Methods of Teaching Mathematics: Inductive-Deductive Method

Inductive method:

 

Inductive method is based on the process of induction or method of development. It leads from known to unknown, simple to complex, easy to difficult, concrete to abstract, particular to general, definite to indefinite, empirical to rational, psychological to logical, parts to whole, near to far, analysis to synthesis, actual to representative etc. In this method, we first take a few examples and then generalize. Thus, it is a method of constructing a formula or generalization with the help of a sufficient number of concrete examples. Induction means to provide a universal truth by showing that if it is true for a particular case, it is true for all such similar cases. A formula or generalization is thus arrived at through a convincing process of reasoning and solving problems.

The specific elements of this method are as follows:

 

a.                       Observation

b.                      Collection of facts

c.                       Experiment

d.                      Generalization

e.                       Verification

 

Process of inductive method:

 

a.                       Observation of the given material

b.                      Discrimination and analysis of differences and similarities

c.                       Classification

d.                      Abstraction and generalization

  e.                       Application after verification    

Example: 1

If we wish to frame the formula: (a-b)² = a²- 2ab +b²

Let the students actually multiply (a-b) × (a-b) and find out the product. They may then be asked to find the answer for (p-q) ², (1-m) ² etc. by actual multiplication. After this, they are asked to observe results and be helped to make a generalization to get the required formula.

Example: 2

 

Let X₁, X₂, X₃.......... X_n be the scores

 

and the N₁, N₂, N₃,........... N_n the number of scores in a series or group.

Calculate the Arithmetic mean of the following scores: 85, 70, 10, 75, 500, 8, 42, 250, 40, 36.

Σ X = 85+ 70+ 10+ 75+ 500+ 8+ 42+ 250+ 40+ 36 = 1116 Σ N = 10

Therefore, 𝑴𝒆𝒂𝒏 =    𝚺 𝐗   =        𝟏𝟏𝟏𝟔      = 111.6

  𝚺 𝐍                   𝟏𝟎

 

Merits:

 

1.     It is easy to understand an economic/mathematics principle/theory which is established through a number of simple examples. The doubts about how and why of a formula/theory can be clarified in the beginning.

2.     It gives the opportunity of active participation to the students in the discovery of a formula.

3.     It is based on actual observation, thinking and experimentation.

4.     It reduces dependence on memorization and homework.

5.     When a new rule is to be taught, inductive approach is the best.

6.     It gives freedom from doubt and helps in understanding.

7.     It is a psychological method and, in this method,, the interest of the student is sustained till the end.

8.     It is a natural method of making discoveries, majority of discoveries have been made inductively.

9.     Since it is a logical method, it suits the teaching of economics/mathematics.

10.  This method is found to be suitable in the beginning stages. All teaching in economics is inductive in the beginning.

11.  It increases the teacher-pupil interaction.

12.  It promotes self and permanent learning among the students.

13.  It promotes curiosity and interest among students.

14.  It discourages cramming of learning material by students.

 

Demerits:

 

1.     It is limited in range and it is not suitable for all topics. It contains the processes of discovering the formula with the help of a sufficient number of cases, but ‘what next’ is not provided in it. The discovery of a formula does not complete the study of the topic. A lot of supplementary work and practice is needed to fix the topic in the minds of the learners. Certain complex and complicated formulae cannot be generalized in this manner.

2.     It is not absolutely conclusive. Three or four cases are picked up to generalize an observation. Therefore, the process established a certain degree of probability which can of course, be increased and made more valid by increasing the number of cases, but still may not hold good in all cases.

3.     It is likely to be laborious and time-consuming.

4.     At the advanced stage, it is not so useful as some of the unnecessary details and explanations may become dull and boring.

5.     Sometimes the students make wrong generalization because of insufficient data.

 

Conclusion: Its application has to be restricted and confined to the understanding of rules in the early stages. This method is quite useful in lower classes where certain simple generalizations are to be made. At the advanced stage of research, it is useful.

Deductive method:

It is the opposite of the inductive method. It is based on deduction. In this method, we proceed from general to particular and from abstract to concrete etc. In this method, the rules are given at the very onset. The students are asked to apply these rules to solve more problems. In this method, the formula is accepted by the students as pre-established and well-established truth.

Example: 1

 

𝐒𝐢𝐦𝐩𝐥𝐞 𝐈𝐧𝐭𝐞𝐫𝐞𝐬𝐭, 𝐒𝐈 =    (𝐏𝐫𝐢𝐧𝐜𝐢𝐩𝐥𝐞 (𝐏) 𝐱 𝐑𝐚𝐭𝐞 (𝐑) 𝐱 𝐓𝐢𝐦𝐞 (𝐓)) ÷𝟏𝟎𝟎

 

 Merits:

1.                      1.       This method is short and time-saving.

2.       It glorifies memory, as the students have to memorize a considerable number of formulae and definitions.

3.       If we combine with the inductive method, it removes the incompleteness and inadequacy of the deductive method.

4.       It enhances speed and accuracy (efficiency) in solving problems.

5.       This method suits all types of students.

6.       This method is suitable for many topics of economics/mathematics.

7.       This method provides sufficient practice in the application of various formulae and rules in economics.

Demerits:

 

1.     It is very difficult for a beginner to understand an abstract formula if it is not preceded by a number of concrete instances.

2.     Pure deductive work requires a formula for every type of problem and an extensive use of this method will demand the blind memorization of a large number of formulae. It will thus cause an unnecessary and heavy burden on students.

3.     Here memory is more important than understanding and intelligence.

4.     If the pupils forget the memorized formula which is very likely in cramming, s/he cannot recollect and reconstruct the formula easily.

5.     The students cannot become active learners.

6.     It is not suitable for the development of thinking, reasoning and discovery.

7.     It is not a psychological method.

 

Combination of inductive and deductive method:

 

The two approaches, inductive and deductive, aim at establishing the validity of the thought process. Deduction can only give us formal validity because the rule is taken for granted. This may be misleading if the general statement is wrong. It is only induction which tests the material validity i.e. whether the application of deduction is actually real or not. Thus induction must supplement deduction to complete the thought process. Induction

is to be the forerunner or predecessor of deduction. The deduction will give a good follow- up; if the understanding is earlier obtained through induction. The loss of time, due to the slow speed of induction, can be covered up through a quick and time-saving process of deduction. There may be a number of arguments against deduction but it cannot be driven out of teaching. It serves as the complement of induction. Induction leaves the learner at a point where s/he cannot stop; the after-work has to be completed by deduction. The two methods complement each other. Deduction is a process particularly suitable for a final statement and induction is most suitable for the exploration of new fields.

The modern teaching always starts with induction, leads to deduction, where the knowledge learnt is verified and then ends in induction where the knowledge is applied to further examples. It should be Induction → Deduction → Induction. There is no question of ‘either-or’. Both are required and work well with average students and teachers. 

Unit test, Half yearly examination, Annual examination & Semester System

Unit test, Half yearly examination, Annual examination & Semester System

 Unit Test:

Teaching and testing are the integral part of educational system. Testing is implicit in teaching some of the stages, which may be properly marked for testing procedures.

1.      During teaching.

2.      At the end of teaching a daily lesson.

3.      At the end of teaching a unit.

   A test at the end of a teaching unit is known as the unit test.

Usually the test / Examinations are held based on the entire syllabus. A unit test is not a random assessment of questions. It is preplanned, systematic and scientific test.

A unit test is a test which is constructed, administered and assessed by a teacher after teaching a particular unit to the students.

Characteristics of Unit test:

1.      Unit test is an Evaluation tool for measurement of pupils and knowledge achievement and to improve by giving feed  back.

2.      It is based on a single unit.

3.      It is administered at the completions of the unit.

4.      It is clear from the above evidence that a unit testing is an integral part of teaching process. Unit testing makes instruction a dynamic process through continuous evaluation and regular feed back.

Half yearly examination:

As per the latest terms, the two periodic tests or unit tests conducted during the course of the year carry 10 percent weightage each. The half-yearly exams carry 30 percent weightage. And the remaining 50 percent weightage is carried by the Session ending examinations.

Annual examination:

A final examination, annual, exam, final interview, or simply final, is a test given to students at the end of a course of study or training. Although the term can be used in the context of physical training, it most often occurs in the academic world. Most high schools, colleges, and universities run final exams at the end of a particular academic term, typically a quarter or semester, or more traditionally at the end of a complete degree course.

The purpose of the test is to make a final review of the topics covered and assessment of each student's knowledge of the subject. A final is technically just a greater form of a "unit test". They have the same purpose, finals are simply larger. Not all courses or curricula culminate in a final exam; instructors may assign a term paper or final project in some courses. The weighting of the final exam also varies. It may be the largest or only factor in the student's course grade; in other cases, it may carry the same weight as a midterm exam, or the student may be exempted. Not all finals need be cumulative, however, as some simply cover the material presented since the last exam. For example, a microbiology course might only cover fungi and parasites on the final exam if this were the policy of the professor, and all other subjects presented in the course would then not be tested on the final exam

Semester System:  

It  is divided into two major parts-school education and college education. The school education consists of elementary, secondary and senior secondary stages. The college education also consists of three stages-graduation, post graduation and research degree level. First public examination is conducted at the end of secondary stage (class X), second at the end of senior secondary education (class XII), and after this examination are held every year. After one year of instructions a public examination is conducted by the institution. Every student has to appear in an examination after one year or academic session. An academic session is the duration of teaching, approximately equivalent to one year between two successive annual examination.

Importance of semester system:

Carrying out the teaching work more regularly and comprehensively by using interdisciplinary approach, mutual  corporation and new instructional strategies.  Making the syllabus more broad-based with main subjects including core and elective subjects.

 Providing greater variety of subject area to suit the intellectual, motivational requirements of the students.

 Reducing unnecessary stress and strain before and during teaching and examination period and making learning purposeful, pleasant and joyful activity.

  Making teaching-learning more broad-based by including class discussion, tutorials, assignments and library work.

 Introducing a system that can receive continuous feedback for improvement of learning experience so as to make them more effective. The implementation of semester system

In the traditional system, the whole educative process is aimed at fulfilling the requirements of examination and teacher has no time for development activities due to pressing demand of completing course. The system provides for introduction of greater flexibility in the curriculum and students have variety of courses open to them. This strengthens their knowledge base and provides flexibility in making future educational and vocational choices.

The teaching-learning process is constantly monitored by teachers through tests, assignments, classroom discussion and personal interaction with the students.

 The concept of credit hours is more  good thing to engage students in learning.

 Students develop the habit of regular study sense of punctuality and work ethics.

 Semester system requires a flexible and broad based curriculum to allow interdisciplinary approach of teaching and learning. The teachers are not in the habit of taking pain.

The teaching schedule will have to be adjusted to suit the requirements of flexibility of courses. Teaching not only requires well planned time-table, but also expert teachers capable of organizing such programmes.