Microcomputers || INTRODUCING COMPUTER SIMULATION INTO THE HIGH SCHOOL: AN APPLIED MATHEMATICS CURRICULUM

INTRODUCING COMPUTER SIMULATION INTO THE HIGH SCHOOL: AN APPLIEDMATHEMATICS CURRICULUMAuthor(s): NANCY ROBERTSSource: The Mathematics Teacher, Vol. 74, No. 8, Microcomputers (November 1981), pp. 647-652Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27962646 .

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INTRODUCING COMPUTER SIMULATION INTO THE HIGH SCHOOL: AN

APPLIED MATHEMATICS CURRICULUM

By NANCY ROBERTS Lesley College

Cambridge, MA 02138

The foci for mathematics education in the 1980s have been clearly stated by the

public, as reported in the Priorities in School Mathematics project (Suydam and

Higgins 1980). Two areas high on the

agenda are problem solving and integrating computers into the curriculum. New sec

ondary level materials, Introduction to Sim ulation: The System Dynamics Approach, funded by a grant from the U.S. Office of Education (Grant #G007903439), address both of these issues and are currently being negotiated with commercial publishers.

Simulation is a way of analyzing prob lems by using a representation or model of a situation and then exercising the model to see how it behaves under different circum stances. Originally, the word simulate

meant to imitate or feign. A simulation im itates a real system by using some kind of

model. As a simplified representation of a

system, the model aids one in understand

ing how the system operates. A simulation model may be a physical model, a mental

conception, a mathematical model, or a

computer model.

Many simulations involve physical mod els. The United States Army Corps of En

gineers has constructed a physical model of the Mississippi River to study ways of less

ening the impact of flooding. Wind tunnels and wave tanks are other forms of simula tion in which a physical model is used to imitate a larger system.

Since physical models are often rela

tively expensive to build and unwieldy to

move, mathematical models are often pre ferred. In a mathematical model, symbols

or equations represent the relationships in the system. To perform a simulation, the calculations indicated by the model are

performed over and over. If these calcu lations have to be performed by hand, sim ulation can be time consuming and costly.

In the last forty years, computer simula tion has replaced simulation using hand calculation. With the development of com

puters, the cost of arithmetic computations has halved approximately every two years and is likely to decline at this rate for at least another decade. This decline means

that simulation, once a rare and expensive way of solving problems, is now very in

expensive.

Computer Simulation

A computer simulation model is a model

represented as a set of instructions to a

computer. Equations, representations of how a system is believed to work, are used to instruct the computer on how to manip ulate the numbers. For some complex problems, computer modeling is not only less expensive than building a physical

model or experimenting with the actual

system, it may be the only way to attempt to understand the system. Much of what is known about the likely behavior of nuclear reactors during acccidents is derived from

computer models.

Computer simulation is used in a wide

range of fields, from management to the

physical sciences. Meteorologists use com

puter simulation to forecast the weather.

Computer simulation is used to design many of the moving subsystems of a car.

Social and economic systems are studied

using computer simulation models. A num

ber of companies use computer models to

study their operations. Some are interested

November 1981 647



in forecasting future sales, others in under

standing why certain problems occur.

Computer models are used in research.

Agronomists in Mississippi are using com

puter simulation to study ways of improv ing cotton yields. An improved under

standing of the possible outcomes of

experiments allows researchers to devote their resources to the most promising ex

periments.

The High School Curriculum

High school curriculum materials have been developed to introduce mathematical

modeling and computer simulation to a

high school audience with a mathematical

background of intermediate algebra and no

prior computer experience. The computer language employed in this curriculum was

developed especially for writing simulation models. DYNAMO (DYNAmic MOdels) was designed to be understandable by managers who, it is assumed, know little mathematics and dislike computers?as sumptions convenient to accept for high school classes. DYNAMO has been found to be easier to learn than BASIC, the lan

guage currently taught to students from about the fifth grade. DYNAMO is avail able on almost all mini and larger comput ers and is now being developed for the

Apple II personal microcomputer. (For fur ther information on DYNAMO or the high school curriculum, contact the author at

Lesley College, 29 Everett Street, Cam

bridge, MA 02138.) However, simulation models may also be written in BASIC. The teachers manual includes instructions on

building simulation models in BASIC for those who do not have DYNAMO avail able.

The following is a synopsis of each of the

six, self-teaching curriculum packages. The thrust of the material is to take students from a qualitative approach to problem analysis through a variety of skill steps that enable them to become more and more

quantitatively oriented. At the same time, the basis for the quantitative problem anal

ysis, or mathematical model, is only as sound as the original qualitative under

standing of the problem. This learning path is geared to help the students think more

clearly and therefore have a deeper under

standing of the problem under study. The

approach is iterative. As the students move

through the booklets, their earlier problem statements are continuously clarified with the use of new skills.

Learning Package I, Basic Concepts: Sys tems, Models, and Causal Relations pro vides students with the necessary concep tual understandings. This package introduces students to many different sys tems, how models are used to better under stand systems and their behavior, and how

identifying cause-and-effect relationships can aid one in developing models. Figure 1 indicates the degree of understanding reached by the end of package I.

Fig. 1. Tired-sleep feedback loop and graph of pos sible behavior of such a system

Figure 1 suggests that your body has au tomatic controls, or feedback, built-in, that determine your sleeping patterns. The

causal-loop diagram suggests that the more tired you are, the more you will sleep, be

coming less tired, needing and then getting less sleep, eventually becoming tired again. The graph in the bottom of figure 1 is an other way of illustrating this feedback be havior that, over time, tends to be self

regulating. This is a rather qualitative problem statement. However, quantitative

648 Mathematics Teacher



thinking is introduced by the suggestion that causal-links represent a change in be

havior, either an increase or a decrease.

Learning Package II, Structure of Feed back Systems focuses on helping students master the technique of causal-loop dia

gramming as an aid to understanding the behavior of complex issues over time. The first part of the learning package introduces students to the use of the signing symbols + or ? to indicate more precisely the direc tion of influence of a causal link and 4),

@, or G, to indicate the direction of

change over time of a closed feedback loop. The second part of Learning Package II

teaches students to take a written descrip tion of a problem and develop a causal

loop diagram that represents the under

lying dynamics of the problem statement.

Figure 2 suggest the depth of understand

ing that students are able to achieve by the

Fig. 2. Causal-loop diagram representing the di lemma of Maine lobster fishermen

end of package II. Figure 2 is a possible diagram developed from a newspaper clip ping as one of the last exercises in the pack age. The newspaper clipping suggests that as unemployment in Maine increases, more men turn to fishing for a living. This causes more lobsters to be harvested, depleting the

supply and making it more difficult for the lobstermen. As the number of lobsters de

creases, the government attempts to in crease the number of fishing regulations. Since this is not a generally agreed upon strategy, the passing of regulations is slowed down, allowing the continual de crease in available lobsters.

Learning Package III, Analyzing and

Graphing the Behavior of Feedback Systems introduces students to drawing meaning from data through the use of graphs as well as to representing dynamic behavior graph ically. The students are taught to think

more quantitatively in preparation for

building computer models. Figure 3 illus trates how the last chapter of package III,

"Linking Causal-Loops and Graphs," strengthens one's ability to understand the

dynamic feedback behavior of a system.

Figure 3 suggests first in causal-loop form and then in graphical form the nature and effects of the time delay from planting a tree to harvesting that tree. The graph in dicates much more quantitatively the im

pact of the growth rate of trees on the har vest rate.

Learning Package IV, Analyzing Less

Structured Problems integrates the skills in troduced in the first three learning pack ages and provides the student with a frame work for problem solving. This framework is composed of four elements: perspective, time frame, problematic behavior, and policy choice. Perspective refers to the point of view of the concerned person. Time frame refers to the time period of interest in

studying the behavior of a given system. Identifying the problematic behavior sug gests making a careful statement of exactly which aspects of the changing pattern of the system are to be isolated as undesirable. This "statement" is usually made with the aid of a causal-loop diagram. Finally, pol icy choice refers to the particular mode of attack that one might choose for more de tailed analysis of how to help alleviate the

problematic behavior.

Having completed the first four learning packages, the students have been taught a structure for attacking a system problem as

well as for writing a paper describing the

problem and their proposed policies to

November 1981 649



PLANTING MATURATION RATE v / RATE

MATURATION /RATE

WOWSER Of MEDIUM TREES

+^*UMB?R OF HARVESTABLE

Fig. 3. Causal-loop diagram and time graph of some aspects of the lumber industry.

eliminate the problem. These skills will be invaluable to the student in any future aca

demic or professional setting. Learning Package V, Introduction to

Simulation teaches the student the skills

necessary for converting problems under

study from causal-loop diagrams to com

puter models and for simulating these models over time with the aid of a com

puter. At this point students actually begin to use calculus, although they only have had and needed a mathematics background of intermediate algebra. The first equation in figure 5, the level (L) equation, is a first order difference equation representation of a differential equation. Further, the rate

equation (R) is an example of applying the students' years of writing rate equations from written arithmetic and algebra text book problems. DYNAMO equations may be written in any order, depending on how each student thinks through the process. DYNAMO internally orders the equations appropriately during the code-compiling stage, prior to calculating the simulation re sults.

The yeast model, illustrated in figures 4,

5, 6, and 7, shows the causal-loop diagram for the model (fig. 4), the DYNAMO equa tion listing of the model (fig. S), the hand simulation that students are requested to

carry out so they know exactly what the

computer is doing (fig. 6), and its baseline

computer simulation run (fig. 7).

Fig. 4. Causal-loop diagram for the yeast growth model

* YEAST GROWTH

L YEAST. =YEAST.J+(DTXBUDDNG.JK) YEAST=10

NOTE YEAST CELLS (CELLS) R BUDDNG.KL-(YEAST.KXBUDFR) NOTE BUDDING (CELLS/HOUR) C BUDFR-0.25

NOTE BUDDING FRACTION (1 /HOUR) PLOT YEAST =? Y/BUDDNG -

PRINT YEAST, BUDDNG

SPEC DT * 1 /LENGTH - 20/PLTPER -1 /PRTPER - 5

Fig. 5. DYNAMO equation listing of the yeast growth model




TIME (HOURS)

CHANGE IN YEAST (CELLS)

YEAST (CELLS)

BUDDING (CELLS/HOUR)

Fig. 6. Hand simulation of the yeast growth model

Learning Package VI, Formulating and

Analyzing Simulation Models presents the students with six problems that they can

develop into simple computer models. These problems are the impact of the inter

ference of people on an ecosystem?The Kaibab Plateau Model; a study of epidem ics?Influenza; the growth and decline of a

city?Urban Growth; an energy model? Natural Gas; heroin addiction in an inner

city environment?Heroin and Crime; and economic cycles?Hog Cycles.

The first chapter of package VI begins by taking the student very slowly through the

steps of model building and simulation?

defining the problem; drawing the causal

loop diagram; drawing the flow diagram; writing the equations; running the model on the computer; and testing, analyzing, and using the model for decision making. In addition to the six learning packages there is a teachers manual and answer

books.

Pilot Test Results

The project was pilot tested with stu dents from grade 9 through grade 12 in both public and private schools. The stu

dents' backgrounds ranged from no com

puter experience to extensive experience. Both students and teachers found that the DYNAMO simulation language was easier to learn than BASIC and did not require previous knowledge of computers. The ma

terial was used in a variety of courses?ap

plied mathematics, history, environmental

studies, a new course created for the cur

November 1981 651



riculum, a computer course, and independ ent study.

The teachers generally felt that a prereq uisite of either first- or second-year algebra should be required and that the first few

learning packages could be used with jun ior high students but the later packages should be reserved for senior high students. When asked for general evaluative com

ments about the project, the teachers made

highly favorable comments. They were de

lighted to be able to use materials they saw as new, innovative, and solidly academic in

quality. These materials have the potential to enable the mathematics teachers to help create a mathematics and computer literate

high school community by introducing a

problem-solving strategy that can have

meaningful applications in many areas.

REFERENCE Suydam, Marilyn ., and Jon L. Higgins. "PRISM:

Shedding Light on NCTM's Recommendations for the 1980s." Mathematics Teacher 73 (December 1980): 646-47.

Whole Numbers

.98

HOW TO .

SJevelop ?rroblem

^Solving llsing aUSalfulator

ft ft

Geometry 24 ready-to-use activities dedicated to the proposition that calculators free children to think out problem solutions. 42 pp 1981 ISBN 0-87353-175-2 $4.00

Individual NCTM Members-Discount 20%

See NCTM Educational Materials Order Form in "Professional Dates"

(Continued from page 629)

position of the symbol in the message to be coded.

Depending on the background of your students, you may wish to discuss the al

phabet spiral in terms of remainder (modu lar) arithmetic. Numbers that appear in the same relative position on the spiral, such as 13 and 42, have the same remainder when divided by 29. A discussion of how remain der arithmetic with modulus 29 can be used in conjunction with matrix multiplication to create more complex codes may be found in Peck (1961). Perhaps a few of

your students could study this material and then present it to the class as a follow-up activity.

BIBLIOGRAPHY Feltman, James. "Cryptics and Statistics." Mathe

matics Teacher 72 (March 1979):189-91.

Joshi, Vijay S. "Coded Events in American History." Mathematics Teacher 69 (May 1976):383-86.

Peck, Lyman C. Secret Codes, Remainder Arithmetic, and Matrices. Reston, Va.: National Council of Teachers of Mathematics, 1961.

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Microcomputers || INTRODUCING COMPUTER SIMULATION INTO THE HIGH SCHOOL: AN APPLIED MATHEMATICS CURRICULUM