Methods of teaching mathematics

Inductive-Deductive method
Dr. Jaganmohana Rao Gurugubelli
Assistant Professor
MITE, KOHIMA

INDUCTIVE METHOD:
It is based on the process of induction; it leads from
concrete to abstract, particular to general and from examples to
the generalization.
Induction means; ―to provide the universal truth by
showing that if it is true for a particular case, it is true for all
such cases. In this method we arrive at a formula or
generalization through reasoning and solving problems.

E.g: To arrive at the generalization that: The sum of all the angles of
triangle is 180º.
1. The teacher draws different types of triangles and asks the students
to measure all the angles of the triangles and state the sum of the
three angles in each case.
2. Every time the students realize that the three angles measure up to
180º.
3. They observe the results and can make the generalization that ―The
sum of all the angles of triangle is 180º
Ex-2: Square of an odd number is odd &
Square of an even number is even.
Ex-3 : Sum of two odd numbers is even

Inductive approach proceeds from:
 Particular cases to general rules of formulae
 Concrete instance to abstract rules
 Known to unknown
 Simple to complex
Key Elements in Inductive Method:
Inductive approach is advocated by Pestalaozzi and Francis Bacon
Inductive approach is based on the process of induction.
In this we first take a few examples and greater than generalize.
It is a method of constructing a formula with the help of a sufficient number
of concrete examples.
Inductive approach is psychological in nature.
The children follow the subject matter with great interest and understanding.
This method is more useful in arithmetic teaching and learning.

Steps in Inductive Method:-
(a) Presentation of Examples (Number of Cases)
In this step teacher presents many examples of same type and solutions
of those specific examples are obtained with the help of the student.
(b) Observation:
After getting the solution, the students observe these and try to reach to some
conclusion.
(c) Generalization
After observation the examples presented, the teacher and children
decide some common formulae, principle or law by logical mutual discussion.
(d) Testing and verification
After deciding some common formula, principle or law, children test
and verify the law with the help of other examples. In this way children
logically attain the knowledge of inductive method by following above given
steps.

Merits & Advantages:
It enhances self confident
It is a psychological method.
It is a meaningful learning
It develops scientific attitude.
It develops the habit of intelligent hard work.
It helps in understanding because the student knows how a particular
formula has been framed.
Since it is a logical method so it suits teaching of mathematics.
It is a natural method of making discoveries, majority of discoveries
have been made inductively.
It does not burden the mind. Formula becomes easy to remember.
This method is found to be suitable in the beginning stages.

Demerits & Disadvantages:
Certain complex and complicated formula cannot be generated so this
method is limited in range and not suitable for all topics.
It is time consuming and laborious method
It is lengthy procedure.
It’s application is limited to very few topics
It is not suitable for higher class
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.
Applicability of inductive method: Inductive approach is most suitable for
Rules are to be formulated
Definitions are be formulated
Formulae are to be derived
Generalizations or law are to be arrived at.

DEDUCTIVE METHOD:
It is the exact opposite of the inductive method. Over here we
proceed from general to particular, abstract to concrete. In this method
the rule or generalization or formula is given at the very beginning. Thus
the students are expected to apply these rules or formulae to solve the
sums. The formula or rule is accepted as the universal truth and the
student uses them to solve the problems.
E.g:The teacher states that: “The sum of all the angles of triangle is 180º”
1. In triangle PQR, measure of angle P is 50◦ the measure of angle Q is
75◦.Find the measure of angle R?
▲P+Q+R=180◦ i.e 50+75+R=180◦ R=180-125=55◦
2. In triangle XYZ measure of angle Z is 57◦ the measure of angle X is 97◦. Find the
measure of angle Y?
▲X+Y+Z=180◦ i.e 97+Y+57=180◦ Y=180-154=26◦

Deductive approach proceeds from:
 General rules to specific cases
 Abstract rules to concrete instance
 Unknown to Known
 Complex to Simple
Key Elements in deductive Method:
Deductive approach is advocated by Aristatile and Comenius
Deductive approach is based on the process of deductive reasoning.

Steps in deductive Method:-
(a)Clear Recognition of the problem
(b)Search for tentative hypothesis (Solutions)
(c)Formulation of a tentative hypothesis (choose the proper solution)
(d)Verification
Merits & Advantages:
Short and time saving method.
Method suits all types of students.
Suitable for all topics.
Provides sufficient examples for practice.
Speed and efficiency of doing sums increases.

Demerits & Disadvantages:
 It is not a psychological method.
 It encourages cramming.
 Taxes the pupil‘s mind.
 Students are passive learners and thus lose interest.
 It puts more emphasis on memory.
 Not suitable for the development of thinking, reasoning and discovery.

Analytic & Synthetic
Method
Dr.G.Jaganmohana Rao
Faculty of Education
MITE, Kohima

Analytical Method:
The meaning of the word analysis is to ―separate things that are
together. In this method we start from what is to be found or proved.
Thorndike says that, Analysis is the highest intellectual performance of
the mind.
Analysis also means, ―Breaking up of a given problem, so that it
connects with what is already known. In analysis we proceed from,
Unknown to Known.
 It is derived from the word analysis,
 Its means breaking up.
 It leads to conclusion to hypothesis
 It leads to unknown to known
 It leads to abstract to concrete

Merits/Advantages:
 It is a psychological method.
 Facilitates understanding, as we discover facts.
 Each step has reason and justification.
 Student gains confidence and understanding.
 Method suits the learner and the subject.
 It develops the power of thinking and reasoning
 It develops originality and creativity amongst the students.
 In this method student’s participation is encouraged.
 It develops self-confidence and self reliant in the pupil.

Demerits:
• It is time consuming and lengthy method, so it is uneconomical.
• Difficult to acquire efficiency and speed.
• Not applicable to all topics.
• Not suitable for students with weak conceptual knowledge.
• Every teacher cannot use this method successfully
Conclusion: So this method is particularly suitable for
teaching of Arithmetic, algebra and Geometry as it analyses the
problem into sub-parts and various parts are reorganized and
the already learnt facts are used to connect the known with
unknown. It puts more stress on reasoning and development of
power of reasoning is one of the major aims of teaching of
mathematics.

Synthetic Method:
The word synthesis simply means, ―To place things together or to
join separate parts. It is the process of relating known bits of data to a
point where the unknown becomes true.
To synthesis is to combine the elements to produce something new.
Actually it is reverse of analytic method. In this method we proceed “from
know to unknown.” So in it we combine together a number of facts,
perform certain mathematical operations and arrive at a solution.
That is we start with the known data and connect it with the
unknown part.
• It leads to hypothesis to conclusion
• It leads to known to unknown
• It leads to concrete to abstract

Merits:
• It is Short and precise method.
• It Saves time and energy.
• It Suits the needs of majority of the students.
• It Can be applied to a majority of topics in mathematics.
• Omits trial and error as in analysis method.
• Accuracy is developed by the method
Demerits:
 Teacher–centered method, students are passive listeners.
 Students rely on rote memory. (There is a scope for forgetting)
 No opportunity to develop the skills of thinking and reasoning
 Students lack confidence to do other type of sums.
 There is no scope of discovery.
 The recall of each step cannot be possible for every child.

Analytic Vs Synthetic
Analytic Method Synthetic Method
Meaning: Analysis means breaking up into
components
Meaning: Synthesis means combining the
elements to get something new.
Leads from:
Unknown to known, Conclusion to hypothesis,
Abstract to concrete, Complex to simple
Leads from:
Known to unknown, Hypothesis to conclusion,
Concrete to abstract, Simple to complex
Method: A method of discovery and thought. A
psychological method
Method: A method for the presentation of
discovered facts. A logical method
Time: Lengthy, laborious and time consuming Time: Short, concise and elegant.
Learning: Encourages meaningful learning Learning: Encourages rote learning
Encourages: Encourages originality of thinking and
reasoning
Encourages: Encourages memory work
Thinking: Process of thinking Thinking: Product of thinking
Participation: Active participation of the learner Participation: Learner is a passive listener
Sequence: Valid reasons to justify every step in the
sequence.
Sequence: No justification for every step in the
sequence.

HEURISTIC METHOD
Dr. G. Jaganmohana Rao
MITE, Kohima

Introduction:
The word ‘Heuristic’ has been derived from the Greek word
‘Heurisco’ which means ‘I find’ or ‘I discover’ . This method implies that
the attitude of students shall be that of the discoveries and not of
passive recipients of knowledge. H.E. Armstrong originally introduced
this method for learning of science.
This method emphasis experimentation as the teacher becomes
on looker and the child tries to move a head independently without any
help. This method makes the student self-reliant and independent. But
the teacher should develop the heuristic attitude by making a lot of
preparation. The question should be so planned that it may be possible
for the students to find the solution independently by proceeding in the
proper direction.

Definition:
According to H.E.Armstrong, “This is the method of
teaching which places the pupils as far as possible in the
attitude of a discoverer.”
According to westaway, “the heuristic method is intended
to provide training in method. Knowledge is a secondary
consideration altogether.

Merits:
 This is a psychological method as the student learns by self-practice.
 It creates clear understanding & meaningful learning
 It develops scientific attitude & critical thinking
 It develops self-confidence, self-discipline in the students
 The students acquire command of the subject. He has clear
understand and notions of the subject.
 It gives the student a sense of confidence and achievement.
 The methods make them exact and bring them closer to truth.
 It inculcates in the student the interest for the subject and also
develops willingness in them.

Demerits:
• It is not suitable for lower classes as they are not independent thinkers.
Discovery of a thing needs hard work, patience, concentration,
reasoning and thinking powers and creative abilities.
• It is very slow & lengthy method. That is time consuming method.
• The students have to spend a lot of time to find out minor results.
• The teacher may find it difficult to finish the syllabus in time.
• It does not suit larger classes.
• It suits only hard working and original thinking teachers.
• A method is successful if well-equipped libraries, laboratories and good
textbook written in heuristic lines but such facilities are lacking in our
school.

Example:
Prove that aº = 1
Teacher : what is 10/5? Student : 2
Teacher : what is a/a ? Student : 1
Teacher : what is am/am ? Student : 1. (Equation I)
Teacher : how do you get the result?
Student : if we divide a number by itself , we will get 1.
Teacher : how can you write am X 1/am ?
Student : am-m = aº (Equation II)
Teacher : what do you infer?
Student : aº = 1

PROJECT METHOD
MITE, Kohima

Introduction:
The base of this method lies in the philosophy of pragmatism. This
method emphasizes on building a comprehensive unit around an activity
which may be carried out in school or outside. The essence of this
method lies in the fact that a group of students do a purposeful task.
This implies the students undertake the activity in a group or
individually over a period of time. It may include a number of activities
and the end product is in the form of written report or a display.
Project method is of American origin and is an outcome of
Dewey’s philosophy or pragmatism. However, this method is developed
and advocated by Dr.Kilpatrick.

Definitions:
“Project is a bit of real life that has been imported into school” – Ballard
“A project is a unit of wholehearted purposeful activity carried on
preferably in its natural setting” – Dr.Kilpatrick
“A project is a problematic act carried to completion in its natural
setting”. – Stevenson.
Thus, project is a purposeful activity and planned activity which is
achieved in social, natural situations created in schools.

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 Project Method:
Step-1: 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.
Step-2: Proposing and choosing the project: while choosing a
problem teacher should stimulate discussions by making suggestions.
The proposed project should be according to the rear need of students.
The purpose of the project should be well defined and understood by
the children.
Step-3: Planning the project: for the success of the project, planning of
project is very import. The children should plan out the project under the
guidance of their teacher.

Step-4: Execution of the project: every child should contribute actively
in the execution of the project. It is the longest step in the project.
Step-5: 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.
Step-6: Recording of the project: the children maintain a complete
record of the project work. While recording the project some points like
how the project was planned, what discussion were made, how duties
were assigned, how it was evaluate etc. should be kept in mind.

Merits:
 This is based on various psychological laws and principles.
 It develops self-confidence and self-discipline among the students
 It provides ample scope for training.
 It provides score for independent work and individual development.
 It promotes habits of critical thinking and encourages the students to
adopt problem-solving methods.
 This method the children are active participants in the learning task.
 This is based on principle of activity, reality, effect, and learning by
doing etc.
 It develops discovery attitude in the child.
 It provides self-motivation as the students themselves select plan and
execute the project.

Demerits:
• It takes more time & not economical.
• The knowledge is not acquired in a sequential and systematic manner
• It is very difficult to complete the whole syllabus by the use of this
method.
• Textbooks and instructional materials are hardly available.
• The project method does not provide necessary drill and practice for
the learners of the subject.
• The project method is uneconomical in terms of time and is not
possible to fit into the regular time table.
• Teaching is disorganized
• This method is not suitable for a fixed curriculum.

PROBLEM SOLVING
METHOD
MITE, Kohima

Introduction:
The child is curious by nature. He wants to find out solutions of
many problems, which sometimes are puzzling even to the adults. The
problem solving method is one, which involves the use of the process of
problem solving or reflective thinking or reasoning. Problem solving
method, as the name indicated, begins with the statement of a problem
that challenges the students to find a solution.
Definitions:
“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” –
Ausubel
Cont..

“Problem solving is a planned attacks upon a difficulty or perplexity for
the purpose of findings a satisfactory solution”– Risk,T.M.
Steps in Problem Solving / Procedure for Problem solving:
Identifying and defining the problem: The student should be able to
identify and clearly define the problem. The problem that has been
identified should be interesting challenging and motivating for the
students to participate in exploring.
Analysing the problem: The problem should be carefully analysed as
to what is given and what is to be find out. Given facts must be identified
and expressed, if necessary in symbolic form.

Formulating tentative hypothesis: Formulating of 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 solution to the problem.
Testing the hypothesis: Appropriate methods should be selected 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 of the result or checking the result: No conclusion should
be accepted without being properly verified. At this step the students are
asked to determine their results and substantiate the expected solution.
The students should be able to make generalizations and apply it to their
daily life.

Merits:
 This method is psychological and scientific in nature
 It helps in developing good study habits and reasoning powers.
 It helps to improve and apply knowledge and experience.
 This method stimulates thinking of the child
 It helps to develop the power of expression of the child.
 The child learns how to act in new situation.
 It develops group feeling while working together.
 Teachers become familiar with his pupils.
 It develops analytical, critical and generalization abilities of the child.
 This method helps in maintaining discipline in the class.

Demerits:
• This is not suitable for lower classes
• There is lack of suitable books and references for children.
• It is not economical. It is wastage of time and energy.
• Teachers find it difficult to cover the prescribed syllabus.
• To follow this method talented teacher are required.
• There is always doubt of drawing wrong conclusions.
• Mental activities are more emphasized as compared to physical
activities.

Role of the Teacher:
Teacher must work as a facilitator.
Teacher must keep in mind that if in a child-directed learning not
teacher-directed.
Teacher must provide situation for all students to come formed and
contribute towards the success of the activity.
He must be alert and active to arouse interest among students.
Teacher must provide democratic atmosphere.
He must be initialize, tactful and we experienced.
Problem solving method can be an effective method for teaching
mathematics in the hands of an able and resourceful teacher of
mathematics.

Role of the Teacher:
Teacher must be a friend, guide and working partner.
Teacher must have through knowledge of individual student and allot work
accordingly.
Provide democratic atmosphere.
He should learn with students and should not claim to know everything.
Some projects for mathematics: A few projects suitable for high school
mathematics are listed below
Execution of school bank.
Running stationary stores in the school.
Laying out a school garden.
Executing the activities of mathematics clubs
Collection of data regarding population, death rate, birth rate etc.

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