Mathematical ModelingintheUSMACurriculumDavidC.Arney,KathleenG.SnookIntroductionMathematicalmodelingisanimportantandfundamentalskillforquantitativeproblemsolvers.Increasingly,Armyofficersarecalledupontosolveproblemsthat canbe modeled mathematically.It is therefore importantthat theUnitedStatesMilitaryAcademy(USMA)curriculum includesmathematicalmodelingand problemsolving.Manyoftheprinciples,skills,and attitudesnecessarytoperform effective mathematical modeling canbe presented in the coremathematics courses taken by all cadets.AfterprovidingsomebackgroundabouttheUsMAprogram,webrieflydiscussthemathematicalmodelingprocess.Wethengiveexamplesof coursetopicsdevelopmentalproblems,and studentprojectsthatareusedtoaccomplishearlypresentationofmodelingand problemsolving.This“earlyand often"approachtoteaching modeling results in undergraduates who are able to apply theirmathematicstosolvequantitativeproblems.Thispaperalsoservestodemonstratehowmodeling supports the innovative mathematics curriculumpresentedatUsMA.Wediscussandexplainlarge-scalemodelingprojectscalled Interdisciplinary LivelyApplicationProjects(ILAPs)in thecontext of thecurriculum.Background"A mind is not a vessel to be filled,but a flame to bekindled"-- PlutarchUSMA was founded in 1802 as the nation's first engineering school and the firstnationaleducational institution.In1817,newlyappointedAcademySuperintendentSylvanusThayer,previouslyanassistantprofessorofmathematics,returnedtoUSMAfromavisittotheEcolePolytechniqueinParisand instituteda rigorous four-yearengineering curriculum. Heplaced greatemphasis on modelingand appliedmathematics.Thebasis of civil engineeringat the time was descriptive geometry, with algebra, trigonometry, and somecalculus supporting thenecessary mechanics ofengineering.Thayerand hisProfessorof Mathematics,Charles Davies,convertedthetheoreticalapproachoftheFrenchattheEcolePolytechniquetotheappliedmodelingapproachneededinAmerica.Modeling becamethethreadholdingtogetherthe rigorousandsubstantial undergraduate core mathematics program at West Point.In turn thisappliedmathematicsandmodelingapproachwasexportedfromUSMAtoothertechnical schools around the country during the 19th century.13
13 Mathematical Modeling in the USMA Curriculum David C. Arney, Kathleen G. Snook Introduction Mathematical modeling is an important and fundamental skill for quantitative problem solvers. Increasingly, Army officers are called upon to solve problems that can be modeled mathematically. It is therefore important that the United States Military Academy (USMA) curriculum includes mathematical modeling and problem solving. Many of the principles, skills, and attitudes necessary to perform effective mathematical modeling can be presented in the core mathematics courses taken by all cadets. After providing some background about the USMA program, we briefly discuss the mathematical modeling process. We then give examples of course topics, developmental problems, and student projects that are used to accomplish early presentation of modeling and problem solving. This “early and often” approach to teaching modeling results in undergraduates who are able to apply their mathematics to solve quantitative problems. This paper also serves to demonstrate how modeling supports the innovative mathematics curriculum presented at USMA. We discuss and explain large-scale modeling projects called Interdisciplinary Lively Application Projects (ILAPs) in the context of the curriculum. Background “A mind is not a vessel to be filled, but a flame to be kindled”- Plutarch USMA was founded in 1802 as the nation’s first engineering school and the first national educational institution. In 1817, newly appointed Academy Superintendent Sylvanus Thayer, previously an assistant professor of mathematics, returned to USMA from a visit to the Ecole Polytechnique in Paris and instituted a rigorous four-year engineering curriculum. He placed great emphasis on modeling and applied mathematics. The basis of civil engineering at the time was descriptive geometry, with algebra, trigonometry, and some calculus supporting the necessary mechanics of engineering. Thayer and his Professor of Mathematics, Charles Davies, converted the theoretical approach of the French at the Ecole Polytechnique to the applied modeling approach needed in America. Modeling became the thread holding together the rigorous and substantial undergraduate core mathematics program at West Point. In turn this applied mathematics and modeling approach was exported from USMA to other technical schools around the country during the 19th century
Bythemiddleofthe2othcenturylesscurriculumtimewasavailableformathematics,yetmoresophisticated mathematicaltopicswerebeing required ofcadets.Thecorecurriculumdilemmaoffittingseventopics(DifferentialCalculus,IntegralCalculus,Multi-variableCalculus,DifferentialEguationsLinear Algebra, Probability and Statistics, Discrete Math) into the four allottedsemestersofcoremathematicswasasubstantialchallengeforUSMAandmanyothertechnicallybasedschoolsWestPointinstitutedanewintegratedmathematicscurriculumin1990.Thefoursemestercourses;DiscreteDynamicalSystemsandIntroductiontoCalculus, Calculus 1 (with differential equations), Calculus 2 (multivariable), andProbability&Statistics,satisfiedthe“7into4"demandsatWestPoint.Modelingwasanimportantthreadlinkingthefourcoursestogetherforstudentgrowthanddevelopment.The integration ofthefour core coursesmadeforamoresophisticatedmultipleperspectiveofmodeling.Students investigateandtraverse behavior,models,and solutionmethods using discrete orcontinuous,linearor nonlinear,anddeterministic or stochasticmathematics.Thefollowingfigureshowsthepossibilitiesofthemodelingflowjustconsideringdiscreteandcontinuousclassifications.Similarflowperspectivesarepossibleforlinear-nonlinearandstochastic--deterministicclassificationsSOLUTIONMETHODBEHAVIORMODELDISCRETEDISCRETEDISCRETECONTINUOUSCONTINUOUSCONTINUOUSFigure1:ModelingflowpossibilitiesAnotherfeatureoftheUSMAmathematicscorecurriculumistheuseofInterdisciplinaryLivelyApplicationsProjects(ILAPs).Theseprojectsarebroader and, in some ways, more realistic than previous educational modelingprojects used at West Point. ILAPs are co-authored and co-presented bymathematiciansandexpertsfrompartnerdisciplinesintheUSMAprogram(forexample,engineers,scientists,andeconomists).Thedisciplinaryexpertspresentbackground informationtothestudents.Studentsthenmodel and solve14
14 By the middle of the 20th century less curriculum time was available for mathematics, yet more sophisticated mathematical topics were being required of cadets. The core curriculum dilemma of fitting seven topics (Differential Calculus, Integral Calculus, Multi-variable Calculus, Differential Equations, Linear Algebra, Probability and Statistics, Discrete Math) into the four allotted semesters of core mathematics was a substantial challenge for USMA and many other technically based schools. West Point instituted a new integrated mathematics curriculum in 1990. The four semester courses; Discrete Dynamical Systems and Introduction to Calculus, Calculus 1 (with differential equations), Calculus 2 (multivariable), and Probability & Statistics, satisfied the “7 into 4” demands at West Point. Modeling was an important thread linking the four courses together for student growth and development. The integration of the four core courses made for a more sophisticated multiple perspective of modeling. Students investigate and traverse behavior, models, and solution methods using discrete or continuous, linear or nonlinear, and deterministic or stochastic mathematics. The following figure shows the possibilities of the modeling flow just considering discrete and continuous classifications. Similar flow perspectives are possible for linear- nonlinear and stochastic-deterministic classifications. BEHAVIOR MODEL SOLUTION METHOD DISCRETE CONTINUOUS CONTINUOUS CONTINUOUS DISCRETE DISCRETE Figure 1: Modeling flow possibilities. Another feature of the USMA mathematics core curriculum is the use of Interdisciplinary Lively Applications Projects (ILAPs). These projects are broader and, in some ways, more realistic than previous educational modeling projects used at West Point. ILAPs are co-authored and co-presented by mathematicians and experts from partner disciplines in the USMA program (for example, engineers, scientists, and economists). The disciplinary experts present background information to the students. Students then model and solve
theproblem in small groups, writetheir solutionandpresenttheirresults.Finally,disciplinaryexperts critiquetheproblemandexplaintheuseofmathematics intheapplieddiscipline.Studentsbenefitbyseeingamorerealistic,multidisciplinaryproblem,andfacultybenefitbyworkingtogethertodevelop educational projects using modeling andwriting askeyelements inproblemsolvingIntegratedCurriculum"The greatest good you can do (for students) is not just to share your riches, buttoreveal (tothem)theirown..."--BenjaminDisraeliTheeducationphilosophyoftheDepartment of MathematicalSciencesatWestPointisthatundergraduate students shouldacquire important andfundamentalknowledge,developlogicalthoughtprocesses,and learnhowtolearnSuccessful studentscanformulateintelligent questions,reason andresearchsolutions,andareconfident and independent in theirwork.Modelingplays a vital role insupportingthis philosophy.With this philosophy in mind,the core mathematics curriculumintegrates concepts of the sevenfundamental engineering-basedmathematicstopics intothecoreprogram.Akeytothiscorecurriculumeffortisafirstcoursecontainingdiscretedynamicalsystems(differenceequations)includingsystemsofequations(matrix/linearalgebra)and a sequentialapproachto differential calculus.ThemajoringredientsofthiscoursearesimpleproportionalitymodelsandapplicationsThesecondcourse,CalculusI,coversintegralcalculus,differentialequationsandcalculus-basedmodelsofasinglevariable.Manyoftheapplicationsandmodelscoveredinadiscretemodeinthefirstcoursearerevisitedinacontinuous mode in the second core course. Bythe end of the first year,studentshavetheabilitytounderstandtheperspectiveofproblemsolvingpresented in Figure 1. The third course, Calculus Il, includes topics in15
15 the problem in small groups, write their solution and present their results. Finally, disciplinary experts critique the problem and explain the use of mathematics in the applied discipline. Students benefit by seeing a more realistic, multidisciplinary problem, and faculty benefit by working together to develop educational projects using modeling and writing as key elements in problem solving. Integrated Curriculum “The greatest good you can do (for students) is not just to share your riches, but to reveal (to them) their own.”- Benjamin Disraeli The education philosophy of the Department of Mathematical Sciences at West Point is that undergraduate students should acquire important and fundamental knowledge, develop logical thought processes, and learn how to learn. Successful students can formulate intelligent questions, reason and research solutions, and are confident and independent in their work. Modeling plays a vital role in supporting this philosophy. With this philosophy in mind, the core mathematics curriculum integrates concepts of the seven fundamental engineering-based mathematics topics into the core program. A key to this core curriculum effort is a first course containing discrete dynamical systems (difference equations) including systems of equations (matrix/linear algebra) and a sequential approach to differential calculus. The major ingredients of this course are simple proportionality models and applications. The second course, Calculus I, covers integral calculus, differential equations, and calculus-based models of a single variable. Many of the applications and models covered in a discrete mode in the first course are revisited in a continuous mode in the second core course. By the end of the first year, students have the ability to understand the perspective of problem solving presented in Figure 1. The third course, Calculus II, includes topics in
multivariableandvectorcalculus.Studentsencountermoresophisticatedmodelsinhigherdimensionswithmorecomplexgeometries.ThefourthcoremathematicscourseatUSMAisProbability&Statistics.Inthiscourse,studentsusestochasticmodelingtorevisitpreviousproblemswithnewmathematicalperspectives andnewfundamental concepts,andto solvenewapplicationproblems.Throughoutthesefourcorecourses,themajorcontentthemes ofundergraduatemathematicsarestudiedusingnewanddifferentperspectives.Thesecontentthemesarefunctions,limits,change,accumulation,vectors,approximation,visualization,representationofmodels,andsolutionmethods.Inadditiontomodeling and writing,a computation thread ties course content together.Theroleoftechnologyinthisprogram includesandextendsbeyondusingtoolforcalculations,explorationanddiscovery.Technologyaidsstudentsinvisualization andgraphics,dataanalysis,communication,and integrationofvariousmeansofproblemsolving.Mathematical ModelingMathematical modelingistheprocessofformulating and solving real-worldquantitativeproblems using mathematics.Inperformingthisprocess,wenormallyneedtodescribea real-world phenomenonorbehaviorinmathematicalterms.Often,theproblemsolveris interested inunderstandinghowasystemworks.thecauseofitsbehavior, the sensitivity of the process tochanges,predictingwhatwill happen,ormakingadecisionbased onthemathematicalmodeldeveloped.Thefourbasic steps in the mathematicalmodeling process are as follows:Step1:IdentifytheProblemStep2:DevelopaMathematicalModelStep3:SolvetheModelStep4:Verify,Interpret,and Use theModelWebriefly discuss eachof thesesteps which arenot always soclear anddistinctin everyproblem-solving endeavor.However,byusing thismathematicalmodelingprocess,problemsolverscangainconfidencetoapproachcomplexanddifficultproblemsandevendeveloptheirowninnovativeapproachestosolvingproblems.16
16 multivariable and vector calculus. Students encounter more sophisticated models in higher dimensions with more complex geometries. The fourth core mathematics course at USMA is Probability & Statistics. In this course, students use stochastic modeling to revisit previous problems with new mathematical perspectives and new fundamental concepts, and to solve new application problems. Throughout these four core courses, the major content themes of undergraduate mathematics are studied using new and different perspectives. These content themes are functions, limits, change, accumulation, vectors, approximation, visualization, representation of models, and solution methods. In addition to modeling and writing, a computation thread ties course content together. The role of technology in this program includes and extends beyond using tool for calculations, exploration and discovery. Technology aids students in visualization and graphics, data analysis, communication, and integration of various means of problem solving. Mathematical Modeling Mathematical modeling is the process of formulating and solving real-world quantitative problems using mathematics. In performing this process, we normally need to describe a real-world phenomenon or behavior in mathematical terms. Often, the problem solver is interested in understanding how a system works, the cause of its behavior, the sensitivity of the process to changes, predicting what will happen, or making a decision based on the mathematical model developed. The four basic steps in the mathematical modeling process are as follows: Step 1: Identify the Problem Step 2: Develop a Mathematical Model Step 3: Solve the Model Step 4: Verify, Interpret, and Use the Model We briefly discuss each of these steps which are not always so clear and distinct in every problem-solving endeavor. However, by using this mathematical modeling process, problem solvers can gain confidence to approach complex and difficult problems and even develop their own innovative approaches to solving problems
Step1:ldentifytheProblemTheproblemneedstobestatedinaspreciseaformaspossible.Onemustunderstandandconsiderthescopeoftheproblemwhenwritingthisstatement. Sometimes,this is aneasy step,whileothertimes thismaybethemostdifficult stepoftheentiremodelingprocess.Step2:DeveloptheMathematicalModelDeveloping themodel entails both translating the natural languagestatementmadeinStep1toamathematicallanguagestatementandunderstandingthe relationshipsbetweenfactors involved intheproblem. Tofurtherunderstandanddefinetheserelationships,simplifyingassumptionsareusuallyneeded.Inthis step,theproblemsolverdefinesvariables,establishesnotation,andidentifiessomeformofmathematicalrelationshipand/orstructureThemathematical modelissometimestheequivalentoftheproblemstatementin mathematical notation.Step 3: Solve the ModelThis stepis usually the mostfamiliar to students.Themodel from Step2is solved, and theanswer understood in the context of the original problem.Theproblemsolvermayneedtofurthersimplifythemodel ifitcannotbesolved.Thesolutionproceduremanytimes involvesanalytic,numeric,and/orgraphictechniques.Step4:Verify,Interpret,and UsetheModelOnce solved, the model must be tested to verify that it answers theoriginal problemstatement,itmakessense,anditworksproperly.Afterverifyingthemodel,theproblemsolverinterpretsitsoutputinthecontextoftheproblemIt ispossiblethatthemodel worksfine,but themodeler coulddevelopor needstodevelopabetterone.Itisalsopossiblethatthemodelworks,butit'stoocumbersome ortoo expensiveto use. Onceagain,theproblem solver returnstoearlierstepstoadjustthemodeluntilitmeetsthedesiredandnecessarycriteriaOncethemodel meetsthedesignneeds,oneusesthemodel tosolvetheproblem.As indicated, the modeling process is iterative in the sense that as the problemsolverproceeds,heorshemayneedtogobacktoearlierstepsandrepeattheprocessorcontinuetocyclethroughtheentireprocessorpartofitseveraltimes."Simplifyingthemodel"referstotheprocessofgoingbacktomakethemodelsimplerbecauseitcannotbesolvedoristoocumbersometouse."Refining the model"indicates the process of going back to make the modelmorepowerfulortoaddmorecomplication.Bysimplificationandrefinement,onecanadjusttherealism,accuracyprecision,and robustnessneededforthemodel.17
17 Step 1: Identify the Problem The problem needs to be stated in as precise a form as possible. One must understand and consider the scope of the problem when writing this statement. Sometimes, this is an easy step, while other times this may be the most difficult step of the entire modeling process. Step 2: Develop the Mathematical Model Developing the model entails both translating the natural language statement made in Step 1 to a mathematical language statement and understanding the relationships between factors involved in the problem. To further understand and define these relationships, simplifying assumptions are usually needed. In this step, the problem solver defines variables, establishes notation, and identifies some form of mathematical relationship and/or structure. The mathematical model is sometimes the equivalent of the problem statement in mathematical notation. Step 3: Solve the Model This step is usually the most familiar to students. The model from Step 2 is solved, and the answer understood in the context of the original problem. The problem solver may need to further simplify the model if it cannot be solved. The solution procedure many times involves analytic, numeric, and/or graphic techniques. Step 4: Verify, Interpret, and Use the Model Once solved, the model must be tested to verify that it answers the original problem statement, it makes sense, and it works properly. After verifying the model, the problem solver interprets its output in the context of the problem. It is possible that the model works fine, but the modeler could develop or needs to develop a better one. It is also possible that the model works, but it’s too cumbersome or too expensive to use. Once again, the problem solver returns to earlier steps to adjust the model until it meets the desired and necessary criteria. Once the model meets the design needs, one uses the model to solve the problem. As indicated, the modeling process is iterative in the sense that as the problem solver proceeds, he or she may need to go back to earlier steps and repeat the process or continue to cycle through the entire process or part of it several times. “Simplifying the model” refers to the process of going back to make the model simpler because it cannot be solved or is too cumbersome to use. “Refining the model” indicates the process of going back to make the model more powerful or to add more complication. By simplification and refinement, one can adjust the realism, accuracy, precision, and robustness needed for the model