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.