Opportunities and Limitations of Computer Algebra in Education

Opportunities and Limitations of Computer Algebra in

Education

Arthur Nunes-Harwitt

Nassau Community College

One Education Drive

Garden City,NY11530

nunesa@https://www.360docs.net/doc/dd6579920.html,

Abstract

Computer algebra systems are frequently used for research.In addition,some instruc-tors have based entire advanced courses around these systems.One bene?t is that they allow

students to become familiar with the methods of calculus by individual experimentation.How-

ever,instructors have generally seen computer algebra systems as unsuitable for introductory

algebra since the software is capable of doing all the math for the students.That barrier could

be overcome if a suitable user interface were developed.Furthermore,embracing computer

algebra could make for a completely different type of introductory course that shifts the focus

from calculation to https://www.360docs.net/doc/dd6579920.html,puter algebra systems could also make life easier

for math instructors as problem generators for tests and as automatic on-line graders,even of

non-multiple choice https://www.360docs.net/doc/dd6579920.html,puter algebra systems offer many new opportunities for

math education.

1Introduction

It is widely understood that computers can perform numerical computations quickly.It is less well known that computers can also perform algebraic computations https://www.360docs.net/doc/dd6579920.html,puter algebra systems

are computer programs that can perform algebraic manipulation and computation.Mathematica and Maple are popular examples;others include Macsyma,Reduce,Derive,and Axiom.All of these systems are capable of advanced graphics,numerical calculation,and symbolic manipulation. Many professors have found these features useful for instruction as well as research,and have integrated computer algebra systems into the curriculum of their advanced courses.However,these systems have potential for more:computer algebra systems could also be useful for the instructors themselves and for students in introductory courses.

This paper begins with the current applications of computer algebra systems to advanced courses.Various useful features are discussed as well as certain limitations.The next section introduces ideas for how these systems might pro?tably be used in introductory courses.Finally, the last section shows how instructors could use computer algebra for both exam question genera-tion and correction.

2Current Uses of Computer Algebra

New curricula based around computer algebra systems have been designed for advanced courses, such as calculus and linear algebra.There are many textbooks available now for such courses[4,1, 13].Computer algebra systems offer a number of features that are advantageous for both instruc-tors and students.Most of these features fall into two categories:advanced graphics processing and advanced symbolic processing.Being able to visualize what is happening in a mathemati-cal setting is often a crucial prerequisite for understanding.Advanced interactive graphics make it easy to visualize mathematical objects and phenomena and allow experimentation and explo-ration.Students can make progress on a problem by having the system make a graph and then take measurements,rather than having to laboriously make the graph themselves or having to proceed directly to symbolic calculation before gaining insight into the problem.Interactive graphics make it easy to plot functions,vector?elds,and tangent lines,and to examine points on these objects.

Experimentation and exploration are also quite valuable in the symbolic realm,which will be

the focus of the rest of this paper.For example,rather than jumping to a deductive derivation of how to compute the derivative,an instructor could?rst have the students experiment on various ex-pressions and induce the rules.A relatively recent and exciting result is that computer algebra sys-tems can always decide if an expression consisting of elementary functions has an anti-derivative expressible using elementary functions(i.e.,functions like log,sin,e x)[2,3,9],so students can ex-pect to get answers to integration questions from the computer when they exist.However,there are some caveats regarding this result.While formally correct,symbolic integrals may not be suitable for numerically computing the area under the curve[8].Also,for expressions that do not consist of elementary functions(e.g.,Bessel functions)computer algebra systems may have less success[6]. Despite these limitations,students have found computer algebra systems extremely useful when studying advanced math such as calculus.

3New Courses for Computer Algebra

While computer algebra systems are put to good use in advanced courses,it is less clear how to insert them into the curriculum for introductory algebra and other lower-level courses.There are two immediate concerns.One is that the software may simply be too dif?cult for these students to use.One might expect calculus students to have a certain sophistication that would allow them to learn quickly how to use the software,but one has no such expectation of introductory algebra students.The other concern is that computer algebra systems are so powerful that students will learn only how to type in equations,effectively eliminating the content of the course.

These concerns are valid;however,they do not preclude the use of computer algebra systems. These concerns could be addressed by modifying the software to better suit the needs of less advanced students:a simpli?ed user interface could make the system accessible to these students. General purpose computer algebra systems are very powerful;introductory algebra students do not need that power.The issue of the software doing too much can also be overcome by designing an appropriate interface.The student will not get much out of the interaction if the software simply

responds with40when a student types in the command to solve the equation w=1

w+20.

2 However,the student can get a lot out of a richer interaction that shows more of the steps of the solution and requires more student participation.

There has already been some experimentation with computer algebra systems suitable for in-troductory algebra students.An early example is the Macintosh Graphing Calculator[7].This software was capable not only of making excellent graphs,but also of helping students manipu-late equations.One could click on an expression and move it about within the equation;if moved to the other side of the equation it would automatically change sign.There currently exist other sophisticated algebra tutorials.These systems can show all the steps of an algebra problem,and some attempt to do cognitive modeling to better help a student learn the material[10,12].

I would like to suggest using computer algebra systems as“training wheels.”Algebra can be decomposed into two skill sets:arithmetic-type activities such as multiplying numbers or adding variables,and equation-solving activities such as adding or multiplying on both sides of an equa-tion.These activities are fundamental for both simple and complicated equation solving,including equation solving of systems of equations and quadratic equations.The system that I am advocating would do the arithmetic-type activities for the student,allowing him or her to focus on building equation-solving skills.In a sense this system is only a minor extension of what students already have available.Generally,calculators are allowed to aid purely numeric arithmetic activities;I am merely suggesting expanding this technical aid to algebra.Equation-solving would begin when a student typed in an equation.Beside each equation line would be a menu with options to add or subtract something on both sides,multiply or divide something on both sides,or simplify a highlighted expression.A heuristic that might be used to determine whether or not the student is off track could involve determining that the complexity of the equation is being reduced.For the solution of quadratic equations,once the student has reduced the equation to standard form (ax2+bx+c=0),he or she could be prompted to identify a,b,and c,and to type in the quadratic formula to complete the solution.

There is another more radical approach to incorporating computer algebra systems into intro-

ductory algebra courses.Instead of modifying the system so that it does not merely give the answer, we could modify the curriculum so that equation-solving is no longer the focus of the course.The new focus would be problem-solving.In most introductory algebra courses,relatively little time is spent on word problems and practical applications;this new course would reverse that trend. Questions concerning,for example,ages of individuals,geometry and applied geometry,distribu-tion of resources,and simple physics problems would be posed.The students would have ample time to learn how to convert such problems into systems of equations.These equations would then be fed into a computer algebra system,which would solve them.Again,the content of the course would be on translating problems into equations,and not on solving equations.Advanced physics courses often take that approach—students learn to turn problems into differential equations, which can only be solved by machine.Interestingly,there is already a precedent for this approach in a lower-level math course:introductory statistics[5]is frequently taught this way.Students are given word problems that they must analyze in order to extract the relevant information,and de-termine the appropriate calculator or computer command to get the answer.Students coming from an introductory algebra course are often weak on word problem skills;perhaps a word problem course would better serve them.

4Computer Algebra for Instructors

While using a computer algebra system for research is not new,and basing certain math courses around a computer algebra system is not uncommon,I know of no discussion regarding the use of computer algebra systems as tools for https://www.360docs.net/doc/dd6579920.html,puter algebra systems can be as useful to instructors as they are to researchers and students.There are at least two areas where a computer algebra system could simplify the instructor’s task:exam construction and exam correction.

The job of constructing an exam is a tedious and time-consuming task.The instructor has in mind that he or she would like to test certain concepts at a particular level of dif?culty,but crafting many such suitable questions requires a lot of effort.Previously constructed question banks are

makeQuad:=proc(rng)

local f,r1,r2;

f:=rand(rng);

r1:=f();

r2:=f();

[expand((x-r1)*(x-r2))=0,r1,r2]

end proc;

>makeQuad(1..10);

[x2?15x+54=0,6,9]

Figure1:Maple problem generator code

not always available,and even if they are,they might be deemed unsatisfactory for some reason. Because they are programmable,computer algebra systems allow instructors to construct classes of questions by writing problem-generating procedures.Consider the following example.Suppose an instructor wanted to test students on their understanding of quadratic equations.One might try to describe the levels of dif?culty by articulating what the kind of numbers the coef?cients a,b, and c are.The spectrum of complexity might range from c being zero while a and b are small positive integers,to the coef?cients being arbitrary integers,to the coef?cients being arbitrary rational numbers,up until the coef?cients possibly being irrational numbers.However,such a hierarchy is not adequate for characterizing the levels of dif?culty.The equation x2+1=0has non-negative integer coef?cients,yet the solutions are imaginary.It is necessary to articulate what kind of numbers the solutions are.Now the spectrum of complexity ranges from zero,to small positive integers,to arbitrary integers,to rational numbers,up until real numbers.A procedure could be written to generate problems for each level on this spectrum.The input parameters would be the range of integers or rational numbers.The procedure would generate two random numbers (r1,r2)in the desired range,and multiply the binomials(x+r1)and(x+r2)together to get the trinomial for the quadratic equation.The procedure would return the quadratic equation together with its solutions.Figure1shows an example of Maple code that generates sample problems with their solutions.

The only job more tedious than constructing exams is correcting them.If one chooses to give

multiple-choice exams,it is possible to automate the grading with a Scantron machine.If one wishes to give an exam that is not multiple choice,then there are no automated methods for cor-recting the exams.It may be that computer algebra systems can help out here.If one is willing to have an on-line exam,where the students type their answers into a computer,then in many cases the computer can determine whether an arbitrary entry is equivalent to the correct result.Matching precise simple forms such as1/2or true is easy;this feature can be implemented without much dif?culty using any programming https://www.360docs.net/doc/dd6579920.html,puter algebra systems can match patterns such as the form y=mx+b,and the system can check that m and b are the correct numbers.Fur-thermore,the system could be even more?exible and allow a variety of answers.For example,if an instructor wants a student to come up with an equation for a line given a point and a slope,the instructor might be willing to allow any form of the line as long as it is correct.The system could easily be made to accept any correct equation of the line.This?exibility could be implemented by having the system put the equation into standard form(ax+by+c=0)and then it verify that the coef?cients(a,b,c)are correct.It is worthwhile noting that computer algebra systems do have limitations:once expressions become too complicated,the system can no longer determine equiv-alence.If the expression language allows the expression of log(2),π,e x,sin(x),and something equivalent to|x|,then the question of whether two arbitrary expression are equivalent becomes formally undecidable[11].Additionally,while the question of whether two expressions involving radicals are equivalent may be decidable,there do not yet exist algorithms to always decide[3,9]. Nevertheless,systems have no trouble determining the equivalence of fractions,polynomials with rational coef?cients,and rational functions with rational coef?cients.Thus,a wide range of answer checking could be easily automated.

5Conclusion

Although computer algebra systems have existed for over thirty years,there are still new opportu-nities for their use.Professors who use computer algebra for research may not be taking advantage

of it for the courses they teach.This paper shows two practical ways that instructors can bene?t from using these systems.Also,while calculus students have had the help of these systems for over a decade,these tools are still mostly unknown to introductory algebra students.With the right software and the right curriculum,these students can bene?t as well.

6Acknowledgments

I would like to thank Melissa Nunes-Harwitt for carefully reading previous drafts of this paper.

References

[1]W.C.Bauldry,B.Evans,J.Johnson.Linear Algebra with Maple.Wiley,1994.

[2]M.Bronstein.Symbolic Integration I:Trancendental Functions.Springer Verlag,1997.

[3]J.H.Davenport,Y.Siret,https://www.360docs.net/doc/dd6579920.html,puter Algebra:Systems and Algorithms for Alge-

braic Computation.Academic Press,1993.

[4]B.Davis,J.J.Uhl,H.Porta.Calculus&Mathematica:Approximations:Measuring Near-

ness.Addison-Wesley,1994.

[5]C.DeSanto,M.Totoro.Introduction to Statistics6th edition.Pearson,2003.

[6]T.H.Einwohner,R.J.Fateman.Searching Techniques for Integral Tables.Proceedings of

the1995international symposium on Symbolic and algebraic computation,1995.

[7]T.Erickson.Introducing dynamic algebra with NuCalc:investigating symbols,functions,

and graphs.Key Curriculum Press,1995.

[8]R.J.Fateman,W.Kahan.Improving Exact Integrals From Symbolic Algebra Systems.UC

Berkeley Tech Report PAM386,1986.

[9]K.O.Geddes,S.R.Czapor,https://www.360docs.net/doc/dd6579920.html,bahn.Algorithms for Computer Algebra.Kluwer Academic

Publishers,1992.

[10]K.R.Koedinger,B.A.MacLaren.Developing a pedagogical domain theory of early algebra

problem solving.CMU-HCII Tech Report02-100,2002.

[11]D.Richardson.Some Unsolvable Problems Involving Elementary Functions of a Real Vari-

able.Journal of Symbolic Logic,V ol.33,No.4,511-520,1968.

[12]L.J.Shapiro,E.Sueker,W.Hadley.Quantitative Literacy for Algebra Using Cognitive Tu-

toring Technology.Proceedings of the International Conference for Technology in Collegiate Mathematics,1998.

[13]I.P.Stavroulakis,S.A.Tersian.Partial Differential Equations:An Introduction With Math-

ematica and Maple.World Scienti?c,2000.