Steps in Inductive Method
1. Selection
of a number of cases.
2. Observation
of the case under given conditions.
3. Investigation
and analysis.
4. Finding
common relations.
5. Arriving
at generalization.
6. Verification
or application.
Merits of Inductive method
1.
It helps in understanding.
2.
It is logical method and develops critical
thinking.
3.
It encourages active participation of the
students in learning.
4.
It provides ample opportunities for exploration
and observation.
5.
It sustains the students’ interest as they
proceed from known to unknown.
6.
It curbs the tendency to rote learning as it
clears the doubts of the students.
7.
It facilitates meaningful learning.
8.
It enhances self-confidence.
9.
It is helpful for beginners as it provides a
number of concrete examples.
10.
It encourages experimentation, observation,
analytical thinking and reasoning.
11.
It helps in increasing the pupil-teacher
contact.
12.
It does not burden the mind. Formula becomes
easy to remember.
13.
It discourages cramming and also reduces home
work.
Demerits of Inductive method
1.
This method is limited in range and is not
suitable for all topics. Certain complex and complicated formula cannot be generalized
in this manner.
2.
It is lengthy, time consuming and laborious
method.
3.
Inductive reasoning is not absolutely conclusive
because the generalization made with the help of a few specific examples may
not holds good in all cases.
4.
We don’t complete the study of a topic simply by
discovering a formula but a lot of supplementary work and practice is required
for fixing the topic in learner’s mind.
5.
This method is not suitable for higher classes
because higher order mathematical principles cannot be generalized through the
observation of concrete cases.
6.
It is not suitable for gifted
students as unnecessary details and too many examples make the teaching dull
and boring.
DEDUCTIVE METHOD
Deductive method is based on deductive
reasoning. Deductive reasoning is the process of drawing logical inferences
from established facts or fundamental assumptions. Contrary to inductive
method, in deductive method we begin with the formula, or rule or
generalization and apply it to a particular case. In this method, the teacher
presents the known facts or generalization and draws inferences regarding the
unknown, following a network of reasoning. Therefore, deductive method proceeds
from:
i)
General rule to specific instances.
ii)
Unknown rule to known case.
iii)
Abstract rule to concrete instance.
iv)
Complex rule to simple example.
Steps in Deductive
method
Deductive method of teaching
follows the steps given below for effective teaching.
1. Clear recognition of the problem:
A clear recognition of the problem statement provides the basic link for the
thinking process and the solution to the problem.
2. Search for a tentative hypothesis:
The second step in deductive method is the search for tentative hypothesis, a
tentative solution to the problem.
3. Formulation of tentative hypothesis:
The 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.
4. Verification: Finally the
hypothesis that has been formulated is to be verified as the right solution to
the problem at hand.
Problem solving
technique:
The problem-solving technique is one
involves the use of the process of problem-solving or reflective thinking or
reasoning. Problem-solving method/technique, as the name indicates, begins with
the statement of a problem that challenges the students to find a solution. In
this process of solving the problem the students may be required to gather
data, analyse and interpret the information, to arrive at a solution to the
problems.
Definitions of
Problem-solving
1).”A problem occurs in a
situation in which a felt difficulty to act is realized. It is a difficulty
that is clearly present and recognized by the thinker. It may be a purely
mental difficulty or it may be physical and involve the manipulation of data.
The distinguishing thing about a problem, however, is that it impresses the
individual who meets it as needing a solution. He recognizes it as a
challenge”. -----Yokam and Simpson
2).
“Problem solving is
a planned attack upon a difficulty or perplexity for the purpose of finding a
satisfactory solution”. ---- Risk. T.M
3).”Problem-solving is an
Educational device whereby the teacher and the pupils attempt in a conscious,
planned, purposeful manner to arrive at an explanation or solution to some
educationally significant difficulty”. --- James
Ross.
From the above definitions,
problem solving involves the following,
·
A
goal to be reached
·
A
felt difficulty to reach the goal
·
Challenging
the felt difficulty through conscious, planned and purposeful attack
·
Reaching
the goal or arriving at a satisfactory solution to the problem at hand
Main objectives of
Problem-solving technique
·
to
stimulate reflective and creative thinking of the students.
·
It
involves the thought process the result from a doubt, a perplexity or a
problem.
·
The
approach leads to the formulation of generalisations that are useful in future
situations involving the solution of similar problems.
·
The
solution of a problem, whatever be its nature, practical or informational
involves the process of reflective thinking.
Steps in
problem-solving
1. Identifying and
defining the problem
The problem arises out of a felt need and out of existing
student activities and environment activities. The students 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.
2.
Analyzing 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.
The relationships are to be clearly stated. Relations that are not explicitly
stated may be supplied by the students.
3.
Formulating tentative hypothesis
The focus at this
stage is on hypothesising-searching for a tentative solution to the problem.
Analysis of the given data, and analysis of interrelationships among the given
facts help the students in formulating hypothesis or educated guesses as the
solution to the problem at hand.
4.
Testing the hypothesis
Appropriate method
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.
5.
Checking the results or verification of the result
At this step
the students are asked to determine their results and substantiate the expected
solution. The students should be able to make generalisations and apply it to
their daily life.
Approaches
and Techniques to Problem-solving
Problem solving
advocates the following approaches
·
Analytic
and synthetic methods.
·
Inductive
and deductive methods.
·
Method of analogies:
In analogy, problems are solved by
comparing them with similar problems that have been solved before. Thus the
method of solution becomes explicit and clear.
·
Restatement method:
Problem solving becomes easier if the
student is able to redefine the given problem using his own language and
symbols. This approach is known as restatement method.
·
Method of Dependencies:
In this method, the problem is solved by
focusing on mutually dependent components in the problem. The analysis of the
problem into its constituent elements throws light on the mutually dependent
elements in the problems. The interrelationships among the elements can be made
use of for reaching the correct solution of the problem.
·
Graphic Method:
In this method, the problem is represented
using diagrams and figures. The graphic representation aids the students in
determining fundamental relationships that exist among the given data and to
look for further details and relationships necessary for solving the given
problem. This method is very helpful in proving theorems, solving riders,
problems relating to mensuration etc.
Teacher’s
role in Problem-solving technique
The teacher plays a significant role in
problem solving method. The teacher’s role is to:
1.
Ensure
an atmosphere of freedom in the class.
2.
Create
the problem situation.
3.
Assist
the students in accepting, defining and stating the problem.
4.
Helps
the students in analyzing the problem and in breaking up the problem into
simple units.
5.
Help
the students keep their attention focused on the main problem all the time.
6.
Guide
the students in locating relevant source materials.
7.
Encourage
the students in seeking important relationships in the data.
8.
Helps
the students develop an attitude of open mindedness and critical enquiry.
9.
Exhibit
spirit of enquiry and discovery.
Characteristics of Good Problem
1.
The
problem should be real rather than an artificial one.
2.
It
should facilitate the integration of old and new process.
3.
The
solution of the problem should result in learning new higher order rules.
4.
The
solution of the problem should help in transfer of knowledge.
5.
The
problem should be educationally significant, productive of important and
worthwhile learning.
6.
It
should be possible of solution. The students should be equipped with background
information and skills which are prerequisite for solving the given problem.
7.
It
should be related the sub-unit, the unit and the course.
8.
It
should form the basis for further learning.
9.
It
should be clear and free from ambiguities.
10.
It
should be interesting and challenging.
11.
It
should arouse the curiosity of the students.
12.
It
should occur frequently in life situations.
13.
It
should provide best mental discipline to the students.
14.
It
should facilitate realization of the objectives of teaching mathematics.
Reasons for Difficulties in Solving Problem
1.
Lack
of interest and motivation.
2.
Lack
of language clarity in understanding the problem.
3.
Inability
to analyse the problem thoroughly.
4.
Lack
of focus on the key relationships.
5.
In
ability to identify the interrelationship among the given data.
6.
Lack
of fluency in the mental visualization or diagrammatic representation of the
problem.
7.
Inability
to recall and apply appropriate rules and formulae.
8.
Lack
of skill and practice in solving problems.
9.
Lack
of proficiency in the fundamental arithmetic operations.
10.
In
adequate knowledge of fundamental mathematical concepts, rules and formulae.
11.
Difficulty
in reading, identifying and using mathematical symbols.
Merits of Problem-solving method
1.
Problem-solving
provides a real life like experience to the children.
2.
It
develops in pupils good habits of planning, thinking, reasoning and independent
work.
3.
It
develops initiative and self-responsibility among the students.
4.
It
takes into account individual differences.
5.
It
helps the students to develop reflective thinking.
6.
It
helps the students to approach future problems with confidence.
7.
It
builds a mental attitude for effective learning based on critical thinking.
8.
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.
Demerits of Problem-solving method
1.
Not
all students are problem solvers.
2.
The
problem solving method becomes monotonous if used too frequently.
3.
It
is time consuming and consequently it is not possible to cover the syllabus on
time.
4.
The
success of this method depends upon mathematics teachers who are well versed in
critical thinking and reflective thinking. Not all mathematics teachers are
well versed in these type of thinking.
5.
Reference
and resource materials may be difficult to come by.
6.
Only
a skilled and resourceful teacher will be able to make an effective use of this
method.
7.
All
topics in mathematics cannot be taught through this method.
8.
Lack
of interest and motivation on the part of the students can spoil the effectiveness
of this method.
