By Ellington, Aimee J Hardin, Jill R
INTRODUCTION Representation is one of the Process Standards that the Principles and Standards for School Mathematics  lists as useful for obtaining and applying mathematical skills and concepts. The ability to clearly and accurately represent mathematical ideas is key to analyzing real world problem situations. The National Council of Teachers of Mathematics (NCTM) advocates that K-12 mathematics courses should incorporate all aspects of mathematical representation. One significant aspect is to “use representations to model and interpret physical, social, and mathematical phenomena” [4, p. 67]. This is fundamental to understanding the connections between concepts and applications and is the heart of mathematical modeling. The Committee on the Undergraduate Program in Mathematics (2004)  has recommended that a mathematics degree include a course that focuses on real-world mathematical applications. They recommend that the course emphasize all aspects of the solution process from the development of a model to the interpretation of the numerical values the process yields.
Mathematical Modeling is a semester-long course at Virginia Commonwealth University that emphasizes building accurate models and analyzing model solutions. With a focus on problems that reflect real-world situations, this college course incorporates many of the NCTM (2000) guidelines on the development and use of mathematical models. To meet state certification requirements, this is a required course in the degree tracks for pre-service middle and high school mathematics teachers. Each semester, the class is also comprised of students preparing for careers in operations research, engineering, or other mathematics-related fields. This diverse population provides for a dynamic learning environment and an opportunity for all students to realize the importance of mathematical modeling to a variety of career paths.
To present students with a more accurate picture of the role technology plays in modern applications of mathematical models, student versions of state-of-the-art software play an integral role in our course. This use of technology adds its own set of challenges to course instruction, however, and previous offerings of the course have shown that emphasis unintentionally shifts from the modeling process to software specifics. Certain subgroups of the student population tend to have more difficulty with the technical aspects of working with software programs than others. For example, in our experience pre-service teachers tend to exhibit higher levels of computer-related frustration than Engineering majors. With an interest in keeping the focus of our course on the important aspects of the modeling process, we designed a series of video tutorials to help students master the software details (i.e. the entry locations for numerical values, the steps to perform a certain function, etc.) that must be understood before they can successfully use the software to develop mathematical models. The modeling course, the video tutorials, and an assessment of the usefulness of the tutorials are described below. We hope that our work will inspire others to design tutorials for courses in which software plays a significant role and there is a need to get students up-to-speed quickly on the technical details of how to use a software tool.
MATHEMATICAL MODELING COURSE
The textbook for the course is Spreadsheet Modeling and Applications – Essentials of Practical Management Science1 . As the title reflects, a spreadsheet software program – Microsoft Excel – is the tool used to develop mathematical models. The course is taught in a computer lab where all class presentations, examples, and assignments take place with the use of technology. Each student has access to a computer for the entire class period. Occasionally the instructor uses Microsoft PowerPoint presentations to highlight important aspects of the mathematical content being covered in a class session. However, lecture plays a small role as most class time is spent with students engaged in developing and evaluating mathematical models. Specific details about the syllabus and topics covered can be found on the course website: www.people.vcu.edu/ ~jrhardin/MathematicalModeling.htm.
By design, the course emphasizes model development and solution analysis, and properties of various model classes are discussed. Students have ample opportunity in other mathematics courses to study methods of problem solving; often in these courses, there is insufficient time for detailed model development and solution analysis. We place emphasis on these things to balance students’ perspectives. Nevertheless, solutions cannot be analyzed until they are provided. This is where software enters the picture.
Microsoft Excel and Excel add-in programs are used for all examples as well as for in-class and out-of-class assignments. Solver is an add-in that is included in the Excel software but before use must be loaded through the Tools menu. It allows the user to optimize a target function subject to constraints on the variables provided within the spreadsheet. The other addins used in the course (PrecisionTree and @Risk) were designed by Palisade Corporation, and student versions are included on a CD-ROM that accompanies the textbook. They are also available for purchase at www.palisade.com. PrecisionTree generates a tree diagram with probabilities and payoff values for each stage of a decision process. With user-provided information for a given scenario, the tool generates an analysis of the mathematical model and provides the user with information on the ideal decision making path through the tree. @Risk is a tool used to perform Monte Carlo simulation generating a sample of possible outcomes based on a user-provided probability distribution. These add-in modeling tools are powerful for analyzing and building mathematical models. While they represent the state of the art in solution methodology, they are not the only software options available. Thus it is important to note that the course is not dependent on using these specific software packages, but rather on building the kinds of models they are intended to solve and on interpreting the model solutions they provide. Since these were the tools we wanted students to be able to use with minimal difficulty as soon as they were needed in the course, they were the programs for which video tutorials were constructed.
When teaching Mathematical Modeling in spring 2003, the instructor (also an author of this paper) noted several technical aspects of Microsoft Excel and the add-ins with which students repeatedly had difficulty. Working with undergraduates who have not had much exposure to the course software, the instructor found she was spending excessive amounts of time answering software questions – valuable time that could have been used to help students develop the skills to build mathematical models or to understand modeling concepts. Furthermore, many of the questions were repetitious and focused on basic software functions (e.g. “Which button starts the simulation?”) rather than on modeling issues. A set of video tutorials were developed to cover the technical skills students need to successfully run the software and free class time for the important aspects of mathematical modeling inherent in the recommendations provided by various mathematical organizations [2,4].
Four tutorials were designed, one for each of the Excel add-ins (Solver, PrecisionTree, and @Risk) and one on building a data table. Many of the course assignments required students to build data tables – an advanced Excel capability that requires several user steps and does not have its own “wizard”. Students were having trouble with the Excel commands needed to build a data table, and thus this skill was covered in a tutorial. Each tutorial was created using freely available software, Microsoft Producer, which is an add- in to PowerPoint. This software allows the user to enhance a PowerPoint presentation with video screen capture and audio voice- over. Readers may view the tutorials at
In the case of the tutorials, video screen captures of the creation of a data table or use of an Excel add-in were added alongside a series of PowerPoint slides listing the steps needed to perform the skill upon which the tutorial was based. To tie the images together, a script describing the important details of each stage of the tutorial was written. The script was recorded as an audio voice-over and added to the presentation to provide explanation and support for the visual details provided by the slides and the video images. When incorporating all of the pieces (PowerPoint slides, video screen shots, and audio voice-over), thoughtful consideration was given to the ideal order for the flow of information and the timing of the audio and video pieces. A detailed discussion of the design and limitations of the tutorials can be found in Hardin and Ellington .
Figure 1 is a screen shot of the Solver tutorial that was constructed with Microsoft Producer. The PowerPoint slides appear on the left of the screen. The video runs on the right side of the screen. With respect to the Solver tutorial, the video is of the Solver Excel add-in being used to work through a particular problem. The tutorials were posted on the Mathematical Modeling course website. The students were required to access a particular tutorial one week before the skill covered by mat tutorial was needed in class. Students completed the tutorial before they used the software in class. Therefore, when they began the tutorial, they were completely unfamiliar with the software and the modeling concepts the software would help them explore. In fact, they did not need to be familiar with any type of mathematical modeling for the tutorial to be useful. For example, before viewing the Solver tutorial they were not expected to be able to build a linear program, but the tutorial was based on an example of a linear program and worked through the steps of how to enter it into Solver with the latter being the primary purpose of the tutorial. The tutorials remained available throughout the semester so that students could refer to them as needed. Data tables were required early in the semester, so the data tables tutorial was available the first week of class. The other Excel add-in tutorials were introduced throughout the course, each at least a week before the corresponding software was used in class. Since technical difficulties occasionally arise with different computer platforms (i.e. Windows-based machines vs. Macintosh) students had access to a computer laboratory from which the tutorials could be viewed any time campus buildings were open.
As is often the case when using freely available software, the tutorials created with Microsoft Producer are not as polished as they might be if they had been designed with an expensive program or by someone with extensive experience in instructional technology. We had no budget for the design of these tutorials. In spite of the program’s limitations, we were pleased with the finished product and with the students’ experiences in using the tutorials. With access to these same tools (Microsoft Producer and PowerPoint), we believe that other educators wishing to create similar tutorials for their courses would find the results useful and effective. In the case of our project, one person designed and constructed the tutorials. Once she was comfortable with the tools involved, it took two days to complete a tutorial from constructing the PowerPoint slides, to writing and recording the audio segments and putting all of the pieces together.
To further illustrate how a tutorial might be useful, we describe the Solver tutorial in relation to an example given to students in the mathematical modeling course. Students were given the following example of such a problem:
Suppose you go to your local hamburger restaurant for lunch. You have determined that, in order to stay awake for your afternoon classes, you need at least 1150 calories and 35 grams of protein. A burger costs $3 and has 450 calories and 25 grams of protein. A serving of fries costs $1 and has 350 calories and 5 grams of protein. You are a college student on a budget, so you want to spend as little money as possible. How many burgers and how many servings of fries should you buy?
If the number of burgers and servings of fries to buy are labeled x^sub 1^ and x^sub 2^, respectively, then the resulting linear program is:
Minimize 3x^sub 1^ + x^sub 2^
Subject to 450x^sub 1^ + 350x^sub 2^ >/= 1150
25x^sub 1^ + 5x^sub 2^ >/= 35
x^sub 1^, x^sub 2^ >/= 0
Solving this linear program graphically and algebraically would allow a student to arrive at an optimal solution. However, if any of the numerical values change, then all of the work would need to be done again. After several iterations to explore the changes that result in the solution set, the process has been reduced to an exercise in algebraic manipulation. By using Solver, students quickly see differences in solutions based on changes to the mathematical model.
As is the case with many software programs, Solver requires that the information be provided in a specific format. While the algebraic method requires solving a set of equations in two variables, finding the optimal solution to this situation using Solver involves changing cells (instead of labeling variables), a target cell (i.e. the objective value for the situation), and the declaration of problem constraints in a Solver dialog box (instead of constructing and solving linear inequalities). These details are covered by the Solver tutorial which students view before the class period in which this example is worked and discussed.
When first viewing this tutorial, students do not know how to build a linear program. This is their first exposure to taking a scenario and translating it to a mathematical model. They do not do this on their own. The tutorial guides them through the process and covers the example outlined above. As the course progresses, students continue to have access to the tutorial in case they need to refer to it whenever Solver is appropriate for working a problem. Since the tutorial helped students become familiar with how to designate changing cells and the target cell in Excel and how to use the Solver dialog box to specify the constraints, the instructor is able to spend less time on these details in class. As a result, more examples are covered during the class period and there is more time for discussion of the properties of linear programs and how to interpret resulting solutions. We believe others will have similar results designing tutorials for other software products that students typically have difficulty mastering.
ASSESSMENT OF THE USEFULNESS OF THE TUTORIALS
Anecdotal evidence of the amount of time spent fielding software- related questions revealed that the tutorials provide students with the technical details needed to develop a basic data table and run the Excel add-in tools. However, we conducted a formal analysis with an assessment instrument designed to determine whether the tutorials were effective in helping students learn the basic features of the software. The questions were developed to assess if students understood the software features covered by the tutorials, recognized output from specific point-and-click commands, and understood the purpose of the Excel add-ins they were using. They did not assess student understanding of mathematical modeling concepts or other course content. Two examples of the assessment questions are:
1) A Target Cell
a) Contains the value you are trying to attain with your solution
b) Holds the quantity you are trying to maximize or minimize
c) Is an optional part of a Solver model
d) None of the above
2) A Two-way Data Table
a) Computes the value of two outputs for any number of unknowns
b) Computes the value of an output for two different unknowns
c) Computes the value of an output for two different values of the same unknown
d) All of the above
All 22 multiple choice questions were given as a pre-test during the first week of the semester. Students were encouraged to answer the questions to the best of their ability in spite the fact that the questions addressed software components which were most likely unfamiliar to them. After being given time to view a tutorial, students were given the same pretest questions related to that particular tutorial as a post-quiz. The post-quiz was given before the software features covered by the questions were used or discussed in class. A code word was embedded in the audio of the tutorial and students were asked to provide it on the post-quiz to ensure that they had viewed the tutorial.
Table 1 contains a statistical summary of the number of questions answered correctly on the pre-test and post-quiz for each tutorial. This assessment was conducted with students in the spring 2004 and spring 2005 sections of Mathematical Modeling. Data for a student was used in the calculation of these statistics if the student answered all of the questions on the pre-test and on each of the post-quizzes. Of the students enrolled in each section, 80% completed all assessment items in spring 2004 and 67% completed all items in spring 2005 which resulted in complete data for 38 students. Five questions each were devoted to the Data Tables and Solver tutorials. The remaining two tutorials (Precision Tree and @Risk) were each evaluated with student responses to six questions.
Students began the semester with the ability to answer approximately one question related to each tutorial correctly. A /- test of the pre-test/postquiz differences at the 1% level of significance was used to determine whether the students correctly answered more than one question after viewing the tutorials. The statistical analysis revealed that for each tutorial the mean number of correct post-quiz questions was significantly larger than the mean number of correct pre-test questions. Based on this assessment, the tutorials were successful at helping students become familiar with the software features for which the tutorials were developed.
While the results reflect a significant increase in pre-test/ post-quiz responses for the Data Tables tutorial, the average number of questions answered correctly on this tutorial was smaller than the average number of post-quiz responses for the other three tutorials. This was the first tutorial the students viewed. Therefore, we suspect that students were uncertain of what was expected of them on the post-quiz and this resulted in the lower number of correct responses.
A solid understanding of mathematical modeling requires a blend of mathematical skills and conceptual understanding. The technology available today may reduce, but does not eliminate the need for mathematical skills. The mathematical modeling course described provides an appropriate balance of skills and concepts and places significant emphasis on the use of technology. To ensure that they are equipped with a well balanced mix, this type of course is important for mathematics majors. Future analysts need a solid understanding of mathematical modeling when they enter the workforce. Future teachers will need to prepare lessons that – in this technological era – are heavily application and concept-based. They also need the ability to appropriately demonstrate mathematical modeling for their students. We found that the design and use of video tutorials to help students with the technical skills needed to successfully use the software were helpful with time management and skill practice issues. As a result, the course instructor was able to focus on the important aspects of mathematical modeling – the development and study of models. We believe that teachers at all levels of the education hierarchy with software skill development needs will find similar tutorials helpful for their situations. Every instructor will be presented with unique difficulties based on the software used in a course, as well as the student population. However, designing a tutorial will provide students a reference for their technical questions and free the instructor to focus on the more important aspects of the course including concept building and problem solving. 1 The textbook for the course when this project began was Practical Management Science: Spreadsheet Modeling and Applications by Winston & Albright (2001). We note that it is very similar to the textbook we discuss here in its choice of mathematical models, its choice of software packages, and its emphasis on the use of spreadsheets.
1. S. C. Albright and W. Winston, Spreadsheet Modeling and Applications – Essentials of Practical Management Science, Duxbury Press, Pacific Grove, CA, ISBN: 0534380328 (2005).
2. Committee on the Undergraduate Program in Mathematics, Undergraduate Programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide 2004, Mathematical Association of America, Washington, DC, ISBN: 0883858142 (2004).
3. J. R. Hardin and A. J. Ellington, “Using Multimedia to Facilitate Software Instruction in an Introductory Modeling Course,” INFORMS Transactions on Education, Vol. 5, No. 2, http:// ite.pubs.informs.org/Vol5No2/HardinEllington/(2005),
4. National Council of Teachers of Mathematics. Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA, ISBN: 0873534808 (2000).
5. W. Winston and S. C. Albright, Practical Management Science: Spreadsheet Modeling and Applications, Duxbury Press, Pacific Grove, CA, ISBN: 053442435X (2001).
Aimee J. Ellington, Department of Mathematics
Jill R. Hardin, Department of Statistical Sciences and Operations Research
Virginia Commonwealth University
P. O. Box 842014
1001 West Main Street
Richmond, Virginia 23284-2014
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