Mathematical Modeling

This topic will discuss the use of mathematical modeling, spreadsheets, and a programming language. With the emergence of new data analysis tools and new types of data problems, novel types of models are consequently required

to deal with them. Examples of these models include rule-based knowledge repre- sentations as used in expert systems, neural networks, genetic algorithms, multi- variate adaptive regression splines, computer-intensive estimation methods such as the Markov chain, Monte Carlo methods, and statistical databases, which need op- erations and applications such as a datacube.

The modeling process is evolutionary. There are always other influences or fac- tors, not anticipated in the first model or prototype, that interfere with prediction ac- curacy. Reiterating steps in the modeling process, making improvements to define the problem, constructing the model, and improving data collection techniques enhance not only the product but also the conceptualizing process. The refinement of a model may progress ad infinitum if using a trial-and-error approach and still achieve only an arbitrary degree of accuracy. However, in the context of data analy- sis, an extremely sophisticated data analytic technique can data mine the informa- tion and achieve more predictive power. See Chapter 12 for the introduction to data analysis and problem solving.

A strategy for data analysis descriptively plans the steps, decisions, and proce- dures to follow when analyzing data to build a model or answer a question. Gener- ally stated, the steps are

1. Define the
2. Extract and identify the most important
3. Construct a model to describe the shape of the reduced

Effective procedures are called algorithms. Flowcharts are used to graphically rep- resent algorithms, which are detailed sequences of simple steps needed to solve a problem. A problem is unsolvable if no algorithm can solve it and, if a decision prob- lem, undecidable if no algorithm can decide it.

There are different types of models. For instance, a miniature replica of a spacecraft used to study design features is a physical model. Examples of this model type can be found at science fair exhibits in which students represent the solar system using a box with balls of various colors and sizes hanging from strings. Another, and per- haps the most useful, type is a conceptual model. For instance, a diagram of the wa- ter cycle is used to describe the behavior of the system, in this case water. When the language of mathematics is utilized to describe the system’s behavior, a mathematical model is created. For example, representing the shape of the Earth as a sphere pro- vides a mathematical model. In this case, elementary geometry is used to describe the system.

Models, both physical and conceptual, are problem-solving tools because they are easily manipulated and changed as needed. Models assist experimentation, which aims to achieve an understanding and generate new information about the modeled system. Some systems may be too complex or too inaccessible to explore in their orig- inal settings. In many business and scientific situations, the design and use of mod- els in planning and production processes are critical. The process of modeling follows these steps:

1. Observing a system or problem
2. Constructing a mental image or model
3. Identifying important variables and constants (parameters)
4. Determining relationships
5. Developing equation(s) to express relationships
6. Testing
7. Revising

Modeling is used when simulations are not a feasible approach. Modeling activ- ities can be used to demonstrate change. Additionally, modeling is a technique of ex- pression and explanation. For an excellent learning activity for the secondary grade level, see “Chaos and Beyond” in ISTE’s National Educational Technology Standards for Students (Thomas & Bitter, 2000).

Problem Activity 1

Mathematical modeling can occur in nonmathematical contexts. Develop a mathe- matical model of how you plan to read this textbook. Share your model in a small group (no more than four). Work together to create a group model. Diagram or make a flowchart of your group process. How did your group achieve consensus?

Problem Activity 2

Consider the following real-world problem:

What is the cost of illiteracy to a community? In groups of three or four, brain- storm the possible cost areas using a brainstorming tool such as Inspiration to develop a concept map. Identify the important variables and parameters. Decide on a data collection plan. After the data are collected and graphically represented, construct a mathematical model by developing an equation representing the rela- tionships between variables and parameters. Test using what-if questions and refine your model. Would using a different definition of illiteracy change your model?

Mathematical modeling can be applied to additional real-world problems. Some examples are predicting wildlife populations, costs of long-distance phone calls, ir- rigation flow rates, and deciding on the fastest checkout line at a grocery store.

Modeling is effective in education. Due to changes and shifts in population, it is important for educational institutions to assess their immediate future needs, as well as long-term ones. By asking what-if questions based on census information and other known statistics, school districts, colleges, and universities can be better pre- pared to offer quality education. In the classroom, probability situations can be mod- eled with a computerized framework such as a spreadsheet. A spreadsheet model is created and its figure displayed; the user then asks the computer “what if” a specific value were to be changed. The value can be a change in actual numbers or an in- crease or decrease by a specified percentage rate. By experimenting with various probabilities, students can examine the basic theory of probability.

Spreadsheets are very useful for modeling activities (Scaife & Wellington, 1993) and are fast becoming programming environments (Carson, 1997). As explained in Chapter 2, the spreadsheet’s array of rows and columns may be organized in such a way as to be easily understandable to the reader. Because each piece of data entered into the spreadsheet is considered as it relates to the other data, any change made in one part affects the whole spreadsheet. Subsequently, information can be manipu- lated to reflect potential change or what-if questions in any or all sections of the spreadsheet. This capability makes spreadsheets a useful tool to detect mathemati- cal change. Having students graph and chart using spreadsheets helps them visual- ize change and learn the advantages and disadvantages of different graphical representations.

More on Spreadsheets: Their History and Use

The electronic spreadsheet made its debut in 1979, a creation of two Harvard Business School graduate students, Daniel Bricklin and Robert Frankston. They named their program VisiCalc, and it became the prototype for many other software programs that formed the first generation of electronic spreadsheets.

The second generation came on the scene with the introduction of Lotus 1-2-3 in 1982. This program was unique in being the first integrated software program. An in- tegrated software program blends several different programs so that information can be presented in various forms. For example, a graphics option allows the user to cre- ate visual aids, such as line graphs, vertical and horizontal bar graphs, and pie charts. In this way, the figures from the spreadsheet can be converted into graphs that show at a glance what the user wishes to communicate about the financial picture.

Another option contained in Lotus 1-2-3 is database management. A database is an electronic filing system in which large volumes of data (information) can be stored and organized. For instance, names, addresses, phone numbers, and account num- bers entered into a database can then be sorted alphabetically, numerically, or in any other way that will be helpful to the user. In an integrated program, any of the needed stored information can be retrieved from the database and inserted into the spread- sheet. The time saved in locating and adding data to a spreadsheet made this new generation of software a welcome addition to the electronic spreadsheet family.

The most recent addition to this family, and the beginning of the third genera- tion, is the integrated software program with extended capabilities. It adds word pro- cessing, expanded spreadsheet size, and communications between computers. Word processing allows text to be added to the report. In this way, an explanation of the spreadsheet figures or the graphic pictures can complete a financial report.

Another improvement is the potential size of spreadsheets. Spreadsheet pro- grams vary in the numbers of rows and columns available, but the average size is much larger now than what was originally offered. Any number of these columns and rows may be used, depending on the needs of the user and the memory capac- ity of the computer. Although only a small portion of this total picture appears on the computer screen at one time, the user may move around within the spreadsheet to display any part of it.

Yet another improvement in this latest generation of spreadsheet programs is the ability of computers to communicate with each other. Through the new technology of telecommunications, computers in different locations can communicate by tele- phone lines and exchange data as needed. Examples of programs combining all of the preceding options are AppleWorks and Microsoft Office. Chapter 2 discussed the use of spreadsheets in education. The following section presents additional practi- cal applications related to problem solving and data analysis.

Analysis and Projection

Spreadsheet information is essential in analyzing the financial condition of any organization.  Spreadsheets  also  make  analysis  of  statistical  information more manageable and much faster. Analysis, then, is one of the more important uses of electronic spreadsheets. A natural outcome of analysis is projection for the future. Let’s look now at the value of spreadsheets in these two areas.

In business and other organizations, analysis is the examination and evaluation of data to measure their impact on the particular organization for which it is being performed. Spreadsheets are an excellent way to compare data. To accomplish this, spreadsheets can be combined, both in part and in total. By defining a range of cells, rows, or columns, any part of a spreadsheet may be retrieved and made a part of others. In this way, very complicated data can be displayed for comparison.

As in education, planning for other public and social programs is dependent on knowing what will happen if certain changes take place within society. Changes in the political climate, for instance, have an impact on almost every area of life. The educational community will obviously benefit quite as much as private business and industry by using the analytic properties of electronic spreadsheets.

By combining the information gained from analysis of a spreadsheet with ex- pectations for the future, it is possible to project what may happen. Projection is the predicting, or forecasting, of what may be logically expected to occur in the future based on what we know about the past. There are three basic types of planning for the future: operational, tactical, and strategic. There are also two basic areas wherein planning is necessary—expense and income.

In the area of expense, operational planning is the simplest and covers a period of only a few months at a time. It deals with day-to-day operations. Usually done by lower-level managers, this type of planning obviously is short-term and therefore re- quires less complex spreadsheet information than do other types. An example of op- erational planning is the adjustment of teaching staff and classroom space.

Tactical planning is for an intermediate period, perhaps covering a time span of 1 to 3 years. This includes planned stages of development working toward long-term goals. An example of this type of planning is the revision of school budgets in re- sponse to projected increased enrollment.

Strategic planning occurs at the highest level of management and addresses the overall long-range welfare of the organization. Examples of this level of planning are direction of growth, curriculum development, and expansion of school districts.

Future income also requires planning and projection on the part of school ad- ministrators. For instance, expected changes in income due to school bond issues can be included in planning.

For higher-level educational institutions, administrators must predict potential changes in student population and project income based on tuition and fees. For in- stance, funds allotted from other agencies may change from time to time. For this and other reasons, planning for the future is essential to a smooth operation.

Government agencies have a similar need to make projections based on changes in population, income, and societal needs. One of the most obvious areas of public service is transportation. How many freeways to build? What is the projected need for public transportation? This is only one of many services provided by local, state, and national government agencies that may be better analyzed through the use of electronic  spreadsheets.

Problem Activity 3

In small groups, look at census data from your state. (Visit http://factfinder.census.gov/ home/saff/main.html for Bureau of Census data or www.census.gov for data sets.) Look at the available spreadsheet data files from the Crime & Justice Electronic Data Abstracts’ website (www.ojp.usdoj.gov/bjs/dtdata.htm). Select a data set to down- load. Create a spreadsheet to compare the downloaded files. Look for a mathemati- cal change to detect a pattern. Try out different types of representations such as maps or graphs. Based on exploratory data analysis, develop a question to explore using this sentence stem: Is there a relationship between xxx and yyy? Write a formula that represents the relationship. Test the formula. What happens numerically? Graph the change. Create what-if questions to further explore the changes. Present your find- ings to the class and discuss the results.