How Should We Prepare the Students of Science and ... · ing how everything works. Technology is the application of scientiﬁc models for developing devices capable of increasing

How Should We Preparethe Students of Science and Technologyfor a Life in the Computer Age?Hans Petter Langtangen · Aslak Tveito

1. Models in Science and Technology

1.1 Computer Modeling

Science is modeling; we derive models of Nature for the purpose of understand-“We are still in the beginningphase of the computer revolution.This revolution can be regarded asthe second half of the scientific-industrial revolution. Machinerywill not only relieve the drudgeryof our aching backs, but of ourminds as well.”“We can expect rapid progress on abroad front in science and technol-ogy. The progress will be driven bybetter hardware, better algorithmsand better theory.”

J. GlimmProc. Symp. P. Math. 50 (1990)

ing how everything works. Technology is the application of scientific modelsfor developing devices capable of increasing the quality of life in general.

Certainly, “model” is a very general term, and its content vary throughoutthe branches of science. In the natural sciences, the mathematical models haveplayed an important role during the last centuries, and there is no doubt thatthe computer has dramatically increased the scientific and technological po-tential of applying mathematical models. For the last fifty years scientists havewitnessed a dramatic progress in computer power, numerical methods, and soft-ware solutions. These three fields are truly crucial for the successful applicationof mathematical models. Indeed it is fair to argue that von Neumann’s vision ofreplacing physical experiments with computer simulations has been realized,or at least, is about to become reality. Today, experimental scientists and com-putational scientists join forces in order to derive realistic computer modelsfor complex problems. As von Neumann predicted, such computer models doreplace tedious and expensive experiments allowing the experimental scientiststo focus on even more complex situations where models are not yet applicable.Furthermore, computer simulations allow us to study problems that is out ofreach for physical experiments.

In the next subsections we review a couple of characteristics of the devel-opment of hardware, numerical algorithms, and software during the last fivedecades, and thereafter we outline some likely future trends.

“Indeed, to a great extent, experi-ments in fluid mechanics are car-ried out under conditions where theunderlying physical principals arenot in doubt, where the quantitiesto be observed are completely de-termined by known equations. Thepurpose of the experiment is not toverify a proposed theory, but to re-place a computation from an un-questioned theory by direct mea-surements.”

J. von NeumannCollected Works, vol. V.

1.2 Hardware Versus Numerical Methods

The first electronic computer, ENIAC1, was able to perform about 330 floatingpoint operations per seconds (flops)2. For the solutions of linear systems arisingfrom implicit finite difference discretization of partial differential equations,banded Gaussian elimination was the standard method, and the implementation

1 Eniac was put into use at the University of Pennsylvania in 1946; for further informa-tion, consult http://homepage.seas.upenn.edu/~museum/

2 The speed is in discussed in Goldstein and von Neumann’s paper “On the Principlesof Large Scale Computing Machines”, cf. von Neumanns Collected Works, vol. V.

Page: 1 job: langtang Engquist/Schmid (eds.) Mathematics Unlimited – 2001 and Beyond date: 1-Sep-2000

2 H. P. Langtangen · A. Tveito

was done in a programming environment very close to the hardware of thecomputer.

The Poisson equation on the unitcube, −∇2 = f , with suitableDirichlet or Neumann boundaryconditions, can be discretized bya second-order standard finitedifference method on a uniformlypartitioned grid with m = n×n×ngrid points. The work of a Conju-gate Gradient method with, e.g.,one V-cycle multigrid iteration aspreconditioner is O (m), whereasthe work of banded Gaussianelimination is O (b2m), whereb = O (n2) is the matrix band-width. The fraction of these twowork estimates is hence O (n4).Notice that the work estimatesassume that the problem size fitwithin the available RAM. Thestorage demands are O (bm) forbanded Gaussian elimination andO (m) for the iterative method.For n = 100, the storage demandsof banded Gaussian eliminationis O (80Gb) RAM, which is farbeyond the limits of today’s PCs.Numerical experiments on a 600Mhz Pentium III processor showthat the CPU-time for bandedGaussian elimation is about10−7n7 and 1

2 10−4n3 for theConjugate Gradient method usinga multigrid V-cycle as precondi-tioner. State of the art computinguses a value of n ranging from 100to 1000. In the case of n = 1000,the modern method is about2 · 109 faster than the old method(provided that sufficient memoryis available).

At the time of this writing, our desktop computers are approaching one gi-gaflop with gigabytes of fast memory. Hence, a single processor in year 2000 isabout three million times faster than ENIAC. In addition, most scientists can logon to even more powerful parallel computers in super-computing centers. Manyreaders may not be aware of the fact that the field of numerical methods haswitnessed a similar development in the algorithmic speed of numerous solutionprocedures. For example, today one can apply optimal multigrid methods forsolving linear systems arising from partial differential equations. These mod-ern methods outperform the typical method at von Neumann’s time (bandedGaussian elimination) by a factor of the order n4 when solving a Poisson-likeequation with finite differences on an n × n × n cube.

Students meet the simple Gaussian elimination algorithm already in thefirst year of their university education. The multigrid method, on the otherhand, is a mathematically quite complicated topic. Understanding this methodand adapting it successfully to new problems require knowledge of advancednumerics and mathematical analysis. This is only one out of many examplesshowing that the advances of the computer revolution is a combined effect ofbetter hardware and better algorithms, and that the development and applicationof better algorithms increases the need for mathematical knowledge.

1.3 Software

The art of creating software for new numerical methods utilizing the new fea-tures of modern computers has changed completely over the years due to majorprogresses in computer science. The era of personal “don’t change – it works”-code is gone, and vast collections of old, monolithic, “spaghetti” codes in pureFortran are heading for the museums. Currently, all serious developers of sci-


How Should We Prepare the Students of Science and Technology? 3

entific software rely heavily on the use of software standards and techniques toimprove reliability, maintenance, and reuse of code. “It has been proposed that problems

with producing complex computercode in debugged and reliable formmay provide an outer limit for theuse of computers to solve certaintype of problems.”

J. Glimm and D. SharpIBM J. Res. Dev. 31 (1987)

We believe that further major progress is to be expected in the field ofscientific software. There is currently a giant gap between methods and software.We are able to formulate – in mathematical terms – well-defined methods forextremely complicated problems, but the path from such methods to softwareis long and complex. A striking fact is that software implementation frequentlyinvolves extensive work at abstraction levels that are much more primitive thanthose of mathematics (as an illustration, it is still very hard to read a scientificcode even if you know the method it is based on very well). A serious implicationof the relatively primitive state of software is that the time needed to transforma mathematically well-defined method for a complicated physical problem intorunning code is far too long. The solution to this problem must be computer toolscapable of handling formulations much closer to the language of mathematics

In the training of programming forscientific computation the empha-sis has historically been on squeez-ing out every drop of floating pointperformance for a given algorithm.. . . This practice, however, leads tohighly tuned racecarlike softwarecodes: delicate, easily broken anddifficult to maintain, but capable ofoutperforming more user-friendlyfamily cars.

B. Smith, P. Bjørstad, and W. GroppDomain Decomposition,Cambridge, 1996.

The processor speed and memory capacity of computers will continue toincrease. Nevertheless, of more importance now is the scientists’ ability tobuild their own supercomputers through affordable standardized components;see http://www.beowulf.org, where it is carefully explained how to build a su-percomputer based on standard PCs. In a few years it is expected that evenstandard PCs come with a number of CPUs. The average engineer or scientistutilizing comprehensive mathematical models must hence have parallel algo-rithms and software easily accessible to fully utilize the power of mainstreamcomputers. As parallel computing has mainly been restricted to expert groupsin numerical simulation until now, a dramatic progress in adapting methods forparallel computing environments must take place in the near future. We believethat this represents the main algorithmic challenge to the scientific computingcommunity, and we do not believe that the problem will be solved through smartcompilers.

2. The Educational System

2.1 Paper and Pencil; Ready for the Museums?

The future importance of computers in science and engineering is obvious tomost of us. We would therefore expect that the entire educational system hadchanged accordingly. This has, however, not been the case so far. Browsingthrough virtually any recent textbook in physics, geophysics, petroleum engi-neering etc., the picture is the same; over-simplified models are analyzed withpaper and pencil methods throughout. Where are all the books that reflect theimportance of computers in these applied fields? Even in applied mathematics,which by nature is close to the computer revolution, the curricula have changedvery little for the past 30 years. It seems obvious that a complete revision ofthe basic education in both science and engineering is necessary to meet thedemands of modern candidates and their employers.

Much of the current focus on algebraically challenging, lengthy, error-pronepaper and pencil work can be significantly reduced. In fact, we seriously doubtthat there will be space for this type of activity at all in a few decades, at least



not in the mainstream education. The reason for such an evolution is that thecomputer is simply much better than humans on any theoretically phrased well-defined repetitive operation.

2.2 Do You Still Use Numerical Tables to Evaluate the Sine Function?

The authors remember from their high-school days how they had to learn com-puting sines and cosines from tables. The calculator was there, actually weall had our own, but the educational programs had not yet adapted to the newtechnology. Basically, the same effect hits the universities when we ignore theexistence of Matlab, Mathematica, Maple and similar software, which solvesvirtually any exercise in basic calculus and linear algebra. Sometimes it appearsthat many teachers in mathematics regard such software as a threat towards theirprofession. That is in our view a tragic misunderstanding; proper applicationof software would allow us to increase the level of calculus, by enabling thestudents to learn more and focus on what are really the difficult issues ratherthan wasting their time on repetitive trivialities.

2.3 We Are Out of Phase with the Rest of the World

It is the authors’ opinion that the undergraduate education at most universities isout of phase with the modern professional application of mathematical models.Considering computationally oriented research projects in science or industry,successful problem solving in such contexts normally consists in combiningthe best tools and knowledge from all relevant fields. In many occasions thisincludes physical and mathematical modeling, adapting numerical methods ap-propriately, design and implementation of software, design of computationalexperiments, implementation of physical experiments if possible, and valida-tion of the model. These ingredients are often repeated in an iterative process.Unfortunately, such real-world problem-solving strategies are seldom reflectedin the educational system.

Not only pure mathematicians seem to neglect the importance of doingcomputer-based mathematics and the need to adapt the education accordingly.Also in classical subjects, like physics and the geosciences, the role of math-ematics and computers are kept at a moderate level with little impact on theculture or courses. Some trivial observations explain the slow progress in in-corporating modern computing tools. Students are sent like ping-pong ballsbetween university buildings. Each building has its own traditional culture andtheories – and its own budget that must be protected. Each building gets itsshare of courses in a program, and the professors in the building put in mucheffort to preserve the traditions of their particular subject. The result is a setof “pure” subjects and strong conservatism – two characteristics that are notwell correlated with a multi-disciplinary and rapidly developing technologicalworld.



2.4 The Revolution Has Started

The reformation we think is demanded in mathematical education, and whichwe describe in the present chapter, has already been initiated at some univer-sities around the world. Comprehensive reformed calculus projects have beenexplored in the United States during the last two decades, see e.g. Murphy’s re-view of projects and measured achievements [1]. Several universities in Europehave recently launched new programs, mainly at the master’s or doctoral levels,with increased emphasis on computers in science and engineering education.Some examples are ETH in Zurich [2], KTH in Stockholm [3], Chalmers inGothenburg [4, 5], and NTNU in Trondheim [6]. Despite these scattered ef-forts, the mainstream mathematics education still follows the traditional tracks.We use this opportunity to explain why it is both strategically and scientificallyimportant for the mathematical world to adapt more thoroughly to the computerage.

In the last couple of years, numerous initiatives in the form of conferencesand university programs have been launched under the heading “computationalscience and engineering”, frequently abbreviated as CSE. CSE appears as amixture of well-established disciplines, but where the subjects are more inte-grated and computational tools are used to a much larger extent. It is a strikingfact that CSE very often emerges from computer science3 or applied/industrialmathematics departments and not from classical pure mathematics, science, orengineering departments. One reason might be that professors in computer sci-ence and applied mathematics departments are closer to the computer and itsapplications and can more clearly see the revolution that is going on. Anotherreason could be the heterogeneous composition of professors in a new subjectlike computer science and the lack of a long and strong scientific tradition, orput in another way: they do not have centuries of traditions to defend. CSEprograms try to fill the gap between the traditional university education and thecomputer-based problem-solving techniques that the candidates will meet inthe real world.

Later in this chapter we will argue that the view on problem-solving pursuedin the CSE programs we see today is the right view to implement also in thebasic mathematics education.

“It is impossible to exaggerate theextent to which modern appliedmathematics has been shaped andfueled by the general availabilityof fast computers with large memo-ries. Their impact on mathematics,both applied and pure, is compara-ble to the role of the telescopes inastronomy and microscopes in bi-ology.”

P. LaxSiam Rev. 31 (1989)

3. Will Mathematics Leave the Mathematicians?

3.1 Mathematics Used to Be Scientific Computing

Traditionally, mathematics is introduced to students by pure mathematicians.Since pure mathematicians, in general, seem to ignore or sometimes fight againstthe possibilities offered by computers, it is appropriate to ask whether they willbe able to keep their monopoly on teaching mathematics. Traditionally, mathe-matics played the role of scientific computing, i.e., the role of computing solu-tions to mathematical problems arising in science. Today, scientific computing

3 Computer science, as used here, also includes mathematically oriented subjects suchas scientific computing, numerical analysis, and cybernetics.



is a field of its own closer related to computer science and applications than tomathematics.

One common tradition in classical subjects is to present well-developedfields through building elegant theories, followed by simplifications and ex-amples, and then perhaps addition of some heuristics to approach a real-worldproblem. At the introductory mathematics level most of the theoretical exposi-tion quickly collapses to teaching recipes for the exam exercises, while at higherlevels in theoretical sciences, reproduction of the theories usually pays betteroff at the exam than good real-world problem-solving skills. If one presentsmathematically oriented theories in an abstract way, and one ignores practicalapplications of these theories, the educational system has one fortunate feature:students are well trained in abstract thinking. Earlier, when only a small portionof the population attended university studies, the fraction of good students wasof course higher than in today’s mass education. Consequently, the need forpedagogical presentation was lower in the past. Moreover, the technologicallevel in the job market was also lower so demands to practical applications ofthe theories were not as crucial as today. In contrast to this, modern univer-sity education needs to ensure that a large number of people are able to solvepractical problems using advanced mathematical models and software tools.

3.2 Mathematics Must Interact with Other Disciplines

The previous arguments imply that many mathematically oriented theories nolonger live an academic life on their own. Continuum mechanics is one strikingexample. A few decades ago, the general equations of continuum mechanicsneeded substantial simplifications before reaching any practical engineeringimpact. Today, numerous codes offer robust engineering modeling based onvery complicated mathematical models from continuum mechanics. Thousandsof engineers run these codes every day as a part of industrial design procedures.Simulations of metal forming, turbulent flow, or electromagnetism have madethe theories a practical fundament for our modern technology. Any decision-making process based on such simulations needs to view the results in light ofthe applicability and the built-in assumptions of the underlying theories.

A natural consequence would be that applied and industrial mathematicsas well as e.g. theoretical mechanics exploded in popularity and also enteredmost parts of the engineering education. This has not happened, despite all ex-pert predictions of the future need for knowledge of mathematical modeling.In fact, many engineering programs cut down on mathematics and theoreticalmechanics with the argument that computer packages now solve the problemsin these areas so there is less need for detailed courses on the classical subjects.Such a conclusion is partly correct and partly wrong. Surely, engineers runningadvanced simulation codes need to be better trained in the models that are im-plemented in the codes. Hence, courses in classical subjects need to go intomore depth. On the other hand, the classical education still follows the traditionof heavy analytical work and have not yet adapted to the new world of simu-lations, questioning the importance of classical courses in modern engineeringeducation.



Unless the professors of mathematical subjects adopt a new vision for thecomputer age and redesign their courses, there is no future of the mathematicswe know today across a wide range of programs in science and engineering.Mathematical thinking and mathematical concepts will definitely play an in-creasingly important role in the future as we increase the use of computers, butthe corresponding education will take completely different forms and probablynot be dominated by scientists from pure mathematics.

4. Computational Mathematics from Day One?

Forty years ago the Noble Prize Laureate, Richard P. Feynman at Caltech, ini-tiated a major revision of the introduction to physics. One of his objectiveswas to maintain the enthusiasm students have as they enter the university. Webelieve that the situation is somewhat similar today; students entering univer-sities today expect to find a modern environment fully utilizing the power ofcomputers, which they know from their own experience and from what theyread in newspapers etc. Instead, the situation today is that many students do notseriously use a computer beyond text processing. Modern concepts like parallelcomputing are only presented for a small group of students at advanced levels inscience and technology. We should work more in the spirit of Feynmann; startday one with computing, also in parallel (see section 5.3), and let the studentsexperience how powerful mathematical models are to predict the behavior ofNature and technical devices.

“The special problem we tried toget at with these lectures was tomaintain the interest of the veryenthusiastic and rather smart stu-dents coming out of high schooland into Caltech. They have hearda lot about how interesting and ex-citing physics is – the theory ofrelativity, quantum mechanics, andother modern ideas. By the end oftwo years of our previous corse,many would be very discouragedbecause there were really very fewgrand, new, modern ideas presentedto them. They were made to studyinclined planes, electrostatics, andso forth, and after two years it wasquite stultifying. The problem waswhether or not we could make acourse which would save the moreadvanced and excited student bymaintaining his enthusiasm.”

R. P. FeynmanThe Feynman Lecture on Physics,Addison-Wesley, 1963.

At the research level in engineering, economics, and physical sciences, suc-cessful problem solving proves to be a sound combination of classical theories,computational tools, software, and experimentation. Such a mixture of subjectsfor the solution of a particular problem seldom appears before the graduate orpost-graduate level. The reason is quite obvious as most university programsare built on courses from classical disciplines. It takes at least an undergraduatestudy to collect enough classical material for an interdisciplinary graduate study.This is also the reason why most CSE programs are placed at the Ph.D. level.If our aim is to produce candidates with multi-disciplinary problem-solvingexperience and strong focus on modern computational tools, is it then optimalto let the student spend several years bumping back and forth between variousdepartments teaching classical subjects in the classical way? We do not thinkso. We believe that students should meet multi-disciplinary, problem-orientedmaterial saturated with computer experiments from the very first day, followedby theoretical or classical subjects as part of a specialization. In other words,first get acquainted with modern life, then learn to know the roots.

5. Intuition Motivates Rigor

5.1 Increasing the Understanding by Computing

Professional scientists and engineers frequently utilize computations and exper-imentation to build intuition in relation to a specific project and thereafter collecttheories to organize and explain typical observations made in the more empir-



ical approach. Our justification of spending time on theory is that it increasesthe understanding of the practical work. Instead of first saturating the studentwith rigor and then hoping that intuition will develop, the intuition should growthrough computer experimentation and prepare for rigor [7], as we shall exem-plify. Rigor, with a firm basis in intuition and experience, can hopefully bringacademic theories out to a larger audience. This strategy means that we thinkthe ideas of CSE are well suited for the undergraduate study and not only as aPh.D. program.

The discussion of intuition versus rigor in mathematics is classical; shouldcalculus be a thorough understanding of what integration and differentiation re-ally means, should it be a bag of useful tricks, or should it be a careful derivationof fundamental mathematical results about functions? Often, rigorous proofs aregiven quite some attention in the textbooks, where as the lectures and the examsfocus on the techniques. But what do the students really understand? And howdo they develop this understanding? Obviously, very good students developunderstanding from reading proofs. They understand both the idea on whichthe proof is based and they understand the proof itself. Weaker students, i.e.,the masses in mathematics education, are probably satisfied when they masterthe techniques without really grasping the fine print of the proofs. Anyway,mastering the techniques is usually sufficient for the exam.

We believe that all students can develop deeper understanding through sim-ple computational studies, and calculus is very well suited for such experiments.Let us mention some of examples.

Differentiation: In any computer language the student can generate a completeprogram in less than ten lines, which illustrates the limiting process of computingthe derivative.

Integration: Implementing Riemann sums and performing numerical experi-ments is easy, even on a calculator, and will enhance the students understandingof the concept of integration.

Ordinary differential equations: The simplest numerical schemes for ordinarydifferential equations can be implemented in just a couple of lines using anyreasonable computer language.

Limits: Very simple programs can be used to study limits of sequences and thesum of series. Limits also naturally bring in the effects of the computer’s finitearithmetic.

Algebraic equations: Newton’s method and similar algorithms for solving non-linear algebraic equations are very simple to implement and give answers toequations that are otherwise considered “impossible” to solve.

Graphing: Students’ understanding of functions can be greatly enhanced byextensive use of the plotting features found in modern computer tools.

Different approximation rules for integration or differentiation also demon-strate different speeds of convergence towards a limit, a point of great practicalinterest (the teacher must of course be careful in order to avoid confusing diffi-culties with round-off errors).



5.2 More on Integration

Allow us to be a bit more specific on how basic integration of functions can betaught in a CSE setting and how the pedagogical strategies differ from thoseof classical calculus. The classical exposition of integration, at least as experi-enced by the authors, draws a set of rectangles approximating the area under acurve, and presents the integral as the limit of these rectangles as their widthsapproach zero. Many students hear the teacher say that this limit process is afundamental topic of the course without understanding why; students gain littleunderstanding of what integration really means from this derivation. What theyunderstand is the practical calculation of some anti-derivatives using variousrecipes (substitution, integration by parts, partial fractions etc.). For some stu-dents the motivation for learning the recipes is to get through a compulsorycourse, for others the integration exercises act as fun puzzles. That integrationis a fundamental and practical tool in a wide range of jobs out in the real worldis not obvious.

What is important to learn about integration? First of all, the understand-ing of what integration really means. In addition one must be able to computeintegrals by the methods that professionals use. Such methods cover numericalapproximations and symbolic manipulation software like Maple and Mathe-matica. We believe that discrete formulation of a real-world problem involvingintegration, programming numerical approximation schemes, and experiment-ing with these provide good means for developing an understanding of limitprocesses and what integration is about. Both differentiation and integration areeasier to understand from a discrete viewpoint than from the continuous one.Working with discrete quantities and generic algorithms should therefore be thenatural starting point, not classical calculus. Analytical techniques are of courseessential as these will be met in all types of mathematical modeling literature,but the techniques should be taught after the discrete counterparts and based onavailable computer tools, like Maple or Mathematica. This pedagogical strat-egy is probably much more difficult than training integration tricks in classicalcalculus and will hence be more demanding to teach. Unfortunately, the amountof tricks is constant or perhaps even increasing when utilizing Maple and Math-ematica, but the tricks are of different nature and of more practical value if theaim is to find closed-form anti-derivatives.

5.3 Integration Illustrates Concurrency

Another fundamental ability in which modern scientists and engineers shouldbe trained is thinking in parallel. As we have pointed out, parallel computingwill be essential for utilizing cheap hardware in the near future, hence requiringstudents to become familiar with formulating mathematical tasks in a parallelfashion. Sequential algorithms have a strong tradition in mathematics as wellas in our everyday life, and this tradition makes parallel thinking demanding.However, its importance deserves attention already in the introductory math-ematics course. Numerical integration is perhaps the simplest example of amathematical problem whose basic operations (function evaluations) can berun in parallel and where broadcasts are necessary to arrive at the final result.



Simple hands-on experiments with load balancing, speed-up etc. are also easyto accomplish. Through numerical integration and very short programs, writtenin almost any computer language, the student can discover many fundamentalaspects of mathematics and modern computing.

Learning from one’s own experience is generally regarded as a good wayof developing an understanding of a phenomenon. Nevertheless, knowledge ofsome facts is also required for successful solution of new problems. When itcomes to integration, not everything can be learned from a numerical approachand the use of symbolic software. The students must of course learn by heartwhat the integral of sine, cosine, exponential, and logarithmic functions is. Thiscould be accomplished by a study of the most important fundamental functions,covering their graphs, derivatives, integrals, and other properties that must beknown as hard facts. Formal testing of such hard facts can to a large extend beautomated on the Web.

5.4 Differential Equations

Simple differential equations and associated numerical algorithms form a nat-ural topic of continuation, which extends the basic ideas of integration anddifferentiation, and in addition introduces solution of linear and nonlinear equa-tions, systems of algebraic equations etc. Furthermore, important concepts suchas stability and the solution’s dependence on input data are easily illustrated.More demanding challenges for parallel thinking are also obvious. The studentcan experiment with small programs and discover the importance of all thesemathematical concepts while trying to find solutions to physical or financialapplications that seem meaningful.

When students from the reformed teaching of integration and differentialequations reach the basic course of, e.g., physics, which usually concerns parti-cle and rigid body mechanics, they will probably not be well enough prepared forthe integration tricks and lengthy algebraic hand-calculations that are requiredfor solving some of the mechanics problems in closed form. What is requiredis a reformation of the mechanics course as well. A computationally orientedintroductory mechanics course has many benefits. First of all, analytical toolsput strong limits on the type of problems that can be considered. Numerical inte-gration of ODEs, e.g. in Matlab, Maple, Mathematica, or in one’s own program,constitutes a tool that performs the mechanics of producing and visualizing thesolution of a problem. The course can then concentrate on developing mathe-matical descriptions of a physical problem, the topic that should appear in mainfocus all the time, but which is unfortunately often drowned in tedious algebraor “local tricks” for solving the equations in the model. With the visualizationcapabilities of modern software one can also spend much more time on inves-tigating the solutions’ response to variations in physical input parameters andthereby develop the physical understanding and intuition to a larger extent thanwhat we think is possible with pencil and paper.

Another benefit from extensive use of computations in mechanics and physicscourses is that the formulation of the mathematical model and the solution pro-cess becomes decoupled, a fundamental requirement when utilizing numericaltechniques and simulation software. The classical literature and education in



these fields often mix the problem formulation with special mathematical solu-tion tricks. One frequent example is to first mention boundary conditions whenthey are needed to determine integration constants.

5.5 Statistics

Integration, differentiation, and differential equations are just the most obvi-ous examples of computer utilization when introducing mathematical methods.Statistics is a topic that offers even more exciting and useful application of com-puters. Almost all theoretical higher educations have an introductory course instatistics. Looking at the enormous amount of texts written for such courses, itappears that the contents are virtually the same; strong emphasis on probabilityalgebra, Gaussian models, (mathematical) estimation, hypothesis testing, F andt distributions, two-sample tests, large samples, and single-variable regression.Computer exercises are integrated in most courses now, but only as an illustra-tion on how to apply the classical statistical procedures to practical problemswith collected data.

What would be the most “useful” set of topics to be covered in an intro-ductory statistics course? The most fundamental topic is clearly to learn aboutstochastic models. Associated computer-based simulation methods for inves-tigating stochastic models, with relaxation of the common assumptions, likenormality, provide a widely applicable tool for later courses throughout sci-ence, engineering, medicine, or business. Hypothesis testing and two-sampletests have of course great practical value in clinical medicine and pharmacol-ogy, but one can ask why these topics together with probability algebra andtedious mathematical derivations always form the core part of the first statisticsencounter.

In business, engineering, or science, we claim that estimation and predictionrepresent the most important aspects of stochastic models. Focusing at useful-ness and wide applications of the developments in statistics, it seems fruitfulto give high priority to algorithms for estimation and prediction in introduc-tory courses. This involves least-squares problems, maximum likelihood esti-mation, numerical optimization, generation of random variables with specifiedcorrelations and distributions, bootstrapping, multi-variable linear and nonlin-ear regression, as well as visualization of higher-dimensional data. The overallgoal should be to use the rich collection of powerful tools based on proba-bilistic thinking to find structure in seemingly unstructured datasets. This isuseful in any discipline where the underlying mechanisms of Nature or societyare too complicated to be adequately represented by deterministic models. To-gether with training in building stochastic models (an art that requires extensiveknowledge of other subjects than statistics), such a collection of tools will havemuch wider applicability throughout engineering, science, and business thantoday’s compulsory subjects like hypothesis testing and F tests.

Bringing in a simulation-based methodology when working with stochasticmodels also involves many of the mainstream numerical methods for determin-istic problems. Such an evolution will therefore integrate statistics tighter withother disciplines of computational mathematics. Another interesting aspect ofsimulation-based statistics is the possibility of generating large samples on the



computer and analyzing these samples by the very well developed asymptoticlarge-sample theory. This is one of several examples demonstrating that classi-cal topics of mathematics do not necessarily lose their relevance in a moderncomputerized mathematical world; they just become important in another way.

5.6 Focus on Applications

More extensive use of computers and widely applicable algorithms is one partof the reformation in the mathematical education. Of equal importance is astronger focus on applications. The “theory first, examples later” tradition inbasic mathematics courses must be broken. A guiding rule is that, where itis possible, theoretical concepts should be clearly motivated by applicationsbefore diving into the finer mathematical details. Fortunately, recent texts onmathematical methods are tighter integrated with applications, although there isstill a long way to go before one can easily find a wealth of motivating exampleswith associated computer experimentation for various mathematical conceptsin the textbook literature.

Constructing suitable applications for mathematical teaching is in fact verychallenging. The required background from the application area must be lim-ited, special jargon must be removed, the principal issues must not drown inthe explanations of lots of parameters, and so on. Fortunately, an applicationcan be made availabe through a ready-made computer program, which allowsthe teacher to focus on principal issues, while the internals of the program in-corporate enough details of the application to make it relevant for real-worldproblems.

Computer exercises will naturally make extensive use of graphics whoseimportance in teaching can hardly be over-estimated. A visual approach tomathematical concepts is effective in developing intuition and understanding.This might be even more important in advanced physical, statistical, and engi-neering topics than in basic calculus. We believe that the graphical capabilityof the computer is perhaps the strongest arguments in favor of reforming thescience and engineering education.

To summarize, the pedagogical strategy we advocate is to learn mathematicsthrough a CSE approach. This supports development of intuition before rigor.Too many university programs focus on rigor and leave intuition as a by-product.Intuition is a basic requirement for any successful career involving mathematicalmodeling outside academic systems. The best students develop intuition fromstudying and using theories, while the not-so-good students only manage torepeat theories from books, i.e., rigor without intuition often pays off well atthe exam.

6. Computerized Mathematics: Easy or Hard?

A widespread view among theoreticians is that CSE is some kind of “soft-ening” of classical, “hard” mathematically oriented subjects, constructed forthe not-so-bright students. Indeed, CSE is softer in the sense of being morepractical and less abstract when introducing new topics, since the approach



is algorithmic, computational, and experimental. This property could increasethe understanding and interest among a larger portion of the students. On theother hand, CSE projects are multi-disciplinary and make strong demands tooverview and understanding of principal ideas in several fields. In that sense,passing a CSE project is much harder than passing a classical exam, where youonly need to carry out some portion of tedious algebraic manipulations of anexercise puzzle. It might well turn out that combining numerics, mathematics,physics/engineering, and computer science in the CSE philosophy is generallytoo demanding for most students. Narrowing the width of CSE projects wouldthen be needed. Obviously, creating intellectually challenging courses is as easyin CSE as in the classical university disciplines.

Another objection against CSE and “computerization of mathematics” isthat the computer algorithms are black-box recipes that one can just type intothe computer without understanding what is going on. Our view is quite theopposite. A computer program is completely useless if it calculates the wrongnumbers – even one small error is unacceptable. The verification and debug-ging process require the programmer to carefully go through all the detailsof the solution method and investigate how each piece contributes to the fi-nal answer. Frequently, one has to think more precisely than in a proof. Asmany of us have experienced, the understanding of a numerical algorithm oftenarises during debugging, and the simulation itself may reveal effects that werenot considered initially. Computer implementation is therefore an indispensabletool for increasing the understanding. Many students have major problems with,or spend an unreasonable large amount of time on, programming and debuggingmathematical problems, reflecting that such work is “hard”.

Computerization of classical topics will appear equally hard for the teachers.Successful teaching of CSE requires a strong interest in applications, hands-onexperience with modern computer algorithms and tools as well as knowledgeof classical mathematical subjects. Few teachers have this broad background,and especially not in university systems where narrow interests usually paybetter off. We therefore think that a criterion for success of CSE is to place thecore of the teaching in scientific computing groups having close collaborationwith groups in applied sciences (physics, engineering, economics, medicine).Finding the right balance between the various ingredients in CSE programs isessential for producing successful candidates for the outside world.

7. The Central Role of Software

Extensive use of computers, as outlined above, requires appropriate software.For the simplest examples of numerical integration, differentiation, solution ofdifferential equations, or stochastic simulation, a page of code in almost anycomputing environment (e.g. Matlab, Maple, Mathematica, Fortran, Java, C,C++) suffices. An important part of the student’s learning process is to writeand debug such programs. More comprehensive software development projectsare also necessary to improve the programming abilities; numerical codes arein general huge and complex.



The way we teach students to write programs is important. The field ofnumerical computing has and will always have a strong emphasis on efficientcode. The tradition of optimizing every statement for the problem and hardwareat hand has unfortunately produced large and extremely complicated codesthat are hard, and occasionally impossible, to extend and maintain. Prematureoptimization should hence be avoided, and the focus, at least in education, shouldbe on writing code that is sufficiently general and well designed for extensions,easy maintenance, and reuse by others. Experience shows that this type of codespends most of its CPU time in just a few functions. A standard profiling pointsout the functions that are candidates for special optimization. Such optimizationmight involve careful rewriting of statements or a change of algorithm. As wementioned in the introductory section, the speed of a code might be very sensitiveto the choice of numerical algorithms. Therefore, developers of numerical codemust have a balanced view on algorithms, choice of data structures, and theefficiency of a particular set of statements. Recall that in research, the humanefficiency of numerical code development is often of greater importance than thepure number crunching efficiency, a view that is supported by the observationthat computers get cheaper while humans only become more expensive.

When studying advanced numerical methods or non-trivial applications inscience and engineering, writing the appropriate software from scratch is far tootime consuming. Moreover, developing a huge code for one single applicationusually introduces a lot of errors ranging from trivial bugs to serious misunder-standings. The modern approach to software development is to reuse, combine,and extend existing components. Such software components can frequently befound on the Internet or in commercial libraries.

The ability to build software by assembling “black box” solutions is impor-tant to develop, but the traditions for such kind of working strategies are weakin academic institutions. In an ideal world, a scientist or engineer should havecomplete knowledge of all computational details, including the statements inthe applied software, when solving a problem. However, this level of control isunrealistic. Most of us rely on the “sin” button on the calculator or the sin(x)function in any programming environment for computing the sine of a number,although the inner details of the sine computation are unknown to us. Only inapplications that spend most of their time on sine calculations, and where aperformance boost is important, we need to be concerned with algorithms forevaluating the sine function. When using environments like Matlab, Maple, orMathematica numerical solution to problems are often obtained by black-boxfunctions. Many of the functions also offer computations with a specified reli-ability, i.e., error tolerance. One common example is the solution of ordinarydifferential equations (initial-value problems); today we have easily accessiblesoftware that solve a wide range of systems of nonlinear differential equationswithout requiring an expert numerical analyst as user. The argument againstwidespread use of such black-box software is that the implementations andchoice of algorithms are frequently far from optimal with respect to compu-tational speed. For the average user, the implementation may be fast enoughon today’s computers, and users also often want to trade computational speedfor increased reliability and user-friendliness. It is therefore important that we



teach students throughout science and engineering to take advantage of black-box solutions as a natural part of any problem-solving process.

It is fair to say that we now have software with sufficient reliability for thetopics of basic mathematics, like calculus, linear algebra, ordinary differentialequations, and optimization. A major software industry is working together withacademic research groups to improve the quality of such computations further.For more complicated mathematical problems, partial differential equationsbeing one example, the access to reliable software is not that easy. Scientificand practical work is therefore needed to develop software tools to narrow thegreat gap between the mathematical language and the computer languages wecurrently use. One way to go is to construct higher-level abstractions using com-mon computer languages with support for user-defined abstractions (Fortran 90,C++, and Java are examples). Alternatively one can develop programming lan-guages tailored to large-scale numerical computing and associated softwareengineering (Fortran 2000 is one such example). Still another approach is todefine user-friendly, application-specific languages to a confined set of mathe-matical problems, following e.g. the successful style of Matlab (Python-Fortranintegration [8, 9] is one example). During the 1990s we have seen increasedinterest in research related to better software tools for scientific computing. Itwill, nevertheless, take more investigations over many years to reach consensusand to converge towards widely accepted standards. Meanwhile, the productionof software might be the most important limiting factor for taking full advan-tage of complicated mathematical models throughout a wide range of appliedsubjects.

We expect the central role of software to be manifested in other ways aswell. As scientific fields mature, techniques of analysis tend to converge to-wards a state where further research on the techniques themselves is not ofgeneral interest except in very special or demanding cases. This is a signal thatit is time to wrap the techniques in black-box software components and makethem available to a large community of potential users. At present, only a fewresearch groups around the world have an emphasis on migrating theory intowidely applicable software. Software development with such goals is of coursea difficult and resource-consuming task, but in far too many groups much timeis spent on writing software that is difficult or impossible to reuse. Paying moreattention to software design and making it reusable within the group itself andits collaborators accelerates the research and is a first step towards a tighterconnection between mathematical research and development of high-qualitysoftware modules for computational science and engineering. Many of the fa-mous American universities, like MIT, Stanford, and Berkeley have long andsuccessful traditions in this respect.

8. Concluding Remarks

In this chapter we have discussed why and how university education in math-ematics and mathematically oriented subjects should be completely reformedthrough a computer-based and algorithmic approach. This provides the op-portunity to teach concepts and applications before the traditional, algebraic,



manipulative skills. Our assertion is that the resulting students will have moregeneric and powerful tools at their disposal and produce solutions to a wider setof mathematical problems in shorter time compared with students from tradi-tional paper and pencil courses. On the other hand, the new students’ abilitiesto perform lengthy hand calculations on their own will without doubt decrease,but they will have more experience in using computer tools for doing suchcomputations.

Reforming the mathematics education will have impact on other subjectsthat traditionally make extensive use of mathematical methods and replace thecurrently strong focus on tricky hand calculations of simplified problems bynumerical simulation and visualization of more demanding applications. Webelieve that this change of focus will produce better candidates in accordancewith future demands in engineering and science. Moreover, it emphasizes math-ematics and computing as a profession and not only as a science (which is thewidespread view in current education).

Many readers will argue that examples and computer exercises already playan important part of modern mathematics education. This is true, but we believethat many lecturers’ attitude is to use examples to illustrate a theory and not letthe applications form the core of a mathematics course. In our opinion, a changein the philosophy of mathematics teaching should take place. Topics like calcu-lus and linear algebra, which act as practical tools for scientists and engineers,should be taught as problem-solving tools and not as polished mathematicaltheories when introduced to fresh students.

Looking at the complete education, proofs and rigor must not disappearas a consequence of computationally oriented, problem-based learning. Under-standing and constructing proofs can appear as more attractive and useful if thetopic is tightly integrated with computations; numerical experiments can sug-gest a more general theorem and the students can be guided in a formal proofof the theorem. Perhaps there is also a psychologically attractive effect of usingthe expression “rigorous explanations of why something is correct or wrong”instead of the word “proof”.

The impact of computerized mathematical education on research shouldnot be under-estimated. Teachers will be pushed to learn more about computer-based methods and experimental investigations. Using the same techniques inscientific work will most probably uncover new insight and open up challengingproblems for basic research.

The name of the kind of mathematics we suggest to teach is always a majorissue. We have chosen to speak about “computational science and engineering”to emphasize the importance of computations and applications. Surely, one canargue that this is “mathematics”, but we doubt that it is fruitful to change the con-tents dramatically and still keep the old label. “Applied mathematics” is clearlya good label literally, but these two words have strong traditions in scienceand also mean different things to different people. “Industrial mathematics” hasbeen a popular term for more applied activity, but with its roots mostly in puremathematics. The very important and comprehensive work of SIAM has firmlyestablished “applied and industrial mathematics” as a major and wide scientificdiscipline. The SIAM organization has also recently started a conference series



in computational science and engineering, emphasizing this topic as “a crucialthird mode, along with theory and experiment, of scientific investigation and en-gineering design” [10]. In SIAM’s view, computational science and engineeringis a branch in applied and industrial mathematics with main focus on satisfyingthe demands for computer-based models in engineering and applied sciences.

The realization of a more computerized education in mathematically ori-ented subjects appears to be very resource consuming. Design of illustratingprojects, preparation of suitable software, and merging separate disciplines likenumerics, calculus, and physics takes a lot of time and energy. Fortunately, withthe Internet and the significant interest among publishers in this topic, seriouswork can quickly be made available to the rest of the scientific and engineeringcommunity, thus accelerating the process.

Acknowledgements. The authors thank Are Magnus Bruaset, Erik Bølviken, Xing Cai,Helge Holden, Linda Ingebrigtsen, Bjørn Fredrik Nielsen, and Geir Pedersen for fruitfuldiscussions and many valuable comments on this manuscript.

References

1. http://www.mste.uiuc.edu/murphy/Papers/CalcReformPaper.html2. http://www.rw.ethz.ch/main 01e.htm3. http://www.nada.kth.se/kurser/master/index-eng.html4. http://www.md.chalmers.se/Centres/Phi/education/new.html5. K. Eriksson, D. Estep, P. Hansbo, and C. Johnson. Computational Differential Equa-

tions. Cambridge University Press, 1996.6. http://www.math.ntnu.no/cse/7. C. can Loan. An Introduction to Computational Science and Mathematics. Jones

and Barlett, 1996.8. http://pyfortran.sourceforge.net/9. http://cens.ioc.ee/projects/f2py2e/

10. http://www.siam.org/meetings/cse00/


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