity and purposefulness in learning.
• Provide variety, freshness and reality to learning.
• Cater to individual needs and abilities.
• Leave long lasting impressions on the minds of the learner.
• Motivate the children and stimulate their mental activity.

Projected Aids
A projected aid is one in which items to be perceived are projected on screen using mechanical devices. The major projected aids that could be used in teaching are given below.

1. Over head projector (OHP)
This device helps in projecting transparent images on to a screen under normal light condition.
Uses:
OHP can be used to explain topics and at the same time teacher can retain eye contact with the students
OHP can be used to avoid the monotony of continuous audio perception.

2. Episcope and Epidiascope:
Episcope is used for the projection of opaque objects.
Uses:
It can be used for the projection of pictures, photographs, drawings or any opaque material with in the age of the platform.
It can be used to enlarge book diagrams and to illustrate complex materials.

Epidiascope:
Epidiascope is used for the projection of opaque as well as transparent objects. It is actually a combination of episcope and diascope.

Uses:
Used for projecting slides
It is possible to produce enlargement of diagrams, maps or illustrations.
It can be used to reproduce the original colours of the diagram or picture during projection.

Slide projector
Slide projector is an instrument equipped with a powerful light source and carrier for holding slides of suitable size.

Uses:
This can be used when a continuity of a theme is required.
Slides of different size can be projected.

Film strip projector
Film strip projector are used for projecting film strips on the screen when a series of slides are to be presented in a sequence, it is preferable to prepare a film strip.
Uses:
It can be used to stimulate imagination and sustain interest.
It can be used to provide vicarious visual experiences.

Non-Projected aids
Visual aids which do not help in their projection on to a screen.
Non-projected aids can be classified in to
(i) Graphic aids (ii) Display boards (iii) 3-dimensional aids and (iv) Audio aids

(i) Graphic aids:
Graphic aids are visual aids such as graphs, diagrams, charts etc and are presented on plane surface. The different categories of graphic aids are described below:

a. Graphs
The nature of the relation of two dependent variables could be very easily presented by graphical representation.

Uses
It is used to study the relationship between two variables.
It can be used to get clear picture and immediate information.

b. Diagrams
Diagrams could be used to explain many facts at the same time, by using a variety of symbols and  labels.
Use
For getting clear and vivid information at a glance.
Used for motivating the students

c. Charts
The chart is a systematic arrangement of key facts or ideas in a logical sequence or representing ideas and facts in a pictorial or graphic form.

Uses
•   Motivating the students
•   Introducing a lesson
•   Depicting various geometrical figures and their properties
•   Comparison of properties of different geometrical figures
•   Highlighting key points in a lesson
•   Concretising abstract mathematical concepts into visual forms.

(ii) Display boards
The stereo typed atmosphere of the traditional classroom, libraries and corridors of schools is recently giving way to attractive and educative surrounding with a wealth of display materials. A brief description about various types of display boards are given below.

a. Chalk board
Chalk board is, perhaps the most important visual aid for teaching mathematics. It is the most indispensable tool in the hands of a mathematics teacher. Mathematics cannot be taught effectively without the use of chalk board.

Uses
• Drawing geometrical figures and illustrating construction of geometrical figures.
• Developing various steps of the solution of a problem or proof of a theorem.
• Writing important points to draw the attention of the students.
• Writing problems for assignment and drill.
• Summarizing the day’s lesson.

b. Roll-up chalk board
Roll up chalk board is that which can be conveniently rolled up. These arte usually made of think rexin cloth.
Uses
The difficulties due to spot writing or drawing can be solved.

c.Flannel board.
Flannel board have a fibre surface to which similar materials will adhere.

Uses
•   Showing the sequence of steps in the derivation of a formula.
•   Re arranging the steps in the proper sequence
•   Moving certain parts of a geometrical figure to another position so as to get a new geometrical figure.

d. Magnetic board
Magnetic board is a modification over the flannel board. Instead of  flannel, a metallic sheet is fixed on the plywood. Small magnets are used to hold the materials to be exhibited. It serves the same purpose as a flannel board does.

3. Three-dimensional aids:
Models
Models are three dimensional representation of an idea and therefore they are replicas of the original thing. Models provide contrived experiences where reality is altered or simplified for teaching purpose. Thus models simplify the reality and enable the teacher to reduce or enlarge objects to any desirable size.

Uses
•  For teaching the properties and areas of plane figures such as cubes, prisms the teacher can make use of these shapes cut out from chart papers or card board sheets.
•  While deriving the expansion of algebraic identities such as ( a + b )3, ( a + b + c )3and so on the teacher can make use of cubes to get the required terms in the expansion.

4. Audio aids
The audio aids like radio and tape recorder depend on a single sense – hearing to convey its rich meanings. When matter is presented through radio and tape recorder, listening remains the only means of communication.

Radio adds to variety to learning experiences. It is a powerful medium for mass communication.
Uses
•  It can be used to teach large mass.
•  It will help the teacher to supplement classroom instruction
•  Cost effective and reaches faster to people than any other device.

b. Tape Recorder
A tape recorder is used to record sound on magnetic tape which can be reproduced at will as many times as required
Uses
The recorded lessons by eminent teachers can be played.

Improvised aids
Improvised aids are the aids which can be designed and produced from simple articles found in one surroundings. These aids can be prepared by teachers involving students and community resources. Some such community resources are sticks, leaves, stones, shells, broken bangle pieces etc., which are helpful in making learning aids.

Uses
•  It develops useful habits and dignity of labour
•  It develops skill and clear understanding of basic principles
•  Utilization of leisure
•  To concretize abstractions
•  Self motivation towards making new and creative things
•  It makes the school self sufficient
•  Students gain confidence to face problems
•  It helps in hand-head co-ordination.