HOW TO TACKLE THE PROBLEM OF MATHEMATICS IN SCHOOLS

Mathematics in general is construed as a subject which is very difficult and thus students fear this subject resulting in low scores. However the fact is otherwise. In order to tackle this problem, the teaching of mathematics has to change. It should be child centred.

Firstly, the child should be made to master tables up to 19. For this quizzes may be designed by the teachers by dividing the students into two groups and asking tables randomly. A game may also be designed on this and may be played on knockout basis.

Next the child should be made to have command over addition, subtraction, multiplication and division and the BODMAS rule.

It is found that the child gets confused as to whether his/her addition, subtraction etc is correct or not? For this the method of 9’s for checking addition, subtraction and multiplication should be introduced to the students (Details are available in the accompanying CD).

The teacher should divide the students into bright, average and under-achievers. The teacher should make the seating arrangement in his class in such a way that the under-achievers should be made to occupy the first two rows in the class. Each bright child should be made to sit beside each under-achiever and the average students and the remaining bright students should be made to occupy the subsequent rows. In this way the teacher will have an eye to eye contact with his/her under-achievers. Further, the under-achievers will receive a dual guidance while a class is on- one, directly from the teacher and two, from his neighbour who is a bright student. This will certainly help him to learn the topic/subject more effectively.

The teacher while teaching the subject/topic should at no point of time say “This topic is very difficult” or, “This topic can not be learnt easily” so on and so forth as such remarks being highly negative in character, de-motivate the students in general and under-achievers in particular. Thereby they loose interest in the subject as a consequence. Rather if it is really so, then the teacher must devise ways-invent memory aids, codes etc so that the formula can be remembered and the topic may be learnt in an effective way.

The teacher must divide must divide his/her syllabus in such a way that each branch of mathematics is taught alternatively by him/her. For example, in a week the teacher may teach algebra for first two days followed by arithmetic in the next two days and geometry in the remaining days. This way the students will be in touch with all the branches of mathematics. Otherwise, if instead the teacher teaches algebra first, then arithmetic and finally geometry, then, it is every possibility that by the time the teacher finishes geometry, algebra is forgotten by most of the students particularly the under-achievers because, they will have no time to revise what is learnt earlier.

The teacher should teach limited number of concepts in a single period, say one or two only. This will help the average and under-achievers to grasp what is being taught.

The teacher should give few problems say 5 or so in the class to solve and should ensure that the notebooks of the under-achievers be corrected during the same period. Such an exercise should follow the completion of each concept. For this, the teacher should come prepared with the number of questions he/she intends to give in order to test the clarity of a particular concept. Similar problems should be given in home-work and the students be asked to complete the same during the supervision periods as far as possible. The notebooks of the under-achievers should be invariably corrected the very next day. The correction should not be only a mere tick mark to indicate the problem is correct or incorrect, but, it must be thorough. The detailed solution to the problems must be given by the teacher in the notebook(s) of the under-achievers in his/her own handwriting. This will motivate the under-achiever to scan his notebook in detail and not to repeat the mistake in future.

The students should not be rebuked, scolded or ridiculed for providing a wrong answer to a question asked. Rather, he/she should be told that why the answer provided by him/her is wrong and its correct solution. If the student provides partially correct answer, or fully correct answer, he/she should be encouraged by patting, remarks like-very good, well done etc. This will motivate the student to a great extent and encourage him/her to learn more.

The teaching of mathematics should be black-board oriented. The students should be asked to come to the black-board to solve problems frequently. While adopting this practice, care should be taken that the under-achievers are given easy problems to solve, while, the average/bright students should be asked to solve tough problems. This way the students will have much needed confidence-an important ingredient for success in mathematics.

While teaching geometry, the teacher should always use demonstration geometry box where ever needed. The teacher should emphasize the use of sharp and pointed pencils while drawing figures. The use of eraser should be as minimum as possible.

The students must be encouraged to write complete statements while writing answers. Further, they must be told to write the units along side the quantities, while writing answers.

In order to enable students to perform calculations quickly, the students must be taught the methods of calculating squares of two digit numbers, methods of calculating percentages quickly, quick multiplication and quick division etc.

For example, to calculate 10% of a number, just remove a zero or place a zero or place a decimal just before the digit in the units place. To calculate 20% first calculate 10% and then double it and so on. To multiply by 10 just place a zero after the units digit or shift the decimal point one place towards the right and so on.

While taking unit tests the teacher must prepare three sets of question papers, one for bright children, one for average children and one for under-achievers. The result of the under-achievers should not be compared with that of the bright or average children, but, should be compared with the scores of other children classified in their group or with his/her own previous score in tests. This way they will know as to how much improvement they have made without getting de-motivated.

The child who has improved must be praised in the class and given intensives in the form of token gifts. This will pay rich dividends later. The records of their score must be maintained by the teacher.

The teacher must prepare the matrix (marks-sheet question-wise), for under-achievers and analyse the same. This analysis will help him/her to know as to which is the topic the student has not been able to understand. On knowing the same, the teacher must revise the topic for them during supervision classes and re-test the students. Such a process should continue till the student has a considerable hold over the topic.

Role of Remedial Teaching:

In order to improve mathematics, effective remedial teaching is a must. Let us discuss.

Remedial teaching is not re-teaching. Any remedy however costly or sophisticated is useless unless it cures the disease.

A remedial teacher should have a mentality of a sympathetic doctor who has love and care for his/her patients (students).

A. Identification:

a) Through academic achievement:

i) Class interaction: An under-achiever will give wrong answers frequently to the questions asked. He will appear to be confused. He may probably not respond to the questions asked in the class at all.

ii) Home assignment: An under-achiever will not do the homework. If pressurised to complete the work, he may resort to copying, which may be easily detected.

i)                   Unit tests and term tests: He will show poor performance consistently in tests. He will either not attempt the question(s) at all or, will do cuttings and overwriting. He may even try to copy the solution to the problems from his peers.

b) Through behavioural aspect:

i) Attitude towards academic activities: He will be disinterested in such activities. He will try to refrain himself from such activities. He will try to avoid discussion about academics with his peers or teachers.

ii) Class escapism: He will try to bunk classes for one reason or another. He will give excuses for not attending classes.

ii)                Fiddle with notebooks instead of studying: He will be found to fiddle with notebooks and books instead of studying.

Once the under-achiever has been identified, the next step is the diagnosis of deficiencies.

B. Diagnosis of deficiencies:

a) Learning of concepts: His concept(s) related to a particular topic or formula is not clear. For example, the difference between 2×2 and (2x)2 may not be clear to him.

b) Computational Skill: He may not be good at computations and thereby may gives erroneous results frequently while performing basic arithmetical operations and simplification.

c) Procedure of solving problem: He is not clear about the procedure of solving problems and so he/she often gets wrong answers.

d) Application of knowledge: He may not be able to apply the learned knowledge in different situations. For example, in word problems, he may fail to translate sentences into equations or identify the variables.

Once, the deficiency has been diagnosed, let us explore the possible causes for the same.

C. Causes:

a) Memory: Individual capacity of memorising facts and figures.

b) Understanding: Lack of comprehension-he does not follow what he reads.

c) Presentation: Finds difficulty in expressing views-vocabulary is not sufficient.

d) Knowledge Gap: Incomplete coverage units in the previous class-long absence.

e) Parental background: Socio-economic status; education

f) Parental attitude: Indifference of parents towards studies; over-expectation.

g) School Based: Lack of suitable equipment and environment in school-overcrowded class.

h) Medium of instruction: Language problem.

i) Physical factors: Poor eyesight; poor audibility; illness and other problems.

j) Individual factors: Good in oral tests but does not prepare notes and does not do home work regularly; not sincere in studies; very anxious but is unable to concentrate on studies; lacks self confidence; inferiority feeling; fear of failure; wants company of students who avoid classes; emotional instability.

k) Teacher based: Lack of confidence in teacher; lack of time at teacher’s disposal; faulty method of teaching; does not encourage student participation in class; inadequate home assignments and problems for practice; improper way of correction of homework and of guidance to students at appropriate time and stage.; knowledge of the subject is not thorough; unable to clarify difficult concept; lacks in expression; unable to provide secure and affectionate climate in classroom and lack of understanding and acceptance for each individual child.

The causes having known let us now discuss about the possible cures and remedies.

D. Cures and Remedies:

a) Category wise remedial-not more than 5 to 10 students in each class.

b) Personal and individual attention by teacher.

c) No humiliation.

d) Special carefully devised UAA (under achiever’s assignment) – Simpler-Simple-Complex.

e) Read-Re-read-Write-Re-Write-Reproduce-Drill.

f) Group studies; group learning.

g) Micro-notes.

h) Teaching selected portion of syllabus only.

I now propose an action plan to be undertaken by a remedial teacher.

THE ACTION PLAN:

Out of two approaches of evaluation in vogue today, i.e. the process approach focusing on the performance of the teacher and the product approach focusing on the performance of the students with regard to specific objectives-here to get high score in the examinations in terms of marks and subject average, the latter is preferred for sure for obvious reasons. This process is based on the principle that what ever the teacher might have done in the class room is irrelevant unless the objective (of obtaining a high score in the examinations in terms of marks and subject average) is achieved. This then is the primary criteria of evaluation of both the teacher and the taught at all levels.

Herein lies the importance of diagnostic and remedial teaching, which is therefore, the primary responsibility of the teacher. This type of teaching involves:

i)                   Diagnosis of the specific difficulty of the student by conducting a suitable diagnostic test.

ii)                Providing suitable remedial measures

iii)              Providing ways and means for preventing them from reoccurring in future.

If a teacher is able to do justice to his primary responsibility then it may safely be presumed that the teaching profession has a bright future in store for sure.

For the benefit of teachers in general, I am now suggesting an action plan on these lines:

a)     Be an innovative and imaginative teacher with an open mind.

b)    Apply suitable diagnostic test to identify the weakness of each child.

1. For this split the topic into several subtopics. For example, a topic in class X Mathematics “Linear simultaneous equations in two variables” –solution of equations can be split as:

i)                    Adding the two equations directly to find the value of the variables.

ii)                   Changing the sign and adding the equations to find the value of one variable.

iii)                 Making coefficients equal and using i) or ii) above to find the value of the variables.

iv)                 Substituting the value of one variable in the equation to find the value of the other variable.

1. II. . Set at least 20 questions on each subtopic (They should preferably be      knowledge based)
   1. Take a test of each child. One subtopic to be tested at a time.
   2. As far as possible uniformity is to be maintained while evaluating the test.
   3. A student scoring less than 35% marks in this test is surely having difficulty in the subtopic.

c)     Explore the causes of weakness which may be:

i)                   Lack of understanding/misconceptions.

ii)                Faulty teaching method.

iii)              Fear of the subject

iv)               Bad work study habits.

v)                 Physical and emotional factors like poor health, some mental shock etc.

vi)               Teacher’s attitude.

d)     The cause(s) having been identified, suitable remedial measure (depending upon the cause) should be suggested which may be:

i)                   Re-teaching of the subtopic—should be resorted to only if the student has completely failed to understand the subtopic due to one reason or the other.

ii)                Computer Aided Teaching—should be resorted to if the student has a vague idea about the subtopic and therefore finds it difficult to answer questions relating to it.

iii)              Drilling of Problems—Should normally be prescribed to the weak child during examination times. For this the teacher should be able to design an effective study material containing objective questions, knowledge based problems; the practice/drilling of which will cure the weakness.

iv)               Other Measures:

The work of the teacher does not end here. He/She must ensure that the student continuously practices upon them to ensure that the weakness does not reoccur in future.

To conclude, it may be said that this is indeed a gigantic task with immediate rewards a remote possibility; therefore requires zeal, enthusiasm and a sense of commitment on the part of the teacher to undertake this project.

Last but not the least; the institution has to play a pivotal role to achieve the ultimate objective. The difference between supervised study (study under the supervision of a teacher) and remedial teaching be clearly understood. The supervised study time table be framed in such a way that a teacher should be assigned at least two periods a week in Maths, Science, English and Social Studies (the subjects where maximum weakness is found) as is done in JNV Longowal, Distt: Sangrur, Punjab. The teacher on his part should not just while away his/her time but should perform these activities as suggested above in letter and spirit and then and only then the ultimate objective can be achieved. He/She must remember that if a student fails then: the teacher has failed; the examination system has failed; the evaluation system has failed and by and large the education system as a whole has failed.

Let us now discuss as to how mathematics teachers can help students improve in problem solving?

A teacher can help students become oriented to problems by teaching them the meaning of the problem. He/ She is a psychology teacher. Orientation to problems depends up on a well organised body of knowledge pertaining to a problem. Individual differences exist in problem solving. He can encourage them to verbalize; diagrammatize; and construct models. The teacher should try to create a climate in the class friendly to questions. He should tell him not to persist with an unsuccessful search model. Students should be advised to abandon temporarily the attempt to solve a problem on which they have worked unsuccessfully for a long time and return to it later. Recognition should be given to the student who solves a problem in more than one way and also to a student who is able to find a particularly neat solution.

As the students test hypotheses, some of them ill-conceived, the teacher must be patient and objective least the student test the hypotheses not by trying them out rather by watching teacher’s reactions.

” A bad teacher complains,

An average teacher explains,

A good teacher inspires,

An excellent teacher motivates”

Mathematics derives its main strength from its following features:

1)      Abstractness

2)    Generalization

3)    Logical Consistency

4)    Depth

5)    Precision

6)    Seriousness

7)    Elegance

8)    Economy of thought

9)    Thoroughness

10) Significance

11)  Clarity

12) Permanence.

Now, I present some ideas if followed by teachers; students and administration will result in sure and certain improvement in teaching and learning of mathematics.

1)      Monitoring of effective teaching and learning of mathematics from class VI itself.

2)    Students should write all important formulae chapterwise/topicwise in a separate notebook and repeatedly apply them to solve problems so as to learn them by heart.

3)    Mathematics club/Mathematics Laboratory should be formed in the school to promote creative thinking.

4)    Methods such as problem solving; question-answer; discussion and demonstration should be used for teaching and learning.

5)    Proper method should be carved out to transact difficult; complex; abstract concept in a simple and concrete manner.

6)    Teacher should give only a few model solutions on every topic.

7)    Students should be made to solve problems by themselves in the class and teacher should hint or prompt only those students who need help.

8)    Students should be encouraged to participate maximum in classroom activity.

9)    Students should be encouraged to ask questions.

10)  Teaching should be made interesting and absorbing.

11)  Teachers’ interaction with students should stimulate their adventurous thoughts and divergent thinking to solve a problem in more than one possible ways.

12)  Teaching and A-V aids should be used wherever possible for teaching.

13)  Clarity of fundamentals and concepts must be ensured at every stage of learning right from class VI.

14)  Every student should solve at least 10 different problems by himself every day.

15)  Students should practice solving mathematics problems of their own for at least 2 hours every day as a rule.

16)  Students should practice enough calculations. They should acquire mastery over quick calculation techniques.

17)  Writing skills such as neatness; sped; and accuracy should be developed.

18)  Skills to read problems with comprehension and logic should be developed.

19)  Home assignments should be preferably typed on a sheet and every student should get a sheet of assignment to solve problems.

20) Students should be counselled and explained with reasons why not to copy solutions of homework questions from their friend(s) to enhance learning.

21) Teacher should correct home-assignments properly and promptly and give immediate feedback.(re-enforcement).

22) Exchange of teacher(s) having mastery over topic(s) from the neighbouring JNV’s for a few days should be done to augment instructional inputs.

23) Unit test(s) and term test(s) should be reliable.

24) Evaluation should be done honestly and immediately, clearly pin-pointing mistakes and their corrections.

25) Test should have concept based question(s) as far as possible.

26) Chapter-wise test should be taken during re-vision and proper feedback should be given for improvement.

27) Continuous Comprehensive Evaluation (CCE) should be done in letter and spirit.

28) Diagnostic remediation should be done timely and immediately after every test, simultaneously teaching and learning of next unit.

29) The Principal and teacher should together talk to the under-performer (achiever) after every test to know his/her difficulty and find workable ways to solve them to build confidence and boost morale of the child.

30) Peer group tutoring for under-achievers can be tried.

31) After every test, question-wise analysis should be done by teachers to short list common and usual mistakes done by the students and their correction should follow.

32) Syllabus should be thoroughly completed as per schedule and intensive re-vision should be taken up for enough practice and assimilation.

33) The teacher should analyse the question matrix to diagnose any flaw with his/her technique and should suitably revise the topic to teach it effectively in future.

34) Absentees in any test should be strictly checked. Any absentee in a test for any reason should be given test as early as possible.

35) Test or examination should be regarded as a compulsion for effective teaching and assimilated learning.

36) Teacher should love and care for every under-achiever; give him individual attention.

37) There should be close rapport between the teacher and every under-achiever. He/She should feel at-ease to approach the teacher to solve his/her difficulty.

38) Teachers and students should be punctual; sincere and regular and work willingly with joy of teaching and learning.

39) Teacher should never discourage any child.

40) There should be good discipline, serene and conducive academic climate pervading in the school.

Role of Reading in Mathematics :

Reading in mathematics play a significant role in improvement of the subject. Let us discuss.

In today’s competitive world, it is said; one must be well read in order to succeed. It is because, ‘it is the basic skill which opens up the window to the world of knowledge which is available in writing form’, according to-“Developing Reading Skills-Series2” a monograph published by Navodaya Vidyalaya Samiti.

At the primary level (up to class VIII), it is agreed that most of the portion consists of problem solving, but even then, importance of reading cannot be undermined. For example, let us consider the statement, “State the converse of Pythagoras theorem”. In case of casual reading, the word “casual” can be overlooked and in that case, the statement supplied shall be just the reverse of what is asked for.

At the lower level,(classes IX to XII), apart from geometry, we require reading in some portions of algebra,mensuration,probability, permutations and combinations etc, else it is too difficult to comprehend the problem, leave alone  solving them.

For example, in algebra, the problem can be-“State and Prove the Binomial Theorem for (positive integral index; any index). Now the bracketed part being important here must be carefully read as the statement and conditions of validity of the theorem differ in both cases.

In mensuration, the construction of the correct figure, on which the correctness of the formula to be applied to solve the problem is based, can only be done, when the problem is read carefully. For example, let the problem be-“Calculate the volume of the cylinder with the square base.” A casual reader will just apply ?r2h, r and h having usual meanings, which is an error. In fact the volume has to calculated using the formula (side)2 x h; where “side” stands for the side of a square.

In permutations and combinations, extreme care has to be taken to read a problem to judge whether it is a problem concerning arrangement or selection, as the former is related to “permutations” and the latter is related to “combinations”, and in both the cases, different sets of formula have to be applied altogether.

As for as probability is concerned, the correct solution of any problem requires effective reading and in fact it is the only key to success in this branch.

At the higher level, wherein a student is subjected to a feast of theorems many of them by the same mathematician in one or more branches, a thorough reading is a must or else it may lead to Confusion.

At the research level, reading has no substitute as the technique of reading the research papers only enables one to see what others do at times. Thereby one comes up with a problem to ponder on. Thus reading forms the very basis of research work.

The teachers of Mathematics in the Vidyalayas perhaps fail to take note of this aspect of the subject and train the students for problem solving only. This is to my mind one of the most powerful reasons for dismal performance in the subject.

Hence, efforts should be made by them to see this guide of the problem as well and it is assured that the performance in the subject will improve.

That the students have really improved in mathematics can only be said with certainty if it is verified. The tool for it is the examination. The marks obtained in the examination will certify the claim. Therefore, thorough preparation is a must for examination. As such, I am suggesting certain guidelines which if the examinee follows scrupulously, will surely reap rich harvest in terms of marks.

The guidelines may be divided into two parts: Those which should be followed while preparing for the examination and those which should be followed while writing the same.

Guidelines for preparation:

a)     Be a systematic and determined person to have a strong urge to succeed.

b)    Check your level of preparation periodically till last moment both topic wise and marks wise.

c)     Identify your weak areas and derive strategies to work on them.

d)     Give equal importance to all topics

e)     Do not waste your valuable time in repeating problems.

f)     Revise and re-revise the course focusing on weak areas.

g)     Collect the tricky problems and mention their trick (s) against them—frequently revisit these problems.

h)     Categories the problems so that you may have command over large number of problems in less time.

i)       Utilize each and every second in preparing yourself for the ultimate ordeal.

Guidelines for writing the Examination:-

a)     Revise each problem carefully after solving it.

b)    Attempt a problem after reading it at least twice. Check whether you have copied down the problem correctly before attempting it to solve.

c)     Attempt first your familiar and precise problems practiced thoroughly.

d)     Next, do the problems which appear solvable. If you fail to do the problem in your fist attempt, revise it at once. Do it as rough work (provided you are sure that your method is correct). In this way you will be able to complete most of the problems.

e)     Next, attempt the problems which are unknown to you write whatever you know about the problem e.g. the formula to be used, first few steps of its solution etc.. This will fetch you extra marks.

f)     Finally always keep some time for final revision and you must invariably revise the whole paper before submission. This helps you to detect minute mistakes which might have gone unnoticed in the absence of such a revision.

g)     Always maintain neatness while writing the answers as this will have a very favourable impression on the examiner.

h)     If you do proper study and still the question paper appears to be tough you must understand that the same must have been felt by al the examinees appearing in the examination. You should not therefore be nervous. What you should have to be cautious about is that no mark is deducted from your solved problems.

i)       Never resort to copying in the examination as this will have a telling below on your self-confidence which will not be in god stead for you.

Finally, you must know that the paper is set in such a way that it can be completed in 3 hours time by an above average examinee. Therefore, one must not worry unnecessarily about the myths that the paper was lengthy, it involved lot of calculations, the paper was tough etc.