Interdisciplinary Programs Involving Mathematics

[Dr. Mary George, Selection Grade Lecturer in Mathematics,

Mar Ivanios College, Trivandrum, India-695 015.

Dr. P. G. Thomaskutty, Reader in Economics,

Mar Ivanios College, Trivandrum, India-695 015.]

1. Introduction

Mathematics occupies a very important position in the Modern World. It may

be remarked that Mathematics plays a vital role in technical professions and latest

researches. Years ago, people believed that Mathematics is a classroom discipline.

Now we realize Mathematics is a tool, rather than a discipline. This argument comes

because; Mathematics is now the main ‘ingredient’ of any ‘finished good’ in Pure

Sciences or in Applied Sciences. Modern technologies in Medicine, the recent

developments in Communication, the fast growing of Engineering, are owed to

Mathematics for a great extend. Thus Mathematical tools have allowed many

advances in the present time.

Mathematics is many things to many people. To many students, it is simply

one of the obstacles that must be overcome to obtain a degree; to most undergraduate

students, it is a subject they wish they had studied more diligently; to some of us it is

a constantly used tool or a language- a very concise language that makes exact and

logical statements easier to form. To others, Mathematics is a logical development

made up of undefined terms, principles of logic, hypothesis and conclusions. But, to

Mathematicians, it is a pleasant way of living. (Rees, 1965). The importance of

Mathematics, from Babylon and Egypt to the present, as the primary source of

workable approximations to the complexities of daily life is generally appreciated.

(Bell, 1940)

But, even now, most of the Mathematics classrooms are boring, especially, in

the school level. Students either hate Mathematics, or fear it. The blame for this plight

is partly to the teachers and the rest to the curriculum. Students get no interest in

studying this subject, because neither the teacher, nor the syllabus points out the

practical use of the prescribed portions. Here comes the need of coining Mathematics

with other disciplines. There should be an interdisciplinary approach in teaching

Mathematics.

Several attempts have been done among experts and educators to make

connections between abstract mathematical ideas and the everyday material world.

Almost all sorts of simple everyday materials offer great scope for a variety of

interesting and mathematically rich activities. Connecting mathematical concepts

includes linking new ideas to related ideas learned previously, helping students to see

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mathematics as a unified body of knowledge whose concepts build upon each other.

Major emphasis should be given to ideas and concepts across mathematical content

areas that help students see that mathematics is a web of closely connected ideas.

Mathematics is also the common language of many other disciplines and students

should learn mathematical concepts used in those disciplines. Finally, students should

connect their mathematical learning to appropriate real-world contexts.

In this paper, the authors wish to discuss some areas where Mathematics can

be used fruitfully and interestingly, so that students may enjoy the study of

Mathematics. Certain interdisciplinary programmes are mentioned and discussed. The

paper is based on a study made in certain schools in our locality, and also among

experts of Mathematics Curriculum.

2. Defects in the Present-Day Mathematics Teaching

We have to admit that the present day teaching of mathematics is not up to

satisfaction (Anice James 2005). Everybody complaints that the methodology adopted

for teaching Mathematics is not right. It is too far from life to catch the interest of

students. Students feel it dull, boring, difficult and even useless. Mathematics is as

such an art form as is art and music. It deserves be taught by teachers who love it, just

as music teachers love music and art teachers love art. Instead, we ask our children to

learn Mathematics from teachers, many of whom, themselves are bored and tired-and

even afraid of it. (Clawson, 2004). The elements of novelty, usefulness and sheer

intellectual curiosity are the primary stimuli for the awakening of interest. Hence

motivation of the work in Mathematics has two aspects; viz. that of creating or

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arousing interest and that of maintaining the interest after the novelty of the work in

hand has worn off. It is of the greatest importance that work in Mathematics be so

organised and conducted as to emphasize the values and the inherent intellectual

challenge of the subject and to ensure understanding and a reasonable degree of

competence by keeping the subject matter and the activities at a level of difficulty

appropriate to the intellectual maturity of the students (Benjamin, 1960).

Even though the students and the public blame teachers for the predicament,

the teachers have their own justifications and grievances. Their blame often goes to

the excessive syllabus, lack of teaching aids and reluctance of students for hard work.

Hence it is the collective responsibility of all concerned to bring about necessary

improvements and appropriate changes in the Mathematics curriculum so as to make

it popularised at all costs. The organisers of the syllabus, being experts and authorities

in the subject, are expected to suggest the centres of correlation of different topics, the

use of aids and devices, the connected practical and project works, etc. The syllabus

not only should be a collection of topics, but should also deal with the actual

procedures to be adopted for their effective teaching. Their application and utility in

actual life should also be mentioned side by side. The connected games and activities

should be referred to.

The authors approached a school in Trivandrum (Government School,

Kattachakonam, Trivandrum, Kerala, India) to hear directly from students, the

problems they face in studying the subject. We interviewed 33 students from 10th

standard. There were 3 students who failed in all subjects for their last term

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examinations, while 6 students were there who passed in all subjects. The table below

shows that the least percentage of pass is for Mathematics.

Table

Subjects No. of students passed

Percentage of pass

English 9 27.3

Language-1 27 81.8

Language-2 24 72.7

Gen. Science 18 54.5

Social Studies 22 66.7

Mathematics 6 18.2

When we asked the students about their difficulty in studying Mathematics, their

responses were not the same, but they all admitted that they are not interested in

studying Mathematics. This response of the students point to the need for revitalizing

our Mathematics programmes to make them more appealing, relevant and cutting

edge. We have to broaden the scope of uses of Mathematics, as it appears more

relevant to the learners. An interdisciplinary approach in Mathematics will facilitate a

lot in this catastrophe. As the above Table shows, 54.5 per cent of students passed the

examination in General Science, while the pass percentage for Mathematics is just

18.2. We have to note that 36.3 per cent students could not make through

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Mathematics, even though they could pass in General Science. As we know, no

advancement in General Science can be attained without Mathematics, why can’t we

coin Mathematics with General Science in the school syllabus. These 54.5 per cent

students who are interested in General Science will develop an interest towards

Mathematics if we present Mathematics before them in such a way.

The following section analyses the subjects and topics where we could

incorporate Mathematics more beautifully and fruitfully.

3. Mathematics and Other Subjects

Connecting mathematical concepts and linking new Mathematical ideas to

related ideas learned previously, helps students to see mathematics as a unified body

of knowledge whose concepts build upon each other. Major emphasis should be given

to ideas and concepts across mathematical content areas that help students see that

mathematics is a web of closely connected ideas. Mathematics is also the common

language of many other disciplines and students should learn mathematical concepts

used in those disciplines. Students should connect their mathematical learning to

appropriate real-world contexts.

Mathematics is the language of science, and is greatly utilised in industry and

business. Mathematics gives us not only great power to solve difficult real world

problems, but helps us to understand how the universe operates. It bestows the power

for problem solving upon us. It enhances our understanding of the most basic

processes we encounter as inhabitants of this universe. It has been very well said that

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Mathematics is the Science of all Sciences and the Art of all Arts. It is the pivot of all

Sciences (Sidhu, 1995). So it is well connected with all science subjects. While it is an

essential constituent in Science subjects, it adds brevity, logic and charm to subjects

in Social Science and in Humanities. The basic ideas and relationships in the physical

sciences have been expressed in Mathematical terms for a very long time, and in

recent years the use of Mathematics in the biological and social sciences has increased

tremendously.

For example, if we take the case of Physics, it is related with Mathematics to

a great extend. Mathematical efficiency gives more confidence to learners of Physics.

Each rule and principle in Physics takes Mathematical form and Mathematics gives

them their final shape. Mathematical calculations occur at every step in Physics. The

laws of motion, friction, expansion of solids, liquid pressure are explained using

Mathematics. All the measurements in Physics need Mathematics. The coefficient of

linear expansion of different metals, cubical expansion of liquids, expansion of gases

and conversion of scales are a few to mention. We can have plenty of similar

occasions to prove the dependence of Physics on Mathematics. The most important

equations of mechanics, astronomy and the physical sciences are differential or

integral equations, both outgrowths of Calculus. Of all the exact sciences, mechanics

has probably the most influential in the development of modern Mathematics. In its

early period of invention, Quantum mechanics used an enormous amount of

Mathematics, from special functions to modern algebra.

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As another example, consider the case with Biology. There is an erroneous

belief that Biology is free from Mathematics. In fact, modern Biology needs

Mathematics in a great amount (http://www.dundee.ac.uk). The Life Sciences will be

for Mathematics in the forthcoming century what Physics was for Mathematics in the

previous century. New, exciting challenges in the Life Sciences can and are being met

using mathematical modelling with a direct impact on improving people's quality of

life in health, social and ecological issues. Knowledge of Mathematics is considered

essential for a biologist for two reasons: firstly, biological study depends largely on its

branches Bio-Physics and Bio-Chemistry, which have attained a rank almost equal to

that of independent Sciences, which cannot exist without Mathematics (Kulshrestha,

2005). Again, Bio-Mathematics is also growing as an important field of study.

Secondly, Mathematics helps the Biologist to perform his investigations easily and

correctly. Experimentation in Biology requires analysis and isolation of the particular

character that is to be experimented upon. Biological phenomena are so complex and

the required analysis and isolation are so difficult that it is impossible to bring many

of them under control without applying mathematical formulae. At every stage of

classification, or comparison or generalisation, the investigator needs the help of

Mathematics. Apart from this, simple Mathematics is used everywhere in Biology.

The Calorie and Nutritive values of food articles are calculated using Mathematics. To

find the rate of respiration and transpiration we need the knowledge of Mathematics.

Study of living cells, composition of blood, age and category of plants and animals

are studied using Mathematics. Mathematical process and calculations have been

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applied to advanced studies in heredity, nutrition, growth, maturation, fatigue and

many other branches of Biology and Physiology.

Just like the two cases described above, Mathematics is nowadays very much

used in all Science subjects. In Chemistry, all chemical combinations and their

equations are governed by certain Mathematical laws. Also, Mathematics is the

foundation of all Engineering Sciences, including IT. We know that Engineering

Sciences deal with surveying, lending, construction, estimation, designing,

measurement, calculation, drafting, drawing etc. All these need a fair knowledge of

Mathematics. The branch IT specially owes to Mathematics, without which it cannot

exalt. There are many aspects in Agriculture, where Mathematics is directly applied.

Measurement of land, average investment, average return, production per unit area,

cost of labour, time and work, seed rate, manure rate are name a few.

Again, Mathematics is applied in a great amount in Social Sciences. For

example, let us have a look into the case of Economics. Mathematics plays a very

important role in Economics. This role has been significant for almost a century, and

has been increasing in importance particularly in recent years. A comparison of

academic journals now with, say, fifty years ago reveals a tremendous increase in

mathematical expression. Researchers in Economics, both theoretical and empirical,

are using more mathematical tools in their research work and the growing importance

of Econometrics speaks for itself. Increased use of Algebra is more prevalent in

research in Economic research (Grubel and Boland, 1986). The same is true also of

textbooks of Economics at all levels. Mathematics is increasingly important in terms

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of the expression and communication of ideas in Economics. This in itself is a matter

of interest, particularly with respect to the public understanding of Economics.

Further, to the extent that public understanding of mathematics is limited, so too will

be the public understanding of Economics. This applies at a variety of levels, from

school pupils making subject choices to policy makers’ understanding of policy

advice. In Economics, it is constantly necessary to choose the best possible solution.

In such cases the Economists make use of techniques of Calculus or Operations

Research. (Tikhomirov, 1998). Mathematical terms like Relations, Functions,

Continuity, etc., are very much used in Economics. To explain marginal concepts like,

marginal utility, marginal cost, marginal revenue, etc., method of Calculus is best

used today. Difference and Differential Equations are used in a great deal in

Economics to solve problems.

Not only in Economics, Mathematics is used in almost all Social Science

subjects. Mathematical knowledge is applied in History to know the dates, time, etc.,

of various historical events. In Geography to study the shape and size of earth, to

measure area, height and distance, to study about latitude or longitude we need

mathematical knowledge. To study the rivers, mountains, canals, population, climate,

etc. all these studies need the tools of Mathematics in one way or other. In short, any

geological or geographic study cannot be envisaged keeping Mathematics away.

Everybody very well knows the relation between Commerce and Mathematics. The

basis of banking and accountancy is nothing but Mathematics. Only with a fair

knowledge of Mathematics, one could become an efficient accountant. Shares,

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debentures, mutual funds, interest, are all based on Mathematical calculations.

Experimental Psychology is much based on Mathematical calculations and

applications (Guilferd and Fruchter, 1970). Various Mathematical techniques are used

to collect, analyse and interpret psychological data. Also, subjects like Demography,

Actuarial Science, Statistics etc., are mostly depend on Mathematics to develop their

theory.

There is a close relation between Mathematics and fine arts and drawing. It

is evident that good drawing is needed to draw good geometrical figures. Exactness of

a figure, shape etc., can be measured using Mathematical tools. The Mathematical

knowledge is applied in drawing and painting with symmetry, making right ratio and

proportion, etc. In Music, almost all musical notes and system work on Mathematical

principles.

Thus we can see that Mathematics is the main component of any subject that

a student learns in his classroom. It seems to be fun that the student is not reluctant to

use Mathematics in other subjects (knowingly or unknowingly), and still he fears

Mathematics. To make the students unafraid of the subject is the major challenge

faces by a Mathematics teacher everywhere in the globe (Tannee and Jones, 2000). He

should convince the students the usefulness of learning Mathematics in their daily life

and for higher studies. He should be able to correlate the content of Mathematics with

other classroom subjects. Here comes the need of providing students with

interdisciplinary programmes in Mathematics.

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4. Interdisciplinary Approach in Mathematics

Many people tend to use the word interdisciplinary synonymous to

multidisciplinary (Mercykutty, 1996). But, interdisciplinary means a higher theme,

which means blending more than one discipline in a right proportion. The dictionary

meaning of the word is ‘involving two or more academic, scientific or artistic

disciplines’ (Webster’s New Collegiate Dictionary). It is not just the combination of

two or more disciplines, but one discipline, which is facilitated by one or more

disciplines. Even though interdisciplinary approaches in Mathematics can arouse

students’ interest in learning Mathematics, it is not advisable in school curriculum. In

schools, it is difficult to implement interdisciplinary programmes. But, the teacher of

Mathematics can make the student aware of the topics in different subjects, where the

studied Mathematical knowledge can be applied. The teachers of the concerned

subjects should also generous to point out the topics in Mathematics, which are used

in developing their topics under study. There should be a co-ordination with

Mathematics teacher and other subject teachers. Since Mathematics is a type of

subject, which is most needed to develop other subjects, it can be used as a

component in many interdisciplinary combinations.

Although we have seen that Mathematics is an essential tool for other

disciplines, it is an entirely different kind of subject. Mathematics is completely

abstract, while other disciplines are closely tied to the physical world. As our nature,

we may shudder at the thought of anything, which is abstract, and consequently, some

of us may have a mental block against Mathematics. We may consider this with due

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importance while moulding interdisciplinary courses involving Mathematics. Some

work should be done in convincing them the true nature of Mathematics and in

making them aware of the fact hat there is nothing so terrifying about the abstractness

of Mathematics.

5. Some Specific Interdisciplinary Programmes

The authors gathered the opinion of many Academicians and Experts in

curriculum planning about the interdisciplinary Mathematics programmes for Under

Graduate Courses. All of them suggested that the syllabus of Mathematics should be

selected with immense care, so that the student might feel that the proposed portions

of Mathematics are indispensable for him to study his optional subject. For example

let one student opts an interdisciplinary programme in Commerce-Mathematics; the

programme is intended for him to study Commerce using the Mathematics that he is

getting from the course itself. The syllabus of Mathematics should be so designed that

it should give all the fundamental ideas needed in studying the prescribed portions in

Commerce. But, it is not necessary to limit the subjects in two, can be more than that.

When selecting a third or fourth subject, the experts who are preparing the syllabus

should bear in mind that the selected subjects will go with Commerce and

Mathematics.

We can propose a number of interdisciplinary programmes involving

Mathematics like the one above. A list suggested by experts is given as Appendix. In

all those programmes, the portions should have to be selected with utmost care and

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deliberation. For example, when we select portions of Mathematics to include in a

programme Mathematics-Physical Science, we could not avoid Complex Analysis,

Calculus, Fourier analysis, Laplace Transforms, Matrics, Vector spaces, etc.

Continuous and discrete Fourier transforms, the Fast Fourier Transform, Wavelet

Transforms, Mathematical theories of Space and Time may also be included. When

we select portions of Mathematics to include in a programme Mathematics-

Economics, we have to include Relations and Functions, Limits and Continuity,

Compact sets, Convex sets, Calculus, Difference and Differential Equations,

Separating hyperplanes, Lower and upper hemi-continuous correspondences, Fixed

Point Theorems, Optimal Control, etc (http://www.ucalgary.ca). Likewise, in a

programme Mathematics-Actuarial Science, the portions from Mathematics are to

be selected so that a student will not feel difficulty in studying the portions prescribed

in Actuarial Science. In Actuarial Science, the student have to cover portions in Life

Contingencies like the survival function, force of mortality, life tables, analytical laws

of mortality, life insurance, continuous and discrete life annuities, recursion equations,

benefit premiums, insurance and annuity models, multiple life functions, Multiple

decrement models, pension funding cost method, retirement and salary components,

etc. For studying these concepts, they student should have attained certain knowledge

in Mathematics. Hence in an interdisciplinary programme in Mathematics-Actuarial

Science should contain the portions from Mathematics which makes the study of the

programme easy. For a programme in Mathematics-Demography, the student

should get the Mathematical knowledge to study portions prescribed in Demography,

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such as, measures of mortality, measures of fertility, measures of morbidity,

Demographic characteristics and trends, evaluation of demographic data, projections

for stable and stationary populations, actuarial applications of demographic

characteristics and trends. We can mould a number of such interdisciplinary

programmes involving Mathematics which makes the learning process of

Mathematics more appealing and fruitful.

6. Conclusion

the world today which leans more and more heavily on Science and

Technology, demands more from Mathematics. No, doubt, the world of tomorrow

will make still greater demands from Mathematics. Even though Mathematics is one

of the most practical subjects of study, learners everywhere feel it more impractical

and dull. The interdisciplinary programmes in Mathematics will make the subject

more attractive and meaningful.

Acknowledgement

The authors wish to thank University Grants Commission of India, for the

financial support in the form of Post-Doctoral Research Award to the first author in

pursuing this research.

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References

1. Anice James, Teaching of Mathematics, (First Edition), Neelkamal Publications,

Hyderabad, India, 2005

2. Bell, E. T., The Development of Mathematics, McGraw Hill, New York, 1940

3. Benjamin, H., The Teaching of Secondary Mathematics, (Ed.), McGraw Hill,

USA, 1960.

4. Clawson, C. C., Mathematical Sorcery, Viva Books, India, 2004.

5. Grubel, H.G. and Boland, L.A. , On the Effective Use of Mathematics in

Economics, Kyklos 39: 419–42, 1986

6. Guilferd, J. P and Fruchter, B. , Fundamental Statistics in Psychology and

Education, (Sixth Edition), McGraw Hill Ltd. , 1970

7. Kulshrestha, A. K. , Teaching of Mathematics, (Third Edition), Surya

Publications, Meerut, India, 2005

8. Mercykutty, A., Developing and Testing Models of Teaching Mathematics

Using Environmental Resources, Doctoral Thesis (Unpublished), University of

Kerala, India, 1996

9. Rees, P. K., Principles of Mathematics, Prentice Hall, NJ, 1965.

10. Sidhu, K. S. , The Teaching of Mathematics, (Fourth Edition), Sterling

Publishers, New Delhi, India, 1995

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11. Tannee, H. and Jones, S., Becoming Successful Teacher of Mathematics,

Routledge Falmer, 2000

12. Tikhomirov, V. M., Stories about Maxima and Minima (translated),

Universities Press India, 1998.

Web sites:

1. http://www.ucalgary.ca/ Jan, 2007

2. http://www.dundee.ac.uk/ Jan, 2007.

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