Teaching Mathematics Concepts via Computer Algebra Systems


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1 Iteratioal Joural of Mathematics ad Statistics Ivetio (IJMSI) EISSN: 4767 PISSN: Volume 4 Issue 7 September. 6 PP Teachig Mathematics Cocepts via Computer Algebra Systems Osama Ajami Rashaw, Jaso D. Johso (Departmet of Mathematics ad Sciece,Ajma Uiversity of Sciece & Techology, UAE) (Departmet of Mathematics ad Statistics,Zayed Uiversity, UAE) ABSTRACT: Most articles eamie computer algebra systems (CAS) as they relate to the teachig ad learig of mathematics from advatages to disadvatages. This paper will eplore juior udergraduate studets ability to solve distiguish tricky eamples usig various CAS techologies. Additioally, a uderstadig for how CAS techologies are adopted ad applied i professioal eviromets is valuable, both i guidig improvemets to these tools ad idetifyig ew tools which ca aid mathematicias. KEYWORDS: Computer Algebra Systems (CAS), Mathematics Cocepts, Limits, Itegratio, Differetial Equatios I. INTRODUCTION Sice the eed for tools i mathematical problem solvig, may computer sciece researchers have created tools to support mathematical problem solvig. Oe way to ehace mathematical problem solvig is to use computer algebra system (CAS) techology. CAS is a program that maipulates both symbols ad umbers to reduce the algebraic maipulatio of algebra; i the same way calculators (i.e., scietific ad fourfuctio) ca reduce computatio for time ad work ivolved i arithmetic. Some CAS techology combie a wide rage of mathematical fuctios with graphig capabilities. As such they have the potetial to reduce the time studets sped calculatig algorithms, allowig more time for cocept developmet, which isehaced by the ability of the program to cocurretly represet mathematical fuctio graphically. The poit of view of the user will vary depedig o the applicatio ad the iteded outcome. Such as, a statisticia may have differet cosideratios tha a physicist, which may i tur be differet from that of a mathematicia, ad/or a mathematics educator. Each type of user has certai goals ad desires for CAS techology ad how this techology will aid i his or her quest to achieve a certai solutio to a problem. II. METHODS AND FUTURE CONTENTS OF MATHEMATICS COURSES We argue that it is importat to let juior udergraduate studets, especially those whose major is ot STEM related, solve distiguish tricky mathematics problems first with CAS. We defie distiguish tricky mathematics problem to be a problem where a CAS solutio is ot accurate ad the studet would eed to kow more about the mathematics cocept ad CAS techology to make a clear determiatio for the solutio. I other words, the studets must recogize that he/she must maipulate the CAS i order to arrive to a correct solutio. By allowig studets with ostem related majors opportuities to solve distiguish tricky mathematics problems first with CAS, allows the studet time to eplore the cocepts, rather tha, the tedious mathematical maipulatios to solve the problem. Additioally, such studets typically have questios regardig the use of CAS, which ca help drive istructio ad lead studets to a greater uderstadig of the mathematical cocept. For the purpose of this article, four differet CAS techologies were used. The four CAS techologies are: Teas Istrumets, Matlab, Mathematica, ad Microsoft Mathematics Calculator. Matlab, Mathematica, ad Microsoft Mathematics Calculator are all computerbased CAS techologies. [] (Rashwa, 4) gives some eamples solved by Matlab ad Mathematica. However, Teas Istrumets (TI) is the oly hadheld CAS techology eplored (TINspire CX CAS). As oted by [] (Johso, ), the TINspire CX CAS has advaced features as, symbolic maipulatio, costructig geometric represetatios, ad eplorig multiple represetatios (i.e., algebraically ad graphically) dyamically all o oe scree (p. 44). I additio, Microsoft Mathematics Calculator is the oly free CAS techology, while the others require a oetime fee. Agai, the missio of this article is to eplore the idea of solvig distiguish tricky mathematics problems usig CAS. Further ivestigatig the eed to iterrupt the solutio with more advaced mathematics skills. III. Distiguish Tricky Problem : Is cotiuous? LIMIT AND CONTINUITY OF ONE VARIABLE FUNCTION f ) 5 ( ) ( ( ),, Page
2 Caot help solvig this problem Remark :The mai poit here is to fid the sum of 5 ( ), ad the fid ( ) lim. ( ) The fuctio is ot cotiuous whe the value of the limit is ot equal to the value of the fuctio at.the limit ivolves L Hopial s rule but we caot do it before havig the sum of 5 ( ). Distiguish Tricky Problem :  IV. DEFINITE INTEGRAL 4 Page
3 Remark :The fuctio f ( ) must be cotiuous o [a,b] to fid f ( ) d ad its value is the area uder the curve f ( ) o the iterval [a,b]. The fuctio b a f ( ) is ot cotiuous at [,]. By igorig the cotiuity coditio, the result is d l( ) l( ) l( ) l( ) which is udefied value. Distiguish Tricky Problem : d Remark :Completig this without techology would give aswer. Improper Itegral Distiguish Tricky Problem 4:The improper itegral e d. ad this is a icorrect 5 Page
4 Remark 4:Let us graph e by the Matlab commad, So, e e,, the it is easy to complete without techology [ ] e d e d e ( ). Imagiary Itegratio Distiguish Tricky Problem 5: l d Remark 5:To graph of f ( ) l, by 6 Page
5 Based o the output warig provided by Matlab, the solutio refers to igorig the imagiary parts of f ( ) whe ad. Itegratio Test Distiguish Tricky Problem 6:To check the coverget of the series l( ) l(l( )) by the itegratio test, is etremely difficult without techology. It is recommeded to use CAS to make sese of the eample. Remark 6: (I) The result is as = If meas which meas the series is diverget. But the series is coverget whe the itegratio value is a fiite value. (II) The cocept ivolves the defiite itegral which could be discussed ad eplored by CAS. For istace, the area uder a curve i a iterval i its domai. ) R V. DOUBLE INTEGRAL AND POLAR COORDINATES f (, y da is the volume of the solid regio betwee a bouded rectagular or o rectagular regio R i the plae R ad the surface Z f (, y ). It is the etesio to the defiite itegral but for a cotiuous fuctio of two variables f (, y ). Distiguish Tricky Problem 7: y e y ddy ad e y dyd. 7 Page
6 Remark 7: Some of CAS tried to compute the itegratio of two variable fuctio i the Cartesia coordiates, the importace of trasformig ito the polar coordiate ad the area elemet i it r ad da = rdrd comes to accout. The boud of the itegratio for are foud from the relatio betwee the Cartesia ad the polar coordiates. Noe of the CAS techologies provided a correct solutio. For istace, trasformig ito polar coordiates by CAS, by usig Mathematica the output is: ddy ( e ) The itegratio is possible usig e r rdrd, that ca be doe by ocas techology. However, ( e ) Teas Istrumet provided the same result as Reversig the itegratio order 6 Distiguish Tricky Problem 8: 4 y cos( ) ddy 8 Page
7 Remark 8: SomeCAS techology caot evaluate some double itegratiowithout reversig it ad ca help after doig so. Net the importace of reversig the itegratio is show: VI. DIFFERENTIAL EQUATION Distiguish Tricky Problem 9:The followig first order differetial equatio: dy y ( y ), y ( ) d Caot solve differetial equatios Remark 9: (I) the two solutio are y, ad y satisfyig the differetial ( e ) ( ) equatio ad its iitial coditio. (II) Without the iitial coditio Matlab gives: Where, C = or C = by usig the iitial coditio y ( ), therefore the solutio is coicide with VII. CONCLUSION. CAS is ot oly used for solvig timecosumig mathematics problems but also is used to eplore mathematics cocepts by solvig distiguish tricky problems.. Sometimes CAS will ot help studets i solvig some mathematics problems ad the studets must be aware whe to use ad ot use CAS.. Use of CAS features to focus o mathematics cocepts ad how to adopt the solutio techique avoidig the theoretical calculatio steps. 9 Page
8 4. Mathematical backgroud ad mathematical software skills are ecessary i solvig mathematical problems whe usig CAS techology. 5. Mathematics course descriptios should iclude three parts. First, teachig of mathematics should lik to the studets major. Secod, CAS should be a essetial compoet i mathematical problem solvig. Third, some mathematical problems should relate to the studet s major. 6. A uderstadig of how these tools are adopted ad applied i professioal eviromets is valuable, both i guidig improvemets to these tools ad i idetifyig ew tools which ca aid mathematicias. REFERENCES []. Rashwa, Osama A. (4), Teachig Egieerig Mathematics Dilemma.Joural of Mathematical Scieces ;556 []. Johso, J. D. (). Prospective middle grades teachers of mathematics usig hadheld CAS techology to create rich mathematical task. Joural of Mathematical Scieces ad Mathematics Educatio. 5; Page
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