METHODS OF TEACHING MATHEMATICS

METHODS OF TEACHING MATHEMATICS

•  INDUCTIVE- DEDUCTIVE METHOD

    it is a combination of two methods.These two methods are complementary to each other  .

Inductive method

     It leads from concrete to abstract, particular to general and from examples to formula. It is the method  of constructing a formula with the help of a sufficient number of concrete examples. It is based on induction which means proving a universal truth by showing that if it is true for a particular case and is further true for a reasonably adequate number of cases, it is true for all such cases.A formula or generalization is thus arrived at through a convincing process  of reasoning and solving problems. After a number of concrete cases have been understood , the student successfully attempts generalizations

Example

        Ask students to draw a few sets of parallel lines with two lines in each set. Let them construct and measure the alternate and corresponding angles in each case . They will find them equal in all the cases . This conclusion in a good number of cases, will enable them to formulate the relevant generalizations.

        Ask them to construct a few triangles. Let them measure  and sum up the angles in each case . The sum will be the same in all the cases. Thus they can safely conclude that the sum of angles of a triangle is equal to two right angles.

Merits

• It helps easy understanding of mathematical principles established through a number of simple examples& How and why the formula created in every attempt.
• it is logical method.
• it gives the opportunity of active participation to students in the discovery of formula.
• It is based on actual observation ,thinking, & experimentation
• It curbs the tendency to learn things by root, and also reduce homework.
• As it gives freedom of doubts, and helps in understanding, it suits the child.

Demerits

• It is limited in range.
• Inductive reasoning is not absolutely conclusive.
• It is likely to be more laborious and consuming .
• At the advantage stage it is not so useful as some of the necessary details and explanation may make teaching dull and boring.
• Its application has to be restricted an confined to understanding of rules in the early stage.

Deductive method

   It is the opposite of Inductive Method. Here the learner proceeds from general to particular , abstract to concrete, and formula to examples . A preconstructed formula is told to the students and they are asked to solve the relevant problems with the help of that formula. The formula is accepted by the learners as a pre-established and well-established truth.

Example

Immediately after announcing the topic for the day, the teacher gives the relevant formula. To explain further the application of the formula to problems, he solves a number of problems on the blackboard. The students come to understand how the formula can be used or applied. Like the formula

Area of a rectangle= Length x Breadth

Merits

• It is a short and time-saving method
•  It glorifies memory , as students have to memorize a considerable number of formulae.
  
•  At the “ Practice and Revision “ stage , this method is adequate and advantageous.
  
•  It combines with the inductive method to remove the incompleteness and inadequacy of the later.
  
•  It enhances speed and efficiency in solving problems.

Demerits

·       It is very difficult for a beginner to understand an abstract formula if it is not proceeded by a number of concrete instances.

·       Pure deductive work requires a formula for every type of problems and an extensive use of this method will demand blind memorization of a large number of formulae.

·       It will thus cause an unnecessary and heavy burden on the brain. It may even result in brain fag.

·       Memory becomes more important than understanding and intelligence , and that is educationally unsound

·       It the pupil forgets the memorized formula, which is very likely to happen in case of blind cramming , he is at a loss and cannot recollect and reconstruct the formula easily.

·       The students cannot become active learners.

·       It is not suitable for the development of thinking, reasoning and discovery.

        

• ANALYTIC - SYNTHETIC METHOD

  The two strategies analysis and synthesis are considered to be complementary stages involved in the same process. ie. meaningful internalization of a problematicc situation and its systematic solution.

This method is most applicable in arriving at the solution to a problematic situation and in recording this process of solution in systematic and orderly manner. 

Analytic method 

     It proceeds from unknown to known . " Analysis" means " breaking up" of the problem in hand to so that it ultimately gets connected with something obvious or already known. It is the process of unfolding of the problem or of conducting its operation to know its hidden aspects. Start with what is to be found out. Then think of further steps and possibilities which may connect the unknown with the known and find out the desired result. In its original sense the verb " to analyse" means to loosen or separate things that are together. About analysis, Thorndike says that all the highest intellectual performance of the mind is analysis.

Merits

• It is a logical method.
• it facilitates understanding
• the steps in its procedure are developed in a general manner.
• the method is suits the learner  and the subject

Demerits

• It is a lengthy method
• With this method , it is difficult to acquire efficiency and speed.
• It may not be applicable to all topics equally well.

Synthetic method

     It is the opposite of the analytic method here one proceeds from known to unknown . Synthesis is the complement of analysis . To synthesize is to place together things that are apart. It starts with something already known and connects that with the unknown part of the statement. It starts with the data available or known and connects the same with the conclusion. It is the process of putting together known bits of information to reach the point where unknown information becomes  obvious and true.

Merits

• It is short and elegant method
• It glorifies memory
• It sits the teacher
• It follows the same process as given in the textbooks.

Demerits

• It leaves many doubts in the mind of the learner
• without  a satisfactory answer to so many questions that arise in synthesis, the ppupil is perplexed when a new problem is put to him.
• It does not provide full understanding.
• There is no scope of discovery and thinking in this method
• Memory work and home work are likely  to become heavy
• It does not suit the learner and the subject.

Example of analytic -synthetic method

    A cylinder has abase with circumference 31.4 cm and a height of 20 cm .of the cylinder

Calculate the volume of the cylinder.

The analysis of this problem, as is made by asking oneself heuristic question helpful for meaningful  analysis and them identifying the exact nature of the problem situation well as the data and principles that could be used for arriving at the solution.

1. What is to be calculated ?( Volume of a given cylinder)
2. How can we calculate the volume of a cylinder? ( Using the formulae  V=.....)
3. What data are required for that? (r and h of the cylinder)
4. Are these given in the problem? ( h is given but r is not given)
5. Is any hint helpful to find r is available? ( yes the circumference is of the base is given )
6. How can we determine r from this? (...... then r= 5 cm)
7. Is there any other data misssing ? ( No )
8. Now how can we calculate the volume? (using the formula v=.... then v= 1570 cubic cm)

comparison

Analytic method	Synthetic method
It proceeds from Unknown to known facts	It proceeds from known to unknown facts
It starts from the conclusion and goes to the hypothesis	It starts with the hypothesis and ends with the conclusion
It is process of thinking	IT is a product of thought
It is a process of explanation and demands thought	It is a process of presentation of previously known facts
It pulls apart or analyses the statement under solution	It puts together or synthesizes  known facts
It is a general method	It is a special device
It is lengthy , awkward, slow, roundabout and involves trial and error	It is concise, elegant ,quick straightforward, and does without trail and error.
It answers satisfactorily any question that may rise in the mind of an intelligent pupil	It does not satisfy the doubts and questions arising in the mind of the learner
It is a method for the thinker and discoverer	It is a method for crammer
There are close contacts between the teacher and taught	There are no such intimate facts between them
The students can recall and reconstruct easily any steps if forgotten.	It is not easy to recall or reconstruct any forgotten steps
It develops originality	It develops memory
It is informal	It is formal
It is psychological	It is logical
It is formational	It is informational
It is based on heuristic lines	There is no heuristic approach in it
It is the fore-runner of synthesis	It is the follower of analysis

• PROJECT METHOD

   Project method is of American origin and is an outcome of Dewey's philosophy pragmatism. However, this method is developed and advocated by Dr. Kilpatrik.

"Project is a plan of action "-- Oxford advanced learner's dictionary.

" Project is bit of real life that has been imported into school"--Balllard

" A project is a unit of whole hearted purposeful activity carried on preferably in its natural setting "--Dr. Kilpatrik

" A project is a problematic act carried to completion in its most natural setting"--Stevenson

Basic principles of project method

  Psychological principles of learning

• Learning by Doing
• Learning by living
• Children learn better through association, co-operation and activity

 Psychological laws of learning     

• Law of readiness
• Law of exercise
• Law of effect

Steps involved in the project method

1. Providing /creating the situation
2. Proposing and choosing the project
3. Planning the project
4. Execution of the project
5. Evaluation of the project
6. Recording of the project

• Creating the situation

           The Teacher creates problematic situation in front of students while creating the appropriate situation. Student's interest and abilities should be given due importance

• Proposing and choosing the project

             While choosing a problem teacher should stimulate discussions by making suggestions. The proposed project should be according to the real need of students. The purpose of the project should be well defined and understood by the children.

• Planning the project

             For the success of the project , planning of project is very important . The children should plan out the project under the guidance of their teacher.

• Execution of the Project

             Every child should contribute actively in the execution of the project . It is the longest step bin the project.

• Evaluation of the Project

            When the project is completed the teacher and the children should evaluate it jointly discussed whether the objectives of the project have been achieved or not 

• Recording of the project

             The children maintain a complete record of the project work. while recording the project some points ;ike how the project ws planned, what discussion were made, how duties were assigned, how it was evaluated etc., should be kept in mind.

Example

Running of a Hostel Mess

     STEPS

1. The number of hostelers will be recorded
2. The expected expenditure will be calculated.
3. Expenditure on various heads will be allocated to the students
4. Budget will be prepared with the help of the class 
5. The account of collections from amongst the students will be noted.
6. Actual expenditure will be incurred by the students
7. A chart of " balanced diet"for the hostelers will be prepared .
8. The time of breakfast , lunch,tea and dinner will be fixed and notified
9. Execution of different programes stated above will be made.
10. Weight of each hostelers will be checked after regular interval, and the same will be put on record.
11. Punctuality in all the activities of the hostelers will be recorded.
12. Evaluation of the entire programme and then it will be typed out for the information of all concerned.

Some projects for mathematics

1. Execution of school bank
2. Running stationary stores in the school
3. Laying out a school garden
4. Laying a road
5. Planning and estimating the construction of a house
6. Planning for annual camp.
7. Executing the activities of a mathematics club
8. Collection of data regarding population , death rate, birth rate...

            Though project method provides a practical approach to learning. It is difficult to follow this method for teaching mathematics.However this method may be tried along with formal classroom teaching without disturbing the school timetable. This method leads to understanding and develops the ability to apply knowledge. The teacher has to work as a careful guide during the execution of the project.

Merits of Project method

• It is based on Psychological laws of learning
• it upholds the dignity of labour
• it introduces democracy in education
• it brings about concentration of studies and correlation of activities and subjects.
•  it emphasis es problem solving rather than cramming or memorizing
• it inculcates social discipline through joint activities
• it develops self- confidence& self- discipline
• A project tends to illustrate the real nature of the subject and produce a spirit of enquiry.
• Projects can be used to arouse interest , justify the study of topics , encourage initiative and give the students joy at the successful completion of the given work.
• Teaching becomes incidental 
• it challenges the capacities and abilities of the child and puts him on the track to think and act
• There is an opportunity for mutual exchange of ideas.

Demerits of Project method

• sometimes it does not suit for the topic
• there is no saving of time , energy and effort
• systematic and continuous teaching is not possible.
• costly
• Planning and execution of project method can be adopted as a co-curricular activity

• LABORATORY METHOD

    To make mathematics more interesting and meaningful, laboratory method is used in teaching of mathematics. In this method students get the opportunity to acquire facts through direct experience individually. It is the experimental portion of the inductive method or practical form of the heuristic method. Therefore in this method one proceeds from concrete to abstract. It is based on the psychological principles of learning such as "learning by doing" "learning by observation" and so on. Laboratory method is quite component to relate the theoretical knowledge with the practical base.

• The teacher clearly explains the aim of practical work to be carried out by the students.
• The students are provided with necessary materials and instruments
• The teacher explains the procedure of the experiment to be carried out by the students.
• The students carry out the experiment
• The teacher themselves observes the students working from time to time and guides them whenever needed
• The students are required to draw the conclusions as per the aims of the experiment.

Merits of Laboratory Method

• An activity involves both the mind and hands of the student working together which facilitates cognition
• It helps to build interest among the students in learning the subject.
• This is based on the sound psychological principle such as learning by doing and learning by observation
• The knowledge gained by this method is long lasting and solid
• It helps in developing the habit of discovery and self-study
• It provides opportunities for social interaction and co-operation among the students
• It helps to develop self-confidence and self-reliance among the students.
• It provides opportunity for students to correlate mathematics with daily life and other subject
• It develops observation and logical power among students
• It helps to develop positive attitude towards mathematics
• It helps to develop problem solving ability
• This method presents mathematics as a practical subject
• It helps to develop scientific attitude among students
• The child get opportunity to use different equipments in the laboratory

Limitations of Laboratory Method

• It is a very expensive method
• A lot of time is wasted in conducting experiments
• This method requires laboratory equipped with different apparatus
• This method is not suitable for all topic
• Individual attention is not possible when number of students is large
• Lack of textbook and materials written the lines of laboratory
• Only an efficient and talented teachers can handle this method effectively
• The dull students are often tempted to copy down the results of the brilliant
• It is difficult to the teachers  to check up the apparatus after every period.
• PROBLEM -SOLVING METHOD

   The child is curious in nature. He wants to find out solutions of puzzling ,even to the adults. The problem solving method is one , which involve the use of the process solving or reflective thinking or reasoning.

   Problem solving method, as ass the name indicated , begins with the statement of a problem that challenges the students to find a solution.

  Definition

"Problem solving is a set of events in which human beings was rules to achieve some goals" -Gagne

"Problem solving involves concept formation and discovery learning" - Ausbel

"Problem solving is a planned attack upon a difficulty or perplexity for the purpose of finding a satisfactory solution"-Risk. T.M

STEPS IN PROBLEM SOLVING

1. Identifying and defining the problem
2. Analyzing the problem
3. Formulating tentative hypothesis
4. Testing the hypothesis       
5. Verifying the result or checking the result

• Identifying and defining the problem

                The student should be able to identify and clearly defining the problem. The problem has been identified should be interesting , challenging and motivating for the students to participate in exploring.

• Analyzing the problem

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

• Formulating tentative hypothesis

                 Formulating hypothesis means preparation of a list of possible reasons of the occurrence of the problem.Formulating of hypothesis develops thinking and reasoning powers of the child. The focus at this stage is on hypothesizing- searching for the tentative solutions to the problem

• Testing the hypothesis       

                 Appropriate methods should be selects to test the validity of the 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.

• Verifying the result or checking the result

               No conclusion should be accepted without being properly verifies. At this step the students are asked to determine their results and substantiate the expected solution. The students should be able to make generalizations ans apply it to their daily life.

Example

Define Union of two sets

If A={2,3,5}, B={3,5,6} , C={4,6,8,9,}
Prove that AU (BUC) =(AUB) UC

• Identifying and defining the problem

     After selecting  and understanding the problem the the child will be able to define the problem in his own words that, 

1. The union of two sets A and B is the set which contains all the members of a set A and all the members of a set B.
2. The union of two sets A and B is expressed as AUB and symbolically represented as AUB = {x: x E A or x E B}.
3. The common elements are taken only once in the union of two sets 

• Analyzing the problem

    After defining the problem in his own words , the will analyse the given problem that how the problem that how the can be used

• Formulating tentative hypothesis

   After analysing the various aspects of the problem, he will be able to make hypothesis that first of all he should calculate the union of sets B and C. ie; BUC. Then the Union of set A and BUC. Thus he can get the value of AU (BUC). Similarly he can solve (AUB) UC.

• Testing the hypothesis 

   Thus on the basis of given data the child will be able to solve the problem in the following manner .

In the example it is given that

BUC={3,5,6}  U  {4,6,8,9}= { 3,4,6,8,9,5}

AU (BUC)= {2,3,5} U {3,4,5,6,8,9}= {2,3,4,5,6,8,9}

similarly (AUB) UC

    After solving the problem the child will analyse the result on the basis of the given data and verify this hypothesis whether AU(BUC)= (AUB) UC or not.

• Verifying the result or checking the result

   After tesing the and verifying his hypothesis the child will be able to conclude that 

AU(BUC)= (AUB)UC

Thus the child generalizes the results and apply his knowledge in new situations.

Merits

• Prepares pupil to solve the problems of life
• i involves reflective thinking. it stimulates critical thinking, reasoning, and critical judgement in the students
• it develops qualities of initiative and self- dependence in the students
• specially suit for mathematics which is a subject of problems
• in it there is a strong motivation , tension and mental activity which are the conditions of effective learning
• It serves individual differences.
• it develops desirable study habits in the students
• it is a method of experience based learning.
• the students gets valuable social experience like patience, co-operation, self-confidence,etc.

Demerits

•  its limitations are due to its ineffective use
• it is difficult to organize the contents according to requirements of this method
• it is time consuming
• all topics and all subject areas are cannot be covered  by this method
• not suit for lower classes
• Teacher's burden becomes heavier
• mental activity dominates in this method 
• some times it becomes spoon feeding or artificial method in normal classroom situations.