Interdisciplinary Programs Involving Mathematics

1. Interdisciplinary Programs Involving Mathematics Dr. Mary George Dept. of Mathematics Mar Ivanios College Trivandrum, India

2. What is Mathematics • Mathematics occupies a very important position in the Modern World. • It plays a vital role in technical professions and latest researches. • It is a tool, rather than a discipline. • It is the main ‘ingredient’ of any ‘finished good’ in Pure Sciences or in Applied Sciences.

3. Importance of Mathematics • Mathematics is the common language of many other disciplines • students should learn this language to understand the concepts used in those disciplines • students should connect their mathematical learning to appropriate real-world contexts.

4. Mathematics and Development Mathematical tools have allowed many advances in the present time. Following are some areas which owed to Mathematics for a great extend. • modern technologies in Medicine • recent developments in Communication • fast growing of Engineering

5. Most of the Mathematics Classrooms are Boring • 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.

6. What is the Remedy? • Highlight the fact that all sorts of simple everyday materials offer great scope for a variety of interesting and mathematically rich activities. • make connections between abstract mathematical ideas and the everyday material world. • make links between new ideas and related ideas learned previously • help students to see mathematics as a unified body of knowledge whose concepts build upon each other. • give emphasis to ideas and concepts across mathematical content areas that help students see that mathematics is a web of closely connected ideas.

7. A Case Study

8. Mathematics & Physics • Physics 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.

9. Mathematics & Physics (Cont.) • The coefficient of linear expansion of different metals, cubical expansion of liquids, expansion of gases and conversion of scales are calculated using Mathematics. • The most important equations of mechanics, astronomy and the physical sciences are differential or integral equations, both outgrowths of Calculus. • In its early period of invention, Quantum mechanics used an enormous amount of Mathematics, from special functions to modern algebra.

10. Mathematics & Biology • 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.

11. Simple Mathematics in Biology 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, growth, maturation, fatigue and many other branches of Biology and Physiology.

12. Bio-Mathematics Knowledge of Mathematics is considered essential for a biologist mainly because - 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.

13. In Biological Research • 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.

14. Mathematics & Chemistry • All chemical combinations and their equations are governed by certain Mathematical laws • Graph theory is much used to study the structure and properties of organic compounds

15. Mathematics & Engineering All Engineering Sciences need a fair knowledge of Mathematics as they deal with the following processes: • Surveying, Lending and Construction • Estimation and Designing • Measurement and Calculation • Drafting and Drawing

16. Mathematics & Agriculture Mathematics is directly applied in measuring many aspects in Agriculture, like • Measurement of land • average investment and average return • production per unit area and cost of labour • time and work • seed rate and manure rate

17. Mathematics & Economics • 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. 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.

18. Research in Economics • Researchers in Economics, both theoretical and empirical, are using more mathematical tools in their research work • The growing importance of Econometrics speaks for itself. • Increased use of Algebra is more prevalent in Economic research

19. Relation with more Subjects 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. • Any geological or geographic study cannot be envisaged keeping Mathematics away. In Geographyto 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.

20. Relation with more Subjects (Cont.) • Everybody very well knows the relation between Commerceand 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 Psychologyis much based on Mathematical calculations and applications . Various Mathematical techniques are used to collect, analyse and interpret psychological data. • Subjects like Demography, Actuarial Science, Statistics etc., are mostly depend on Mathematics to develop their theory.

21. Relation with Fine arts • There is a close relation between Mathematics and fine arts. • 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 systems work on Mathematical principles.

22. Challenge for the Teacher • To make the students unafraid of the subject is the major challenge faces by a Mathematics teacher everywhere in the globe. • 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.

23. Interdisciplinary Approach in School Curriculum • 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.

24. Some Interdisciplinary Programmes involving Mathematics Commerce-Mathematics • The programme is intended for its aspirants to study Commerce using the Mathematics that he has learned 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. • 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.

25. Mathematics-Physical Science The students should get knowledge in • Complex Analysis and Calculus • Fourier analysis and Fourier transforms • Wavelet Transforms • Mathematical theories of Space and Time • Laplace Transforms • Matrices and Vector spaces

26. Mathematics-Economics The students should get knowledge in • Limits and Continuity • Relations and Functions • Compact sets and Convex sets • Calculus • Difference Equations and Differential Equations • Separating hyperplanes • Lower and upper hemi-continuous correspondences • Fixed Point Theorems and Optimal Control

27. Mathematics-Actuarial Science A programme in Mathematics-Actuarial Science should contain the portions from Mathematics which makes the study of the programme easy. In Actuarial Science, the student have to cover portions in Life Contingencies like: • the survival function and force of mortality • life tables, analytical laws of mortalityand life insurance • continuous and discrete life annuities • recursion equations and benefit premiums • insurance and annuity models • multiple life functions and Multiple decrement models • pension funding cost method, retirement and salary components

28. Mathematics-Demography The student should get the Mathematical knowledge to study portions prescribed in Demography, such as, • measures of mortality, fertility and morbidity • Demographic characteristics and trends • evaluation of demographic data • projections for stable and stationary populations • actuarial applications of demographic characteristics and trends

29. Conclusions • Results from the study show the effectiveness and need of interdisciplinary approach in Mathematics education • In class rooms, emphasis should be given on mathematical connections both among mathematical topics and between mathematics and other disciplines of study • Making learning more meaningful by intentionally illuminating connections among various parts of the curriculum is a compelling idea.

30. Mathematics Curriculum The mathematics curriculum should include investigation of the connections and interplay among various mathematical topics and their applications so that all students can use and value the connections among mathematical topics; and the connections between mathematics and other disciplines.

31. Curriculum Integration • Issues arising from this study suggest the need for a greater emphasis on building and using models of curriculum integration including more and more programmes with mathematics. • Curriculum integration is one important facet of the mathematics community's response to calls for the reform of mathematics education.

32. The Debating Issue • Should mathematics remain as pure course of study, or • can it be meaningfully taught as part of a block of interdisciplinary courses.

33. References • Anice James, Teaching of Mathematics, (First Edition), Neelkamal Publications, Hyderabad, India, 2005 • Bell, E. T., The Development of Mathematics, McGraw Hill, New York, 1940 • Benjamin, H., The Teaching of Secondary Mathematics, (Ed.), McGraw Hill, USA, 1960. • Clawson, C. C., Mathematical Sorcery, Viva Books, India, 2004. • Grubel, H.G. and Boland, L.A. , On the Effective Use of Mathematics in Economics, Kyklos 39: 419–42, 1986 • Guilferd, J. P and Fruchter, B. , Fundamental Statistics in Psychology and Education, McGraw Hill Ltd. , 1970

34. References (Cont.) • Kulshrestha, A. K. , Teaching of Mathematics, (Third Edition), Surya Publications, Meerut, India, 2005 • Mercykutty, A., Developing and Testing Models of Teaching Mathematics Using Environmental Resources, Doctoral Thesis, University of Kerala, India, 1996 • Rees, P. K., Principles of Mathematics, Prentice Hall, NJ, 1965. • Sidhu, K. S. , The Teaching of Mathematics, Sterling Publishers, New Delhi, India, 1995 • Tannee, H. and Jones, S., Becoming Successful Teacher of Mathematics, Routledge Falmer, 2000 • Tikhomirov, V. M., Stories about Maxima and Minima (translated), Universities Press India, 1998.

35. Reference: Web Sites • http://www.ucalgary.ca/ • http://www.dundee.ac.uk/

36. A Word of Thanks… The author would like to thank the organizers, especially Dr. Mohamed B. Rhouma, Sultan Qaboos University, and Dr. Lotfi Hermi, Arizona University for inviting her to this International Conference

37. Thank You