Mathematics: A Teacher's Challenge*

PREAMBLE

Quality, in educational parlance stands for that whole child who emerges as a fulsome man* capable of weighing each situation and forming correct decisions without losing sight of societal values. This requires along with analytical and diagnostic abilities an irrevocable conviction about right or wrong. Mathematics is a discipline that has full potential to aid growth in this direction.

But yet, "Mathematics is not everybody's cup of tea", is a majority opinion. It is perhaps that, garbed in nobility as the Queen of Sciences, it has insulated itself thus. According its Royal Status, a few of its exponents, nimble-witted as they are, have succeeded in inspiring awe for the subject in the lay man. But what one should realize is that wise men have held easy accessibility a royal virtue as well.

Here we would endeavor to address a two-pronged issue:

1. What makes Mathematics a rare talent?

2. How do we master these "what's" to make it everybody's cup of tea?

To explore the first aspect is to delve into the very origins of this discipline that would unfold its scope as well as clear the myth that surrounds it.

To evolve strategies for the second aspect is to understand what educationists of the formal schooling should do and could achieve.

SCOPE OF MATHEMATICS

It would not be an exaggeration if we say, "God made the world for man* and man made Mathematics for the world". Bestowed with that wonderful contraption called the brtain, endowed with unique sixth sense that could see beyond the finite, perceive beyond the senses, he gave expression to every message he received as nature unfolded itself around him.

By inventing Mathematics, he was able to effectively communicate every worldly phenomenon to his fellow beings. What started with numbers for micro-level needs of counting, extended to measurement of vast stretches of land adding to the scope of the new discipline. It did not stop with earthly connections but literally reached astronomical heights while exploring into the heavenly bodies. In this process, a unique language evolved that correlated to ordinary language leading on to higher-order communicative skills. With the fusion of language components, the expansion knew no bounds. The domain of logical reasoning established a huge network that enabled the discipline to wield an infinite reach. There is not a single discipline where Mathematics has not made its entry. Be it Social Sciences, or Science, be it Music or Art, Medicine or Engineering, Astronomy or Astrology, Mathematics is inseparably intertwined with them all. No wonder it is the Queen of Sciences. One could even call it the King of Humanity if one were to consider its far-reaching tentacles.

WHAT MAKES THE SUBJECT DIFFICULT IN A CLASSROOM

Thus, the finished product "Mathematics" is awesome, is as complex as the path it has traversed in the process of its evolution, as complex as the human brains that had contributed to it. And yet, it belies a simplicity not obvious to a cursory inspection. If one were to look at a Thermal Power Station, it is so complex. But it works on a very simple concept!

Many a time, a teacher in a classroom introduces the student directly to the finished product without taking him through the logical order of its evolution. This makes the teaching an incoherent, disjointed effort, the outcome of which is tantamount to unreasonable dumping of information. But human brain prefers to receive and store organized information. Those few who are self-motivated and who chew the cud, organize information randomly received later to a digestible form and assimilate. But majority do not work that way. Thus, progressing through the vertical ladder of schooling, the student has to contend with accumulated ignorance. It is like a multi-storied building with shaky foundations.

WHAT IS THE REMEDY?

To prescribe a remedy, the first step is to diagnose and identify the factors that contribute to this malady. This is possible if we ponder over the attributes that the discipline has gained during its developmental process. Broadly, these can be classified as:

• Every concept has arisen out of day-to-day needs. Thus, it is down to earth and has direct relevance to the physical world. IF the introduction of a concept is not need-based, the interest of the child dwindles. The concept takes a backseat in the process of understanding and comprehension. This has an adverse effect on the child's ability to correlate it with similar situations or see dissimilarities in very close situations rendering his/her application skills very low or almost nil.

• In giving the discipline a formal structure, a unique language has emerged which has its basic roots in the normal human language but yet has woven the intricacies and nuances of the logical reasoning that had made the mathematical language most precise, communicating more than what meets the normal eye. A good mathematician is one who masters this language requiring analytical skills of critical thinking and interpretation. But unfortunately, the message is yet to reach the large number of Mathematics teachers who believe in remaining taciturn in the classroom letting only the chalk talk on the board. They defend this under the adage, "Mathematics has to be learnt by doing". The word "doing" is confused with writing down the solution whereas it stands for the actual doing of the concept to get hands-on experience. This has led to two basic faults in the teaching learning process:

> It has reduced mathematics to a lifeless collection of symbols and solutions to

problems.

> It has killed effective communication of one’s experience in realizing a concept.

WHAT MAKES ‘GOOD TEACHING’ IN MATHEMATICS?

A teacher is said to have taught only when the student has learnt. This is the essence of child centered pedagogy. ‘Learning’ of a concept is a process which has a number of experiences that culminates into the product which is ‘realization’ of the concept. The word ‘Realization’ means feeling one with the concept that it becomes a part and parcel of one’s being and results in total internalization. The state of realization automatically ensures retention and continuity and hence permanence; recognition and application are also its byproducts.

A TEACHER’S STRATEGY

A teacher is a facilitator who provides the various learning experiences in a logical sequence never putting the cart before the horse. The logical sequence in a mathematical concept involves dividing the concept into its component stages and placing them in perspective. Each stage will be a value addition:

1. The first stage is to create need – placing the child in the actual environment or in a simulated one that generates the need. The environment’s suitability is to be concomitant with the child’s existing experience.

2. Then gradually to lead on to every value addition from previous experience till the culmination point.

3. To sum up all these stages and provide a holistic view.

4. To reinforce the experiences through similar but different situations during the problem solving.

The following pedagogy is evolved:

I. Teaching/Learning

1. Known to the Unknown.

2. Simple to Complex.

3. Concrete to Abstract.

4. The algorithm of problem-solving that would accord the power of

i. Observation,

ii. Correlation and establishing connection,

iii. Pattern formation,

iv. Induction and inference,

v. Stating the hypothesis,

vi. The mathematical parlance,

vii. Application and invention.

II. Evaluation

Has the following steps:

1. Application of the concept in its perspectives during the formative phase,

2. Addressing those areas which fail to meet expected standards after assessment of performance in (I.i) and enriching with new angles that may emerge and omitted during planning inadvertently or otherwise.

3. A summative evaluation for a holistic appraisal.

ILLUSTRATIVE EXAMPLES FOR THE STRATEGIES

I. Teaching/Learning

1. Known to Unknown - There are two characteristics of this technique:

a. Proceeding from known to unknown determines the learning stages for any concept.

b. It leaves open hooks for future expansion into related/derived concepts.

For instance, let us consider the topic “Formation of Numbers and Numerals” in Class I. It comprises of natural numbers [1 - 99] and assigning numerals.

Simple as it may seem, in general very few teachers emphasize the distinction of the numbers as concept as against the numerals which are just symbols and have only that much sanctity. Often the numerals are taught as the numbers themselves and the emphasis is confined to the formations and the sequence.

The concept of numbers is not something that a child can fully comprehend in class I itself. It takes lot of mature philosophy that life’s experience can give. This is precisely why mathematicians took so long to invent zero.

However the teacher can strive to take the child through a logical path. The need for counting arose when man witnessed larger vs smaller groups of objects. A child who enters the school has experienced only groups of several objects and has developed the faculty of identifying and grouping (sorting) like objects. So the process of isolating single typical unit from groups of like objects and associating the concept ‘one’ item, like one pen, one pencil etc. is the first level. The very formation of two to nine is mechanical but unfolds a pattern that defines an operation addition which forms the basis of all operations. These facts should be emphasized when the addition operation is introduced in later stages. This illustrates 2nd characteristic of this technique mentioned above. The logic behind the rest of the stages is for the teacher to work out in a similar vein.

The stages of learning in this are:

i. Generation of need.

ii. The number one and its symbol: 1.

iii. The numbers two – nine their symbols: 2, 3 ... 9.

iv. The number zero and its symbol: 0.

v. Number ten and place value.

vi. One ten, two tens, three tens … nine tens and their symbols: 10, 20 ... 90.

viii. Numbers eleven, twelve ... ninety-nine and symbols: 11, 12 ... 99.

2. Concrete to Abstract -

Just as idol worship helps to concretize the abstract concept of God and facilitates focus, concrete experience helps conceptualization. The sensory organs experience the concrete. The mind applies on these and perceives the abstract. This technique is most suitable to the middle classes VI – VIII wherein a child is in the transitional stage entering into their teens (adolescence). Psychologically this adolescence is attributed with development of reasoning, critical and logical thinking and comprehension of the physical through the medium of language.

This is very useful in verifying and forming conclusions in respect of geometrical theorems, algebraic identities, and derivations of formulae in Mensuration etc.

The term verification is limited in nature. A child may verify for a given triangle that the sum of the three angles is 180 degree. But to conclude the same, he has to exhaust all the triangles. He has to justify that there is no triangle where this is not true.

The teacher’s task is therefore to provide infinity of experiences to convince the child about the truth of the statement.

How does the teacher achieve this?

The first step is to involve a very large group of students to verify in a random situation. The group is then allowed to observe all the outcomes of the entire group and observe the pattern. Gradually the mind adjusts intuitively towards being convinced of the phenomenon in all situations.

Is it possible to create indefinitely many situations in a classroom? Luckily, technology can step in. A computer can achieve what a human being is unable to achieve. For example, consider the formulae for the area of circle understood as under.(fig. given below)

Experiment:

Simple-to-Complex:

This is very effective when solving a problem involving more than two concepts. A teacher should give adequate training for problems using each of the concepts one at a time working gradually with progressive combinations of these. Thus

• The constructivist approach to conceptualization in its process trains the child to observe phenomena aiming at comparison, correlation so as to identify the pattern that emerges.

• Inducting the particular to general and forming inference is the next step that describes the physical through the medium of language which is rich in critical and analytical component. A statement of Hypothesis is therefore precise comprehensive and exhaustive.

• Some of these which are too basic and intuitive in nature and constitute the origins in terms of which all other hypotheses can be explained are called axioms. The axiomatic approach is so much of language correlated to the physical phenomena that teachers should make best use of the middle school period to correlate the language components with appropriate concrete situations and help to shift dependence gradually to the nuances of the languages than material experience. This is so since language is the medium of the abstract.

Mathematical Language:

In-depth analysis of the choice of the words in a definition and emphasis on learning this without alteration or omission is very essential. Since each definition is like a capsule of a fixed weight. When the doctor prescribes a 50 mg capsule one cannot consume either less or more. Further each concept should be understood along with its various implications.

For instance

• the word if and only if has a two way implication.

• If we say “in a triangle when a side is produced an exterior angle is equal to the sum of the interior opposite angle” it also means the exterior angle is greater than each of the interior opposite angles.

• If we say (x-2) (x-3) is greater than 0 it means the product of (x-2) and (x-3) is positive which means both are either positive or negative.

• An ‘or’ stands for union and an ‘and’ stands for intersection.

• An article may turn the scale. ‘the’ implies the number one whereas ‘a’ or ‘an’ is one of many. Such components are important in permutations & combinations, probability etc.

Further, it is essential to store mathematical results in human language simultaneously correlating with its symbolic representation. One should realize that human brain is more amenable to storing information in the normal language than the language of symbols. To that extent the translation of the lifeless formulae to life situations will be hastened.

All these only form the tip of an iceberg. The teacher has to emphasize many such language intricacies during the course of learning.

Algorithm of Problem-Solving:

The consummation of all these efforts is possible only when a student is able to apply in any problem situation. What are the rudiments of problem-solving? Without loss of generality, we shall assume that the problem is in the ordinary human language.

Read and translate the ordinary into mathematical parlance in the following sequence:

Working towards the answer is the secret of original thinking to solve a problem.

II. Evaluation

This is an integral part of teaching process. It has two major functions in pedagogy. It helps

• to determine the level up to which learning has taken place in an individual

• to diagnose and p0rescribe the kind of assistance required to achieve the expected level of understanding.

Viewed this way it is clear that evaluation is an ongoing process that should be interspersed with teaching.

For every concept/competency, there should be formative and summative tests. If a concept has n functions, each of the n functions should be tested separately and cumulatively. The testing tools should be simple and least time-consuming for the teacher who examines them. Concepts could be clubbed for a cumulative test. The features of such a test will comprise of evaluating

• the stages of learning each individual concept,

• inter-relation if any between these concepts,

• ability to handle interference (if any) of one concept on the other.

Oral tests have a way of removing inhibitions, facilitating fluency of expression of ideas. Being less time-consuming, more can be achieved in a given time-frame. Practicing becomes easier since the brain takes full control minimizing the material participation of the sensory organs. Concentration, Retention and Memory are enhanced. Especially in the formative stages of a concept a teacher should resort to oral tests.

In short, questions should be process-oriented and hence graded through the stages of learning. This kind of training will enhance a student to answer any kind of question paper since he is trained to divide the question into the stages and address each stage. It helps assess the student in the ladder of learning rather than in a peer group.

Typology of Questions may be:

1. Translation items like word problem in algebra to mathematical equations, Geometry riders to figures, Heights and distances to figures, statement problems in arithmetical operations at primary level to mathematical statements, geometrical situations in terms of co-ordinates.

2. The reverse process of constructing life problems for given set of equations, figures etc. This will promote creativity as well as the ability to anticipate analogues.

3. Locating the problems vis-à-vis the concept, theorem, formulae or operations.

4. Jargon-related problems including definitions. The same problem with a minor alteration in the language and analyze its impact on the answers.

5. Writing negative statements and examining implications.

6. Effective use of multiple-choice questions in one or more of all the above areas.

CONCLUSION

The task of a math teacher is therefore to stay with a child throughout from day one to the end of the schooling gradually initiating the child to think the mathematician’s way which is the very essence of constructivism. ‘Association’ which is a basic instinct in every human being should be encouraged. No child should be curbed from asking why? If by chance he misses the question the teacher should facilitate towards it. There is no short cut to Mathematics. Each child has to go through several experiences before internalization. The concrete experiences should converge to the abstract and such transition should be absolutely smooth. Mathematical language should be mastered to overcome slavery to symbols. Breaking complex to simple human language and reconstructing is the best way to attain enlightenment.