Unit 4 - Approaches, methods and techniques of teaching mathematics

APPROACHES, METHODS AND TECHNIQUES OF TEACHING MATHEMATICS

A mathematics teacher has a variety of methods, approaches and techniques available for use in teaching mathematics. The selection of a suitable method depends upon the objectives of the lesson, needs of the learner and nature of the content.

APPROACHES OF TEACHING MATHEMATICS

BEHAVIOURIST APPROACH

The approach which describes learning as a connection between stimulus and response is the behaviouristic school of thought. This approach based on the idea that behavior can be controlled and modified based on the consequences of a behavior.this approach to learning emphasizes that behaviours begins with reflexes.

The major tenets of the behaviouristic approach

·         Learning brings about change in behavior

·         Learning is stimulus-response

·         The resultant behavioural change are objectively observable.

Characteristics

·         Behaviorist believe in the objective study of behavior

·         Product oriented approach

·         Learner receives knowledge

·         Teacher centred approach

·         Child is an empty vessel

·         Emphasis repetition

·         Child is only a listener

·         Rigid classroom atmosphere

·         Reward, punishment and reinforcement make learning effective

·         Teacher as commander

·         The chief method of leaning is conditioning.

PROBLEM BASED LEARNING

Problem-based learning (PBL) is a student-centered approach in which students learn about a subject by working in groups to solve an open-ended problem. This problem is what drives the motivation and the learning. 

Why Use Problem-Based Learning?

• Working in teams.
• Managing projects and holding leadership roles.
• Oral and written communication.
• Self-awareness and evaluation of group processes.
• Working independently.
• Critical thinking and analysis.
• Explaining concepts.
• Self-directed learning.
• Applying course content to real-world examples.
• Researching and information literacy.
• Problem solving across disciplines.

Considerations for Using Problem-Based Learning

Students generally must:

• Examine and define the problem.
• Explore what they already know about underlying issues related to it.
• Determine what they need to learn and where they can acquire the information and tools necessary to solve the problem.
• Evaluate possible ways to solve the problem.
• Solve the problem.
• Report on their findings.

 

Advantages of Problem-based learning

·         Enhance student-centred learning

·         Upholds lifelong learning

·         Prominence on comprehension not facts

·         In-depth learning and constructivist approach

·         Augments self-learning

·         Reinforces interpersonal skills and teamwork

·         Self-motivated attitude

·         Enriches the teacher-student relationship

·         Higher level of learning

 

CONSTRUCTIVIST APPROACH

Constructivist believed that young children learn by doing, constructing knowledge from experiences rather than from adults telling them about their world. Constructivism assumes that knowledge cannot be transmitted to learner but is constructed by him/her. It is constructed by the learner on the basis of experiences. In the process of knowledge generation new experiences, talking to others and reflective thinking could be helpful.

Characteristics of constructivist approach

·         Constructivist teachers invites students for questions and ideas

·         Process oriented approach

·         Child centerd approach

·         Learner constructs knowledge

·         Teacher as a facilitator

·         Child is active participant

·         Accept and encourage student’s invented ideas

·         Scaffolding makes learning smooth

·         Flexible classroom atmosphere

·         Encourage student’s leadership, cooperation, seeking information and the presentation of the idea.

·         Teachers modifies their instructional strategies based upon student’s thought, experience and interest.

·         Encourage free discussions for new ideas

·         The teacher help the students to test their ideas.

·         Encourage the students to challenge the concepts and ideas of others.

·         Teachers may use cooperative teaching strategies

 Role of teacher

·         He/she should have a clear idea about learners’ previous knowledge.

·         Teacher can use appropriate strategies to assess the previous knowledge since this will be very important for designing suitable activities for working.

·         Teacher should provide assistance or support in the process of knowledge development.

·         Teacher should be act as a scaffolder

·         Stimulate knowledge construction.

·         The important task of a teacher is to design appropriate activities

 

HEURISTIC APPROACH

The term heuristic derived from the Greek word “Heurisco” which means I find or finding of oneself. Prof. Henry Edward Armstrong dept. of chemistry Imperial College London was introduced this method. In this method child put in the place of discover. The method involves finding out by the students, instead of merely telling of everything by the teacher.

“This is the method of teaching which places the pupil as far as possible in the attitude of discover”- Prof. Henry Edward Armstrong

Objectives of Heuristic method

·         Make the student a researcher

·         Develops the habit of enquiry

·         Prepare the child for self-learning

·         Inculcate the spirit of scientific attitude

Principles of Heuristic method

·         Principles of activity

·         Principle of logical thinking

·         Principle of proceeding from known to unknown

·         Principle of purposeful experience

·         Principle of self thinking and self study

Essential conditions

·         Freedom should be given to the students

·         Providing responsive environment

·         Teacher should entirely in the background

·         Teacher made arguments should not be forced on him

·         Teacher should act like a guide

·         Provide supplementary materials

·         Encourage students to continue heuristic questions

METHODS OF TEACHING MATHEMATICS

“Method refers to the way of delivering knowledge and transmitting mathematical skill by the teacher to his pupils”.

v  INDUCTIVE-DEDUCTIVE METHOD

Inductive method

Inductive method advocated by Pestalozzi & Francis bacon. It is based on induction. Induction means providing universal truth or theorem by showing that if it is true of any particular case, it is true in the next case in the same serial order and hence true for any such cases.

Steps in Inductive method

§  Presentation of concrete examples

§  Observation of cases under given conditions

§  Finding common relations

§  Generalization

§  Verification

Example : Angle sum property in a triangle

The sum of the angles in a triangle is 180

When a student measures the angles of several triangles and finds that in each case the some of the angles approximates to 180, he has the background to generalize that some of the interior angles in a triangle is 180.

Merits of inductive method

§  It helps understanding

§  It is a logical method

§  It develops critical thinking

§  It encourages active participation of the students in learning

§  It provides ample opportunities for exploration and observation

§  It facilitates meaningful learning

§  It enhances self confidence

Demerits of inductive method

§  This method is very slow, lengthy and laborious method

§  This is not applicable all topics in mathematics

§  Only an experienced and able teacher can handle this, method

§  It is not suitable for mathematically gifted students as unnecessary details and too many examples make the teaching dull and boring

  Deductive method

It is opposite of inductive method, it is a method of explanation of a non-principle and verification of the result. Here there is no scope for investigation. In the deductive method rule, generalization and principles are provided to the students and then they are asked to verify them with the help of particular examples

Steps in deductive method

§  Clear recognition of the problem

§  Search for a tentative hypothesis

§  Formulation of a tentative hypothesis

§  Verification

Merits of deductive method

§  This method is very economical

§  It is a short and elegant method

§  It saves time and labour for both the teacher and the student

§  It helps to develop speed, skill and efficiency and solving problem

§  It is useful for higher classes

§  It is useful for advanced study

Demerits of deductive method

§  It is not a scientific method

§  It is not suitable for beginners

§  It encourages rote memory therefore knowledge gained by this method is not long lasting

§  It fails to develop self confidence among students

§  It does not encourage student’s involvement in learning

 

Inductive method	Deductive method
Based on inductive reasoning Proceeds from Particular → General Concrete → abstract A psychological method A method of discovery that stimulates intellectual powers Emphasis on reasoning Encourages meaningful learning Most suitable for initial stages of learning Suitable for lower classes Lengthy-time consuming and laborious	Based on deductive reasoning Proceeds from General →  Particular Abstract →  Concrete An unpsychological method Method of presentation that does not develop originality and creativity. Emphasis on memory Encourages rote learning Suitable for practice and application   Most suitable for higher classes Short-concise and elegant.

 

v  ANALYTIC-SYNTHETIC METHOD

Analytic and synthetic method is a combination of analytic method and synthetic method. In the beginning we use analytic method to find solutions and then we use synthetic method for presentation of the discovered solutions.

Analytic method

The word analytic is derived from analysis which means to separate or breaking up things that are together. Analysis refers to the ability to break down material into its component parts so that its organizational structure may be understood.

Merits of analytic method

·         It leaves no doubts in the minds of the students as every step is justified

·         It is a psychological method

·         It is based on heuristic approach

·         It develops scientific attitude

·         Knowledge gained by this method is long lasting and solid.

·         It develops thinking and reasoning power among the child.

Demerits of analytic method

·         This method is very slow and lengthy

·         It is not suitable to acquire speed and efficiency

·         This is not applicable for all topics in mathematics

·         In this method information is not presented in a well-organized manner.

Synthetic method

The word synthetic is derived from the word synthesis which means to combine together. It reverse of the analytic method. In this method we start with something already known and connect it with the unknown part of the statement.

Merits of synthetic method

·         This is very short method

·         It glorifies the memory of the child

·         This method is very suitable for slow learners

·         Teachers work is simplified

·         It omits the trails and errors like in analytic method

Demerits of synthetic method

·         In this method there is no scope of discovery and enquiry

·         It creates many doubts in the mind of the learner

·         It leads to rote memory

·         This method does not give full satisfaction to the learner

·         This method is not suitable to the learner and the subject.

Analytic method	Synthetic method
Analysis means breaking up into components. Leads from un known to known A method of discovery and thought   Lengthy, laborious and time consuming Encourages meaningful learning Is formational A psychological method Application of inductive reasoning Process of thinking	Synthesis means combining the elements to get something new. Leads from known to unknown A method for the presentation of discovered facts. Short, concise and elegant Encourages rote learning Is informational A logical method Application of deductive reasoning Product of thinking

 

v  PROJECT METHOD

The project method is the outcome of the pragmatic educational philosophy of John Dewey. This method was developed and applied practically by Dr. William Head Kilpatrick of Columbia University.

“A project is a whole hearted purposeful activity proceeding in a social environment”- Kilpatrick

Types of projects

Individual project & Group projects.

Principles of project method

·         The principle of purpose

·         The principle of activity

·         The principle of experience

·         The principle of  social experience

·         The principle of reality

·         The principle of freedom

·         The principle of utility

Steps in a project method

·         Creating the situation

·         Choosing and purposing

·         Planning

·         Executing the plan

·         Judging

·         Recording

Merits of project method

·         It is based on sound psychological principles and laws of teaching

·         It uphold the dignity of labor

·         It develops self-confidence and self-discipline among the student

·         In this method the children are active participants in the learning task

·         It provides freedom

·         It develops discovery attitude in the child

·         Solves the problem of indiscipline

·         It is based on the principle of individual differences

Demerits of project method

·          This method is uneconomical in terms of time and is not possible to fit into the regular timetable

·         It does not provide any training in mathematical thinking and reasoning

·         Syllabus cannot be completed on time using this method

·         It is expensive method

·         It neglect intellectual work

·         It requires more time

Role of teacher in project method

·         Guide students in selecting the project according to their interest, aptitude and ability

·         Help in creating a friendly and democratic atmosphere in the classroom promoting co-operation and harmony.

·         Help students in planning and allotting activities to each member according to the nature of abilities.

·         Be available to the students and willing to help as and when it is necessary.

·         Check the records maintained by the students

·         Help in the periodic assessment of the progress of the project.

 

v  LABORATORY METHOD

Laboratory method is a procedure for stimulating the activities of the students and to encourage them to make discoveries. It is the experimental portion of the inductive method or the practical form of the heuristic method.

Merits of laboratory method

·         It is based on psychological laws of learning

·         It is based on the principle of learning by doing

·         It stimulates the interest of the students to work with concrete material

·         It promotes self-confidence and self-reliance

·         It provides opportunities for social interaction and cooperation among the students

·         It develops the child a habit of scientific enquiry and investigation.

Demerits of laboratory method

·         It is a very expensive method

·         A lot of time is wasted in conducting experiments

·         This method is not suitable for all topic

·         Individual attention is not possible when number of students is large

·         Only an efficient and talented teachers can handle this method effectively.

 

v  PROBLEM SOLVING METHOD

This method is one which involves the use of the process of problem solving or reflective thinking or reasoning. As the name indicates, begins with the statement of a problem that challenges the students to find a solution.

“Problem solving is a set of events in which human being was rules to achieve some goals”- Gagne

Steps in problem solving

·         Identifying and defining the problem

·         Analyzing the problem

·         Formulating tentative hypothesis

·         Testing the hypothesis

·         Verification of the result

Characteristics of a good problem

·         The problem should be meaningful, interesting and practical

·         It should be well defined

·         It should have some educational value

·         It should have correlated with other subjects

·         The problem should be real

·         It should develop mathematical skills

·         It should develop imagination and critical powers

·         It should be neither too difficult nor too easy

Approaches and techniques to problem solving

·         Inductive and deductive approach

·         Analytic and synthetic approach

·         Method of analogy

Merits of problem solving method

·         It provides a real life experience to the children

·         It  develops in pupils good habits of planning, thinking, reasoning and independent work

·         It develops initiative and self-responsibility among the students

·         It takes into account individual differences

·         It helps the students to develop reflective thinking

·         It builds a mental attitude for effective learning based on critical thinking

TECHNIQUES OF TEACHING MATHEMATICS

Questioning

Questioning is a powerful technique of teaching mathematics. Through this technique, the teacher transacts a lot of learning experiences. The teacher ask questions and the responses given by the students are strengthened and elaborated.

Characteristics of questioning

·         Exploring the entering behaviors of the learners.

·         Provide motivation and encouragement

·         Makes them active in the classroom

·         Provides the direction to the mental process of learning

·         Revise the teaching content

·         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 questions should be lengthy and vague

·         The questions should be relevant to the topic

·         The questions should be interesting

Purpose of questioning

·         To test the previous knowledge of the students

·         To enable them to recall and recognize something

·         To enable them to reason about something

·         To keep the children mentally alert

·         To stimulate the curiosity of the students

·         To  revise the lesson

BRAIN STORMING

Brainstorming is a democratic and problem centred technique. It is based upon the assumption that a child can learn better in a group rather than in individual study. In this technique, the content is largely determined by the teacher. This technique encourages the creativity and originality among the children.

 

 

Steps in brainstorming

1.      Presentation of problem

2.      Provide relevant information

3.      Record the ideas put forth by the participants

4.      Combine similar ideas

5.      Evaluate each idea or solution

Advantages of brain storming

·         It helps to achieve the higher order of cognitive and effective objectives

·         It is a democratic technique of teaching

·         It is highly motivating

·         It makes classroom interaction more effective

·         It develops problem solving ability

·         It develops creativity among 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 at home. It is the work assigned to the student.

Purpose of assignment in mathematics

·         To develop a habit of self-study among the students

·         To give them opportunity to do practice what is done in the school

·         To develop the habit of reading regularly among the students

·         To develop problem solving skill

·         To create interest in mathematics

·         To provide opportunities to the students to utilize their leisure time.

Characteristics of a good assignment

·         It should be related to subject matter under study

·         It should be clear and definite

·         It should remove the difficulties of the children

·         It should give proper reinforcement to the children

·         It should stimulate thought

·         It should be motivating and interesting