Interdisciplinary Interdisciplinary Programs Involving Programs Involving

MathematicsMathematics

Dr. Mary GeorgeDr. Mary GeorgeDept. of Mathematics Dept. of Mathematics

Mar Ivanios College Mar Ivanios College Trivandrum, IndiaTrivandrum, India

What is MathematicsWhat is Mathematics

• Mathematics occupies a very important Mathematics occupies a very important position in the Modern World.position in the Modern World.

• It plays a vital role in technical It plays a vital role in technical professions and latest researches.professions and latest researches.

• It is a tool, rather than a discipline. It is a tool, rather than a discipline. • It is the main ‘ingredient’ of any ‘finished It is the main ‘ingredient’ of any ‘finished

good’ in Pure Sciences or in Applied good’ in Pure Sciences or in Applied Sciences. Sciences.

Importance of Importance of MathematicsMathematics

• Mathematics is the common language of many Mathematics is the common language of many

other disciplinesother disciplines

• students should learn this language to understand students should learn this language to understand

the concepts used in those disciplinesthe concepts used in those disciplines

• students should connect their mathematical students should connect their mathematical

learning to appropriate real-world contexts.learning to appropriate real-world contexts.

Mathematics and Mathematics and DevelopmentDevelopment

Mathematical tools have allowed many Mathematical tools have allowed many advances in the present time.advances in the present time.

Following are some areas which owed Following are some areas which owed to Mathematics for a great extend. to Mathematics for a great extend.

modern technologies in Medicinemodern technologies in Medicine

recent developments in Communicationrecent developments in Communication

fast growing of Engineeringfast growing of Engineering

Most of the Mathematics Most of the Mathematics Classrooms are BoringClassrooms are Boring

Students either hate Mathematics, or Students either hate Mathematics, or fear it. fear it. The blame for this plight is partly to the The blame for this plight is partly to the teachers and the rest to the curriculum.teachers and the rest to the curriculum.Students get no interest in studying this Students get no interest in studying this subject, because neither the teacher, subject, because neither the teacher, nor the syllabus points out the practical nor the syllabus points out the practical use of the prescribed portions. use of the prescribed portions.

What is the Remedy?What is the Remedy?

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

A Case StudyA Case Study

Mathematics & PhysicsMathematics & Physics

Physics is related with Mathematics to a great extend.Physics is related with Mathematics to a great extend.Mathematical efficiency gives more confidence to Mathematical efficiency gives more confidence to learners of Physics. learners of Physics. Each rule and principle in Physics takes Mathematical Each rule and principle in Physics takes Mathematical form and Mathematics gives them their final shape.form and Mathematics gives them their final shape.Mathematical calculations occur at every step in Mathematical calculations occur at every step in Physics. Physics. The laws of motion, friction, expansion of solids, liquid The laws of motion, friction, expansion of solids, liquid pressure are explained using Mathematics. pressure are explained using Mathematics. All the measurements in Physics need Mathematics.All the measurements in Physics need Mathematics.

Mathematics & Physics Mathematics & Physics (Cont.)(Cont.)

The coefficient of linear expansion of different metals, cubical The coefficient of linear expansion of different metals, cubical

expansion of liquids, expansion of gases and conversion of expansion of liquids, expansion of gases and conversion of

scales are calculated using Mathematics. scales are calculated using Mathematics.

The most important equations of mechanics, astronomy and the The most important equations of mechanics, astronomy and the

physical sciences are differential or integral equations, both physical sciences are differential or integral equations, both

outgrowths of Calculus. outgrowths of Calculus.

In its early period of invention, Quantum mechanics used an In its early period of invention, Quantum mechanics used an

enormous amount of Mathematics, from special functions to enormous amount of Mathematics, from special functions to

modern algebra.modern algebra.

Mathematics & BiologyMathematics & Biology

Modern Biology needs Mathematics in a great Modern Biology needs Mathematics in a great amountamountThe Life Sciences will be for Mathematics in The Life Sciences will be for Mathematics in the forthcoming century what Physics was for the forthcoming century what Physics was for Mathematics in the previous century. Mathematics in the previous century. New, exciting challenges in the Life Sciences New, exciting challenges in the Life Sciences can and are being met using mathematical can and are being met using mathematical modelling with a direct impact on improving modelling with a direct impact on improving people's quality of life in health, social and people's quality of life in health, social and ecological issues. ecological issues.

Simple Mathematics in Simple Mathematics in BiologyBiology

Simple Mathematics is used everywhere in Biology. Simple Mathematics is used everywhere in Biology. The Calorie and Nutritive values of food articles are The Calorie and Nutritive values of food articles are calculated using Mathematics. calculated using Mathematics. To find the rate of respiration and transpiration we To find the rate of respiration and transpiration we need the knowledge of Mathematics. need the knowledge of Mathematics. Study of living cells, composition of blood, age and Study of living cells, composition of blood, age and category of plants and animals are studied using category of plants and animals are studied using Mathematics. Mathematics. Mathematical process and calculations have been Mathematical process and calculations have been applied to advanced studies in heredity, growth, applied to advanced studies in heredity, growth, maturation, fatigue and many other branches of maturation, fatigue and many other branches of Biology and Physiology.Biology and Physiology.

Bio-MathematicsBio-Mathematics

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

biological study depends largely on its biological study depends largely on its branches branches Bio-Physics and Bio-Physics and Bio-ChemistryBio-Chemistry

which have attained a rank almost equal to which have attained a rank almost equal to that of independent Sciences, which cannot that of independent Sciences, which cannot exist without Mathematics. exist without Mathematics.

In Biological ResearchIn Biological Research

Mathematics helps the Biologist to perform his Mathematics helps the Biologist to perform his investigations easily and correctly. investigations easily and correctly. Experimentation in Biology requires analysis and Experimentation in Biology requires analysis and isolation of the particular character that is to be isolation of the particular character that is to be experimented upon. Biological phenomena are so experimented upon. Biological phenomena are so complex and the required analysis and isolation are complex and the required analysis and isolation are so difficult that it is impossible to bring many of them so difficult that it is impossible to bring many of them under control without applying mathematical formulae. under control without applying mathematical formulae. At every stage of classification, or comparison or At every stage of classification, or comparison or generalisation, the investigator needs the help of generalisation, the investigator needs the help of Mathematics. Mathematics.

Mathematics & ChemistryMathematics & Chemistry

All chemical combinations and their All chemical combinations and their

equations are governed by certain equations are governed by certain

Mathematical lawsMathematical laws

Graph theory is much used to study the Graph theory is much used to study the

structure and properties of organic structure and properties of organic

compounds compounds

Mathematics & Mathematics & EngineeringEngineering

All Engineering Sciences need a fair All Engineering Sciences need a fair knowledge of Mathematics as they deal knowledge of Mathematics as they deal with the following processes: with the following processes:

Surveying, Lending and ConstructionSurveying, Lending and ConstructionEstimation and DesigningEstimation and DesigningMeasurement and CalculationMeasurement and CalculationDrafting and DrawingDrafting and Drawing

Mathematics & AgricultureMathematics & Agriculture

Mathematics is directly applied in measuring Mathematics is directly applied in measuring many aspects in Agriculture, likemany aspects in Agriculture, like

Measurement of landMeasurement of land

average investment and average returnaverage investment and average return

production per unit area and cost of labourproduction per unit area and cost of labour

time and worktime and work

seed rate and manure rate seed rate and manure rate

Mathematics & EconomicsMathematics & Economics

In Economics, it is constantly necessary to choose the best In Economics, it is constantly necessary to choose the best possible solution. In such cases the Economists make use of possible solution. In such cases the Economists make use of techniques of Calculus or Operations Research. Mathematical techniques of Calculus or Operations Research. Mathematical terms like Relations, Functions, Continuity, etc., are very much terms like Relations, Functions, Continuity, etc., are very much used in Economics. used in Economics.

To explain marginal concepts like, marginal utility, marginal cost, To explain marginal concepts like, marginal utility, marginal cost, marginal revenue, etc., method of Calculus is best used today.marginal revenue, etc., method of Calculus is best used today.

Difference and Differential Equations are used in a great deal in Difference and Differential Equations are used in a great deal in Economics to solve problems. Economics to solve problems.

Research in EconomicsResearch in Economics

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

Relation with more Relation with more SubjectsSubjects

Mathematics is used in almost all Social Science Mathematics is used in almost all Social Science subjects. subjects.

• Mathematical knowledge is applied in Mathematical knowledge is applied in HistoryHistory to know to know the dates, time, etc., of various historical events. the dates, time, etc., of various historical events.

• Any geological or geographic study cannot be Any geological or geographic study cannot be envisaged keeping Mathematics away. In envisaged keeping Mathematics away. In GeographyGeography to study the shape and size of earth, to measure area, to study the shape and size of earth, to measure area, height and distance, to study about latitude or height and distance, to study about latitude or longitude we need mathematical knowledge. To study longitude we need mathematical knowledge. To study the rivers, mountains, canals, population, climate, etc. the rivers, mountains, canals, population, climate, etc. all these studies need the tools of Mathematics in one all these studies need the tools of Mathematics in one way or other. way or other.

Relation with more Relation with more Subjects (Cont.)Subjects (Cont.)

Everybody very well knows the relation between Everybody very well knows the relation between CommerceCommerce and Mathematics. The basis of banking and Mathematics. The basis of banking and accountancy is nothing but Mathematics. Only with and accountancy is nothing but Mathematics. Only with a fair knowledge of Mathematics, one could become an a fair knowledge of Mathematics, one could become an efficient accountant. Shares, debentures, mutual funds, efficient accountant. Shares, debentures, mutual funds, interest, are all based on Mathematical calculations. interest, are all based on Mathematical calculations. Experimental Experimental PsychologyPsychology is much based on is much based on Mathematical calculations and applications . Various Mathematical calculations and applications . Various Mathematical techniques are used to collect, analyse Mathematical techniques are used to collect, analyse and interpret psychological data. and interpret psychological data. Subjects like Subjects like DemographyDemography, , Actuarial ScienceActuarial Science, , StatisticsStatistics etc., are mostly depend on Mathematics to etc., are mostly depend on Mathematics to develop their theory. develop their theory.

Relation with Fine artsRelation with Fine arts

There is a close relation between Mathematics and There is a close relation between Mathematics and fine arts. fine arts. It is evident that good drawing is needed to draw good It is evident that good drawing is needed to draw good geometrical figures. geometrical figures. Exactness of a figure, shape etc., can be measured Exactness of a figure, shape etc., can be measured using Mathematical tools. using Mathematical tools. The Mathematical knowledge is applied in drawing The Mathematical knowledge is applied in drawing and painting with symmetry, making right ratio and and painting with symmetry, making right ratio and proportion, etc. proportion, etc. In Music, almost all musical notes and systems work In Music, almost all musical notes and systems work on Mathematical principles. on Mathematical principles.

Challenge for the TeacherChallenge for the Teacher

To make the students unafraid of the subject To make the students unafraid of the subject is the major challenge faces by a Mathematics is the major challenge faces by a Mathematics teacher everywhere in the globe. teacher everywhere in the globe.

He should convince the students the He should convince the students the usefulness of learning Mathematics in their usefulness of learning Mathematics in their daily life and for higher studies. daily life and for higher studies.

He should be able to correlate the content of He should be able to correlate the content of Mathematics with other classroom subjects. Mathematics with other classroom subjects.

Interdisciplinary Approach in Interdisciplinary Approach in School CurriculumSchool Curriculum

The teacher of Mathematics can make the student The teacher of Mathematics can make the student aware of the topics in different subjects, where the aware of the topics in different subjects, where the studied Mathematical knowledge can be applied.studied Mathematical knowledge can be applied.The teachers of the concerned subjects should also The teachers of the concerned subjects should also generous to point out the topics in Mathematics, generous to point out the topics in Mathematics, which are used in developing their topics under study.which are used in developing their topics under study.There should be a co-ordination with Mathematics There should be a co-ordination with Mathematics teacher and other subject teachers. teacher and other subject teachers. Since Mathematics is a type of subject, which is most Since Mathematics is a type of subject, which is most needed to develop other subjects, it can be used as a needed to develop other subjects, it can be used as a component in many interdisciplinary combinations. component in many interdisciplinary combinations.

Some Interdisciplinary Some Interdisciplinary Programmes involving Programmes involving

MathematicsMathematics

Commerce-MathematicsCommerce-Mathematics

The programme is intended for its aspirants to study The programme is intended for its aspirants to study Commerce using the Mathematics that he has learned Commerce using the Mathematics that he has learned from the course itself. from the course itself.

The syllabus of Mathematics should be so designed that it The syllabus of Mathematics should be so designed that it should give all the fundamental ideas needed in studying should give all the fundamental ideas needed in studying the prescribed portions in Commerce. the prescribed portions in Commerce.

When selecting a third or fourth subject, the experts who When selecting a third or fourth subject, the experts who are preparing the syllabus should bear in mind that the are preparing the syllabus should bear in mind that the selected subjects will go with Commerce and Mathematicsselected subjects will go with Commerce and Mathematics. .

Mathematics-Mathematics-Physical SciencePhysical Science

The students should get knowledge in The students should get knowledge in

Complex Analysis and CalculusComplex Analysis and Calculus

Fourier analysis and Fourier transformsFourier analysis and Fourier transforms

Wavelet TransformsWavelet Transforms

Mathematical theories of Space and TimeMathematical theories of Space and Time

Laplace TransformsLaplace Transforms

Matrices and Vector spacesMatrices and Vector spaces

Mathematics-Mathematics-EconomicsEconomics

The students should get knowledge in The students should get knowledge in

Limits and ContinuityLimits and Continuity Relations and Relations and FunctionsFunctionsCompact sets and Convex setsCompact sets and Convex setsCalculusCalculusDifference Equations and Differential EquationsDifference Equations and Differential EquationsSeparating hyperplanesSeparating hyperplanesLower and upper hemi-continuous correspondencesLower and upper hemi-continuous correspondencesFixed Point Theorems and Optimal ControlFixed Point Theorems and Optimal Control

Mathematics-Mathematics-Actuarial ScienceActuarial Science

A programme in Mathematics-Actuarial Science should contain the A programme in Mathematics-Actuarial Science should contain the portions from Mathematics which makes the study of the programme portions from Mathematics which makes the study of the programme easy. In Actuarial Science, the student have to cover portions in Life easy. In Actuarial Science, the student have to cover portions in Life Contingencies like:Contingencies like:

the survival function and force of mortalitythe survival function and force of mortalitylife tables, analytical laws of mortalityand life life tables, analytical laws of mortalityand life insuranceinsurancecontinuous and discrete life annuitiescontinuous and discrete life annuitiesrecursion equations and benefit premiumsrecursion equations and benefit premiumsinsurance and annuity modelsinsurance and annuity modelsmultiple life functions and Multiple decrement modelsmultiple life functions and Multiple decrement modelspension funding cost method, retirement and salary pension funding cost method, retirement and salary componentscomponents

Mathematics-Mathematics-DemographyDemography

The student should get the Mathematical The student should get the Mathematical knowledge to study portions prescribed in knowledge to study portions prescribed in Demography, such as, Demography, such as, measures of mortality, fertility and morbiditymeasures of mortality, fertility and morbidityDemographic characteristics and trendsDemographic characteristics and trendsevaluation of demographic dataevaluation of demographic dataprojections for stable and stationary projections for stable and stationary populationspopulationsactuarial applications of demographic actuarial applications of demographic characteristics and trendscharacteristics and trends

ConclusionsConclusions

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

Mathematics CurriculumMathematics Curriculum

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

Curriculum IntegrationCurriculum Integration

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

The Debating IssueThe Debating Issue

Should mathematics remain as pure Should mathematics remain as pure

course of study, orcourse of study, or

can it be meaningfully taught as part of a can it be meaningfully taught as part of a

block of interdisciplinary courses.block of interdisciplinary courses.

ReferencesReferences

1.1. Anice James, Anice James, Teaching of MathematicsTeaching of Mathematics, (First Edition), , (First Edition), Neelkamal Publications, Hyderabad, India, 2005 Neelkamal Publications, Hyderabad, India, 2005

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

3.3. Benjamin, H., Benjamin, H., The Teaching of Secondary The Teaching of Secondary MathematicsMathematics, (Ed.), McGraw Hill, USA, 1960., (Ed.), McGraw Hill, USA, 1960.

4.4. Clawson, C. C., Clawson, C. C., Mathematical SorceryMathematical Sorcery, Viva Books, , Viva Books, India, 2004.India, 2004.

5.5. Grubel, H.G. and Boland, L.A. , Grubel, H.G. and Boland, L.A. , On the Effective Use of On the Effective Use of Mathematics in EconomicsMathematics in Economics, Kyklos 39: 419–42, 1986, Kyklos 39: 419–42, 1986

6.6. Guilferd, J. P and Fruchter, B. , Guilferd, J. P and Fruchter, B. , Fundamental Statistics Fundamental Statistics in Psychology and Educationin Psychology and Education, McGraw Hill Ltd. , 1970, McGraw Hill Ltd. , 1970

References (Cont.)References (Cont.)

7.7. Kulshrestha, A. K. , Kulshrestha, A. K. , Teaching of MathematicsTeaching of Mathematics, (Third , (Third Edition), Surya Publications, Meerut, India, 2005Edition), Surya Publications, Meerut, India, 2005

8.8. Mercykutty, A., Mercykutty, A., Developing and Testing Models of Developing and Testing Models of Teaching Mathematics Using Environmental ResourcesTeaching Mathematics Using Environmental Resources, , Doctoral Thesis, University of Kerala, India, 1996 Doctoral Thesis, University of Kerala, India, 1996

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

10.10. Sidhu, K. S. , Sidhu, K. S. , The Teaching of MathematicsThe Teaching of Mathematics, Sterling , Sterling Publishers, New Delhi, India, 1995 Publishers, New Delhi, India, 1995

11.11. Tannee, H. and Jones, S., Tannee, H. and Jones, S., Becoming Successful Becoming Successful Teacher of MathematicsTeacher of Mathematics, Routledge Falmer, , Routledge Falmer, 20002000

12.12. Tikhomirov, V. M., Tikhomirov, V. M., Stories about Maxima and MinimaStories about Maxima and Minima (translated), Universities Press India, 1998. (translated), Universities Press India, 1998.

Reference: Web SitesReference: Web Sites

• http://http://www.ucalgary.cawww.ucalgary.ca//

• http://http://www.dundee.ac.ukwww.dundee.ac.uk//

A Word of Thanks…A Word of Thanks…

The author would like to thank the The author would like to thank the

organizers, especially Dr. Mohamed B. organizers, especially Dr. Mohamed B.

Rhouma, Sultan Qaboos University, Rhouma, Sultan Qaboos University,

and Dr. Lotfi Hermi, Arizona University and Dr. Lotfi Hermi, Arizona University

for inviting her to this International for inviting her to this International

Conference Conference