Unit IV
APPROACHES,
METHODS AND TECHNIQUES OF TEACHING MATHEMATICS
• Behaviourist approach, problem-based
learning, constructivist approach and heuristic approach
•
Methods of teaching mathematics- Inductive-Deductive
method, Analytic-Synthetic method, Project method, laboratory method, problem
solving method
•
Techniques of teaching mathematics- questioning, brainstorming, assignment
•
Teaching for understanding proofs, Kinds of proofs- direct, indirect, by
mathematical induction, by contradiction, by causes, the contra positive and
disproof by counter example.
APPROACHES IN TEACHING MATHEMATICS
Behaviourist approach, problem-based learning,
constructivist approach and heuristic approach
BEHAVIOURIST APPROACH
The traditional
view of learning is forwarded by behaviourists. According to them, “Learning is
modification of behaviour through experience”. For the behaviourists learning
happens through the mechanical invention of the teacher.
The behaviourist
theories explain learning on the basis of stimulus-response. Gestalt
psychology, Psycho- analytical theory and Humanistic Psychology were the
schools which questioned the mechanical nature of behaviourism.
The Behaviourist
movement began in 1913, when John Watson wrote an article entitled
"Psychology as the behaviourist views it, " which set out a number of
underlying assumptions regarding methodology and behavioural analysis.
Basic Assumptions
Behaviourism emphasizes the
role of environmental factors in influencing behaviour, to the near exclusion
of innate or inherited factors. This amounts essentially to a focus on
learning.
We learn new
behaviour through classical or operant conditioning. Therefore when born our
mind is " tabula rasa’(a blank slate).
Behaviourism is primarily concerned with observable behaviour, as opposed
to internal events like thinking and emotion.
The Pioneers of behaviourist approach is Pavlov (Classical
conditioning)- Classical conditioning- Learning by
association, Pavlov discovered classical conditioning by observing dogs
behaviour at meal times.
B.F. Skinner (Operant conditioning)- Learning by reinforcement and
punishment, Skinner developed the theory of operant conditioning experiments
with animals such as rats and pigeons. One of his famous experiments involved a
rat in a skinner box. The rat had to learn to press the lever that would
provide food and avoid the lever that would give an electric shock.
Reinforcement is anything that
increases the likelihood of a behaviour repeating. Reinforcement can be either
positive or negative. Positive reinforcement is reward and negative
reinforcement involves removing a something painful.
Punishment is anything that
reduces the likelihood of a behaviour repeating.
E.L. Thorndike (Trial and error Theory), E. Guthrie
(Contiguous conditioning), and J.B Watson-Father of American behaviourist’s
theory, His works based on the experiments based on Pavlov’s experiments Watson
believed that all individual differences in behaviour were due to different
experiences of learning.
Basic concepts of behaviourist
approach is conditioning, organism (individual) reaction (S-R), stimulus
and reinforcement (positive or negative. All behaviours (normal and abnormal)
are learned. Learned responses from environmental stimuli. Study of
observable behaviour.
IMPLICATIONS
- Emphasis on behaviour
Students
>>>>Active Respondents.
>>>Learning process >>>> Opportunity to actually behave or demonstrate
learning.
>>>> Students
should be assessed by observing behaviour
- Drill and Practice---The repetition of
stimulus response habit
- Breaking habits>>>>Teacher>>
Lead>> New response
Merits
- The teacher controls both
of the children’s behaviour and learning process.
- The teacher has clear lesson plan and step- by step
syllabus.
- Success is reinforced by
praise and reward.
- Focuses only on behaviour
that can be observed and manipulated.
- This approach has proved
very useful in experiments under laboratory conditions, where behaviour
can be observed and manipulated.
- Shapes behaviour quickly.
- The learner adapts to the
environment.
- Behaviour can be measured
Demerits
- Internalized reasoning
may not be an outcome.
- The learner adapts to a
poor environment.
- Behaviour measured may
not be a true picture of understanding.
- It is likely the behaviour
will return after a period of time.
- The behaviourist approach
has been criticized for suggesting that most human behaviour is mechanical
and that human behaviour is simply the product of stimulus-response behaviours.
CONSTRUCTIVIST APPROACH
The cognitivists argue for the
potential of the learner to construct his on knowledge. Hence, they are called
as constructivist and their philosophy as constructivism. To them learning is
biological and social than that of instrumental as perceived by the
behaviourists.
According to
Brader- Araje and Jones (2002), Constructivism can be defined as “the odea that
development of understanding requires the learner actively engage in
meaning-making”.
Constructivism teaching is based
on the belief that learning occurs as learners are actively involved in a
process of meaning and knowledge construction.
Learners are the makers of
meaning and knowledge. Constructivism teaching fosters critical thinking and
creates motivated and independent learners.
The word “construct” comes
from Latin word ‘construere’, which means to give structure. The ongoing
structuring process is the conceptual heart of constructivism.
The following are the fundamental concepts behind constructivist
learning:
Ø Knowledge is constructed through a process of reflective
abstraction.
Ø The cognitive structure within the learner facilitates the process
of learning
Ø The cognitive structures in individuals are in a process of constant
development
Ø Constructivism focuses on construction, reconstruction and re
interpretation of experience
Ø Priority to “How to Learn” than “What to learn”
Ø Nurtures the learner’s instinctive curiosity
Ø Encourages the spirit of enquiry.
Ø Mental models of learning taken into considerations
Ø Accept the rule of exploring and experiencing in learning
Ø Importance to learning contexts.
Ø Supports co-operative learning and collaborative learning
Ø Encourages dialogue and communicating among pupils and teachers
Ø Encourages learning through life situations
Ø Strong foundation of cognitive theories
Ø Provides opportunity for the creation of ideas and concepts
Ø Teacher acts as a facilitator, co-learner, diagnostician. democratic
leader and not an information monger.
Ø Performance is given importance in evaluation
In constructivism,
learners construct their own knowledge by testing ideas and approaches based on
their pre-requisites and experiences.
The following
theories are the underpinning of constructivism
1. Genetic epistemology—Gean Piaget
2. Discovery of learning – Jerome. S. Bruner
3. Social Constructivism – Lev Vygotsky
4. Multiple intelligence – Howard Gardner
Piaget (1977) explains that we accommodate the new information to our old
way of thinking by reconstructing our present knowledge to a higher level of
thinking.. Constructivist teaching depends upon the learner’s level of
cognitive. According to Piaget they are
Sensory Motor stage (0-2 years)
Pre-operational stage (2-7 years)
Concrete operational stage (7-11years)
Formal operational stage (12-15years)
The teacher’s main
focus should be on guiding students by asking questions, it will lead them to develop
their own conclusions on the subject.
Three major roles for facilitators to support students in constructivist
learning environment are
Modelling
Coaching
Scaffolding
Constructivist
classroom environment provides opportunities for students question the material
being presented and explore various topics as their interests. The goal is producing
a democratic classroom environment that provides meaningful learning
experiences for autonomous learners. In a constructivist classroom, by contrast
the teacher and the student share responsibility and decision making and
demonstrate mutual respect.
Merits
§ Children learn more and enjoy
learning
§
Promote divergent thinking
§
Education works best when it concentrates on thinking and
understanding rather than on rote memorization.
§
Boost the confidence of learners
§
Constructivism concentrates on how to think and
understand
§
Promote collaborative learning
§
Constructivism gives students ownership of what they
learn, since learning is based on students.
§
Engaging the
creative instincts develops student’s abilities to express knowledge through a variety
of ways.
§
Constructivism
promotes social and communication skills by creating a a classroom environment
that emphasizes.
§
The learners are
actively involved
§
The environment is
democratic
§
The activities are
interactive and student centered
The teacher
facilitates a process of learning in which students are encourages to be
responsible and autonomous. It encourages active and meaningful learning and
promotes responsibility and autonomy. Constructivist teaching is beneficial in
achieving desirable educational goals for the students.
The teacher acts on the mind and conscience of the students in such a manner
that they are able to unfold their hidden potentials including regard.
Demerits
§ In a situation
where conformity is essential divergent thinking & action may cause
problem.
§ It is time
consuming
§ It is not suitable
for all topics
5E’s phases
of constructivist approach
1. Engage- start the
process
2. Explore- provide
students through a common base of experience. Identify & develop concepts,
processes and skills.
3. Explain-
Opportunities to verbalize students conceptual understanding/ to demonstrate
new skill/ behaviour. Also provides an opportunity for teachers to introduce
formal terms, definitions, & explanations for concept.
4. Elaborate- Extend
student’s conceptual understanding & allows them to practice skills and behaviours
5. Evaluate--encourage
learners and assess their understanding and abilities and lets teachers
evaluate students understanding of key concepts & skill development
PROBLEM BASED
LEARNING
Problem based learning is a process of acquiring and
understanding of knowledge, skills in the context of an unfamiliar situation
and applying that learning to the situation. Problem –based learning (PBL)
describes a learning environment where problems drive the learning.
“Problem based learning is the basic human learning process that allowed
primitive man to survive in his environment “- Barrows and Tamblyn (1980)
“Problem based learning is student centered learning strategy in which students
collaboratelively solve the problems and this reflects on their experience. In
PBL , the starting point is a problem, a query, or a puzzle that the learner
wishes to solve”- D.L. Bound(1985)
“PBL can be explained as “The learning that results from the processes of
working toward the understanding and resolution of a problem”- Barrows (1980)
“ It is a careful inspection of methods, which are permanently successful in
formal education”- John Dewey(1916)
It is an inquiry-based
approach. It is based collaborative learning approach. Begins with an
ill-structured problem ie; based on desired learning outcomes, learner
characteristics, compelling and real-world situations. It is based on
constructivist theory of learning. Promotes active learning by challenging
students to learn to learn . Well-constructed problem stimulates students,
curiosity and engagement. A well –designed problem is constrained to the issues
on which the teacher wants students to focus. Students will need to find
potential solutions to the problem and determine which solution is the best
fit.
Metacognition:
students think about their thinking. What do I think the problem is? Why do I
think that? Has my perspective changed? how?
Role of the teacher
- Designs the problem
- Anticipates teaching
& learning events
- Investigates & gather
resources
- Models and coaches’
students
- Confront the problem
- Determine know/ need to
know
- Define the problem
- Design a plan to solve
the problem
- Gather information
- Construct potential
solutions select & present the “ best fit”
Goals
- Construct an extensive
& flexible knowledge base.
- Foster increased retention of knowledge. Develop
effective problem-solving skills
- Develop self-direction, lifelong. learning skills
- Become effective
collaborates.
- Strengthen the intrinsic motivation to learn
- Recognize, develop &
maintain the personal characteristics and attitude.
Merits
1. To
acquire subject matter knowledge
2. Motivate learners to learn
3. Help learners with retention
4. Develop student’s thinking skills.
5. Developing student’s key skills relevant to
employment such as interpersonal communication skills
6. Fostering professional competencies and
confidence together with professional identity
7. Mirroring the inter disciplinary team
process graduates will use in work and research
8. Linking theory and practice.
9. Encourage learners to integrate knowledge
from different subjects, disciplines and sources.
10. Having a sense of belonging and friendship
11. Having a sense of fun while learning
12. Expressing in operational form a philosophy of learning that
is student –centric and problem focused.
13. Democratic views
14. Teamwork
15. Critical evaluation, skills of inquiry, pear teaching and
peer evaluation
16. Self-directed learning and use if resources
17. Presentation skills
18. Leadership quality
19. Attainment of lifelong learning skills
20. Developing flexible knowledge that can be applied to
different contexts.
21. Increased motivation for learning
22. Promote collaborative learning
23. Moves learning from s positive activity to an active
activity learning becomes the act of discovery.
Demerits
1. It
is very difficult and expensive to use as a teaching technique, when the class
size is large.
2. Students
require orientation to perform the role of a learner in PBL setting
3. Evaluation
is quite difficult and sometimes may be subjective
4. Resource
expensive
5. Teachers and
students may be initially uncomfortable with PBL because they are used to
subject based learning and they do not really understand how to proceed in PBL.
6. Measurement of learning outcomes is difficult.
HEURISTIC APPROACH
`The term
"Heuristic" is discovered from a Greek word , which means ' I find'..
Here the child is put in the place of discoverer. The method involves finding
out by the student, instead of merely telling of everything by the teacher. It
demands complete- self-activity or self- education on the part of the learner.
It is a method by which people learn to reason for themselves. when this method
is applied in the extreme form the teacher stands aside as an onlooker and the
child selects his own path and proceeds according to his own lights. The
teacher is not required to encourage, help or guide him. There is no need of
the teacher's approval or disapproval of his work. Let him to do it in his own
way. For maintaining this approach, the teacher has to arouse in the minds of
the learners desire to discovery by themselves.
Prof. Armstrong was the
originator of this method. He devised it for the teaching of science. And is
also useful for mathematics. Self-confidence originality, independence of
judgement and thinking power are to be developed in the individual to make him
ever successful student.
Example
Take a problem of the
discovery of the characteristics of a parallelogram. Give students many
parallelograms already drawn in sheets of paper and ask them to find out the qualities
of the different elements of a parallelogram.
Naturally the students will
start judging and measuring the elements of different parallelograms.
Merits
- The student becomes an active participant in
the learning process
- The student think, for himself and does
not merely listen for information
- Home study and memorization work become light.
- It is certainly a psychologically sound method
- After discovering something by his own efforts
the student starts taking pride in his achievement .it gives mental
satisfaction
- The subject acquires a real
understanding and clear notion of the subject
- The teacher remains in constant touch with his
students
- It develops in the students the heuristic
attitude or scientific attitude
- It creates in the spirit of enquiry
- The student becomes self-reliant
- The method suits for the learner and the
subject
Demerits
- It demands extraordinary labour and special
preparation from the teacher
- Every teacher may not be able to use it
successfully.
- It is a slow method
- There is very little certainty that, the
child left to himself will make steady and sufficient progress.
- Every child cannot be expected to be a gifted
discoverer
- Specially in early stage the child needs
guidance and hints
- it may not possible to teach all topics by this
method.
PROCESS ORIENTED
APPROACH
A
Process can be defined as a set of interrelated or interacting activities which
transforms inputs into outputs. These activities require allocation of
resources such as pupil and materials.
The application of a system of processes with in an organisation and
interactions of these processes and their management, can be referred to as the
" Process Approach".
METHODS OF
TEACHING MATHEMATICS
Inductive-Deductive
method, Analytic-Synthetic method, Project method, laboratory method, problem
solving method.
- INDUCTIVE-
DEDUCTIVE METHOD
it is a combination of two methods. These two methods are
complementary to each other .
Inductive method
Inductive method advocated by
Pestalozzi & Francis Bacon.
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.
Inductive Method Proceeds From
Ø
Particular to general
Ø
Concrete to abstract
Ø
Known to unknown
Ø
Simple to complex
This method places vital role
in teaching learning process of mathematics because many mathematical formulae
principles and generalisations are based on induction
Steps in Inductive method
Ø
Presentation of concrete examples
Ø
Observation of cases under given conditions
Ø
Finding common relations
Ø
Generalizations
Ø
Verification
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.
- It
helps understanding because knowledge attained by this method on real
facts.
- It
helps to develop critical thinking and logical power among the students.
- It is a
scientific method.
- It
helps to develop self-confidence and self-reliance.
- This is
psychological method.
- Knowledge
gained by this method is long lasting.
- This
method is suitable for lower classes as it provides a number of concrete
examples.
- It
facilitates meaningful learning.
- It is
based on actual observation thinking and experimentation.
- In this
method there is no scope for rote memory.
- This
method helps to develop curiosity and interest in the child to learn
mathematics.
- It
helps to reduce homework.
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.
- This
method is very slow, lengthy and laborious method.
- This is
not applicable to the all topics in mathematics.
- Only an
experienced and able teacher can handle this method successfully.
- This
method is not suitable for higher classes because syllabus is very wide in
higher classes and it is not possible to cover the whole syllabus.
- Deductive
method is need to ensure the value of inductive method hence Inductive
method is not complete in itself.
- It is
not suitable for gifted students as number of concrete examples and
unnecessary explosion make the teaching dull and boring to them.
- Results
drawn by the use of this method are not always true. Their truthfulness
depends upon a number of examples on which they are based.+
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. Here there is no scope of investigation. It is a
method of explanation of a known principle and verification of the result.
Deductive method proceeds from
General to particular
Abstract to concrete
Formula to examples
In deductive method
principles, laws, formulae and relations are presented before the students in
real form. Children learn the laws, principles and formulae by heart and then
ask them to verify the result.
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
enhances speed and efficiency in solving problems.
- It combines
with the inductive method to remove the incompleteness and inadequacy of
the later
- This
method is very economical. It saves time and energy both of the students
and the teachers.
- It
helps to develop speed, skill and efficiency in solving problem.
- It is
very suitable method for lower class students who cannot generalise the
principles, laws and formulae for themselves.
- It is
very suitable at application and revision stages.
- This
method is very useful for teaching of theorems, axioms, tables in arithmetic
etc.
- It is
useful for advanced study.
- Both
teacher and students do not have any difficult in using this method.
- When
heavy syllabus is to be covered, this method is suitable.
- It
supplements inductive method and thus completes the process of inductive-deductive
method.
- It
gives rigorous proof
- It
helps to develop intuition among students.
Demerits
Ø It is very
difficult for a beginner to understand an abstract formula if it is not
proceeded by a number of concrete instances.
Ø 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.
Ø It will thus
cause an unnecessary and heavy burden on the brain. It may even result in brain
fag.
Ø Memory
becomes more important than understanding and intelligence , and that is
educationally unsound
Ø 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.
Ø The students
cannot become active learners.
Ø It is not
suitable for the development of thinking, reasoning and discovery.
Ø It is not a
scientific method
Ø It
encourages rote memory therefore knowledge gained by this method is not long lasting.
Ø It fails to
develop self-confidence among students.
Ø It fails to
develop motivation and interest in learning.
Ø This method
encourages too much of dependence on teacher.
Ø Individual
difference in achievement and performance cannot be met with satisfactorily.
Ø It is not
suitable for the development of thinking reasoning and discovery among pupils.
Ø It does not
encourage student’s involvement in learning.
Ø It doesn’t
clarify the doubts of the student regarding the generalization and hence
learning is incomplete.
Ø Memory
becomes more important than understanding and intelligence and hence it is
unpsychological.
Comparison of Inductive and
Deductive method
|
Inductive
Method |
Deductive
Method |
|
It is based on inductive
reasoning. |
It is based on deductive
reasoning |
|
Proceeds from : Particular to general Concrete to abstract Example to formulae |
Proceeds from : General to Particular Abstract to Concrete Formulae to Example |
|
It is a method of discovery |
It is a method of
verification |
|
It is suitable for teaching
in lower classes |
It is most suitable for
teaching in higher classes. |
|
Emphasis is on reasoning |
Emphasis is on memory |
|
Encourages meaningful
learning |
Encourages rote learning |
|
It is a psychological method |
It is not a psychological
method |
|
It leads to new knowledge |
It does not lead to new
knowledge |
|
It is suitable for
presentation stage of learning |
It is most suitable practice
and application stages |
|
It is very slow method |
It is very quick method |
|
In this method students are
active participants |
In this method students
remains passive listeners. |
§ 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 problematic situation and its systematic
solution.
In the beginning we use analytic method to find
solutions and then we use synthetic method for presentation of the discovered
solutions. Analytic method helps to understand new situations and synthetic
method helps for fixing what has been understood.
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.
In analytic method we proceed
from:
Ø Unknown to
known
Ø Abstract to
concrete
Ø Complex to
simple
Ø Conclusion
to hypothesis
In this method we stat from
what is to be proved and ends with conclusion. Thus, it is the methods of
discovery based on heuristic and inductive approaches. This method can use in
the following conditions:
Ø When we have
to prove any theorem
Ø When
construction work is to be done in geometry
Ø When we have
found out the solution of some new arithmetic problems.
Merits
- It is a
logical method.
- It facilitates
understanding
- The steps
in its procedure are developed in a general manner.
- The
method is suiting the learner and the subject.
- In this
method every step is explained hence it leaves no doubt among the
students.
- It is based
on psychological principles.
- It is
based on heuristic approach.
- It
helps to develop self- confidence self-reliance among the students.
- It
develops scientific attitude.
- Knowledge
gained by this method is long lasting and solid.
- It
leads to spirit of equity and investigation.
- It
promote metal activity on the part of the learner and makes them active in
teaching learning process.
- It
makes teaching learning process interesting by providing challenging
situation to the learner.
- It
helps to develop thinking and reasoning power among the child.
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.
- This
method is very slow and lengthy.
- This
method is not suitable to acquire speed and efficiency.
- Every
teacher cannot use this method successfully.
- This
method is not suitable for all topics in mathematics.
- This
method is not very suitable in the case of small children.
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 of the child.
- It suits
the teacher.
- It
follows the same process as given in the textbooks.
- This
method is very suitable for slow learners.
- Teacher’s
work is simplified.
- It
omits the trials and errors like in analytic method.
- It
develops speed and efficiency among the learners.
Demerits
- It
leaves many doubts in the mind of the learner
- without
a satisfactory answer to so many questions that arise in synthesis, the
pupil 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 sometimes
does not suit the learner and the subject.
- It
leads to rote memory.\homework becomes heavy to the learner. This method
does not give full satisfaction to the learner.
Example of analytic -synthetic method
A cylinder has a
base 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.
- What is
to be calculated? (Volume of a given cylinder)
- How can
we calculate the volume of a cylinder? (Using the formulae V=.....)
- What
data are required for that? (r and h of the cylinder)
- Are
these given in the problem? ( h is given but r is not given)
- Is any
hint helpful to find r is available? ( yes the circumference is of the
base is given )
- How can
we determine r from this? (...... then r= 5 cm)
- Is
there any other data misssing ? ( No )
- 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. William Head Kilpatrick.
"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. Kilpatrick
" A project is a
problematic act carried to completion in its most natural setting"—J. A. Stevenson
“A project is a unit of
activity in which pupils are made responsible for planning and purposing” –
Parker
“A voluntary undertaking which
involves constructive effort or thought and eventuates into subject results”—Thomas
&Long
“Project is a unit of education
work in which the most prominent feature is some form of positive and concrete
achievements”—Snedden
On the basis of above
definitions, the following points have been stressed,
Ø A project is
a problematic act.
Ø A project is
a purposeful activity.
Ø A project is
a whole hearted activity.
Ø A project is
an activity in a natural setting.
Ø A project is
an activity in a social setting.
Ø A project is
a bit of real life introduced in school.
Ø A project is
a problem solving of a practical nature.
Ø A project is
a positive and concrete achievement.
Ø A project is
an activity through which solution of various problems are found out.
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
Other principles of Project
Method
1)
The principle of Purpose : Kowledge of purpose is a
great stimulus which motivates the child to realize his goal. Purpose motivates
learning. Interest cannot be aroused by aimless and meaningless activities.
2)
The Principle of Activity: The pupils are naturally
active. They love activity. Opportunity should be provided for them to be
active and do things for themselves. They must be kept active mentally as well
as physically and must bear the maximum responsible.
3)
The Principle of Experience: Experience is the best
teacher, what is real must be experienced. The children learn new facts and
information through experience.
4)
The Principle of Social
Experience: The child is a social being and we have to prepare him
for social life. Training for a co-operative life must be given to him in his
childhood. In the project method, the child works in group.
5)
The Principle of Reality: Life is real and education to
be meaningful must be real. A child who is to live in a life of reality must be
trained as such through his education. The project method aims at reproducing
real life situation into the school. Pupils are given opportunities to exercise
their powers in real life situations.
6)
The Principle of Freedom: the child should be free of impositions,
restrictions or obstructions so that he may express himself fully and freely.
He must be given the freedom to choose an activity to do an activity according
to his interests needs and capacities.
7)
The Principle of Utility: Knowledge will be worthwhile
only when it is useful and practical. The traditional system of instruction has
very little utility. The project method develops various attitudes and values
which are of great significance from practical point of view.
Types of Projects
Generally projects are two
type: Individual and Group project
According to W.H.Kilpatrick,
there are four types of projects
1)
The producer type: in which the emphasis is
directed towards the actual construction of a material object or article.
2)
The consumer type: where the objective is to
obtain either direct or vicarious experience such as reading and learning from
stories, listening to a musical delectation et.
3)
The problem type: In which the chief purpose
is to solve a problem involving the intellectual processes, such as determining
the density of a certain liquid.
4)
The drill type: where the objective is to
attain a certain degree of skill in a reaction as learning a vocabulary.
Steps involved in the project
method
- Providing
/creating the situation
- Proposing
and choosing the project
- Planning
the project
- Execution
of the project
- Evaluation
of the project
- 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; like how the project was
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
- The
number of hostelers will be recorded
- The
expected expenditure will be calculated.
- Expenditure
on various heads will be allocated to the students
- Budget
will be prepared with the help of the class
- The
account of collections from amongst the students will be noted.
- Actual
expenditure will be incurred by the students
- A chart
of " balanced diet"for the hostelers will be prepared .
- The
time of breakfast , lunch,tea and dinner will be fixed and notified
- Execution
of different programes stated above will be made.
- Weight
of each hostelers will be checked after regular interval, and the same
will be put on record.
- Punctuality
in all the activities of the hostelers will be recorded.
- Evaluation
of the entire programme and then it will be typed out for the information
of all concerned.
Some projects for mathematics
- Execution
of school bank
- Running
stationary stores in the school
- Laying
out a school garden
- Laying
a road
- Planning
and estimating the construction of a house
- Planning
for annual camp.
- Executing
the activities of a mathematics club
- 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.
- This is
based on laws of learning like readiness, exercise and effect.
- The
children remain active throughout the execution of the project.
- It promotes
co-operative feeling and group interaction.
- It
upholds dignity of labour.
- It
brings about a close correlation between a particular activity and various
subjects.
- It is
based on principle of individual differences.
- It
provides the students an opportunity for mutual exchange of ideas.
- It
introduces democracy in education.
- It
develops self-confidence and self- discipline.
- The
gained knowledge becomes solid and durable.
- It
provides freedom
- The
child realizes his responsibilities and duties.
- It
develops discovery attitude in the child
- Solves
the problem of indiscipline.
Demerits of Project method
- it does
not suit for the topic
- There
is no saving of time, energy and effort
- Systematic
and continuous Sometimes teaching is not possible.
- Costly
- Planning
and execution of project method can be adopted as a co-curricular activity
It neglects intellectual work
- Lack of
competent teachers.
- There
is no provision for drill and practice for skills required in mathematics.
- The
knowledge is not acquired in a sequential and systematic manner.
Role of the teacher in Project Method
Ø In the
project method the role of a teacher is that of guide, friend and philosopher.
Ø The teacher
should provide democratic atmosphere in the classroom.
Ø He helps his
students to avoid mistakes.
Ø He should be
alert and active all the time to see that the project is running in its right
lines.
Ø He should
have a thorough knowledge of individual children so as to allot them work
accordingly.
Ø He should
give chance to shy and introvert pupils to come forward and participate
actively.
Ø He should
have thorough knowledge and experiences.
Ø He
encourages his students to work collectively, amicably in the group .
§ 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 equipment’s 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.
- Artificial
correlation.
- 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.
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" - Ausbel
§
"Problem solving is a planned attack upon a
difficulty or perplexity for the purpose of finding a satisfactory
solution"-Risk. T.M
§
“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
§
“Problem solving in teaching refers to the task of
making decisions or doing things that the learner wants to make or to do, the
nature of which he is able to understand but for which at the time he has no
solution” –Hammonds Carsie
§
“a problem occurs in a situation in which a felt
difficulty to act is realised. It is a difficulty that is clearly present and recognised
by the thinker. It may be physical and involved the manipulation of data. The
distinguishing thing about a problem however, is that it impresses the
individual who meets it a needing a solution. He recognises it as a
challenge”.—Yokam and
Simpson
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.
Characteristics of a good
problem
Ä The problem
should be meaningful, interesting and practical.
Ä
It should be well defined.
Ä It should
have some educational value.
Ä As much as
possible the problem should be related with daily life of the child.
Ä It should be
challenging so that the powers of thinking and reasoning can be developed.
Ä It should
have correlation with other subjects.
Ä It should be
related with previous knowledge of the child.
Ä The problem
should be real.
Ä It should be
related to the subunit, the unit and the course.
Ä It should
from the basis for further learning.
Ä It should be
neither too difficult nor too easy.
Ä It should
provide best mental discipline to the students.
Ä It should
develop imagination and critical powers.
Ä It should
develop mathematical skills.
Ä It should
develop scientific attitude among the students.
Ä It should
provide best mental discipline to the students.
STEPS IN PROBLEM SOLVING
- Identifying
and defining the problem
- Analysing
the problem
- Formulating
tentative hypothesis
- Testing
the hypothesis
- 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,
- 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.
- 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}.
- 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 testing 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
- This
method is psychological and scientific in nature.
- This
method stimulates thinking of the child.
- This
method develops reasoning power of the child.
- It
helps in developing good study habit.
- It
helps to improve knowledge.
- It
helps to develop power of expression of the child.
- Children
get training in the art of problem solving in actual life situations. This
method inculcates the problem-solving attitude.
- It is
helpful in making the students resourceful and self-reliant.
- It
inculcates the habit of open-mindedness and tolerance.
- The
child learns how to act in new situation.
- The
child get training in co-operation and fellow feeling.
- It
develops the power of critical judgement.
- This
method helps in maintaining discipline in the class.
- Learning
becomes more interesting.
- It
inculcates better teacher pupil relationship.
- Prepares
pupil to solve the problems of life
- It
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
get valuable social experience like patience, co-operation, self-confidence,
etc.
- 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.
- Sometimes it becomes spoon feeding or artificial method in normal
classroom situations.
- Textbook and written materials on these lines are not available.
- It is not economical
- It needs skilled and talented teachers.
Role of the teacher
The following are the teacher’s role in Problem
solving method:
« Ensure an
atmosphere of freedom in the class.
« Create the
problem situation.
« Assist
students in accepting defining and stating the problem.
« Help the
students in analysing the problem and in breaking up the problem into simple
units.
« Guide the
students in finding relevant materials.
« Encourage
the students in seeking important relationship in the data.
« Help the
students develop an attitude of open-mindedness and critical enquiry.
« Exhibit
spirit of enquiry and discovery.
The teacher should be careful about the following:
Ø How to
identify the problem, ie., experiences in the classroom should facilitate to
sense the problem meaningfully.
Ø How to
represent problems, ie., the students should have thorough knowledge in the subject
matter surrounding the problem as well as a familiarity with the particular
type of problem
Ø How to
compile relevant information, ie., emphasize the importance of retrieving and
using previously learned knowledge.
Ø Several
methods for formulating solutions and use different
combinations of methods to solve problem as there is individual difference in
problem solving ability.
Ø The skills
of evaluation (A habit of verifying the result is to be practiced).
In problem solving, the teacher is only a
facilitator. Teacher has to create an environment in which the problem can be
solved by the students themselves. The teacher should not provide a readymade
solution to the problem. He is only a guide to reach the solution.
Techniques of teaching
mathematics- questioning, brainstorming, assignment
Questioning
Questioning
is a powerful technique of teaching mathematics. Through this technique, the
teacher transacts a lot of learning experiences. The teacher asks questions and
the responses given by the students are strengthened and elaborated. According
to Lorber and Pierce (1990), “ questions can be used to find out how well
students understand a particular block of information, to shift students
attention from one point to another, to increase retention of important points
by isolating and emphasizing them, and to put students in the right direction
before starting assignments”. Questioning facilitates high order thinking skill
like analysis, synthesis and evaluation in the students. Questioning is also an
important teaching skill. An effective teacher employs questioning technique
appropriately and also prepares good questions.
Characteristics of Questioning
Ø Exploring the entering behaviours
of the learners.
Ø Provides motivation and encouragement.
Ø Makes them active in the classroom.
Ø Establishes a chain of learning by
linking previous knowledge with new knowledge.
Ø Provides the direction to the
mental process of learning.
Ø Develops thinking and analytical
abilities of the learners.
Ø Revise the teaching content.
Ø Evaluate the effectiveness of
teaching learning.
Characteristics of Good Questions
« The
language of the questions should be simple.
« The questions should neither too easy nor too
difficult.
« The
form of the question should be straight forward. The should not be any
ambiguity.
« Students should not be lengthy and vague. They
should be clear, brief and to the point.
« The questions should be relevant to the topic.
« The questions should be thought provoking.
« Two question should not be asked in one.
« The question should be interesting as far as
possible.
« The teacher should try to vary the form of his
questions.
« The question once asked should not be repeated
unless the teacher is sure the class has not followed It.
« The question should be addressed to the entire
class.
« The questions should be asked in a pleasing
manner.
« The questions should be put in such a way that
every student think that he will be asked to answer whether he is good or weak.
« Adequate time should be allowed to answer.
Purpose of Questions
· To
test the previous knowledge of the student.
· To enable them to recall something.
· To enable them to recognize something.
· To enable them to think over something.
· To enable them to reason about something.
· To elicit something from students.
· To stimulate interest and effort on the part
of students.
· To keep the children mentally alert.
· To promote initiative and originality.
· To
stimulate the curiosity of the students.
· To ascertain whether they are following the
lesson or not.
· To revise the lesson and those fix the facts
in the minds of the students.
· To secure the cooperation of the students.
· To diagnose the weak points of the students.
· To formulate general rules.
· To
Synthesis.
Type of Questions
Ø
Introductory Questions:
These are
used in introducing new topic or lesson to the students. The Introductory
questions are based on the previous knowledge which is related new knowledge.
These questions employee from known to unknown maxims of teaching.
Ø
Developing
Questions: After introducing new topic, some
questions are asked to develop the new content. It employees from logical to
psychological technique of teaching. The logical sequence of units of new
content should be psychologically workable.
Ø
Recapitulatory Questions: In
the learning process law of practice or law of exercise place a significant
role in teaching new content. These type questions provide the situations to
practice the learned content. The question serves 3 purposes.
a)
To consolidate the learner’s new
knowledge:
b)
To organize the ideas of facts at
proper place.
c)
To provide well organized situation
of the content.
Ø Evaluatory Questions: At the end of the lesson, the
teacher the teacher puts questions to test the knowledge of the learners to
assimilate whatever they could learn the content. Generally oral questions are
asked. These are short answer type. Sometimes these questions are helpful to
diagnose the weakness of learning. Remedial instruction can be provided to
learner by the teacher. It also function a feedback to the teaching learning
process.
Another way to classify questions
Generally, Questions are used for
two purposes
a) Didactic Questions
These
questions are used to providing new knowledge to the students i. e,,
introductory, developing and recapitulatory question. 20 questions involve some
prompt, cues or hint for emitting new behaviours.
b) Testing Questions
These
questions are used assessing and diagnosing purpose. A Hin,t cues or prompts
are not used in these type of questions.
Brainstorming
Brainstorming
is a democratic and problem Centered technique. In this technique the content
is largely determined by the children. Brainstorming creates situations for
students and teacher interaction and both remain active in teaching. This
technique encourages the creativity and in the originality among the children.
Brainstorming is based upon the assumption that a child can learn better in a
group rather than in individual study. By this the higher order of cognitive
and affective objective can be achieved.
Steps
in brainstorming
1.
Presentation of problem.
2.
Provide relevant information.
3.
Record the ideas put forth by the participants
4.
Combined similar ideas.
5.
Evaluate each Idea or solution.
Principles of brainstorming
The following are the four principles of
brainstorming.
1.
Freewheeling:
once started, the brainstorming session should progress like the lubricated
wheel. That is, expression of opinion should be possible without interruption. The
idea may irrational, comical or strange. But all such ideas maybe accepted.
2.
No
criticism.: it must be ensured that expressions and gestures do not reflect a
critical attitude.
3.
Quality breeds quality: The greater number of
Ideas generated, the number of qualitative ideas should be collected. Each
person may be given opportunity to present ideas again. Prompting those
students who remain silent, without entering others is the duty of the anchor
man.
4.
Hitch hiking: it is a way of arriving at
the destination by travelling free by any mode of transportation available on
the way. This can be means of ideation. Once you hear an idea, another related
or unrelated ideas occurs to your mind. This again catalyzes another person's
ideation. Date helps to form Nobel ideas.
Advantages of brainstorming
v It helps to achieve the higher order of
cognitive and affective objectives.
v It is a democratic technique of teaching.
v It is highly motivating.
v It makes classroom interaction more effective.
v It develops problem solving ability.
v It provides more ideas and views of the child.
v It develops thinking and reasoning power of
the children.
v It is efficient and productive technique.
v Habit of decision-making is developed in the
mind of the child.
Assignment
Assignment is
the work given to the students either before the lesson or after the lesson and
it may completed at school or home. Assignment is the work assigned to the
student. It is a sort of undertaking or commitment on the part of the learner. The
child undertakes upon him, self the responsibility of carrying out the work
assigned to him.
“The central
position of the assignmeny in the techniques of teaching has remained
unquestioned: -- N.L.Bossing
“Upon the
proper assignment of the lesson depends much of the success of the recitation,
and also much of the pupil’s progress in learning how to study”—G.H.Betts
“Teaching
generally do not appreciate the importance of assignment, and work of the
pupils probably suffers as much from hasty or careless assignment as from any
other single course”
“The assignment
represents one of the important phases of teaching”-- H.R.Douglass
Purpose of assignment in Mathematics
Ø To develop a habit of self-study
among the students.
Ø To give them an opportunity to do
practice what is done in the school.
Ø To develop habit of reading regularly
among the students.
Ø To provide opportunities to the
students to utilise their leisure time.
Ø To finish the prescribed course in
time.
Ø To provide remedial measures for
backward students.
Ø To develop skill in problem solving
Ø To create interest in mathematics.
Ø To enable the child to revise his
previous lesson and prepare the next one.
Ø To give chance to every child to progress at
his own speed.
Ø To serve as a link in the present
teacher co-operation. It enables the presents to know that regular work is
being done in the school.
Ø To expose students to various
resources of learning.
Ø To correlate the experiences and
previous knowledge of the child.
Characteristics of a good Assignment
ü It should be related to subject
matter under study.
ü It should remove the difficulties
of the children. It should be clear and definite.
ü It should be such so as to suit to
the age, aptitudes and interest of the students.
ü It should give proper reinforcement
to the child.
ü It should be motivating and
interesting.
ü An assignment should be insightful.
ü It should correlate experiences
with the previous knowledge.
ü It should be concise and balanced
which can be finished by student easily and quickly.
ü It should stimulate thought.
Role of the teacher
To make assignments effective and
interesting, the teacher has to do the following
v He should split up the prescribed
course in mathematics into successive and progressive assignments.
v He should prepare a progress chart
for each student.
v He must prepare and provide a list
of reference material required for each assignment.
v To cover the learning gaps he should
prepare remedial assignments.
v He should prepare activity sheets
for laboratory work and experiments.
Co-operative
learning
Cooperative
learning is a successful teaching strategy in which small teams, each with a
student of different levels of ability, use a variety of learning activities to
improve their understanding of a subject. Each member of a team is responsible
not only for learning what is taught but also for helping teammates learn, thus
creating atmosphere of achievement. Students work through the assignment until
all group members successfully understand and complete it. It is a constructivist
strategy.
Definitions
" co-operative
learning is defined as a teaching learning strategy in which the students of a
class engage themselves in a variety of useful learning activities in a
Cooperative and non-competitive
environment by forming a number of teams each consists of a small number of
students of different levels of ability for their understanding of a subject
".
" Cooperative learning is the instructional use
of small groups so that the students work together to maximize their own and
each other's learning ". JOHNSON
From co-operative learning, all
group members:
Ø Gain from each other’s
effort (your
success benefits me and my success benefits you)
Ø Recognize that all group
members share a common fate. (we
all sink off swim together here)
Ø No that one's
performance is mutually caused by oneself and one's team member. (We cannot do it
without you)
Ø Feel proud and
jointly celebrate when a group member is recognized
for achievement. (We
are congratulating
you on your accomplishments).
Elements of
cooperative learning
Johnson, et al (1990) describe, positive
interdependence, face to face promotive interaction, individual and group
accountability, interpersonal and small group skill and group processing
interpersonal and small group skill and group processing are five essential
components of cooperative learning.
Positive
interdependence
Ø One for all,
all for one.
Ø A sentence of
sink or swim together.
Ø Everyone is
committed to others success and accountable for sharing the work to be done to
ensure your success of the group.
Individual
accountability
· Strengthening
the individual strengthens the group.
· Each of us has
to contribute and learn.
· The group must
be accountable for achieving its goals and each member must be at count at able
for contributing his or her share of the work.
· Access
individual performance to identify areas of weakness and support everyone were
they need it most to complete their tasks.
Interpersonal
and small group skills
« Treat others as
you want to be treated.
« Students have
to engage simultaneously in task work and teamwork.
« Team members
must be able to effectively use skills in leadership, decision making, communication,
trust building and conflict management
Face- to -face promotive
interaction
§ Sharing is
caring students need to do real work together in which they promote each other’s
success by sharing resources and helping, supporting, encouraging and applauding
each other’s
efforts to achieve.
§ Activities
include: orally explaining how to solve problems, teaching ones knowledge to
others, checking for understanding, discussing concepts being learned and
connecting present with the past learning
Group processing
· Your opinion matters.
· The team
members reflect on how will the team is functioning and how to function even
better.
Distinction between traditional learning and
cooperative learning
|
Traditional
Learning |
Co-operative
learning |
|
No individual accountability |
Individual Accountability |
|
No Interdependence |
Positive Interdependence |
|
Homogeneous |
Heterogeneous |
|
Based on Behaviourism |
Based on Cognitive/Social Constructivism |
|
Interaction between teacher and students only |
Opportunity for interaction among students |
|
Social skills assumed and ignored |
Social skills directly taught |
|
Emphasises drill and repetition |
Emphasises cognition |
|
Responsibility only for self |
Shared responsibility for each other |
|
Emphasis on teacher’s evaluation of learner |
Emphasis on self- evaluation and peer evaluation |
|
Teacher acts as transmitter of knowledge |
Teacher act as facilitator democratic leader and
diagnostician. |
Advantages of cooperative learning
ü Promote student
learning and academic achievement.
ü Increase
student retention.
ü Enhance student
satisfaction with their learning experience.
ü Help students
develop skills in oral communication.
ü Develop
students Social Skills.
ü Promote student self-esteem.
ü Help to promote
positive race relations.
ü Increases
student motivation by providing Peer support.
ü Encouraged to
learn material in great depth than they might otherwise have done, and to think
of Creative ways to convince the teacher that they have mastered the required
material.
ü Low achieving
students can make contributions to a group and experience success, and all
students can increase the understanding of Ideas by explaining them to others.
ü Improve
relationships among students from different ethnic backgrounds.
ü Cooperative
learning promotes academic achievement.
ü It is
relatively easy to implement,
and is not expensive.
ü Children's
improved behaviour and attendance and increased the liking of school.
Collaborative
learning
Collaborative
learning is an educational approach to teaching and learning that involves a
group of students working together to solve a problem, complete a task, finding
solutions to problems, or create a product. Collaborative learning develops
higher level of thinking, oral communication, self-management and leadership
skills. It is a constructivist strategy.
Collaborative
learning means everyone is involved in the learning. In collaborative learning
students work together in small groups on a structural activity. A group of
students discussing a lecture or students from different schools working
together over the internet on a shared assignment are both examples of
Collaborative learning.it is an informal method of teaching and learning which
allows freedom and creativity.
According to Findley (1981), Collaborative
learning is a situation in which two or more people learn or attempt to learn
something together.
According to Gokhale, individuals are
able to achieve higher levels of learning and retain more information when they
work in a group rather than individually, this applies to both the facilitators
of knowledge, the instructor, and receivers of knowledge, the students.
According to Wintz, “Collaborative
learning as an educational approach to teaching and tearning that involves
groups of students working together to solve a problem, complete a task, or
create a product”
Characteristics/ Advantages of Collaborative Learning
§ Positive interdependence.
§ Individual accountability.
§ Face to face
promotive interaction.
§ Appropriate
use of collaborative skills.
§ Group processing.
§ Development
of higher level of thinking skills.
§ Develops oral
communication skills, self-management and leadership skills.
§ Increase in
student retention, self-esteem and responsibility.
§ Develops social
interaction skills
§ Problem solving
ability.
§ Promotes a
positive attitude towards subject matter.
§ Promotes positive
race relations.
§ Create an
environment of active, involved, exploratory learning.
§ Promotes student
faculty interaction and familiarity.
§ Preparation for
real life and employment situations.
Disadvantages of Collaborative
Learning
·
Some groups may go too fast for one person and that
person feels lost, so he/she just copies.
·
Sometimes the members of groups may work faster then
others which may cause tension.
·
Collaborative learning allows work to get done without
knowing that every person in that group actually understands what was done.
·
People may try to take over the group and dictate what
everyone does. These types of people are not good group workers.
·
When there is one dominant person, others don’t learn how
and why things are done, but only copy the information.
·
Another problem in working in groups is dictatorship. One
person may take control and not allow others to share their knowledge.
·
More quiet people may not feel comfortable in expressing
themselves and their ideas with a group.
·
Some students are shy and secured and they feel awkward
when working with others.
·
Not everyone in the group will participate. Some people
relay on others to do the work for them.
·
In a group may of the hardworking students do all of the work
and the lazy students do nothing and still receive that same grade. Which is
not fair to the people in the group who did all of the work.
·
The advantages of working in a group is that members of
the group don’t cooperate well.
Difference between
Co-operative learning and Collaborative learning
|
Co-operative learning |
Collaborative Learning |
|
The teacher observes,
listens and intervenes in a group when necessary. |
The activities monitored by
the instructor. When the questions are directed towards the teacher, the
teacher guids the sudents to the information needed. |
|
More structured |
Less structured |
|
More teacher controlled |
More learner controlled |
|
Assigned roles |
No assigned roles |
|
Focus more on individual
achievement |
Focus more on collective
achievement |
|
It focuses on the product of
working together |
It focuses on the process of
working togher |
|
The teacher provides extra
materials for learners to analyse. |
The students are in charge
of obtaining the additional source of materials |
|
Students submit the work at
the end of lesson, for evaluation/ assessment. |
Students assess their own
individual and group performance. |
|
Success of the group depends
upon the efforts of everyone involved |
Success depends on
individual strength |
SEMINAR
A seminar is a small group discussion in a formal
setting with a clear agenda. A speaker presents a theme (or a set of papers)
and it is discussed by all those who participate. The chairperson of the
seminar guides, directs,co-ordinates and organizes the discussion in a
systematic way.
Seminar can be used in classroom to develop social
awareness and interaction skills in the learners. A small group of students
come together under the guidance of a teacher to study a problem. The
moderator/ chairperson selected from among the participants conducts the
seminar and every student get a chance to express his views. One of them
prepares a report, which is finalized after the discussion and incorporating
the suggestions by the teacher.
Objectives of Seminar
Ø
To help the students get an in-depth understanding of
the subject matter.
Ø
To develop the habit of tolerance and co-operation
among the students.
Ø
To help the students to overcome the problem of
stage fear.
Ø
To help in developing the ability for keen
attention and to present ideas effectively and answering questions.
Steps to be followed in conducting a Seminar
1. Select a
useful and appropriate topic.
2. Divide the
class into different groups and help them to gather data from different
sources.
3. Select a
group leader.
4. Prepare the
paper for allotted period.
5. Classify and
supplement the data collected within the group.
6. Presentation
of the seminar paper by group leaders.
7.
Discussion
8.
Preparation of seminar report
Advantages of Seminar
§
It helps the students to stimulate and
motivate in learning.
§
It develops the sense of responsibility
co-operation among the students.
§
It develops self-reliance and self-
confidence among students.
§
It develops the ability to raise relevant
and pinpointed questions.
§
It develops analytical and critical
thinking.
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