MATHEMATICAL ANIMATIONS : THE ART OF TEACHING

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1 MATHEMATICAL ANIMATIONS : THE ART OF TEACHING Nur Azman Abu 1, Mohd Daud Hassan 2, Shahrin Sahib 3 Session S1C Abstract Mathematical visualization is the art of creating a tangible experience with abstract mathematical objects and concepts. The advent of high-performance interactive computer graphics systems has opened a new era, its ultimate significance can only be imagined. Computer visualization has become an important tool in several fields of mathematics, helping mathematical understanding and communication. Mathematicians can now use computers to generate pictures that would be tedious or impossible to generate by hand. However, the process of approximating a picture seen on a computer monitor is nontrivial. The paper begins with some general background and then focuses on some of the visualization methods that have been used to bring computer graphics technology to model mathematical problems, from Calculus to Complex Numbers. Examples of computer-generated images are supplied throughout the paper and an overview of selected animations concerned with mathematical visualization will constantly be made. Animation can make mathematics more interesting and stimulating. It features a hands-on exploratory and experiential learning tool, making Mathematics more dynamic and meaningful. This problem solving through visualization illustrates the processes and behaviours of equations and formulas as they are graphically mapped out. Index Terms Calculus, Animation, Visual Mathematics 1. Introduction The aim of this paper is to show the natural interrelationship between Calculus mathematics and computer graphics. This article will focuses on the most part an IT perspective on the progress, techniques, and prospects of mathematical visualization, emphasizing areas of 2D and 3D geometry where interactive paradigms are of growing importance.[2] Due to tremendous paradigm shift that technology has brought in the recent years, instruction in mathematics will have to catch up with the new era or otherwise be increasingly irrelevant. Computer oriented mathematics courses should focus more on cooperative learning, problem solving, and investigative learning as an important part of education. Simultaneously with the aid of computer visualization, the horizon of teaching Mathematics is consistently broaden. Nevertheless it still requires the art of drawing pretty and nice functions to be presented. The scope of choosing clear understandable examples is expanding and still an art of teaching. 2. Computer Algebraic System(CAS) In the mid-eighties the availability of CAS for personal computers attracted mathematics educators to the possibility of using them in the classroom. CAS technology with its powerful combination of numeric and symbolic computation, colourful graphics as in figure 1 and figure 2, and programming facilities is a natural and logical continuation of the scientific and graphical calculators. The fresh approach of CAS has inspired many lecturers of Calculus all over the world. It has, however, as any new system does, brought complications of its own. It requires more reading and more insight from the student than traditional Calculus. This is the very reason why an average student experiences difficulties with this new approach. More guidance should be given to assist the student in improving his reading skills in Mathematics and to cultivate the required insight. 3. Pretty Pictures Since a long time ago, symmetric geometry was a popular pattern design. Recently, fractals has become the object of interest because they look very pretty when generated by computer graphics, even though the vast majority of the people generating them had no idea of its significance. The graphs illustrating the function of a complex variable are also pretty but the concept is more basic and easier to understand. It is hoped that these graphs will be, not only aesthetically appreciated, but also fundamentally understood. They are comparable to graphs of functions of a real variable, which are understood by most people who would have learned it in a pre-calculus course. 1 Nur Azman Abu, Faculty of Information Science and Technology, Multimedia University, Jalan Ayer Keroh Lama, Melaka, MALAYSIA, azman@mmu.edu.my 2 Mohd Daud Hassan, Center of Foundation Studies & Extension Education, Multimedia University, Jalan Ayer Keroh Lama, Melaka, MALAYSIA, daud@mmu.edu.my 3 Shahrin Sahib, Academic Affairs, Kolej Yayasan Melaka, 1 Jalan Bukit Sebukor, Melaka, MALAYSIA, shahrin@kym.edu.my S1C-10

2 However, the clever "trick" here involves using various colors to show the correspondence between domain and range points instead of "strings". So that, if a point in the w- plane is red it is because "it came" from the red point in the z-plane. Therefore, except for the striped pattern maps, every pixel in the domain and its corresponding pixel in the range have its own unique red, green, and blue components.[4]..figure 3a : Getting arccos( from cos( symmetric to the line y = x. figure 1: An example of translation, rotation and dillation of a circle. incorporates interactive graphics into a hypertext system for teaching all courses including Calculus. Besides reading the text, students need to merely click on a highlighted word to obtain definition, follow links to related subjects, and interact with a multitude of 3D visualizations, whose parameters can be manipulated to help understand a wide array of concepts. A typical example permits the user to "fly" on a curve in 3-space. The "interactive book" approach of situating interactive graphics within a written context is an appealing educational paradigm.[1] Information Technology can be potent mathematical tools that engage learners in meta-cognitive reflection. Proponents of computer-aided mathematical visualization argue that visualization can help build the intuition necessary to understand mathematical concept. Computer graphics is a useful tool for teaching simple topics in mathematics.[3] 5. Potential Problems and Challenges figure 2 : The function of complex variable w = z 2. On the left is the image of input z and on the right is the image of output w. Many textbooks show plots of functions of a complex variable but since they are printed in black and white only few points that are labeled to exhibit their correspondence. The day will come when color printing will be cheap enough so that these plots, also called maps, will be printed in color. Color will then be used as another dimension. 4. Multimedia Learning System The immediate goal of the Multimedia University is to create online Multimedia Learning System(MML) which The level of technology has risen. The trend towards making computers easier to use is apparent in mathematical software. On the other hand, mathematicians often point out that computers do not do real mathematics. Using mathematical software is not without difficulties. Posing relevant mathematical questions properly is essential to success, and requires some care. The use of computer programs, which can solve essentially any calculus homework problem, has provoked wide discussion concerning the proper content of the undergraduate mathematics curriculum, which parallels the discussion of the school arithmetic curriculum provoked by cheap calculators. Some worry that students will not learn how to do their homework "manually" if they rely upon automated computation: and that without "manual" skills and practice, understanding of concepts will be hindered. However, software cannot help, if students do not know what to do with it.[5] S1C-11

3 the ability to rip the cover off the process of integrals and let students see the pieces of it in action. The early introduction of numerical integration can both consolidate the conceptual understanding of the definite integral and directs the students attention immediately from Riemann sums to approximations of more practical value as in figure 4. Integrals can also be animated in a much more livelier mode, as visually mapped out in figure 5...figure 3b : Getting a segment of arccosh( from cosh( symmetric with respect to the line y = x. An important concept in mathematics is that of "reversing" a function process. That is, beginning with a function's output, an attempt to recover the original input. This is a difficult concept to grasp for a beginning calculus student. It is indeed too difficult to illustrate manually on the blackboard in any other way than an elementary case. Inverse function can be graphically introduced to the students. In figure 3, the resulting solution is displayed along with the original function. Students can "see" and understand the concept of an inverse in a non-trivial manner. figure 5 : the integral of a piecewise function x, 0 < x 1 f ( x ) = 2 x, 1 < x < 2 0, otherwise 6. Algebraic Functions Function is rapidly becoming one of the organizing concepts within mathematics education[10]. The day is long past when mere mathematical computation is adequate. Students should not only apply and use a particular function but they are also required to create new functions which adequately represent the mathematical problem. In addition to its figure 4 : sin( π f ( = together with its 16 and 32 middleboxes. πx Unfortunately, the software does not handle inverse functions well at times and some signs get reversed; all complicated programs have bugs, and all complicated math programs make mistakes. Mistakes pose a particular hazard: mathematics has long enjoyed a good reputation for its integrity, and many expect the same to be true of mathematical software[5]. Consider the general routine of integration, which provides symbolic definite and indefinite integration capabilities. This can be a convenient way of quickly integrating functions, but it is the ultimate in an operator with respect to the object function. What we really need is figure 6a,b : A sample of algebraic function ( f g)( where the f is zigzag and g is a cosine function. S1C-12

4 mathematical topics, in fundamentally the same ways, that could be taught without technology, does not enhance students' learning of mathematics and rely on the usefulness of technology. Furthermore, using technology to perform tasks that are just as easy or even better carried out without technology may actually be a hindrance to learning. figure 6c : Evolving the curve of ( f g)( about the y-axis mathematical power, the notion of piecewise function is uniquely suited for teaching and learning with technology. The various types of functions may be studied through the analysis of the meaning for various rates of change, zeroes, maximum and minimum values within contextual settings. With an understanding of the various families of functions, coupled with real world phenomena, connections are established, and students begin to see mathematical models as pictorial representations that have meaning. Basic algebraic operations employ two powerful representation of functions, i.e. symbolic and graphical. The use of graphing technology provides both symbolic and graphical representations of functions. The use of graphing technology enhances students' understanding of the function concept. The graphing technology offers an outstanding opportunity for teachers to present the function concept in a meaningful manner. Given two functions f and g, below are 5 basic algebraic operations ( f + g)( = f ( + g( ( f g)( = f ( g( ( f g)( = f ( g( f f ( ( )( = g g( ( f g)( = f ( g( ) The product of 2 functions where one of which is piecewise is a tedious to work on manually, can be illustrated as in figure 6. These are just few examples of how to enhance students understanding of concepts by giving them visual representation to accompany the verbal presentation and to discover properties of algebraic operations on their own. This may aid deeper understanding and longer retention of ideas in students. 7. Take advantage of technology to enhance teaching Learning activities should take advantage of the capabilities of technology, and hence should extend beyond or significantly enhance or facilitate what can be done without technology. Using technology to teach the same figure 7a,b : Another sample of algebraic function get evolved about x-axis for ( f g)( + 1 where the f is the above zigzag function and g is a sinus function. Learning activities should incorporate multiple representations of mathematical topics and/or multiple approaches to representing and solving mathematics problems. Furthermore, use of technology allows students to set up and solve problems in diverse ways, involving different mathematical concepts, by removing both computational and time constraints.[8] Thus, it is possible to show interesting new broader views( from the student s point of view ), thereby extending the teaching of mathematics such as in combining several topics in Calculus as given in figures 6 and 7. The algebraic function of f and g is then evolved about the y-axis to produce surface of an object. 8. Animated images Animation is the result of a creative process to substitute the real thing for computer-generated images sequenced together to give motion. The use of animations in mathematics is both refreshing, as it is complex. It provides a light-hearted relief in a serious and static presentation. It can effectively relay the message of the presentation using a realistic image. It has been used to demonstrate crucial S1C-13

5 concepts that cannot be otherwise explained by text or graphics. Animations fall into two categories: 2D and 3D animations. 2D animations are easier to create and cover a wider variety of motions. 2D animation consists of individual images sequenced together to display motion. Motions are simple to create and give the audience a dynamic experience to the presentation. } ε }ε figure 9 : The graph of the function f(x, y) = cos( sin(y) and its Taylor s polynomials of order 20. [9] figure 8 : The animation on the continuity of a function at a point x = 1. In figure 8 the picture represents the animation of the existences of δ for every given ε at point x = 1 to test the continuity of the function. It is much more realistic and fruitful to utilize 3D animated images to elaborate on theoretical concept as in figure 9. 3D animations can make images look more realistic. Once the object is modeled, textures and colours can be applied to it. Lights and cameras can also be placed to give the 3D object some shadow and shading.[7] The real power of computer graphics lies in its ability to accurately/approximately represent objects of interest within its limited resources combined with its ability to allow the user to interact with simulated worlds. To understand how significant these features are, consider this: our entire common-sense knowledge of the physical world exists only in our mental models and have developed, through interaction, a mental picture that enables us to predict with some accuracy what is physically reasonable. [2] 9. Conclusion Animations and interactive computer graphics methods provide new insights into the world of Mathematics. This paper describes some recent attempts to use computer graphics to understand problems in modern Calculus. Typical geometric problems of interest to mathematical visualization applications involve both static structures, such as complex numbers, and changing structures requiring animation. In practice, it is quite a challenge to construct intuitively useful images. Nevertheless, despite the apparent limitations of visual representations, their utility is far from being completely exploited. Today, there is a lively activity in the area of producing mathematically oriented computer graphics animations that are shown at computer graphics conferences and mathematics seminars. It is time to consider teaching a modern Calculus course that is a lean and lively group of topics from a dynamical systems perspective and uses technology to treat most topics graphically, numerically, and analytically. Computer visualization has broadened the horizon of teaching Mathematics. The art of drawing pretty and nice functions demand skills on the part of the lecturers even more. The scope of choosing clear understandable examples is expanding and still undoubtedly an art of teaching. 10. References [1] Mohammad Khalid Hamza, Bassem Alhalabi, Technology and Education : Between Chaos and Order, First Monday, Peer-Reviewed Journal of Internet. S1C-14

6 [2] Andrew J. Hanson, Tamara Munzner, and George Francis, Interactive methods for visualizable geometry,, IEEE Computer 27, 7:73-83, July [3] Nur Azman Abu, Shahrin Sahib and Rosli Besar, Dynamic images in Calculus and Analytical Geometry, MERA-ERA Joint Conference, Malacca, MALAYSIA, December [4] George Abdo and Paul Godfrey 1999 Functions of a Complex Variable [5] David Hart, Computational Mathematics : Where we are, where we're going, The University Computing Times, Indiana University, November [6] Nur Azman Abu, Shahrin Sahib and Rosli Besar, Use of Maple and Animation to Teach First Year Calculus, National Colloqium: Integration of Technology in Mathematical Sciences, USM, Penang, MALAYSIA, May 27-28, [7] Neo Mai, Dynamic images : Using animations in multimedia, Computimes Shopper Malaysia, pp July [8] Joe Garofalo, Tod Shockey, Holly Drier, Guidelines for Developing Mathematical Activities Incorporating Technology, 9 th International Conference, Society for Information Technology & Teacher Education, March [9] Art Belmonte and Philip Yasskin, Multivariable CalcLabs with Maple, For use with Stewart s Calculus, 4 th Edition, Brooks/Cole ITP, [10] Chieko Fukuda, Kyoko Kakihana, Katsuhiko Shimizu, The Effect of the use of Technology to Explore Functions(2), ATCM2000 Proceeding of the 5 th Asian Technology Conference in Mathematics, Chiang Mai, THAILAND, December 17-21, 2000, pp S1C-15