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Aug 8, 2008 - This paper presents a prototype of a computer learning assistant ILMEV (Interactive Learning - Mathematica. Enhanced Vector calculus) ...
Towards the Development of an Automated Learning Assistant for Vector Calculus: Integration Over Planar Regions By Yuzita Yaacob(1) , Michael Wester(2) and Stanly Steinberg (3) (1) Industrial Computing Department, Faculty of Technology and Information Science, Universiti Kebangsaan Malaysia, Malaysia, 43600 UKM Bangi, Selangor, Malaysia. [email protected] (2) Department of Mathematics and Statistics, Center for High Performance Computing, University of New Mexico, Albuquerque NM 87131-1141 USA. (3) Department of Mathematics and Statistics, Cancer Research and Treatment Center, University of New Mexico, Albuquerque NM 87131-1141 USA. Received: 11 February 2008

Revised: 8 August 2008

This paper presents a prototype of a computer learning assistant ILMEV (Interactive Learning - Mathematica Enhanced Vector calculus) package with the purpose of helping students to understand the theory and applications of integration in vector calculus. The main problems for students using Mathematica is to convert a textbook description of a problem into a form that Mathematica can understand and then choose the correct solution technique. ILMEV is designed to help students with this process. The typical presentation of this material in textbooks is not easily adapted to an interactive interface, so we developed a model of vector integration that allows ILMEV to present a structured overview of this material that helps students choose a correct solution method. Mathematica can solve the translated problem using its cylindrical algebraic decomposition (CAD) algorithm, but does not provide any explanation of what is being done. To overcome this, we implemented a simplified CAD algorithm which is used to reduce integrals appearing in vector calculus to sums of iterated integrals which Mathematica can then compute. This allows students to interactively compute closed form solutions to many two dimensional textbook examples using the ILMEV interface. ILMEV is built on important pedagogical guiding principles – interactivity, visualization, experimentation, White and Black Box Principle, multiple representations, and step-by-step technique with explanations. The user interface was critical for the implementation of the guiding principles. 1

INTRODUCTION

In recent years, there has been much interest in incorporating computers into the teaching and learning of university level science and mathematics courses, especially in calculus, using a computer algebra system (Yasskin & Belmonte, 2003; Tiwari, 2007; Guyer, 2008). While educators believe that computer algebra systems (CASs) have great potential in enhancing mathematics education, the limitations of such software have often made them quite difficult to use in the classroom (Tintarev, 2002). By design, most CASs only give an answer to the students without any explanation of the steps used to obtain the answer. ILMEV is a tool developed to assist students in learning how to perform integration over domains in an

introductory university level vector calculus course at the National University of Malaysia (Yaacob, 2007; Yaacob, et al, 2008). These students have previously completed their mathematics curricula for upper secondary schools in Malaysia: calculus, algebra, geometry and trigonometry. The content of the course is based on the textbook written by Davis and Snider (1995). Vector calculus was chosen because it is critical for solving problems in engineering and science. Many educational programs require a course on this topic and the mathematics involved is difficult, even for talented and well-trained students (Davis, Porta & Uhl, 1999). ILMEV is currently being extended so that it can compute integrals of tangential components of vector fields over curves and integrals of normal components of vector fields over surfaces. This capability will then allow the inclusion of the multidimensional versions of the Fundamental Theorem of Calculus and integration by parts theorems, including the divergence and Stokes' theorems (Wester, Yaacob & Steinberg, 2009). The careful application of pedagogical guiding principles ensured that ILMEV would be useful in the teaching and learning of the subject matter (Leinbach et al., 1997). The key to implementing these principles is a user-friendly interface to Mathematica that allows users access to the full power of Mathematica without mastering the idiosyncrasies of its interface (Tintarev, 2002). 2

A TYPICAL INTEGRATION PROBLEM

Consider a non-trivial problem like many encountered in vector calculus books (Davis and Snider, 1995): The six lines

y = x +1, y = − x +1, y = x − 1, x = − x − 1, y = 1 / 2, y = 1 / 2,

bound a hexagon in the plane. Write the area of this region as a sum of iterated integrals and then evaluate the integrals to find the area of the region. Check your result by finding the area using elementary geometry. The crucial part of this problem is understanding the geometry. ILMEV's interface facilitates this task.

2D ∫ c ds

∫ c F• t ds

∫ c F• n ds

∫∫



dA

International Journal for Technology in Mathematics Education, Volume 16, No 2

Figure 1 Beginning ILMEV session

Next, ILMEV displays the iterated integrals symbolically:

The student must start ILMEV in Mathematica and then choose the type of problem to be solved. Part of the menu is given in Figure 1. For this problem, the student must select the area integral. Next, the student needs to specify the number of boundary equations in the problem and then starts to enter the equations. After entering an equation, ILMEV helps the student provide the appropriate corresponding inequality. Here is the first part of the solution of the problem:

The student can then request that ILMEV evaluate the integrals, which here yields an area of 3/2.

The number n of inequalities: 6 Equation #1: y=x+1 The graph of y = x + 1 where -4 ≤ x ≤3: 4 3 2 1

-4

-3

-2

-1-1

1

2

3

-2 -3

Figure 4 Quadratic Equations

The inequality for y = x + 1: y≤x+1 The graph of y ≤ x + 1: 4

ILMEV can also compute integrals for regions described by quadratic equations in the same way as was used for linear equations. For example the region shown in Figure 4 is given by

3

y≥x , y≤1− x , y≤3/ 4

2

2

ILMEV gives the area as

1 -4 -3 -2 -1 -1

1

2

3

-2

which is easily evaluated by Mathematica, producing

-3

Figure 2 Part of an ILMEV Session The student now inputs the remaining five equations and converts them to inequalities. ILMEV prints out some information about the intersection points of the lines, which gives the corners of the hexagon, and then plots the hexagon (see Figure 3).

0.4 0.2 -1

-0.5 -0.2 -0.4 Figure 3 The Hexagon

2

0.5

1

(4

)

2 −1 / 6

An advantage of ILMEV is that notebooks can easily be reused. For instance, in this notebook the inequality y≥x 2 can be replaced by y≥2x2 −1/2 , and then the notebook can be re-executed. As another example, a student could replace the top three segments in the hexagon by y=1− x 2 and then recompute the area. The graphics capabilities of ILMEV (plotting each inequality separately and then all of them together) will help the student see that the termini of the new curve coincide with the termini of the lower boundary of the hexagon. 3

MULTIDIMENSIONAL INTEGRATION

The first mathematical problem we encountered in trying to implement ILMEV was that the presentation of the material on multidimensional integration in standard texts had to be substantially reorganized to be useful for International Journal for Technology in Mathematics Education, Volume 16, No 2

implementation. The resulting organization was used to design the beginning part of the interface where students choose the type of problem they wish to work on. This reorganization was completed for the spatial dimensions 1, 2 and 3 and is included in the ILMEV interface (Figure 1 shows the possible 2D integrals). Algorithms for solving all of the problems described in the interface have been developed (Wester, Yaacob & Steinberg, 2009) and will be included in future versions of ILMEV. The critical CAS tool for solving integration problems is the cylindrical algebraic decomposition (CAD) algorithm (see Hong, Liska & Steinberg, 1997 for an introduction). The CAD algorithm can break up any region defined by polynomial inequalities into a union of disjoint simple regions called cylinders. The algorithm for computing a CAD was developed in the latter part of the twentieth century and is important for solving many geometric problems using a CAS. Recent versions of Mathematica implement CAD functions. For this version of ILMEV, we implemented a simple CAD that can handle many elementary problems found in common mathematical textbooks. The next version of ILMEV uses the Mathematica algorithm (Wester, Yaacob & Steinberg, 2009). For the above example, the CAD breaks the hexagon into three pieces, a square in the middle and a triangle at each end: -1 ≤ x ≤ -½ and -1-x ≤ y ≤ 1+x -½ ≤ x ≤ ½ and - ½ ≤ y ≤ ½ ½ ≤ x ≤ 1 and -1+x ≤ y ≤ 1-x This decomposition gives all of the information needed to write the iterated integrals given above. 4

ILMEV’S USER FRIENDLY INTERFACE

Some educational packages take a conversational approach, using question-answer dialogs to interact with the student through a specific logical sequencing. Since our goal was to produce a tool which took advantage of pedagogical guiding principles such as interactivity, visualization and experimentation, we elected not to go in this direction. We decided that the task called for direct manipulation and menu-based interfaces using event-based dialogs where the user initiates the dialog sequences (see Figure 4). The area we want to integrate over is formed by the intersection of n linear and/or quadratic inequalities where n is a positive integer greater than or equal to 2. The number n of inequalities: -3 Sorry, invalid input. (Note: Accepted value for n is 2, 3, 4, … only.) Do you wish to enter again the number n of parametric equations? (Y/N) Y Figure 4 Correcting Input

Computer algebra programs are very unforgiving with regard to user errors and demand that the user specify the desired command or input in exactly the required format (syntax of commands as well as the right order of operations), with little or no help from the software (Tintarev, 2002). However, in ILMEV, we help to reduce this inadequacy by providing the following input guidance facilities: i.input mathematical expressions are checked and, if necessary, modified to suit Mathematica syntax ii.conditions and restrictions on the input are noted when applicable iii.error messages are produced for incorrect input iv.suggestions are given on how to improve a particular input This allows users to easily do input and editing of a mathematical expression to follow Mathematica's format. The advantage of this facility is that users do not need to interact with Mathematica directly and need to know very little of Mathematica’s syntax and command structure. All input to ILMEV is done through pop-up dialog boxes that are used to generate mathematical, plotting and animation commands. There are two types of pop-up boxes: dialog and text only boxes. Input such as equations, inequalities and plotting commands are entered into dialog boxes, and error messages for incorrect input and suggestions on how to improve a particular input are in the form of text boxes (no input is allowed). Expressions can be entered in a dialog box by typing at the keyboard. Dialog boxes and text boxes may be moved around the screen and hidden if desired. If the user makes an error, it can be corrected by retyping the information. The results of all operations by the user are pasted in the Mathematica notebook automatically, documenting the solution of the problem. We developed the ILMEV interface to be powerful enough to support the implementation of several important learning principles. 5 PEDAGOGICAL GUIDING PRINCIPLES OF ILMEV ILMEV was developed using guiding principles used in computer-based education: interactivity, visualization, experimentation, White and Black Box Principle, multiple representations, and step-by-step technique with explanations (Kutzler, 1994; Kutzler, 1998; Kutzler, 2008; Yaacob 2007). 5.1

Interactivity

Reeves (2006) defined a learning environment as interactive in the sense that a person can explore through it, select relevant information, respond to questions using computer input devices such as a keyboard, mouse, touch screen or voice command system, solve problems, complete challenging tasks, create knowledge representations, collaborate with others near or at a distance, or otherwise engage in meaningful learning activities. Concepts about interactive learning (Grabinger, 1996) strongly influenced the design of ILMEV. The users International Journal for Technology in Mathematics Education, Volume 16, No 2

can choose their own learning situations such as by moving from one topic to another easily and efficiently (Heugl, Barzel & Furukawa, 1997). Moreover, we also stress the higher-order outcome in interactive learning such as motivation and intellectual curiosity (Drijvers, 1997). It is important that the students feel motivated to learn mathematics and that their mathematical intellectual curiosity be enhanced. ILMEV's interface strongly encourages interactivity. Since all of the students work is in a Mathematica notebook, they can easily modify what they have done. For example, plots that appear in the notebook can easily be improved using the mouse to change the size of the graphs and their positions, as well as zooming in or out. In addition, it is not difficult to modify one or more of the boundary lines and redo the calculation. 5.2Visualization As stated in textbooks (see the beginning of Section 2), many integration problems are difficult to visualize. As shown in Figures 2 and 3, ILMEV uses the powerful and versatile graphics facilities in Mathematica to assist students in understanding the relationship between algebraic formulas and geometry. It is easy to design new graphics or animations in ILMEV, just by modifying the parameters in the existing examples (Heugl, Barzel & Furukawa, 1997). By creating their own visualizations of a particular subject matter, users will discover the mathematical concepts themselves and, consequently, improve their understanding (Amrehein, Bengtsson & Maeder, 1997). Additionally, graphics allow users to view a problem from different perspectives, thus helping them to solve a problem they had difficulty visualizing before, and at the same time, improving their spatial visualization capabilities. For instance, basic regions in ILMEV are described using inequalities in the form of f(x, y) ≤ 0 or f(x, y) ≥ 0. Students frequently have difficulty choosing between ≤ and ≥. Thus, ILMEV first plots f(x, y) = 0 and the students can use this plot to help them choose the appropriate inequality. 5.3Experimentation Many of today’s psychological theories of learning consider learning to be an inductive process in which experimentation plays an important role (Kutzler, 1994; Kutzler, 1998; Kutzler 2008). The learning process in ILMEV is based on experimentation where the user applies known algorithms to generate examples and consequently forming a conjecture through the observation of the examples (Heugl, Barzel & Furukawa, 1997). The typical activities involved are experimenting (by trial and error) with various parameters to produce more examples, form conjectures and plausible reasoning (Heugl, Barzel & Furukawa, 1997). The most important outcome of this learning process is a more pupil-centered, experimental learning (Heugl, Barzel & Furukawa, 1997). Moreover, within the class or tutorial lab period, students and teachers can only observe a limited number of problems and a hefty portion of the student's work could be incorrect due to calculation errors (Kutzler, 1998). By

using ILMEV, they are able to experiment with more problems in the class or tutorial lab period. For instance, they can try different types of examples by changing the values of the parameters in the problem and form conjectures through the observation of these examples. They can also create more complex examples taken from textbooks or their own imagination. Testing and studying the influence of parameters on the behaviour of the solution is an easy task in ILMEV. 5.4The White and Black Box Principle The White and Black box principle was introduced (Buchberger, 1993) to reconcile the extreme views of the “traditionalists” who wanted to ban computers from mathematics education on the grounds that they could prevent deeper understanding of mathematical concepts, with the progressives who disagreed with training the students to use mathematical operations that can be done faster and more effectively by using a computer. In the white box phase, the students are taught the mathematical concepts, algorithms and mathematical theory for solving a particular problem. In addition, the necessary computation skills are developed during the calculation process. ILMEV is used later, in the black box phase, as a tool for experimenting with the algorithms that the students have studied earlier in the white box phase. They can test the conjectures that were made by applying the algorithms with various input values and calculating the results by hand. Although it is certainly possible to solve problems like the area of the hexagon by hand, it is tedious to do so, and this is the case for increasingly more complex geometries. Using ILMEV as a support tool makes this process easy as it has the capability to perform the required computations. 5.5Multiple Representations Understanding mathematics requires students to understand problems written as text and to write clear explanations of their answers. They also need to be able to understand the connections between facts written as formulas and the geometry of plots of quantities related to the formulas. The statement of the problem at the beginning of Section 2 and its partial solution in Figure 3 illustrate how ILMEV helps develop these skills. In ILMEV students can explore symbolic, numeric, text, graphic, sound, animation and still image representations. Figure 3 illustrates the representation of a geometric object as lines that define its boundary and inequalities that define its interior. Importantly, users can construct their own knowledge by exploring these multiple representations (Heid, Sheets & Matras, 1990; Drijvers, 1997). 5.6Step-by-Step Technique Commands to ILMEV are presented in a step-bystep manner as illustrated in Figure 2. For example, users input all the equations and inequalities in dialog boxes, International Journal for Technology in Mathematics Education, Volume 16, No 2

and then ILMEV presents the steps required to obtain the solution along with some discussion. The user has control over when to execute a particular step in a solution sequence by clicking on a particular button (such as “compute”), in order to perform a task. Thus, ILMEV naturally encourages step-by-step solutions of problems and provides explanations (Drijvers, 1997). 6 NEW DEVELOPMENTS Here we describe how the new version of ILMEV will solve more advanced problems of a type commonly found in calculus textbooks. The implicit curve

ILMEV gives the area integral as

The new version of ILMEV can also compute the length of the boundary and the vector flux and work integrals over the boundary of this region (Wester, Yaacob & Steinberg, 2009). The formulas for these integrals are complex. Computing these integrals by hand, especially by students, is tedious and error prone.

f ( x, y ) = − x 2 ( 2 + x ) + y 2 = 0 encloses a region of finite area. Find the area and the length of the perimeter of this region. The initial problem for the student is to visualize the region. The first step in the solution is to use ILMEV to plot the curve as shown Figure 5. Most students will require several attempts to obtain proper ranges for x and y to see the region. All the student needs to do is change the values defining the ranges in four input boxes.

Figure 6 The Region .

Figure 5 A Self Intersecting Curve From the plot, the student can see that the point

( − 2,0) is in the interior of the finite part of the region. Evaluating f ( x, y ) at this point gives -4, so the student might guess that the finite region is given by f ( x, y ) ≤ 0 , which is plotted in Figure 6. In fact, the student gets too much and needs to figure out how to remove the infinite part of the region shown in the figure. This can be done by limiting x to be negative, so the student writes the region as

− x ( 2 + x) + y ≤ 0 ∧ x ≤ 0 2

Figure 7 The Region with Finite Area 7 CONCLUSION

2

Plotting this region gives Figure 7. Discovering the correct description of the region is facilitated by ILMEV's flexible interface in which it is easy to change the details of plots.

The response of students to ILMEV and suggestions from colleagues has encouraged us to to significantly revise the current version of ILMEV. Our experience with the interface points to several improvements. For example, all of the polynomial equalities in the given problem should be plotted together International Journal for Technology in Mathematics Education, Volume 16, No 2

first. A facility that would allow the students to evaluate a polynomial at a point selected by a mouse would help the student better understand polynomial inequalities. Next, ILMEV will be extended so that it can handle boundary integrals and the multidimensional versions of the integration by parts theorems (Wester, Yaacob & Steinberg, 2009). ACKNOWLEDGEMENTS This work was supported by IRPA 04-02-08-10005 grant (Kulliyyah of Engineering, International Islamic University Malaysia) under the Ministry of Science, Technology and Innovation (MOSTI), Malaysia. We also wish to express our sincere gratitude to Azami Zaharin and Nuryazmin Ahmat Zanuri from the Faculty of Engineering, National University of Malaysia for assisting us in the user testing phase. REFERENCES Amrehein, B., Bengtsson, M. & Maeder, R. (1997) Visualizations for mathematics courses based on a computer algebra system, Journal of Symbolic Computation, 23, 447-452. Buchberger, B. (1993) Teaching math by math software: Newton’s method as an example of the white box/black box principle, presented at the 2nd Krems Conference on Mathematics Education, Krems, Austria. Davis, B., Porta, H. & Uhl, J. (1999), Vector calculus & Mathematica (VC&M) (computer program), U.S.A: Math Everywhere, Inc. Davis, H. & Snider, A. (1995), Introduction to vector analysis, seventh edition, U.S.A.: William C Brown Pub. Drijvers, P. (1997). What issues do we need to know more about? Questions for future educational research concerning CAS In Berry J., Monaghan, J., Kronfellner, M. & Kutzler, B. (eds.) The State of Computer Algebra in Mathematics Education, Sweden: Chartwell-Bratt (Publishing and Training) Ltd. Grabinger, R. (1996), Rich environment for active learning: A handbook of research for educational communications and technology, New York: Macmillan Publication. Guyer, T (2008), Computer algebra systems as the mathematics teaching tool, World Appl. Sci. J., 3(1), 132139. Heid, M.K., Sheets, C. & Matras, M.A. (1990), Computerenhanced algebra: new roles and challenges for teachers and students, In Cooney, T. & Hirsch, C. (eds.), Teaching and Learning Mathematics in the 1990s (NCTM 1990

Yearbook), National Council of Teachers of Mathematics: Reston, Va. P., 205-211. Heugl, H., Barzel, B. & Furukawa, A. (1997), The influence of computer algebra in the teaching and learning of mathematics, In Berry J., Monaghan, J., Kronfellner, M. & Kutzler, B. (eds.), The State of Computer Algebra in Mathematics Education, Sweden: Chartwell-Bratt (Publishing and Training) Ltd. Hong, H., Liska, R. and Steinberg, S. (1997) , Logic, quantifiers, computer algebra and stability, SIAM News, 30(6 ), 10. Kutzler, B. (1994), DERIVE – The future of teaching mathematics, The International DERIVE Journal, 1(1), 318. Kutzler, B. (1998), Solving linear equations with the TI-92 (experimental learning/visualization/scaffolding method), Austria: Bk Teachware Pub. Kutzler, B. (2008), Technology and the Yin & Yang of teaching and learning mathematics: the essence of using technology, in particular computer algebra systems (CAS), in education, published by B. Kutzler, Linz, Austria, ISBN 978-3-901769-83-2. Leinbach, C., Aspetsberger, K., Barzel, B., Fuchs, K., Furukawa, A., Heugl, H., Mann, G., Rothery, A., Sato, T. & Schweiger, F. (1997), The curriculum in a CAS environment. In Berry J., Monaghan, J., Kronfellner, M. & Kutzler, B. (eds.), The State of Computer Algebra in Mathematics Education, Sweden: Chartwell-Bratt (Publishing and Training) Ltd. Reeves, T. (2006), Interactive. Journal of Interactive Learning Research. Retrieved Sept 19, 2006. http://www.aace.org/pubs/jilr/scope.html Tintarev, K. (2002), Design of user interface for computer-aided instruction of mathematics, in Mathematics and Mathematical Education, World Scientific eProceedings, S. Elaydi, E S Titi, M Saleh, R Abu-Saris & S K Jain, eds., 291-305. Tiwari, T. (2007), Computer graphics as an instructional aid in an introductory differential calculus course, International Electronic Journal of Mathematics Education (IEJME), 2(1), 32-48. Wester, M., Yaacob, Y. & Steinberg S. (2009), Computing integrals over polynomially defined regions and their boundaries in 2 and 3 dimensions, submitted for publication. Yaacob, Y. (2007), Interactive learning – Mathematica enhanced vector calculus (ILMEV), PhD Thesis, International Islamic University Malaysia, Malaysia. International Journal for Technology in Mathematics Education, Volume 16, No 2

Yaacob, Y., Steinberg, S., Wester, M., Ismail, A. & Salleh, H. (2008), ILMEV (Interactive learning – Mathematica enhanced vector calculus), in Proceedings of the Seminar Pendidikan Kejuruteraan dan Alam Bina (PEKA 2008), Bukit Merah Laketown Resort, Perak, Malaysia, Faculty of Engineering, National University of Malaysia (ISBN 978-983-2982-25-8), 74-83.

of New Mexico. He is an expert on symbolic and numeric computing and their application to continuum mechanics and signal transduction in cell biology.

Yasskin & Belmonte, (2003), vec_calc (computer program), College Station, Texas: Texas A&M University, U.S.A. BIOGRAPHICAL NOTES Dr. Yuzita Yaacob is a senior lecturer in the Department of Technology and Information Science, University Kebangsaan Malaysia. Her main interest is the use of computer algebra in education. Dr. Michael J. Wester is an applied mathematician who works at the University of New Mexico (in Albuquerque) and consults via his company Cotopaxi. He has been involved with computer algebra since 1974. Dr. Stanly Steinberg is Professor Emeritus in the Department of Mathematics and Statistics at the University

International Journal for Technology in Mathematics Education, Volume 16, No 2

International Journal for Technology in Mathematics Education, Volume 16, No 2