Inquiry training model,inductive thinking model

Sem IV Theory Paper Details

 

EDU 13.10 PROFESSIONALIZING MATHEMATICS EDUCATION

Contact Hours: 50 (Instruction) Maximum Marks: 50 (External: 40, Internal: 10)

 

COURSE OBJECTIVES

1. To know the ways of making Mathematics enjoyable.

2. To understand the ways of catering the needs of gifted students, slow learners and under

achievers in Mathematics

3. To know various initiations to nurture Mathematics Talents

4. To be a professional mathematics teacher.

 

Unit I: Mathematics for All

• Causes for poor performance in mathematics,

• Mathematics phobia among learners- Causes and Remedies

• Role of recreational activities in mathematics learning (mathematical games, riddles, quiz,

puzzles, Sudoku etc.)

• Mathematics club- Activities, importance and organisation

• Mathematics fairs

 

Unit II:  Exceptional Children in Mathematics

• Concept of Multiple Intelligences

• Exceptional children in mathematics- Mathematically gifted, slow learners, under achievers-their characteristics; special programmes for each

• Learning difficulty in mathematics (dyscalculia)- characteristics and remedial measures

• Mathematical creativity- characteristics, Role of teacher

• Governmental and non-governmental initiatives in improving mathematics learning; Field

medal, Mathematics Olympiad, NUMATS, NTSE, MTSE.

 

Unit III|:  ICT in Improving Teaching Performance

• E-content development- concepts, formats, steps for preparation, script writing for e-content

• Using internet for accessing information, Websites for authoritative information like ERIC,

INFLIBNET

• E-resources for teaching and learning- Websites, blogs, E-books, E-journals, wikis,

discussion forums, social networking sites & Apps.

 

Unit IV : Professionalizing Mathematics Teacher

• Teaching as a profession, professional ethics in teaching, Traits of professionalism

• Qualities of a Mathematics teacher- General qualities, specific qualities, Personal qualities..

• Soft Skills for teachers

• Professional growth of Mathematics teacher. – Teaching, Research and Extension.

 

Unit V: Techno pedagogy in Mathematics Teaching

• Techno pedagogy – meaning, need and scope

• Technological Pedagogical Content Knowledge (TPACK)

• Techno-Pedagogical Skills of social Mathematics teacher

 

 

MODE OF INSTRUCTION

Lecture, assignment, small group discussion, Seminar

TASK AND ASSIGNMENTS:

1. Prepare a script for an e-content material for any topic in mathematics at secondary level

Semester 1- Unit 4 Approaches, Methods and Techniques Mathematics teaching

 

 

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)

      experiments, Research project, field trips, class discussions and films are the major activities in the constructivist classroom.

     The constructivist classroom the teacher's role is to prompt and facilitate discussion.

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

 

 

Role of 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.

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

comparison

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

• PROJECT METHOD

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

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

• Creating the situation

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

• Proposing and choosing the project

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

• Planning the project

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

• Execution of the Project

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

• Evaluation of the Project

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

• Recording of the project

             The children maintain a complete record of the project work. while recording the project some points; 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

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

Some projects for mathematics

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

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

Merits of Project method

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

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

• Identifying and defining the problem

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

• Analyzing the problem

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

• Formulating tentative hypothesis

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

• Testing the hypothesis       

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

• Verifying the result or checking the result

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

Example

Define Union of two sets  

If A={2,3,5}, B={3,5,6} , C={4,6,8,9,}

Prove that AU (BUC) =(AUB) UC

• Identifying and defining the problem

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

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

• Analyzing the problem

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

• Formulating tentative hypothesis

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

• Testing the hypothesis 

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

In the example it is given that

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

 

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

similarly (AUB) UC

 

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

• Verifying the result or checking the result

   After 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.

Demerits

• Its limitations are due to its ineffective use.
• It is difficult to organize the contents according to requirements of this method
• It is time consuming.
• All topics and all subject areas are cannot be covered by this method
• Not suit for lower classes.
• Teacher's burden becomes heavier.
• Mental activity dominates in this method.
• 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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