The Beam: Journal of Arts & Science, Vol. 9, (2016) ISSN: 1118-5953

APPLICATION OF COMPUTER ALGEBRA SYSTEMS (CAS) IN SOLVING MATHEMATICAL PROBLEMS

Mukhtar Muhammad Sani, Yusuf Usman and Mansur Aliyu Department of Mathematics Waziri Umaru Federal Polytechnic, Birnin Kebbi [email protected] Department of Computer Science Umaru Ali Shinkafi Polytechnic Sokoto ABSTRACT Mathematical software has the potential to facilitate an active approach to learning, this approach can make the study of mathematics more enjoyable, more relevant and more rewarding too. It allow students to become involved in discovery and to consolidate their own knowledge, thus developing a deeper conceptual and geometrical understanding to learning. Mathematics teaching can be made much more interesting, inventive and exploratory using Computer Algebra Systems (CAS). While the use of CAS in many countries in teaching and learning mathematics has made a significant impact, in Nigeria the progress and awareness of these technology has been really very slow. Mostly, it has been confined among the researchers and a handful of university, polytechnic and college lecturers in well- established research institutes and departments of higher learning. It is against this background that this manuscript discussed the meaning of computer software, gives a brief history on computer algebra systems, its advantages and disadvantages in mathematics teaching. Some relevant examples are demonstrated and made easy with MuPad when one is interested in solving problems associated with simple arithmetic calculations, basic algebra and plotting of graphs. The study further stated some challenges faced in the use of CAS in teaching mathematics and also some steps to be taken to overcome the challenges in Nigeria.

Key words: Computer Software, Computer Algebra Systems (CAS), MuPad

1. INTRODUCTION In the long history of education, there have been four intellectual revolutions (Ashby, 1975). The first occurred when the responsibility for educating the child was in large measure passed to the professionalized institution. The second occurred with the invention of writing and the formation of the written word. The third, like its predecessor, occurred with the invention of printing and its potential in disseminating literature. The fourth is to do with the adoption of audio-visual materials and programmed instruction in education. The computer, for now, epitomizes the fourth revolution. The computer also has a long history and a long tradition of computational bias. From its humble primitive beginnings as an Abacus to its mechanical form (Babbage’s Analytical Engine) through to its electro-mechanical and electronic forms the story has been that of a number-crunching machine (Liman, 2006). The use of the software as a tool was found to have a strong impact on the learning strategies adopted and on their confidence towards mathematics. Others have found similar effects of the influence of computing methods on learning mathematics. For example, Galbraith, Haines and Pemberton (1999), found that students with high computer mathematics interaction feel that computers enhance mathematics learning by providing many examples enable user to focus on major ideas by reducing mechanical toil, and find computers helpful in linking algebraic and geometric ideas.

2. COMPUTER SOFTWARE As the computer has no mind of its own, it has to be fed with instructions. In this it is destined to be universal since it is a very obedient servant and would do almost anything it is asked to do from whatever discipline not only social sciences or mathematics. The term ‘Software’ refers to the set of electronic program instructions or data a computer processor reads in order to perform a task or operation. Software therefore can be referred to as a set of instructions loaded into the computer that tells it what to accomplish (and sometimes how to accomplish the task). Software can be categorized according to what it is designed to accomplish. There are two main types of software; systems software and applications software. Systems Software includes the programs that are dedicated to managing the computer itself, such as , file management utilities, and disk operating system (or Dos). The operating system manages the computer hardware resources in addition to applications and data. Without systems software installed in our computer we would have to type the instructions for everything we wanted the computer to do.

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Application software, or simply applications, are often called productivity programs or end- user programs because they enable the user to complete tasks such as creating documents, spreadsheets, databases, publications, doing online research, sending email, designing graphics, running businesses, and even playing games. Application software is specific to the task it is designed for and can be as simple as a calculator or as complex as a word processing application. Example of application software include Web browser, or simply, browser, Microsoft word, SPSS, GeoGebra, MuPad and MATLAB among others.

3. COMPUTER SOFTWARE AND EDUCATION The use of digital technologies to assess the knowledge has long been recognized as effective, despite the fact that it has its own peculiarities (Rowlett, 2013). But to convince teacher to use mathematical packages in the practical classes in mathematics is not an easy task (Weigand, 2013). Research suggests that, adequate training and collegial support boost teachers' willingness to integrate technology into their teaching and to develop successful technology-assisted teaching practices (Becker, Ravitz & Wong, 1999). Preparing future teacher for the appropriate use of technology in their teaching has to be one of the main issues in today's mathematics and science pre-service (and in-service) teacher training. The appropriate use of mathematical software in mathematics teaching can greatly support the teaching and learning of mathematics. On the one hand mathematical software can be supportive for using mathematical knowledge and in learning how mathematical knowledge can be used. On the other hand mathematical software offers valuable support for the appropriate development of mathematical concepts Therefore the use of mathematical software should not be ignored in pre-service teacher training (Kokol-Voljc, 2007).

4. MATHEMATICAL SOFTWARE Mathematical software is a software used to model, analyze or calculate numeric, symbolic or geometric data. Many mathematical suites are computer algebra systems that use symbolic mathematics. They are designed to solve classical algebraic equations and problems in human readable notation. Computer Algebra Systems (CAS) began to appear in the early 1970s, and evolved out of research into artificial intelligence. Pioneering work was conducted by the Nobel laureate Martin Veltman, who designed a program for symbolic mathematics, especially High Energy Physics in 1963. The first popular systems were Reduce, Derive, and which are still commercially available. A free version of Macsyma called is actively being maintained. The current market leaders are , Mathematica, MatLab, SciLab and MuPad. These are commonly used by mathematicians, scientists, and engineers. Some mathematical software focus on a specific area of application; these are typically developed in academia and are free (Kumar & Kumaresan, 2008).

Table 1: List of Some of the Most Popular Free and Commercial Mathematical Software System Creator Year Latest Stable Cost (USD) Notes Started Release Date Axiom Richard Jenks 1977 Aug. 2014 Free Gen. Purpose CAS Cadabra Kasper Peeters 2001 Nov. 2014 Free CAS for Tensor Field Theory. Calcinator George J. Paulos 2013 Feb. 2015 Free Browser-based CAS for desktop and mobile devices. Derive Soft Warehouse 1979 Nov. 2007 Discontinued CAS designed for pocket calculators; it was discontinued in 2007. DataMelt JWork.ORG (Sergel 2005 May, 2016 Free Java-base. Runs on (DMelt) Chekanov) the Java platform. Supports Python, Ruby, Groovy, Java and Octave. Fermat Robert H. Lewis 1986 June, 2016 $70 if grant CAS for resultant money available, computation and otherwise $0 polynomial entries. Macsyma MIT Project MAC 1968 1999 $500 The oldest gen. and Symbolics purpose CAS. Still

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alive as Maxima. Magma University of Sydney 1990 Sept. 2015 $1,440 Gen. purpose CAS for Group Theory. Works with elements of algebraic structures rather than non-typed mathematical expressions. Maple Symbolic 1980 March, 2016 $2,275*c One of the major gen. Computation Group, $2,155*g purpose CAS University of $239*a Waterloo $99*p $79*s MuPad SciFace Software 1989 2008 Discontinued mathWorks has incorporated MuPad technology into Symbolic Math Toolbox. Mathemat Wolfram Research 1986 April, 2016 $2,495*pro One of the major gen. ica $1,095*Ed purpose CAS $295*p $69.95*s Symbolic MathWorks 1989 2015 $3,150 Including Provides tools for Math required solving and Toolbox Simulink manipulating (MATLA symbolic math B) expressions and performing variable precision arithmetic. (Source: Wikipedia, 7th June 2016).

Key: *a = Academic, *c = Commercial, *Ed = Educational, *g = Government, *p = Personal, *pro = Professional *s = Student Note that: More information on these software can be found on their respective websites. Students do not often regard themselves as active participants in mathematical exploration. Rather they are passive recipients of a body of knowledge, comprising definitions, rules and algorithms. Computers offer a number of didactic advantages that can be exploited to promote a more active approach to learning. Students can be- come involved in the discovery and understanding process, no longer viewing mathematics as simply receiving and remembering algorithms and formulae. The power of computer algebra goes beyond routine computation. It has the potential to facilitate an active approach to learning, allowing students to become involved in discovery and constructing their own knowledge, thus developing conceptual understanding and a deeper approach to learning. Bower (2012), illustrated some examples on how simple arithmetic calculations, basic algebra and plotting is made easy using MuPad.

Example 1: Simple Arithmetic Calculations MuPad can be used as a calculator. Try the following commands

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Here, the gamma function and besselJ are special functions – the gamma function is the generalization of the factorial to non-integers, and the Bessel function is the solution to a common . MuPad has lots of built in special functions, which can be very useful. Notice also that, unlike some other CAS, MuPad returns the correct answer for sin(PI). This is because by default, MuPad is not working with floating point numbers. It will return the exact answer to any calculation. It will only start using floating point calculations if you start first, or explicitly ask for a numerical value. For example, contrast

In the second case, MuPad gives a floating point number because you typed in a floating point number (0.5) as the argument to the Gamma function. You can also ask MuPad to compute a numerical value for an expression with the ‘float’ function

Note the use of the % character – this always refers to the result of the last calculation that MuPad has done and MuPad lets you go back and change any line, and will then let you execute the file again with the changed code. The Notebook> menu gives lots of options for re-doing calculations after a correction.

Example 2: Basic Algebra MuPad is as well good at solving algebraic problems. For example, it can solve equations

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Because there are two solutions, they are returned in a set (enclosed by {}). You can extract each one by using the [number] convention. It is important to notice that the ‘equals’ sign is used in two different ways. If you just type a=b, you have created an equation object that you can use in later manipulations (e.g. solve it!). On the other hand, if you type a := b^2 (with a colon) then you have assigned the value b^2 (a symbol) to a variable called a. MuPad will substitute b^2 for a any time it is used later. For example try this:

Notice that a in the ‘eq1’ object has been replaced by b^2. You can clear the value of a variable using the ‘delete’ function.

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If you want to clear all variables, you can use the ‘reset’ function. This completely restarts MuPad from the beginning. This is often useful for starting a new homework problem. MuPad doesn’t simplify expressions by default. But it can do so if you ask it to. This sort of thing is especially handy for trigonometric functions

Example 3: Plotting of Graphs MuPad is also a very good software in plotting graphs. You can have a basic plot and at the same time display multiple plots on the same axis just like the Figures below.

Fig 1.

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You can do pretty 3D plots as well. You can make animations as well, by adding a 3rd parameter to a 3D plot. MuPad can do parametric plots as well, in both 2D and 3D. MuPad is also great at calculus, solving differential equations, vectors and matrices.

4.1 Advantages of using Mathematical Software The following are some of the advantages of using mathematical software: 1. Helps develop visual/geometrical understanding. 2. CAS can help to increase the value of the knowledge and degree of interest of students. 3. Can explore concepts before “hand skills’’ to do so are available. 4. Can explore realistic problems. 5. CAS help to increase student motivation and improve student’s attitudes towards Mathematics. 6. Due to the potential interactivity of these tools, students are able to attain a higher level of abstraction in mathematical problem-solving something which clearly represents a significant didactic accomplishment. 7. Allows students to concentrate on problem formulation and solution analysis. 8. Easy to give math demos and advanced mathematical ideas can be introduced very easily and concretely. 9. It will help teachers to develop innovative, challenging and exploratory teaching modules. 10. Researchers do not need to spend more time on tedious computations rather they can spend more time in analyzing and the computation part can be easily be done using these tools. 11. When CAS are not used, the teacher tends to be the sole center of attention whereas, when they are used, there is an observable increase in student participation, autonomous activity and interaction among students, hereby making the process of acquiring and constructing mathematical knowledge more student-centered. 12. People from other disciplines not having sound mathematical knowledge can very easily solve mathematical problems which they come across.

4.2 Disadvantages of using Mathematical Software Some of the disadvantages of using mathematical software are outlined below: 1. Students tend to use CAS blindly and they do not bother about the validity of answer obtained through CAS. 2. Most often students try to use CAS as an advanced calculator and refuse to learn concepts. 3. Decline of students’ paper-and-pen skills. 4. Difficulties in evaluation of a course taught using CAS. 5. Greater time needed for class preparation. 6. Lack of familiarity with the computer and CAS. 7. Fear of making syntactical errors in class. 8. Lack of administrative recognition of increasing teaching load. 9. CAS syntax can be an unreasonable burden on students. 10. The course can be victimized by equipment failure or inadequate equipment. 11. Students’ algebraic manipulation skills will deteriorate if they are allowed to rely on computer algebra but that these skills are an essential foundation for mathematics. 12. Using CAS can potentially prevent students from making the proper connections between the techniques used and their mental approach to Mathematics.

5. CHALLENGES AND DIFFICULTIES The following are some challenges and difficulties that mathematical software users faces in teaching/learning activities: 1. Availability of computers in the laboratory and to teachers and students is still a distant dream. 2. Most of the CAS are too costly and hence not affordable to college students and teachers. 3. Classrooms are not equipped with relevant hardwires which is required to integrate teaching using CAS. 4. Teachers are not having proper computer literacy and knowledge of CAS. 5. Many teachers are not willing to move from traditional teaching style to CAS-based teaching wherever necessary. 6. Unavailability of innovative and exploratory teaching modules.

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7. Courses are not designed properly. It does not give space, time and opportunity of exploring the subject using CAS.

6. OVERCOMING THESE CHALLENGES AND DIFFICULTIES The following measures are outlined in other to overcome the challenges and difficulties faced by mathematical software users: 1. All colleges/institutes to have proper computer labs and to give students enough opportunities to explore. 2. Use of free mathematical software like MuPad, Scilab, Maxima, octave etc. be encouraged. 3. Development of similar software may be initiated and encouraged. 4. Classrooms should be equipped with relevant computer hardware. 5. A series of teacher-training programmes throughout the country may be initiated in order to make them aware of such tools. 6. Innovative teaching modules and projects be prepared which make students and teachers realize that these tools are not merely advanced calculators but can be used to solve a very complex problems and help them to experiment and explore (one of the vital aspects of learning). 7. Courses may be redesigned to encourage the use of CAS and also provide time for its use.

7. MATHEMATICAL SOFTWARE AND TEACHERS It goes without saying that the classroom teacher is the key to the successful introduction of new methods and new technologies. Of course, it is possible for the student to come across these independently in the case of CAS. With the increasing speed of technological development, it is crucial that teachers keep themselves informed so that they are in a position to make valid judgments and adapt their teaching accordingly. Teachers, of course, have a crucial role in students learning (with or without CAS). Integrating CAS into teaching changes many aspects of class- room practice which teachers will make on the basis of their prior teaching styles and their beliefs about mathematics and how it should be taught. While using CAS to solve problems, students sometime make silly mistakes which produces a totally irrelevant output. Teacher support and appropriate intervention is crucial to correct such mistakes. Judging the right amount of help at the right time is a skill acquired through experience. use in mathematics teaching and learning is in its infancy. Nevertheless there are many teachers and educationalists who have integrated CAS into their teaching or conducted research into student understanding with CAS or who have led curriculum/assessment projects involving CAS use.

8. CONCLUSION A mathematical software is a tool not a self-contained learning package or encyclopedia of mathematical knowledge. Much emphasis these days is placed on student-centered learning and less on the teaching but also teaching/learning are equally important. It is necessary to first understand the learning process, then designing teaching/learning activities so that students can become deep learners. It is against this background that the following conclusions were made: The use of CAS in teaching of mathematics should be channelized to both teachers and students so as to maximize the opportunities offered by CAS technologies; Optimal use should be aimed at improving students’ motivation, autonomy and participation; and students’ centered learning; It is important to note that mathematical software should support learning not substitute good teaching. Traditional teaching methods must be supported with modern tools for problem-solving. This does not imply a reduction in the standard of education rather it is vital that the curriculum is carefully considered and that passive teaching is replaced in favour of new methods which promote active participation of students.

9. RECOMMENDATIONS In order to make the CAS based mathematics teaching a reality, the following recommendations were outlined by the researchers: 1. Develop methodology for teaching mathematics with CAS and redesign the course curriculum. 2. Develop strategies to implement the teaching methodologies. 3. Produce innovative teaching modules using CAS. 4. Organize regular workshops, training programmes for mathematics teachers on how to use mathematical software.

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5. A lot of research is needed to understand the students’ attitude and psychology of learning mathematics using CAS.

10. REFERENCES Ajit, K. & Kumaresan, S. (2008). Use of Mathematical Software for Teaching and Learning Mathematics. Proceedings of the 11th International Congress on Mathematics Education, July, 2008, Mexico Alvarez, B. (2000). Secondary Education: Critical Policy Issues Becker, H., Ravitz, J., & Wong, N. Y. (1999). Teacher and Teacher Directed Student Use of Computers. Teaching, Learning, and Computing: National Survey Report 3. Irvine, CA: Center for Research on Information Technology and Organizations, University of California Irvine. Bower A. F. (2012). Dynamics and Vibrations. MuPad Tutorial. Galbraith, P., Haines, C. & Pemberton, M. (1999). Making the Difference, edited by J. M. Truran and K. M. Liman, M. A. (2006). Uses of Computer in Social Sciences. Readings in Social Science Research, Publication of the faculty of Social and Management Sciences, Bayero University, Kano Nigeria. Page 229-239 Rowlett, A. & Peter J. (2013). Developing a Healthy Scepticism about Technology in Mathematics Teaching. Journal of Humanistic Mathematics, 3(1), 136-149 Vlasta Kokol-Voljc. (2007). Use of Mathematical Software in Pre-service Teacher Training: The Case of DGS Proceedings of the British Society for Research into Learning Mathematics Volume 27 Number 3 Weigand, H. G. (2013). Looking back and ahead Didactical Implications for the Use of Digital Technologies in Next Decade. Teaching Mathematics and its Applications, 33(4), 3-15 Wikipedia the free encyclopedia (2016). List of Some of the Most Popular Free and Commercial Mathematical Software. Retrieved June 7, 2016 at 04:51pm from https://en.m.wikipedia.org/wiki/list_of_computer_algebra_systems#ref_sagedio.5E

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