An Integrated Powerpoint-Maple Based Teaching-Learning Model for Multivariate Integral Calculus

In this paper, we present and evaluate a symbolic package based teaching-learning model for triple integral calculus. The model utilizes presentation and visualization technologies to assist in the teaching and learning of mathematical concepts and methods. The key mathematical concepts and integration techniques for triple integrals are introduced to students utilizing the animation feature of PowerPoint display supported by the graphic visualization function of the Maple software. To enhance students’ learning process, a series of workbooks and symbolic tools are developed utilizing Maple 10 to guide students step-by-step in their studies. The developed symbolic tools also enable students to display the three-dimensional (3D) integration domain graphically and to check their calculation results. An investigation undertaken shows that with the proposed teaching-learning model, a higher percentage of students achieved a better understanding of the concepts and a better capacity in evaluating triple integrals with complicated integration domains, compared to traditional teaching and learning with no technology use. We purport that our technology model supports understanding by showing the 3D figures in integration from different angles, and also facilitates independent learning by reducing the need to draw these figures by the students themselves.

Over the last two decades, a number of symbolic packages for mathematics courses have been developed using MATHEMATICA, MATLAB or MAPLE. These symbolic packages have been used widely as tools in mathematics education. A computer algebra system (CAS) was used to teach double integration (Mathews, 1990), number theory (Cheung, 1996), graph sketching (Kong and Kwok, 1999), matrix multiplication (Cecil, 2002), mathematical analysis of electronic signals and circuits (Røyrvik, 2002). Recently, Cook (2006) developed a MAPLE graphing tool named calcIIIplots for 3D calculus teaching. Man (2007) introduced CAS Maple and its applications in mathematics education at school level. Students were provided with opportunities in using MAPLE to perform mathematics exploration or problem solving. Symbolic packages have also been used in teaching other subjects such as physics and engineering (Beltzer & Shenkman, 1995;Johnson & Buege, 1995;Ward, 2003) and computation (Tonkes, Loch, & Stace, 2005). Tonkes et al. (2005) developed a learning model and designed a workbook for teaching computation using MATLAB to the first year students at the University of Queensland. Throughout the learning model, students work through the workbook to cover all topics to improve their proficiency in MATLAB. They found that the learning model has captured increased student support and has improved learning outcomes. The quality of student questions was improved and students displayed a deeper understanding of the mathematics.
Many researchers have measured and evaluated student learning of mathematics using symbolic packages (Kramarski & Hirsch, 2003;Tall, 1991;Lagrange, 1999;Drijvers, 2000;Perjési, 2003;Heid, 1988;Kendal & Stacey, 2002;Hayden & Lamagna, 1998). It is found that symbolic packages enable students to achieve high level of logical analytical reasoning by visually supporting the concepts and the proofs with graphics presented through the packages (Fuchs, 2001;Drijvers, 2002;Peschek & Schneider, 2002). Maple, based on a powerful scientific programming environments and libraries packages of special routines, is found to be one of the most popular systems because it is well-suited to aid college students to learn mathematics through verifying calculation and plotting complicated graphs, and also combines mathematical capabilities with a text editor.
Via the use of PowerPoint and Maple, we have developed an Integrated PowerPoint-Maple based (IPM) teaching-learning tool to aid in the teaching and learning of multivariate integral calculus, and an introduction of the tool has been published as a conference paper for the "Thirteenth Asian Technology Conference in Mathematics -2008 (Noinang, Wiwatanapataphee, & Wu, 2008)". In this paper, we present further details of the tool and investigate its effectiveness in improving the outcomes of teaching and learning. Briefly, the tool includes two parts: a teaching tool and a learning tool. The teaching tool consists of a series of PowerPoint slides integrated with Maple animations, and is designed for teachers to use in classroom teaching. The learning tool consists of a set of interactive Maplets and Maple worksheets. It is designed to enable students to have self-planned learning and selfassessment, and to reinforce students' conceptual understanding of integral calculus. The rest of the paper is organized as follows. First, the teaching model supported by symbolic calculation and graphic visualization of the teaching tool is presented. Afterwards, the Maple based step-by-step symbolic learning model using the learning tool is given. Evaluation of the teaching-learning model is then given and various maple procedures used in this paper are presented in Appendices.

Powerpoint based Presentation with Computer Graphic Visualization Support
Teaching triple integral calculus is a challenging task. In traditional mode of classroom teaching, students often find it difficult to understand instructors' explanations due to the difficulty in imagining the 3D integration domain. To overcome this problem, a teaching tool is developed utilizing the animation feature of PowerPoint display and the graphic visualization function of the Maple software.
For students to understand the concept of triple integrals and know how to evaluate triple integrals in different coordinate systems, the IPM teaching tool is designed to cover the following four essential components; (a) concepts of triple integrals; (b) evaluation of triple integrals in rectangular coordinates; (c) evaluation of triple integrals in cylindrical coordinates; (d) evaluation of triple integrals in spherical coordinates.
For each of the components, the teaching tool consists of a number of PowerPoint slides and Maple graphs. The PowerPoint slides are used to explain the concepts and theories, while the Maple graphs are used to help students to visualize the concepts and methods graphically.
For the concept of triple integrals (Stewart, 2003), a PowerPoint slide is used to define the triple integral as a triple Riemann Sum, as shown in Figure 1a. Then, based on the definition, a number of examples are given to demonstrate the application of triple integrals. For example, if f(x, y, z) is the material density of the object Ω, then , ,

∑∑∑ ∫∫∫
is the total mass of the object, as ijk ijk f V ∆ is the approximation of the mass of one of the subboxes of Ω and the summations are to add the mass of all the sub-boxes in Ω together. The concept of triple integral, as a triple Riemann sum, can be clearly illustrated graphically by the Maple graphic visualization function as shown in Figure 1b.  To study the method for evaluation of triple integrals over a general domain in rectangular coordinates, we first use Maple graphic functions to demonstrate that integration domains in rectangular coordinates can be classified into x-cap region, y-cap region, z-cap region or combination of them (Wu & Wiwatanapataphee, 2008). Then, we use PowerPoint slides to describe how to evaluate triple integrals over each type of the regions. As an example, Figure 2 gives the PowerPoint slide and the Maple graph used for demonstrating how to evaluate a triple integral over a z-cap region. In the classroom lecture presentation, firstly, the instructor clicks on Figure 2a to start up the animation of the object in Figure 2b demonstrating graphically that a z-cap region is a region bounded by an upper surface 2 ( , ) z z x y = and a lower surface 1 ( , ) z z x y = with a projection D on the xy-plane. Then, through the PowerPoint slide, the instructor shows that a z-cap region E can be visualized graphically as a set of points (x, y, z) for which (x, y) takes values in D and for each point (x, y) in D, z changes from 1 ( , ) z z x y = to 2 ( , ) z z x y = and consequently the evaluation of a triple integral over a z-cap region E becomes first evaluating a single integral with respect to z from 1 ( , ) z x y to 2 ( , ) z x y and then evaluating a double integral with respect to x and y over D. The graphic visualization helps to understand the integration procedure.  For evaluating triple integrals in cylindrical coordinates, first, we use the Maple graphic function to demonstrate graphically that a point in space can be identified by the cylindrical polar coordinates ( , , ) r z θ representing the intersection of the cylindrical surface of radius r, the vertical plane with angle θ from the xz-plane and the horizontal plane with distance z from the xy-plane, as shown in Figure 3a. Then, from the figure, we show on the PowerPoint slide how to relate the ( , , ) r z θ coordinates with the x, y, and z coordinates, and then, we use the Maple graphic function to demonstrate that a differential element (as shown in Figure 3b) can be obtained by cutting the object with two cylindrical surfaces (r = r, r = r +∆r) and two z-planes (z = z, z = z +∆z) as well as two θ-planes ( , ) θ θ θ θ θ = = + ∆ . Consequently, the volume of the differential element can be calculated from its geometry as follows 2 2 where A xy is the area of the projection of the differential element on the xy-plane. As dr is infinitesimal,   Finally, we demonstrate that a domain as shown in Figure 4, defined in cylindrical polar coordinates, can be visualized as a z-cap region and thus the integration over the region becomes evaluating a single integral with respect to z firstly from 1 ( , ) z u r θ = and then evaluating a double integral with respect to r and θ over the region D in polar coordinates.
Similarly, for evaluating triple integrals in spherical polar coordinates, we first use the Maple graphic function to demonstrate that a point in space can be identified by a spherical polar coordinates ( , , ) ρ θ φ representing the intersection of the spherical surface with radius ρ and centre at the origin, the vertical plane with angle θ from the xz-plane, and a cone surface with angle φ with the z-axis, as shown in Figure 5a. Then from the graph, we derive the relation between the x, y, and z coordinates and the ρ , θ , and φ coordinates, and then, we use the Maple graphic function to demonstrate that a differential element (as shown in Figure 5b ). We then proceed to calculate the volume of the differential element by utilizing the formula for the volume of spherical cone where ρ is the radius of the sphere and φ is half of the cone angle as shown in Figure 7.
Let 1 V be the volume of the solid cut by the two spherical surfaces ( ρ ρ = and Similarly, the volume of the solid cut by the two spherical surfaces within the cone φ φ = is Hence, the volume of the solid cut by the two spherical surfaces and the two cone surfaces is 1 2 V V − and the volume of the differential element is Using the Taylor series approximation to the first order, we have We should also address here that the lengths in the r , φ and θ directions of the element are respectively d ρ , d ρ φ and sin d ρ φ θ (as shown in Figure 5b) and the volume of the element is the same as that of a rectangular box with dimension sin d d d Finally, in the PowerPoint slide ( Figure 6), the formula for evaluation of a triple integral in spherical coordinates is given.

Maple Based Step-by-Step Symbolic Learning Tool
For most students, the main difficulty in the study of triple integral calculus is the sketching of integration domain and the subsequent determination of integration bounds (Cook 2006, Wu & Wiwatanapataphee, 2008). To enhance students' self-learning capability, an interactive symbolic learning tool based on Maple is developed for the triple integral calculus. The tool, a Maplet, allows students to interact with computer and solve problems step by step. Basically, the Maplet has three basic functions: (a) display the graphs of functions; (b) define the integral; (c) evaluate the integral and show the symbolic result in an output window. Obviously, the graphic display is important as it helps students to graphically visualize the integration domain bounded by general surfaces defined by functions, so as to determine whether the domain is a z-cap region, y-cap region, or x-cap region and consequently determine the integration order and integration bounds for the definition of the problem. The following gives an example.

Example
Find the mass of the solid with constant mass density ( , , ) 1 x y z ρ = bounded by a cylindrical surface 2 2 16 x y + = and three planes:

Solution
Firstly, in the input window, enter the equations for the cylindrical surface and the three planes, and also enter the display range in the x, y and z directions. Then, click on the plot button to view the surface and planes and their interfaces as shown in Figure 8. Obviously, this is a z-cap region bounded by a lower z-surface 3 2 z y = − − and an upper z-surface 3 z y = + with a projection D on the xy-plane. To determine the bounds of integration in the x and y directions, project the integration domain onto the xy-plane as shown in Figure 9. Obviously, To evaluate the above integral using Maplet, firstly choose the order of integration to be "dzdxdy", then input the bounds of integration, then double click on the "Show integration region" button to display the region of integration and then click on the "Evaluate the integral" button to evaluate the integral for the mass m step-by-step on the result window, yielding 3 72 3 16 4 7 96 arcsin 4 m π = + + + as shown in Figure 10.

Investigating the Effectiveness of the IPM Based Teaching-Learning Model
In general, mathematics is considered, within the community of students and social in general, as a hard subject, and triple integral calculus is particularly the case. With the aim of making the integral calculus simple, interesting and attractive, a teaching-learning model supported by symbolic calculation and graphic visualization has been developed and presented in the previous two sections.
To evaluate the effectiveness of the developed teaching-learning model in improving students' learning outcomes, an investigation was undertaken at Kasetsart University Chalermphrakiat Sakonnakhon Province Campus involving two teachers, one observer and 73 engineering students. The students were divided into two groups: a computer-aided group (IPM) consisting of 32 students, and a traditional teaching-learning (TTL) group with 41 students. Firstly, students in both groups were asked to sit in a pre-test. The pre-test consists of 35 multiple-choice questions, tapping into conceptual and procedural aspects of the 3D integration. For each question, four answers are given including one correct answer and three incorrect answers. If the correct answer is chosen for a question, the student receives a score of one for the question or otherwise zero. Thus, the maximum score available for the pre-test is 35. After the test, students in the IPM group were taught the triple integral by using the IPM teachinglearning model, while the other group was taught by using the traditional teaching learning model. Then, we conducted a post-test. The post-test also consists of 35 multiple-choice questions with a total core of 35. The following is an example question x y x f x y z dydzdx The results obtained by both groups in the pre-test and post-test are summarized in Table 1. It is clear from the results that students in the integrated PowerPoint Maple (IPM) group achieved higher mean score than those in the TTL group. To determine whether the difference of the mean scores between the two groups is due to group membership (effect of teaching-learning model) or others, the one-way ANOVA variance analysis (Peck, 2008) is carried out. Table 2 and Table 3  We also undertook an open-ended student survey to get students' opinions on the IPM teaching-learning model. In the survey, students were asked whether the proposed teachinglearning model helped them to have a better understanding of the mathematical concepts and facilitated their learning of the methods for the evaluation of multivariate integral calculus. From the analysis of the student responses we concluded that the IPM teaching-learning model had advantages over the traditional mode of teaching and learning in two principal aspects. First, the integrating of the Maple animation with the PowerPoint presentation makes it much easier for instructors to explain certain concepts and methods and for students to understand the explanations in the classroom lectures. On the other hand, in traditional mode of lecture presentation, the instructor has to be an expert with artistic skills in graph sketching and had to spend a lot of time in drawing proper graphs to explain concepts and methods. With the Maple graphic function, the instructor can show students the figures from different angles and thus have more time to focus on concepts and methods with the graphic visualization aids. The second major advantage of the developed model is that it enhances students' self-learning capability for the subject. In the traditional mode of self-learning, those students lacking artistic skills found it extremely difficult to sketch the integration domain and subsequently determine the integration bounds even though they understand the concepts and methodology, while with the learning package, students can use the graphic function of Maple to draw the integration region easily and systematically even without any artistic skills. Also, through the exercises, students can gradually develop the skills of graph sketching. In addition, the learning package can provide solution in symbolic form which can then be used to check the solution obtained manually.

Conclusion
An efficient computer aided mathematics teaching-learning model and tool for triple integral calculus has been developed to enhance students' interest in mathematics. It provides students with an opportunity to have self-motivation, self-planned learning and selfassessment. Teacher and students have more time to focus on more practical examples to cover a wider variety of problems in class. Interactive Maplets of triple integrals enable students to analyze mathematics concepts step by step and also to self assess their homework. Maplets provide fast solutions and good visualizations of applied mathematics problems which can help students to identify patterns and see connections. Students can change functions and the values of parameters during the process of learning to understand the influence of different parameters to the output critically. These allow students to deepen their knowledge of multivariable calculus.
The paper has also demonstrated step by step how to use the IPM teaching-learning model for teaching triple integral calculus. Many useful graphs and Maple animation objects are also given in the paper which can be used to assist in explanation of concepts and methods for the triple integral calculus. The paper has thus provided a useful guideline and resources for the teaching of multivariate integral calculus using modern computation technology and graphic visualization.