APPROACHES IN TEACHING MATHEMATICS

APPROACHES IN TEACHING MATHEMATICS

BEHAVIOURIST  APPROACH

          The traditional view of learning is forwarded by behaviourists. According to them, “Learning is modification of behavior 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 Pavlove (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. R.Guthrie( Contiguous conditioning),and  J.B Watson-Father of American behaviorist’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

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>>>>   Opportunity to actually behave or demonstrate learning

>>>> Students should be assessed by observing behavior

• §  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 behavior that can be observed and manipulated
• This approach has proved very useful in experiments under laboratory conditions where behavior 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 bahaviour will return after a period of time
• The behaviourist approach has been criticized for suggesting that most human behaviour is mechanical and that human behavior is simply the product of stimulus-response behaviors

      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), Construcivism can be defined as “ the odea that development of understanding requires the learner actively engage in meaning-making”.

      Constuctivism 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 ‘constrruere’, 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, dignostecian .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 yrs)

Concrete operational stage (7-11yrs)

Formal operational stage (12-15yrs)

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 costructivist learning environment are

Modeling

Coaching

Scaffolding

 Construcivist classroom environment provides opportunities for students question the material being presented and explore various topics as their interests.

The goal is produce 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 opportunities for teachers to introduce formal terms, definitions,& explanations for concept..

4.     Elaborate- Extend student’s conceptual understanding & allows them to practice skills and behaviors

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 stimulate 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”
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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

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 arous 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 become 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".