THE TEACHING OF MATHEMATICS

TEACHING OF MATHEMATICS

Every teacher of mathematics needs to be informed and convinced about the educational values of his subject. His own conviction enables him to convince the students, parents and the society. These are as under:-

a)     Practical value—He cannot do without learning how to count and calculate. Any person ignorant of it is easily cheated counting, notation, addition, subtraction, multiplication, division, weighing, measuring, selling, loving are fundamental processors of mathematics having immense practical value. It has become the basis of world’s entire business and commercial system.

b)    Disciplinary value—It trains and disciplines the mind. It is exact time and point knowledge and therefore creates discipline in the mind. It develops reasoning and thinking powers more and demands less from memory. Reasoning in mathematics possesses certain characteristics which are suitable for the training of learner’s mind. These are:-

1)      Characteristic of simplicity—it teaches that definite facts are always expressed in simple language which are always easily understand.

2)    Characteristic of Accuracy—Accurate reasoning thinking and judgment are essential for its study. Accuracy, exactness and precision compose the beauty of mathematics

3)    Characteristic of certainty of results— The answer is either right or wrong. Subjectivity or difference of opinion between the teacher and the taught is missing. The student can verify his result by reverse process. It is possible for the child to remove his difficulty by self-effort and to be sure of the removal. He develops faith in self-effort which is the secret of success in life.

4)    Characteristic of originality— Most work in mathematics demands original thinking, reproduction and cramming of ideas of others is not very much appreciated. When the child has a new or a different mathematical problem, it is only his originality which keeps him going. The discovery or establishment (derivation) of a formula or conversion of formula in one form to another is also his original work. This practice in originality enables the child to face new and challenging problems with confidence.

5)    Characteristic of similarity to the reasoning of life—Clear and exact thinking is as important in daily life as in mathematics. Before starting with the solution of the problem, the student has to grasp the whole meaning. Similarly in daily life, while undertaking a task, one must have a firm grip over the situation.

6)    Characteristic of verification of results—Results can be easily verified. This gives a sense of achievement, confidence and pleasure. It inculcates the habit of self-criticism and self evaluation.

7)    Power not knowledge—  In this ever advancing society the important thing is not only to learn facts, but also to know how to learn facts. The main thing is not the acquisition of knowledge but the acquirement of the power of acquiring knowledge.

8)    Application of knowledge—Knowledge becomes real and useful only when the mind is able to apply it to the new situations. Ability to apply knowledge to new situations is inculcated in students. They acquire the power to think effectively. It generates the otherwise latent powers of thinking, reasoning, discovery and judgment of the child.

C)          Cultural Value— It is said, “Mathematics is the mirror of civilization”. It helped man to overcome difficulties in the way of his progress. The prosperity of man and his cultural advancement have depended considerably upon the advancement of mathematics. The modern civilization owes its advancements to the progress of various occupations such as agriculture; engineering, surveying, medicine, industry, navigation; road-rail building etc. and contribution of mathematics in their advancement cannot be undermined.

Therefore mathematics shapes culture as a play back pioneer. Some of the important aspects of cultural heritage have been preserved in the form of mathematical knowledge only and learning of mathematics is the only medium to pass on this heritage to the coming generations. Mathematics is also a pivot for cultural arts, such as music, sculpture, poetry and paining.

Apart from these three major values, it has other fundamental values like:

1. Social value.
2. Moral Value.
3. Aesthetic Value.
4. Intellectual Value.
5. International Value.
6. Vocational Value.

In addition to these major and fundamental values, there are few other values, which is of equal importance. These are: -

1. Development of concentration.
2. Art of economical living.
3. Power of expression.
4. Self-reliance.
5. Attitude of Discovery.
6. Understanding of popular literature.
7. Quality of hard work.

II Aims & Objectives of Teaching Mathematics:

a) Aims of Teaching Mathematics:

1. Utilitarian aim.
2. Disciplinary aim.
3. Cultural aim.
4. Adjustment aim.
5. Social aim.
6. Moral aim.
7. Aesthetic aim.
8. International aim.
9. Vocational aim.

10.  Inter-disciplinary aim.

1. Self-education aim.

12.  Educational preparation aim.

13.  Development of powers aim.

14.  Harmonious development aim.

b)    Objectives of Teaching Mathematics:

1. Knowledge and understanding objectives.
2. Skill objectives.
3. Application objectives.
4. Attitude Objectives.
5. Appreciation and interest objectives.

To illustrate these objectives let us take one particular learning unit

(a-b)2 = a2+b2-2ab.

1. Knowledge and understanding objectives:

a)     The student recalls the knowledge of algebraic multiplication and squaring.

b)    He recognizes the meanings of the formula in hand.

c)     He understands and describes the relation ship between the two sides of the above equation.

d)     He understands the relationship between this formula and formulae learnt earlier.

e)     He understands and expresses the formula in the form of a diagram and through various other substitutions.

1. Skill objectives:

a)     He can prove the formula by multiplication.

b)    He can prove its substituted versions.

c)     He can verify its accuracy by various substitutions.

d)     He can draw a diagram to represent the formula.

e)     He can establish relationship between the two formulae:

(a+b)2= a2+b2+2ab

and

(a-b)2 = a2+b2-2ab

1. Application Objectives:

a)     He can solve new problems independently by applying the formula.

b)    He can work out the geometrical proof of the formula.

c)     He can locate the life situations where the formula may be applicable.

d)     He can construct his own problems based on the formula.

1. Attitude objectives:

a)     He proves the formula through systematic steps and objective reasoning.

b)    He solves relevant problems with confidence.

c)     He develops curiosity for the use and application of the formula.

d)     He demonstrates originality and creativity.

1. Appreciation and interest objectives:

a)     He appreciates the nature of the formula.

b)    He appreciates the application of the formula in promptly solving the relevant problems.

c)     He appreciates the use of this formula in learn other topics and branches of mathematics.

d)     He appreciates the recreational value of the formula.

e)     He appreciates various diagrammatic and other versions of the formula.

f)     He develops interest for learning more and more about the formula and its applications.

III)        Methods of Teaching Mathematics:

The methods of teaching mathematics are:-

a)     Lecture Method.

b)    Dogmatic Method.

c)     Inductive—Deductive Method.

d)     Heuristic Method.

e)     Analytic—Synthetic Method.

f)     Laboratory Method.

g)     Project Method.

h)     Topical Method.

i)       Concentric Method.

j)      Problem Method.

a)     Lecture Method:-

1. Procedure— The teacher prepares his talk at home and pours it out in the class. The students sit silently, listen attentively and try to catch the point. He may not even write anything on the black board simultaneously or may not even argue a point with the listeners by cross questioning.
2. When to apply:-

a)     When the number of students in a class is very large. The teacher’s voice is heard clearly even in farthest corner of the class room. All the students are provided with an equal opportunity to listen and learn.

b)    When heavy syllabus is to be covered in a short time. The teacher can teach the topic at his own speed. He need not adjust his speed to the learning speed of the students.

Conclusion— The Method neither suits the subject nor the learner. It goes against the independent and original thinking of the learner. There is no student participation in learning process. Most of the time his face is towards the class and the back is towards the blackboard. This is defective. He should mostly face the black board.

b)    Dogmatic Method—It is based on some dogmas:-

I)                 Procedure: The rules and formulae are given to the class to cram. The teacher tells the pupils what to do, what to observe how to attempt and what to conclude. He works out the model sums on the black board and the pupils have merely to follow the patterns. The steps of the solution of a problem are brought home to the students who then follow them in minutest details. The model or pattern as presented and advocated by the teacher or the book is to be strictly adopted and imitated by the learner.

II)            When to apply:-

a)     It can be adopted with advantage at a stage when pupils are adequately advanced in mental development.

b)    At the revision stage, emphasis on rigour is most desirable and appropriate as it saves time, energy and a good deal of loose or useless thinking. It promotes skill efficiency and speed in the solution of problems.

Conclusion—  This method suits neither the child nor the subject. The mind of the student is stuffed with information and the understanding of the subject finds no place there. Therefore, if popularized, this method will cause stagnation in teaching.

c)     Inductive- Deductive Method:

I)                 Inductive Method—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.

Procedure—It may be illustrated by some examples:

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 cases. This conclusion in a good no. of cases will enable them to generalize that “the corresponding angles are equal; the alternate angles are equal”.

Ask students to construct a few triangles let them measure and sum up the angles in each case. The sum will be same in all cases. Thus they can conclude that the sum of the angles of a triangle=180.

II)            Deductive Method—It is opposite of inductive method. Here, the learner proceeds from general to particular; abstract to concrete; formula to examples.

Procedure: Immediately after announcing the topic for the day, the teacher gives relevant formula. To explain further the application of the formula to problems, he solves a number of problems on the black board. The students comes to understand, how the formula could be used or applied. Then a few problems are given to the students. They solve them on the same lines as have been explained by the teacher.

Conclusion—Inductive method is a predecessor of deductive method. Any loss of time due to the slow speed of induction can be made up through the quick and time saving process of deduction. Deduction is a process particularly suitable for a final statement, and induction is most suitable for the exploration of new fields. Probability in induction is raised to certainty in deduction. The happy combination of the two is most appropriate and desirable. There are two major parts of the process of learning of a topic viz. establishment of formula, and application of that formula. The former is the work of induction and the latter is the work of deduction. Understand it in ductively and apply it deductively.

Thus teaching should begin with induction and end in deduction.

d) Heuristic Method—Here the child is put in the place of discoverer. It involves finding out by the student by complete self-activity. The teacher is only a passive observer.

Procedure—Here the student is induced into discovering the solution of a problem all by himself. In its normal form, the teacher may guide the students to discover by framing them carefully and well-graded manner which will ultimately lead them to the discovery. Questioning has to replace telling in the class room.

When to apply— May be applied when the number of students are less as it requires individual attention to each child.

Conclusion:--At school, use of extreme form of this method is out of question. The teacher’s presence in the classroom should mean something. He is not to behave as indifferent on-lookers, but his presence is to inspire and stimulate the learners. In practice the success of this method depends on good questioning. The teacher no longer teaches, he guides. The learner no longer listens, he finds. It is in reality a scientific and psychological method of learning. He should let the child be his own teacher, and also see that his difficulties are removed in time.

e)     Analytic-Synthetic Method:- These two methods are applicable in combination.

I)                 Analytic Method—It proceeds from unknown to known ‘Analysis’ means breaking up of the problem in hand so that it ultimately gets connected with something obvious or already know. Start with what is to be found out. Then thinking further steps and possibilities which may connect the unknown with the known and find out the desired result.

Procedure:

Example: If a/b=c/d, prove that (ac-2b2 )/b= (c2-2bd)/d

The unknown part is (ac-2b2 )/b= (c2-2bd)/d is true,

if a c d – 2 b2 d = b c2 – 2 b2 d  is true,

if a c d = b c2 is true,

if a d = b c is true

that is, if a/b = c/d is true,

which is known.

II)            Synthetic Method— It is opposite of analytic meted. Here one proceeds from known to unknown. It starts with something already known and connects that with the unknown part of the statement. It is a process of putting together known bits of information to reach the point where unknown information becomes obvious and true

Procedure—Consider the above example.

The known part is a/b=c/d

Subtract 2b/c on both sides (But why and how the child should remember to subtract 2b/c and not any other quantity)

a/b – 2b/c = c/d – 2b/c

or, (ac – 2 b2)/b c = (c2 -2 b d ) / c d

or, (ac – 2 b2)/b  = (c2 -2 b d ) /  d

Conclusion— They should go together. Analysis help in understanding and synthetic helps in retaining knowledge. The teacher should realize that he may offer help for the analytic form of the solution and that he synthetic work should be left to the pupils.

f)     Laboratory Method—It is more elaborated and practical form of the inductive method. It makes the subject interesting as it combines play and activating..

The construction work in geometry is on the whole a laboratory work e.g., the drawing of a line; construction of an angle; construction of a triangle, quadrilateral, parallelogram etc. all involve the use of some equipment and so their nature is that of laboratory work.

When to apply— Especially in lower classes the introduction of this work is more essential, desirable and practical.

Conclusion— It is a difficult and lengthy method, but can prove exceedingly profitable if properly employed. This method should be a ‘must’ where circumstances favour.

The other methods which are not generally practical in schools are not discussed for obvious reasons.

General Comments:-  A number of methods of teaching have been discussed. Some of them have been recommended for use some have been disapproved and some have been recommended for use with caution. Out of all the available methods, every teacher has to make his own choice. It will have to be made in a rational way keeping in view the facilities available and the nature of the work to be done.

A good mathematics teacher should not depend on just one method, but he should try to imbibe the good qualities of all the methods and should so improve his command over them that he can even make the best out of the worst. He will keep his knowledge of all the methods up to date and will exploit their qualities to the maximum while reducing their short comings to the minimum. The teacher will keeps himself on the right side of every method. He will be a master of every method in the real sense. With the passage of time, he will evolve his own method and will formulate a teaching style of his own. His method will be his own individualized and personalized method, which is the result of his rich and varied experience in teaching. His method, an amalgam of all the known qualities, will obviously carry the stamp of his individuality.

Preferably he will adopt heuristic approach as an overall procedure in teaching. His method will be pupil-dominated method. He will give maximum opportunity of participation to the students and will impose himself the least on the class room work. Whatever method he adopts, the heuristic approach should always be made to prevail. Spoon-feeding has to be completely avoided.  The twin combinations of analytic-synthetic and inactive-deductive methods are recommended as his day-to-day methods.

The inductive-deductive combination will be more suitable in case of arithmetic and algebra whereas analytic-synthetic will find greater application in plane geometry, trigonometry and solid geometry.

The teacher need not stick to the same method always. Even the best of the methods will become monotonous with continuous use.  His method must carry in it some variety and newness in order to sustain interest. No doubt he will stick largely to his favorite method but he should occasionally introduce a new method in his method.

When he finds that the students are showing lack of interest towards the best method being adopted by him, he should bring in a different method just for the sake of change.