Interdisciplinary Programs Involving Mathematics
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.
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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 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 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 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 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 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.
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 (). 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 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 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, 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.
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 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 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 (). 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, 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.
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
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.
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