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Vol. 37, No. 4 2016Table of ContentsEditorialCoding for All?Paul J. Campbell .................................................................... 333

ArticlesThe Utility of Recreational MathematicsDavid Singmaster ................................................................. 339

Modeling-First Approach to Teaching Differential EquationsRosemary Farley, Dina Yagodich, Holly Zullo, andBrian Winkel ........................................................................ 381

SIMIODEModeling ScenarioFish Mixing (Teacher Version)Eric Sullivan and Elizabeth Carlson ....................................... 407

Reviews ........................................................................... 417

Index .................................................................................. 421Reviews Index ........................................................................... 423

Errata ........................................................................................ 423

Electronic Copies....................................................................... 423

Acknowledgments..................................................................... 424

Editorial 333

EditorialCoding for All?Paul J. CampbellMathematics and Computer ScienceBeloit CollegeBeloit, WI 53511–[email protected]

IntroductionThe mantra “coding for all,” meaning that all students should learn

computer programming, has becomeamemeof contemporaryU.S. culture.Why? Several rationales present themselves.

• Coding as SocialWelfare: Jobs in computer programmingpay verywellcompared to most others, hence could provide a socioeconomic “escala-tor” for upward mobility of students from disadvantaged backgrounds.

• Coding as Income Insurance: In the wake of the Great Recession, in thefuture only STEM jobs will provide the living standard that Americansexpect, and most such jobs are computer-related.

• Coding as Cure for Educational Inequity: Teaching coding to all mighthelp close achievement gaps among racial and ethnic groups.

• Coding as Educational Fad: Coding is the new literacy.• Coding as Keeping Up: Other countries have adopted “coding for all”(e.g., the U.K. in 2014), and the U.S. needs to do so too in order to becompetitive in world markets of labor, technology, and commerce.

These reasons parallel the efforts of cities (and even countries) to replacelost manufacturing jobs with “high-tech” jobs. Of course, that strategycannot possiblywork for every locality. Andwealready see an emphasis ontechnology-as-cure-all in schools, as in the hope of addressing achievementgapsbyproviding tablet computers toall students (withconsequentsavingson textbook costs butwithout providing for all students the needed Internetaccess at home).

TheUMAPJournal37 (4) (2016) 333–337. c©Copyright2016byCOMAP, Inc. All rights reserved.Permission to make digital or hard copies of part or all of this work for personal or classroom useis granted without fee provided that copies are not made or distributed for profit or commercialadvantage and that copies bear this notice. Abstracting with credit is permitted, but copyrightsfor components of this work owned by others than COMAPmust be honored. To copy otherwise,to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP.

334 The UMAP Journal 37.4 (2016)

Potential Benefits• Coding would indeed be a new opportunity that would enthuse somestudents.

• “Coding for all” fits very well with the prevailing American view of thepurpose of education as being primarily job training. “Computer skills”are viewed as valued “job skills.”

• Coding involves learning basic logic (boolean logic, conditionals), whichhas applications in other aspects of life.

Drawbacks• Do we need that many coders? The Bureau of Labor Statistics projectsthat the demand for computer programmerswill decrease 8%percent by2024 [Galvy 2016].

• Would a career as a coder be meaningful for most students?• Could all students “succeed” at coding, or would it become just anotherresented obstacle (much as mathematics is now)? How low a standardwould any measure of success have to meet?

• Does the software that you use work reliably? Much software is pro-duced by unlicensed professionals who are actually amateurs who can“code a bit” (pun here intended).

• Maybe achievement gaps would just widen, with some students consid-ered “born to code” and others deemed hopeless.

• “Why do I need to learn this? I just want to use technology, I don’t need(or even want) to know how it works.”

• What other educational opportunities would instruction in coding re-place? In grade and high schools, recess has already disappeared, andmusic and art are frequent casualties of budget-cutting. At some colleges(e.g., my institution), computer programming can satisfy a requirementto study a foreign language [Galvy 2016]. In most states, “computer sci-ence” can replace mathematics or science as a high school graduationrequirement [Code.org 2016].

• There are nowhere near enough teachers for such a plan (and therewon’tbe enough as long as teachers’ salaries remain far lower than those ofcoders and others in IT).

• The U.S. does not have a national educational system but instead stateand local control over curriculum and standards. So it wouldn’t be cod-ing for “all,” hence the result might be exacerbation of educational dif-ferences or opportunities.

Editorial 335

• Where is the money for this going to come from? (Hint: Not the school’ssports programs.) Are we going to have endless arguments about “eq-uity” of computer equipment across schools?

• Concentrating on the specifics of a particular programming language orcomputing platform today will not be good job training for tomorrow.Today’s computers and computer languages will become obsolete.

• Despite popular confusion, computer science is more than coding. Iused to give a talk about computer science as a liberal art (Steve Jobs toocalled it that [1995], without elaboration), because writing a computerprogram is about handling complexity, much as the author of a book orthe manager of a project must. You don’t write a million-line computerprogram by writing the first line, then the second, and so on; and youdon’t do it all by yourself. What we emphasize in teaching computerscience is not learning the ins and outs of the syntax and semantics ofa particular programming language, but cultivating the art of solving aproblembybreaking itdownintomanageablechunks thatwork together.But that’s not coding.

PerspectiveYears ago, I taught introductoryprogrammingandmore advanced com-

puter science (operating systems, compiler writing, analysis of algorithms)to college students. My institution now has faculty who do it better.Students found learning computer programming to be

• demanding (apart from getting the logic right for what you want thecomputer to do, the syntax has to be correct);

• frustrating (the logic can be wrong, and the computer demands per-fect syntax [hey! when are we going to get fault-tolerant computer lan-guages?]);

• time-consuming (unlike a student termpaper, with itsfirst andonlydraftwritten the night before it is due, a program is rarely “done” on the firsttry).

One benefit of the course for students was becoming aware of how theyhandled frustration; some strategies are unhealthful, and they could learnto employ others instead.Today, with toy robots and with easy-to-program graphics and anima-

tion, learning programming can be a lot more fun.I think of computer science as the science of information transfer. Its key

question is, what can information-transfer machines do? And computerengineering is about how to make information-transfer machines that cando those tasks.


Computer programming, in turn, is making information-transfer ma-chines actually do those tasks, i.e., it is instructing a machine to get it dowhat you (or another user) want.Programming computers is not unlike using other machines:

• An analog example is driving a car, which involves steering and braking(“programming” its motion) in real time.

• A digital example for an earlier generation was programming a VCR(videocassette recorder) to record TV shows on a certain channel at par-ticular future time; the programming language was button-pushing. Acomparable example today is programming a programmable thermostator programmable calculator.

• Amore contemporary example is creating aWebpage; the programminglanguage is often a high-level GUI that implements underlying HTMLcode.

What Do Students Really Need to LearnThey need to learn how to get machines to do what they want.

Well, what makes a good programmer—and what should to go into“coding for all”—may indeed be educationally valuable for all:

• logical thinking;• persistence (since thefirst instantiationof a program is unlikely towork);• following instructions (so a program does what it was specified to do)(I am always amazed at students who do not, cannot, or will not followdirections!);

• contingency planning (“since the user may type in rubbish, we need tovalidate the input”);

• conscientiousness (you’ve got to design andwork until you are sure thattheprogram is correct and “bullet-proof”) [Kanij et al. 2015; Brooks 2016];

• attention to detail (but beware of overdoing that, to the point of pay-ing too much attention to details that do not matter much—as, in myexperience, too many mathematicians do!);

• sticking to standards (e.g., documentation of code).But are these qualities better taught through other means (art, music,

mathematics, sports. . . )? Is there any transfer of learning from one domainto another? Are some qualities mainly elements of character? If so, canthey be taught/cultivated/enhanced? In public education, dowe even stilltry to “build character”?

Editorial 337

Let’s Hear from YouI welcome hearing your comments.

ReferencesBrooks, Andy. 2016. Review of Kanij et al. [2015]. Computing Reviews 57

(11) (November 2016) 677.

Campbell, Paul J. STEM the tide? The UMAP Journal of UndergraduateMath-ematics and Its Applications 37 (1) (2016) 1–7.

Code.org. 2016. Where computer science counts. https://code.org/action .

Galvy, Gaby. 2016. Some say computer coding is a foreign language.http://www.usnews.com/news/stem-solutions/articles/2016-10-13/spanish-french-python-some-say-computer-coding-is-a-foreign-language .

Jobs, Steve. 1995. Steve Jobsoncomputer science. https://www.youtube.com/watch?v=IY7EsTnUSxY .

Kanij, Tanjila, Robert Merkel, and John Grundy. 2015. An empirical in-vestigation of the personality traits of software testers. In CHASE 15:Proceedings of the 8th International Workshop on Cooperative and HumanAspects of Software Engineering, 1–7. Piscataway, NJ: IEEE Press.

Paul, Annie Murphy. 2016. The coding revolution. Scientific American 315(2) (August 2016) 43–49.

About the EditorPaulCampbellgraduatedsummacumlaude

from the University of Dayton and received anM.S. in algebra and a Ph.D. in mathematicallogic from Cornell University. He has been atBeloit College since 1977. He is Reviews Edi-tor for Mathematics Magazine and has been ed-itor of The UMAP Journal since 1984. He isalsoa co-authorof theCOMAP-sponsoredbookof applications-oriented introductory collegiatemathematics,ForAll Practical Purposes (10th ed.,W.H. Freeman, 2016).


Utility of Recreational Mathematics 339

The Utility of RecreationalMathematics

David Singmaster87 Rodenhurst RoadLondon, SW4 8AFUnited [email protected]

Prelude and VademecumThis article is an amplificationwith illustrations of a write-up ofmy talk

at the University Mathematics Teaching Conference at Sheffield HallamUniversity, 7 September 1999 [Singmaster 2000].That talkwas basedon earlier talks that I hadgivenon this topic, notably

at the European Congress of Mathematicians in 1992 [Singmaster 1992;1996]. Some topics discussed there were skipped in my talk at Sheffield,and some brief remarks on a few topics not mentioned at Sheffield wereadded in the write-up of that talk [Singmaster 2000]. I have now combinedthe material from all of these talks into the following, accompanied bysuitable images.

Les hommes ne sont jamais plus ingenieux que dans l’inventiondes jeux.[Men are never more ingenious than in inventing games.]

—Leibniz to De Montmort, 29 Jul 1715.

Amusement is one of the fields of applied mathematics.—William F. White, A Scrap-Book of Elementary Mathematics, 1908.

. . . [I]t is necessary to begin the Instruction of Youth with the Lan-guages and Mathematicks. These should . . . be taught to-gether, theLanguages and Classicks as . . . Business and the Mathematicks as . . .Diversion.— Samuel Johnson, first President of Columbia University, in 1731.

The UMAP Journal 37 (4) (2016) 339–380. c©Copyright 2016 by COMAP, Inc. All rights reserved.Permission to make digital or hard copies of part or all of this work for personal or classroom useis granted without fee provided that copies are not made or distributed for profit or commercialadvantage and that copies bear this notice. Abstracting with credit is permitted, but copyrightsfor components of this work owned by others than COMAPmust be honored. To copy otherwise,to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP.


IntroductionMy title is a variation on Eugene Wigner’s famous essay “The unrea-

sonable effectiveness of mathematics in the physical sciences” [1960]. LikeWigner, I originally had “unreasonable” in my title and did not come upwith any explanation, but I believe that I have an explanation that makesit reasonable. But first let me describe the background and illustrate thesituation. For about 25 years, I have been working to find sources of clas-sical problems in recreational mathematics. This has led to an annotatedbibliography/history of the subject [Singmaster 2013], now covering about470 topics on almost 1,000 pp, where you can find more details about thetopics discussed in this article.

The Nature of Recreational MathematicsTo begin with, it is worth considering what is meant by recreational

mathematics. An obvious definition is that it is mathematics that is fun.However, almost any mathematician enjoys the work, even in studyingeigenvalues of elliptic differential operators; so this definition would en-compass almost all mathematics and hence is too general. There are two,somewhat overlapping, definitions that cover most of what is meant byrecreational mathematics.

• Recreational mathematics is mathematics that is fun and popular—thatis, the problems should be understandable to the interested lay person,though the solutions may be harder. (However, if the solution is toohard, this may shift the topic from recreational toward the serious—e.g.Fermat’sLast Theorem, the FourColourTheoremor theMandelbrot Set.)

• Recreational mathematics is mathematics that is fun and used pedagog-ically either as a diversion from serious mathematics or as a way ofmaking serious mathematics understandable or palatable. These peda-gogic uses of recreational mathematics are already present in the oldestknown mathematics and continue to the present day.

In both cases, the fun aspect is often accentuated by posing the problem in acontext that is illegal, immoral, or politically incorrect (for one or more rea-sons), as well as being highly unlikely or even downright impossible. Thiswhimsey is actually important, in that it makes the problem memorable;and the artificiality often eliminates unnecessary complications that tendto occur in reality. Further, the problem may be illustrated or even encap-sulated in a physical object that one can see and touch—I am particularlyfond of such problems and will cite several examples below.Mathematical recreations are as old as mathematics itself, and we will

later see some prehistoric examples. The earliest piece of Egyptian math-ematics, the Rhind Papyrus of ca. −1800, has a problem (No. 79) where


there are 7 houses, each house has 7 cats, each cat ate 7 mice, each mousewould have eaten 7 ears of spelt (a kind of wheat), and each ear of speltwould produce 7 hekat (a unit of volume) of spelt. Then 7 + 49 + 343 +2401 + 16807 is computed. A similar problem of adding powers of 7 occursin Fibonacci (1202) [Sigler 2002], in a few later medieval texts, and in thechildren’s riddle rhyme “As I was going to St. Ives.” Despite the gaps inthe history, it is tempting to believe that “St. Ives” is a descendant fromthe ancient Egyptians. Though there is some question as to whether thisproblem is really a fanciful exercise in summing a geometric progression,it has no connection with other problems in the papyrus and seems to beinserted as a diversion or recreation. (See Figures 1–3.)

Figure 1. Rhind papyrus No. 79.

Figure 2. Extract from Fibonacci’s Liber abbaci.


Figure 3. Postcards illustrating the St. Ives children’s riddle rhyme.

The earliest mathematical works from Babylonia also date from about−1800 and they include such problems as the following on AO 8862 (seeFigure 4):


Figure 4. Babylonian tablet AO 8862, Face I, in the Louvre. Reproduced full size in print versionof this issue of The UMAP Journal. Source: Thureau-Dangin [1932, Plate I].

“I know the length plus the width of a rectangle is 27, while the areaplus the difference of the length and the width is 183. Find the lengthand width.”


By no stretch of the imagination can this be considered a practical prob-lem! Rather it is a way of presenting two equations in two unknowns,leading to a quadratic equation, in an effort to make solving the latter moreinteresting for the student.These two aspects of recreationalmathematics—the popular and the ped-

agogic—overlap considerably, and there is no clear boundary between themand “serious” mathematics. In addition, there are several other indepen-dentfields that containmuch recreationalmathematics: games; mechanicalpuzzles; magic; art.Games of chance and games of strategy also seem to be about as old

as human civilization. The mathematics of games of chance began in theMiddleAges, and itsdevelopmentbyFermatandPascal in the1650s rapidlyled to probability theory. Insurance companies based on this theory werefounded in the mid-18th century. The mathematics of games of strategystarted only about the beginning of the 20th century, but soon developedinto game theory, both of the von Neumann-Morgenstern type and later ofthe Conway type.Mechanical puzzles range widely in mathematical content. Some only

require a certain amount of dexterity and three-dimensional ability; oth-ers require ingenuity and logical thought; while others require systematicapplication of mathematical ideas or patterns, such as Rubik’s Cube, theChinese Rings, the Tower of Hanoi, and Rubik’s Clock.Much magic has a mathematical basis that the magician uses but care-

fully conceals—e.g., the fact that the opposite faces of a die add up to 7;binary divination; the fact that the period of a perfect (faro or riffle) shuffleof a 52-card pack of cards is 8.The creation of beauty often leads to questions of symmetry and geom-

etry that are studied for their own sake—e.g., the carved stone balls that wewill see later.This outlines the conventional scope of recreational mathematics, but

there is some variation due to personal taste.

The Utility of Recreational MathematicsHow is recreational mathematics useful?

• Recreational problems are often the basis of seriousmathematics. Themost obvious fields are probability and graph theory, where popularproblems have been a major (or even dominant) stimulus to the creationand evolution of the subject. Further reflection shows that number the-ory, topology, geometry, and algebra have all been strongly stimulatedby recreational problems. (Though geometry has its origins in practi-cal surveying, the Greeks treated it as an intellectual game; and muchof their work must be considered as recreational in nature, even though


they viewed itmore seriously, as reflecting the nature of theworld. Fromthe time of the Babylonians, algebraists tried to solve cubic equations,though they had no practical problems that led to cubics.) There are evenrecreational aspects of calculus—e.g., the many curves studied since the16th century. Consequently, the study of recreational topics is necessaryto understanding the history of many, perhaps most, topics in mathe-matics.Before Aristotle, the Greeks used logic as a game of forcing an op-

ponent to accept your conclusions, but had never formalized the rules.Aristotle began the study of logic in order to formalize the rules of thisgame.

• Recreational mathematics has frequently turned up ideas of genuinebut non-obvious utility. I will mention a few examples later.Such unusual developments, and the more straightforward devel-

opments of the previous point, demonstrate the historical principle of“The (unreasonable) utility of recreational mathematics.” This and sim-ilar ideas are the historical and social justification of mathematical re-search in general and for the study of recreational mathematics, and Iwill return to this point later.

• Recreational mathematics has great pedagogic utility, and this will bethe main theme of my examples.

• Recreational mathematics is very useful to the historian of mathemat-ics. Recreational problems often are of great age and usually can beclearly recognised; they serve as useful historical markers, tracing thedevelopment and transmission of mathematics (and culture in general)in place and time. The Chinese Remainder Theorem, magic squares, theCisternProblem, and theHundredFowlsProblemare excellent examplesof this process.

The original Hundred Fowls Problem, from 5th century China,has a man buying 100 fowls for 100 cash (an old coin). Roosterscost 5, hens 3, and chicks are 3 for a cash—howmany of each didhe buy?

The number of topics that have their origins in China or India is surpris-ing and emphasises our increasing realisation that modern algebra andarithmetic derivemore fromBabylonia, China, India, and theArabs thanfrom Greece.


Some Examples of UsefulRecreational MathematicsI outline examples to show how recreational mathematics has been use-

ful. (I stretch“recreational”abit to includesomeothernon-practical topics.)

From Gambling Bets to the Insurance Industry

The theory of probability and statistics grew from the analysis of gam-bling bets to the basis of the insurance industry in the 17th and 18th cen-turies.Much of combinatorics likewise has its roots in gambling problems. The

theory of Latin squares began as a recreation but has become an importanttechnique in experimental design (and then returned again in connectionwith Sudoku puzzles).

From Euclid to the Moon and to Buckyballs

Greek geometry, though it had some basis in surveying, was largelyan intellectual exercise, pursued for its own sake. The conic sections weredeveloped with no purpose in mind, but 2000 years later turned out to bejustwhatKepler andNewtonneededandwerewhat tookmen to theMoon.Theregular, quasi-regular, andArchimedeanpolyhedraweredeveloped

long before they became the basis of molecular structures. Indeed, the reg-ular solids are now known to be prehistoric. Beginning in 1985, chemistsbecame excited about fullerenes, molecules of carbon in various polyhe-dral shapes, of which the archetype is the truncated icosahedron, with 60carbon atoms at the vertices, named buckminsterfullerene after BuckminsterFuller (1895–1983), a proponent of geodesic domes. Spherical fullerenesare consequently nicknamed “buckyballs” (see Figure 5). Such moleculesapparently are the basis for the formation of soot particles in the air. Theidea of making suchmolecules seems to have originatedwith David Jones,the scientific humorist who writes as “Daedalus,” in one of his humourcolumns. Chemists have also synthesized hydrocarbons in the shapes of acube (cubane, C8H8, in 1964) and a dodecahedron (dodecahedrane, C20H20,in 1982).

From Non-Euclidean Geometry to Geometry of Physical Space

Non-euclidean geometry was developed long before Einstein consid-ered it as a possible geometry for space.


Figure 5. Frame of a truncated icosahedron buckyball, C60, illustrated in the Ambrosianamanuscript of Luca Pacioli’s De divina proportione (1509) by Leonardo da Vinci. Plate XXIIII, folio103 recto. Source: Veneranda Biblioteca Ambrosiana, DeAgostini Picture Library, Scala, Florence.

From River-Crossing Puzzles to Graph Theory

The river-crossing problems and the problem of getting camels acrossa desert, which occur in Alcuin, ca. 800, are considered to be the earliestcombinatorial optimization problems. Such problems are now solved bygraph-theoreticmethods, dynamicprogramming, or integerprogramming.The problem of the Seven Bridges of Konigsberg (Figure 6), mazes,

knight’s tours, and circuits on the dodecahedron (the Icosian Game) (Fig-ures 7 and 8) were major sources of graph theory and are the basis of majorfields of optimization, leading to one of the major unsolved problems ofthe century: Does P = NP? The routes of postmen, streetsweepers andsnowplows, as well as those of salesmen, are worked out by these meth-ods. Further, Hamilton’s thoughts on the Icosian Game led him to the firstpresentation of a group by generators and relations (Figure 9).

From Number Theory to Splicing Phone Cables

Number theory is another of the fields where recreations have been amajor source of problems, and these problems have been a major sourcefor modern algebra. Fermat’s Last Theorem led to Kummer’s inventionof ideals and most of algebraic number theory. There was a famous ap-plication of primitive roots to the splicing of telephone cables to minimizeinterference and crosstalk [Rosen 1984, 280–286; 2005, 397–399]. Primalityand factorization were traditionally innocuous recreational pastimes; but


Figure 6. Coloured map of the city of Konigsberg ca. 1641, showing the seven bridges featured inthe problem solved by Euler. Copper engraving by Matthaus Merian the Elder or Merian Erben(his sons); colourizer unknown. Source for uncoloured engraving: Zeiller [1652].

since 1978 when Rivest, Shamir, and Adleman introduced their method ofpublic-key cryptography (nowknownasRSAcryptography),my friends inthis field get rung up by reporters wanting to know if the national securityis threatened because someone has factored a large number. The factoriza-tion of a big number or the determination of the next Mersenne prime aregenerally front-page news now.

From Buying a Horse to Negative Numbers

A major impetus for algebra has been the solving of equations. TheBabylonians already gave quadratic problemswhere the area of a rectanglewas added to the difference between the length and the width. This clearlyhad no practical significance. Similar impractical problems led to cubicequations and the eventual solution of the cubic. Negative solutions firstbecome common in medieval puzzle problems about men buying a horseor finding a purse.Galois fields and even polynomials over them are now standard tools

for cryptographers.


Figure 7. Advertisementwith instructions forHamilton’s IcosianGame. Only four original boardsof the plane version are known to exist.


Figure 8. The only known remaining instance of the Traveller’s Dodecahedron, a revision byHamilton of his Icosian game with simpler rules. The 30 edges on the head represent roads to usetovisit 20 ivorypegs that represent cities. “Two travellers set off visiting4neighbouringtowns. Onereturns home and the other continues to travel around the world trying to visit all the remainingcities once only.” Photo courtesy of James Dalgety. c©Copyright 2013 Hordern-Dalgety Collection.http://puzzlemuseum.com , http://puzzlemuseum.com/month/picm02/200207icosian.htm.

Figure 9. Hamilton’s mathematics inspired by the Icosian Game, as presented by him in Proceed-ings of the Royal Irish Academy 6 (1858) 415–416. Communicated November 10, 1856. Hamilton’scollected papers notes that “[t]his is the substance of a letter written on 27 October 1856 to the Rev.Charles Graves, D.D.” (The Mathematical Papers of Sir William Rowan Hamilton, Volume III: Algebra,edited by H. Halberstam and R.E. Ingram, 609. New York: Cambridge University Press, 1967.)


Recreational Curves to Analysis

Even in analysis, the study of curves (e.g., the cycloid) had some recre-ational motivation.

From Knots to DNA

Topology has much of its origins in recreational aspects of curves andsurfaces. Knots, another field once generally considered of no possibleuse, are now of great interest to molecular biologists who have discoveredthat DNAmolecules form into closed chains which may be knotted, or notknotted.The Mobius strip arose about 1858 in work by both August Ferdinand

Mobius (1790–1868) and JohannBenedict Listing (1808–1882), Listing beingapparently a bit earlier. Depictions of it occur in Romanmosaics (Figure 10,noticed by Charles Seife in 2002), including a stripwith five half-turns (Fig-ure 11) [Larison 1973]. These two examples, together with other examplesof “early” Mobius strips, are discussed in Cartwright and Gonzalez [2016].

Figure 10. Mobius strip in detail from a floor mosaic from a Roman villa near Sentinum (nowSassoferrato, Umbria), ca. 200 A.D., now in the Glyptothek in Munich, Germany. Depicted isAion, god of Eternity, surrounded by a zodiac wheel, with earth mother Tellus seated. Source:Photograph in the public domain by Bibi Saint-Pol.

By 1890, the Mobius strip was being used as a magic trick, magic beinganother application of mathematics; indeed, some people view all math-ematics as magic! More recently, such strips have served as the basis forworks by M.C. Escher, art being yet another application of mathematics.The Mobius strip has also been patented several times! e.g., as a single-

sided conveyor belt that has double the wearing surface (Figure 12).


Figure 11. Mobius strip with five half-turns in a Roman floor mosaic of Orpheus charming theanimals, ca. 200 A.D., now in the Museum of Pagan Art, Arles, France. Source: Detail fromFinoskov, Creative Commons (CC BY SA-3.0), Wikimedia Commons.

There are a number of other practical uses for the Möbius strip, but themost unusual is as a non-inductive electrical resistor (Figure 13).None of the patents that I have seenmake any reference to any previous

occurrence of the concept. Martin Gardner says it has also been patentedas a non-inductive resistor. Those who still have dot-matrix printers may(or may not) know that printer ribbons commonly have a twist (so theyare Mobius strips!) in order to allow the printer to use both edges. I firstdiscovered this when I found one of our technicians trying to put sucha ribbon back into its cartridge; he had done it several times, but it keptcoming out twisted, which he thought was his mistake!

From Chinese Rings to Binary Codes

Gray CodesThe Chinese Rings puzzle (Figure 14), known as bagenaudier (“time-

waster”) in French, may indeed have originated in China 1,800 years ago.In combinatorics, the pattern of solution of the Chinese Rings puzzle is

the binary coding known as the Gray code, patented as an error-minimisingcode by FrankGray (1887–1969) of Bell Labs in 1953 (Figure 15) and alreadyused in the same way by Emile Baudot (1845–1903) in the 1870s [Baudot1879] in inventing the predecessor of the teletype (it is from Baudot’s codethat the term “baud” arose as a measure of transmission speed).


Figure 12. Image from a patent for a single-sided conveyor belt.


Figure 13. Patent for use of a Mobius strip as a non-inductive electrical resistor, by Richard L.Davis, granted in 1966. U.S. Patent 32674906 A.


Figure 14. Chinese Rings puzzles. The task is to remove all the rings.


Figure 15. Figures from Gray’s patent for pulse code communication.


Chain CodesAnother binary coding, sometimes called a chain code, was used by San-

skrit poets in about 1000 tomemorise all the combinations of long and shortsyllables [Stein 1961; 1976]. The 10 syllables in the Sanskrit nonsense word

ya-ma-ta-ra-ja-bha-na-sa-la-gamcontain in successive groups of three all the triplets of long beats (markedwith a bar over the a) and short beats (unmarked a). Moreover, since thelast two syllables are the same as the first two, if we regard the sequence ofsyllables as wrapping around, we could arrange the syllables in the formof a wheel.Baudot redesigned his printing telegraph to use a chain code [Heath

1961, 540; 1972, 83; Baudot 1895]. The idea of a chain code led to themore general mathematical concept of a de Bruijn sequence [Gurudev 2007;Diaconis and Graham 2012, 42–60]. In a de Bruijn sequence, every possiblesubsequence of a prescribed length from an alphabet of characters appearsexactly once in the sequence, which like a memory wheel cycles back onitself. For example, the de Bruijn sequence

0 0 0 1 0 1 1 1

contains in order all the different subsequences of length 3:

000, 001, 010, 101, 011, 111, 110, 100.

Such codes are painted on factory andwarehousefloors to enable robotsto determine where they are by scanning a small section of the floor. Theyhave also been used as the basis of card tricks—divinations—where the val-ues of cards are determined from a small amount of information [Diaconisand Graham, 25–29, 42–60]. Diaconis and Graham note some confusion ofGray codes and chain codes:

[Magicians]mistakenly call de Bruijn sequences “Gray codes.” . . . Butas far as we know, there has never been a single use [of Gray codes]in magic. [Diaconis and Graham 2012, 25]

The earliest mention of what later became known as chain codes and deBruijn sequences seems to be by Flye Saint-Marie [1894], though Baudot’suse of it for a teleprinter dates from about 1882, with equipment using itexhibited in 1889 [Heath 1961, 540; 1972, 83]. Kerr [1961] offers in brief somemechanical details of a production teletypemachine that used a chain code.


Examples of Recreational Mathematicswith ObjectsSeveral of these examples are based on objects that I passed around at

the lectures.

Neolithic PolyhedraThese “carved stone balls” date from ca. −2500, and occur in eastern

Scotland. Examples are in the Royal Scottish, Ashmolean, Dundee, andAberdeenMuseums. Figure 16 shows a resin model of a carved stome ballfrom Aberdeenshire, made by an artist in Glastonbury. No one knows thepurpose of these.

Figure 16. A resin model of a neolithic carved stone ball (about 9 cm across).

Plimpton 322, ca. −1800This is the famous Old Babylonian tablet listing Pythagorean triples.

Some years ago I persuaded Columbia University to make casts from theoriginal, and Figure 17 is a photograph of one of those.

Roman Dodecahedron, ca. 200–400

Approximately 100 of these are known, from Roman sites north of theAlps. The one shown in Figure 18was found in 1939 in Tongeren, Belgium,and dates to 150-400 A.D. [Huylenbrouck 2012]. Its total height is 81 mm;the height without the balls at the corners is 66 mm. I have seen a some-what smaller example at the Hunt Museum in Limerick. No one knowstheir purpose [Guggenheim2013]. Nevertheless, they admirablymatch thedescription of Hamilton’s Traveller’s Dodecahedron of Figure 8 on p. 350.


Figure 17. Facsimile of the Old Babylonian tablet Plimpton 322 that lists Pythagorean triples.

Figure 18. Roman dodecahedron found at Tongeren, Belgium, in 1939 and now situated in theGallo-RomanMuseum Tongeren. Reproduced full size in print version of this issue of The UMAPJournal. Photo by Guido Schalenbourg, c©Gallo-Roman Museum Tongeren, with thanks to ElseHartoch, Collection Management Coordinator / Research.


Chinese Magic Square

The cast-iron facsimile in Figure 19 (cast at reduced size) is one of thefive cast-iron examples of a 6× 6magic square excavatednearXi’an, China,in 1956. It is inscribed in East Arabic numerals (similar to those still usedin the Middle East) and dates to the Yuan Dynasty (1280–1368) [Li and Du1987, 172].

Figure 19. Cast-iron facsimile of a Chinese magic square dating to the Yuan Dynasty (1280–1368).

Examples of Medieval ProblemsFibonacci Numbers

Figure 20 shows a page from the manuscript ca. 1275 at Siena of Fi-bonacci’s Liber abbaci of 1202 and 1228. This manuscript, which also in-cludes his hand signs for numbers (Figure 21), is apparently the earliestknown extant version of his book. The page shows the Fibonacci sequence1, 2, 3, 5, . . . 377, where each entry is the sum of the two preceding. Fi-bonacci introduced the sequence in connection with a fanciful model forthe number of rabbit pairs in successive generations.


Figure 20. Earliest known presentation of the Fibonacci sequence of numbers, in Fibonacci’s Liberabbaci (ca. 1275).

The Fibonacci numbers were known to ancient Sanskrit poets, from anuncertain date about 2000 years ago. The number of different patterns offixed length of long syllables and short syllables, where a long syllable istwice as long as a short syllable, is a Fibonacci number. For example, thepatterns with total length the equivalent of 4 short syllables are LL, SSL,SLS, LSS, and SSSS, for a total of 5. However, the first Indianwork inwhichmathematical investigation was made of such numbers was not until 1356,where they were related to binomial coefficients [Singh 1985; Knuth 2011,47ff].


Figure 21. Fibonacci’s version of Roman hand signs for numerals.

The Josephus Problem

This is the problem of recursively counting out every k-th person from acircle ofnpeople. Early versions counted out half the group (see Figure 22);later authors and the Japanese are interested in the last man—the survivor.Euler (1775) seems to be the first to ask for the last man in general. Cardan(1539) is the first to associate this process with Josephus; some later authorsderive this from the Roman practice of decimation.

According to Josephus’s account of the siege of Yodfat (in the FirstJewishWar against the Romans, in 66–73), he and his 40 soldiers weretrapped in a cave, the exit of which was blocked by Romans. Theychose suicide over capture and decided that they would form a circleand start killing themselves using a step of three. Josephus states thatby luck or possibly by the hand of God, he and anotherman remainedthe last and gave up to the Romans. —Wikipedia [2014b]


Figure 22. Calandri’s version of the Josephus problem, with 15 each of Franciscans (in brown) andCamoldensians (in white) on a boat, and counted out by k = 9. Where should the standing monkstart counting by 9, and in which direction, so that all the white-robed monks are counted out?


However, Josephus’s own account does not mention a step of three, onlydrawing of lots [Josephus ca. 75].

Right-Triangle Problems

Right-triangle problems date back to Old Babylonian (−1800), Chinese(ca.−150?), and Indian sources. The Indians include Bhaskara I (629), Ma-havira (850),Chaturveda’s860commentaryonBrahmagupta, andBhaskaraII (1150).

The Sliding SpearThe Sliding Spear (= Leaning Reed) Problem goes back to Old Babylo-

nian times (Figure 23).

Figure 23. Diagrams of the Sliding Spear Problem and Leaning Reed Problem.

The Broken Tree ProblemThe Broken Tree (or Bamboo) (= Hawk and Rat = Peacock and Serpent)

Problem goes back to a Chinese source:

A bamboo (or tree) of heightH breaks at heightX from the ground sothat the broken part reaches from the break to the ground at distanceD from the foot of the bamboo;H andD are given andX is sought.


The Two Towers ProblemThe Two Towers Problem goes back to Bhaskara I (629), who attributes

it to earlier writers! In ca. 1370, dell’Abbaco introduces the following vari-ation:

Given two towers of heights H1 and H2, situated a distance D apartwith a rope of length L between the tower tops. How high H fromthe ground does a sliding weight on the rope hang—or does it reachto the ground?? (See Figure 24.)

Figure 24. The Two Towers Problem in Pietro Paolo Muscarello’s AlgorismusMS of 1478.

Dell’Abbaco gives: H1 = 60, H2 = 40, D = 40, L = 110, and claimsH = 0.A general solution can be tedious; but if one finds the solution by brute

force, the form of the solution shows that it is easy to find!


River-Crossing Problems

ThePropositiones ad acuendos juvenes, attributed toAlcuinofYork, ca. 800,contains two classic river-crossing problems: wolf, goat, and cabbage; andthe three couples.

Wolf, Goat, Cabbage ProblemIn the first, a farmer must transport a wolf, a goat, and a cabbage across

the river in a boat that can hold only the farmer and one other item; therestrictions are that the wolf cannot be left on either bankwith the goat, northe goat with the cabbage, unaccompanied by the farmer. (See Figure 25.).

Figure 25. Wolf-goat-grain puzzle from Columbia Algorism, anonymous Italian MS, ca. 1350 [Vogel1977, 130–131, 191; Cowley 1923, 402 and plate opposite].

Three Couples ProblemIn the three couples (or “jealous husbands”) problem, three married

couplesmust cross the river in a boat that can hold only atmost two people;the constraint is that no wife can be in the boat or on either bank unless herhusband is present.

Modern Combinatorial OptimizationThe problems above are among the earliest combinatorial optimization

problems. Martin Grotschel in Berlin uses the wolf-goat-cabbage problem


to teach integer programming; his class found a shorter solution, but itinvolvedhalving the cabbage andhalving thewolf! [Borndorfer et al. 1998].To generalize the second problem requires an island in the river and

remains perhaps unsolved in general, since the improved solution in Press-man and Singmaster [1989] can be criticised if one takes a more stringentjealousy condition than we did.

Examples of Modern RecreationalProblemsLongest Fishpole One Can Post (Mail)

Anancient problem involves afisherman (or hunter or skier)whowantsto post (mail) his 2.5 m fishing rod (or gun or skis) and finds that the postoffice has a maximum parcel length of 1.5 m. The fisherman solves theproblem by making a cubical box of edge 1.5 m and putting the rod indiagonally. The diagonal of the box is 1.5 × √

3 ≈ 2.598. This is veryingenious, but unfortunately there are other postal regulations. The lengthplus the girthmust be at most 3m. The girth is the circumference in a planeperpendicular to the longest dimension, which is the length. For a box ofdimensionsA × B × C, withA ≥ B ≥ C, the girth is 2B + 2C; and so wemust have A = 1.5 and A + 2B + 2C = 3. What is the longest fishing rodthat can be posted under these limitations? Suppose one uses a cylindricalmailing tube?The problemof finding the largest volume that one can post (mail) is well

known, and the maximum occurs for a cylindrical tube.

Crossing a Field

The following seems as if it should be an easy question, but I find it quitemessy and would like to see a solution better than my own.You are on a path which runs south to a road. Along the road is a bus

stop, and you want to get to it as quickly as possible. Between the pathand the road is a field; and you can cut across the field, but your speedmaybe slower than on the path or the road. Is it ever the case that the optimalroute is to go part way along the path, then go obliquely across the field toa point part way along the road, and then go the rest of the way along theroad? Try to convince yourself of the answer before doing any calculations.Determine the optimal route in general for all situations.For standardization. let us assume that you start at the pointA = (0,W )

and are travelling to D = (L, 0) and that you travel from A to B = (0, y),thence to C = (x, 0), then toD (see Figure 26).


1

Figure 26. Diagram for the Crossing a Field Problem.

There are three speeds involved, but only their relative values are im-portant, so you can assume that your speed on the road is a unit speed, yourspeed on the path is v, and your speed on the field is V , with V ≤ v ≤ 1.

There is a common feature of theproblemsof crossingafield andpostingthe longest fishpole that you should discover when you solve them, andwhich is why I like these problems.

Folding a Chain of Cubes

There are several versions of this puzzle on themarket; I first purchasedone in Paris in 1990 (see Figure 27 for one example).

Figure 27. A snake puzzle. Photo courtesy of Eryk Vershen.

They are simple geometric analogues of “transformer” toys. Each hasa chain of cubes, strung on an elastic. Each cube has a hole, either straightthrough or from a face to an adjacent face (effectively a right-angle bend).


For a string of 27 cubes, one obviously wants to make a 3 × 3 × 3 cube.How does one go about solving such problems systematically? Is theremore than one solution? If so, how many solutions are there? The originalversion of this had one solution, but a set of five different versions hasrecently been marketed. The variation can be identified by specifying inorder whether the connection from one cube to the next is straight (S) orturns (T)There are also examples with the string forming a loop; for example, I

have one with 36 cubes in a loop, with all pieces being bends, and makingthis into a 3 × 3 × 4 takes a little effort. There are also examples with 64cubes on a loop, with some straight pieces and some bent pieces—these aregenerally impossible to solve by hand. I also have an example with 125pieces, all bends, which I have never solved, though it seems that a handsolution should be possible. Can you write a program to do this?What can one say about the number of straight and bent pieces in such

puzzles? Could one have a 27-cube string with all bends? Can you haveversions which make several solid (or even plane) shapes?The problem of folding the snake into a cube is equivalent to finding

a hamiltonian path—that is, a path through each “cubie” (cubelet) in thecube—with the turn at each step specified in advance as either (S) or (T).Abel et al. [2013] show that the problem of decidingwhether it can be doneat all is NP-complete, meaning that the time required to do so for an N ×N × N cube may grow more quickly than any polynomial function of N .Knowing that there is a solution, however, would not necessarily tell howto perform the transformation—but would motivate you, on recreationalgrounds, to try to find one!Scherphuis [n.d.] lists all possible 3 × 3 × 3 possible snake puzzles that

are “doable”: Not counting rotations or mirror images of the hamiltonianpaths, there are 11,487 puzzles, of which 3,658 have unique solutions and 1has 142 solutions. Scherphuis shows that it is impossible to have a doable3× 3× 3 puzzlewith only turns (as doRuskey and Sawada [2003]), and hisWebpage points to various pageswith solutions for commercially availablesnake puzzles as well as for the Kibble Cube, a variation in which the cubeshave grooves that allow for greater freedom.A particularly clear solution to one 3 × 3 × 3 puzzle is at Cole [n.d.],

and there arepotentiallyusefulnotationsatWeston [2003] andKoller [1999].You canfind videos of people solving 3× 3× 3 and even 4× 4× 4 puzzles:Search the Internet with a key such as “youtube snake cube 4x4x4”.The snake puzzles have given rise to mathematical research into “bent”

hamiltonian paths and cycles in any dimension, where every connection isa turn [Ruskey and Sawada 2003], and into whichN2 snakes can be foldedinto a flat square (whether this problem is NP-complete is unknown [Abelet al. 2013]). The general field of study is known as combinatorial Gray codes,in which successive objects or positions differ in some prescribed way. Arecreational example is change-ringing of church bells, in which the order


in which bells are rung can change only in specified ways.But that 125-cube-long snake? I have since found the solution that came

with the puzzle, and a correspondent has sent a solution.

Rubik’s Cube R©

Figure 28. A larger-than-usual (5 × 5 × 5) Rubik cube, scrambled; there are even 8 × 8 × 8 and9 × 9 × 9 cubes. Source: Creative Commons Attribution-Share Alike 3.0 Unported, by Maksim.

I spoke very briefly about Rubik’s Cube, describing it as an excellentexample of problem solving (Figure 28 shows the bigger version that issometimes known as the Professor’s Cube). One can identify many of theclassic problem-solving skills:

• understanding the problem;• establishing a notation;• investigating subproblems;• using conjugates (which is a special case of one of the basic problem-solving techniques—transform a problem to a situation one knows howto solve and then transform the answer back to the original situation—this is the idea of logarithms, Laplace, Fourier and other transforms, sim-ilarity transformations, change of basis, mathematicalmodelling, etc., aswell as the idea behind machine shop or factory work (take the item tobe sawn to the saw, then bring the pieces back to your workplace);

• using commutators (a less general technique, but one of great use ingroup theory);

• creating an algorithm;• demonstrating completeness of the algorithm; and• seeking an optimum algorithm (still unsolved).The eminent Dutch puzzle designer, Oskar van Deventer, has designed

and made a 17 × 17 × 17 cube!

[EDITOR’S NOTE: Prof. Singmaster was author of one of the first books aboutRubik’s Cube [1981], and is co-author of others [Slocum et al. 2009, Freyand Singmaster 2010].]


The Penrose Pieces

Penrose’s pieces have led to the discovery of a new kind of solids, “qua-sicrystals.”I will only sketch the ideas here, with some references.The former coat of arms (Figure 29) of London South Bank University

includes “the net of half a dodecahedron,” i.e., a pentagon surrounded byfive other pentagons.

Figure 29. Former coat of arms of London South BankUniversity, designed to include two Thamesbarges above a pentagon surrounded by five other pentagons.

One of the basic results of crystallography is that no crystal structurecan have five-fold symmetry. In 1973, I wrote to Roger Penrose on a Poly-technic letterhead that shows the half dodecahedron. Penrose had longbeen interested in tiling the plane with pieces that could not tile the planeperiodically, and the letterhead inspired him to try to fill the plane withpentagons and other related shapes.He soon found such a tiling with six kinds of shape and then managed

to reduce it to two shapes that could tile the plane in uncountably manyways but in no periodic way. Some of the tilings have a five-fold centreof symmetry, and all have a sort of generalised five-fold symmetry. Theyare now called quasicrystals. These tilings fascinated both geometers andcrystallographers and were extensively studied from the mid-1970s.Penrose’s “kites and darts” shapes were simplified further to “fat and

thin rhombuses” (Figure 30). Rules for putting them together (e.g., a sidecorner of a kite must coincide with the tip or rear of a dart) prevent theshapes from tiling periodically. Figure 31 shows a “Penrose pattern”madefrom the rhombuses of Figure 30.The rhombus shapeswere extended to three dimensions,where they are

related to the rhombic triacontahedron. Though the tilings are not periodic,they have quasi-axes and quasi-planes, which can cause diffraction. Usingthese, crystallographers determined the diffraction pattern that a hypothet-ical quasicrystal would produce: It has a ten-fold centre of symmetry.In 1984, such diffraction patterns were discovered by Dan Shechtman

(b. 1941) in a sample of rapidly cooled alloy now known as shechtmanite;


Figure 30. Penrose’s dart and kite, and his fat and thin rhombuses, with notation of the degrees ofthe angles involved. Courtesy of Robert Austin [2014].

Figure 31. A Penrose pattern made from the rhombuses of Figure 30. Courtesy of Robin Wilson.

and some 20 substances are now known to have quasi-crystalline forms.Shechtman received the 2011 Nobel Prize in Chemistry for his discoveryof quasicrystals [Royal Swedish Academy of Sciences 2011a; 2011b]. In-deed, examples had been found about 30 years earlier but the diffractionpatterns were discarded as being erroneous! It is not yet known whethersuchmaterials will be useful but theymay be harder or stronger than otherforms of the alloys and hence may find use on aeroplanes, rockets, etc. Soa mathematical flight of fancy has led to the discovery of a new kind ofmatter on which we may be flying in the future! (See Gardner [1979; 1997]for expositions of this topic.)


The Educational Value of RecreationsA Treasury of Problems

Recreational mathematics is a treasury of problems which make math-ematics fun. These problems have been tested by generations going backto about 1800 BC. In medieval arithmetic texts, recreational questions areinterspersed with more straightforward problems to provide breaks in thehard slog of learning. These problems are often based on reality, thoughwith enough whimsey so that they have appealed to students and mathe-maticians for years. They illustrate the idea that ”Mathematics is all aroundyou – you only have to look for it.”

An Optimal Learning Experience

“Agoodproblem isworth a thousandexercises” (ancient proverb,madeup by myself). There is no greater learning experience than trying to solvea good problem. Recreational mathematics provides many such problemsandalmost everyproblemcanbe extendedor amended. Hence recreationalmathematics is also a treasury of problems for student investigations.Solvingproblemsnaturallydevelopsproblem-solvingtechniques. Some

of those which arise in recreational problems are:

• The problems often require clarification of the assumptions and onemayvary the assumptions to get different problems.

• One may need to create a notation.• Themathematical or logicalmethods needed are often non-standard andhence one has to use basic ideas in a novel way.

• Theproblems are often open-endedandnatural generalizations are oftenunsolved, so one has to re-examine the problem and ask new questions.

For better orworse, mathematics is one of the only school courseswherestudents are expected to learn how to think! But thinking, like problemsolving, is best learnedbydoingandourproblemsare ideal for encouragingthis.

A Communication Vehicle for History and Culture

Because of its long history, recreational mathematics is an ideal vehiclefor communicating the historical andmulticultural aspects ofmathematics.

A Communication Vehicle for Mathematical Ideas

An additional utility of recreational mathematics is that it provides us away to communicate mathematical ideas to the public at large.


Mathematicians tend to underestimate the public interest in mathemat-ics. (Lee Dembart of the Los Angeles Times wrote that when he told peoplehewas going to a conference on recreationalmathematics, they replied thatit was a contradiction in terms! And we all know the social situation whenyou confess that you are a mathematician and the response is, “Oh. I wasnever any good at maths.”) Yet somewhere approaching 200 million RubikCubeswere sold in three years! Indeed, there have beenmore Rubik Cubessold in Hungary than there are people. The best-known example of a best-selling game is Monopoly R©, which took 50 years to sell about 90 millionexamples.Another measure of the popularity of recreational mathematics is the

number of books that appear in the field each year, perhaps 50 in Englishalone. The long-term best-selling recreational book in English must beMathematical Recreations and Essays by W.W. Rouse Ball (1850–1925), origi-nally published in 1892 and now in its 13th edition. It has rarely been outof print in that time. And there are many older books, such as Problemesplaisants et delectables. . . by Claude Gaspard Bachet de Meziriac in 1612,which had three editions in the late 19th century, the last of which wasreprinted several times in the 20th century.Many newspapers and professional magazines run regular mathemat-

ical puzzles, though this was perhaps more common in the past. HenryDudeney published weekly columns for about 15 years and then monthlycolumns for about 20 years. Martin Gardner’s columnswere a major factorin the popularity of ScientificAmerican andprobably inspiredmore studentsto study mathematics than any other influence. I have heard that circula-tion dropped significantly when he retired. Other major names in the fieldare the following:

• In English: Lewis Carroll (= Charles Lutwidge Dodgson) (1832–1898),Sam Loyd (1841–1911), “Professor Hoffmann” (= Angelo John Lewis(1839–1919) (about magic), Hubert Phillips (= “Caliban”) (1891–1964),ThomasH.O’Beirne (1915–1982),Douglas St. Paul Barnard (1924–1992?),Henry Dudeney (1857–1930), Martin Gardner (1914–2010), Ian Stewart(b. 1945).

• In German: Wilhelm Ahrens (1912–1998), Hermann Schubert (1848–1911), Walther Lietzmann (1880–1959).

• In French: Edouard Lucas (1842–1891), Pierre Berloquin (b. 1939).[I tried to carry on this tradition by contributing to the Daily Telegraph

and to the BBC Radio 4 programme “Puzzle Panel.”]There really is considerable interest in mathematics out there; and if we

enjoy our subject, it should be our duty and our pleasure to try to encourageand feed this interest. Indeed, itmay be necessary for our self-preservation!


Why Is Recreational Mathematics SoUseful?I have been developing an answer to this, and it also answers Wigner’s

question. Mathematics has been described as a search for pattern—andthat certainly describes much of what we do and also much of what mostscientists do. But how do we find patterns? The real world is messy andpatterns are difficult to see. Aswe begin to see a pattern, we tend to removeall of the inessential details and get to an ideal or model situation. Thesemodelsmay be so removed from reality that they become fanciful—or evenrecreational.For example, physicists deal with frictionless perfectly elastic particles,

weightless strings, ideal gases, etc.; mathematiciansdealwith randomsam-ples, exact measurements, negative money, etc. Then such models getmodified and adapted into a large variety of models and techniques aredeveloped to describe and solve them.Now, one of the ways in which a science progresses is by seeing analo-

gies between reality and simpler situations. For example, the idea of thecirculation of the blood could not be developed until the idea of a pumpwas known and somewhat understood. The behaviour of a real systemcannot be developed until one can see simpler models within it. But whatare these simpler models? They are generally among the large variety ofmodels that have been created in the past, often recreational or fanciful.Perhaps the clearest example is graph theory,whereEulermadea simple

model of the reality that he was studying, then later workers found thatmodel useful in other situations. Graphs were then recognised as presentin many early problems: river crossing in ca. 800, knight’s tours in ca. 900,etc.Thus, recreational mathematics helps as a major source of mathematical mod-

els, techniques, and methods, which are the raw material for mathematicalresearch, in the sameway that mathematics in general serves as a source ofmodels for the physicalworld. I think this is the explanation of the utility ofrecreations inmathematics and the utility of mathematics in the real world.

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Zeiller, Martin,, with engravings by Matthaeus Merian. 1652. TopographiaElectorat, Brandenburgici et Ducatus Pomeraniae. uc. das ist Beschreibungder Vornembsten und bekantisten Statte und Platz in dem hochloblichstenChurfurstenthum und March Brandenburg; und dem Hertzogtum Pomeren,zu sampt einem doppelten Anhang, 1 Vom Lande Preußen unnd Pomerellen2 Von Lifflande unnd Selbige beruffenisten Orten. In Truck gegeben unndtVerlegt durch Matthaei Merian Seel: Erben. 1927. Reprint. Frankfurt amMain, Germany: Frankfurter Kunstverein. 1959. Reprint. TopographiaGermaniae. Faksimile-Ausgabe. XIII: Brandenburg — Pommern — Preußen— Livland. Barenreiter-Verlag: Kassel, Germany, and Basel, Switzer-land. 2005. Reprint. TopographiaGermaniae. —BrandenburgmitPreussen,Pommern, Liffland. Braunschweig, Germany: Archiv Verlag.

The map of Konigsberg would appear to be in the first Anhang (ap-pendix).

AcknowledgmentsI would like to thank the editor for many details andmuch information.


About the AuthorDavid Breyer Singmaster studied at Caltech and

received a Ph.D. in mathematics from the Universityof California–Berkeley. He taught at the AmericanUniversity of Beirut, later lived in Cyprus, and thencame to London in 1970—and has been based theresince. He retired from London South Bank Univer-sity in 1996. His interests are in number theory andcombinatorics, and the history of mathematics and ofscience in general.From1978 to about 1984, hewas the leading expos-

itor of Rubik’s Cube. He devised the now-standardnotation for it, wrote the first book on the Cube, and later edited Rubik’sbook into English. Due to revived interest in the Cube, he and some col-leagues produced a new book The Cube: The Ultimate Guide to the World’sBestselling Puzzle—Secrets, Stories, Solutions in 2009 and The Handbook of Cu-bik Math in 2010.Since about 1982, he has beenworking on a history of recreationalmath-

ematics, which has involved reading and studyingmathematics from everyculture and period.He was an invited speaker at the Third Iberian Colloquium in 2012,

where he spoke on “Vanishing Area Puzzles.” He attended the Fourth Col-loquium in Lisbon in 2015, where he spoke on “Early Topological Puzzles.”His book Problems for Metagrobologists: A Collection of Puzzles With Real

Mathematical, Logical or Scientific Content, a collection of over 200 problemsthat he composed since 1988, appeared at the end of 2015.

Modeling-First 381

Modeling-First Approach to TeachingDifferential EquationsRosemary FarleyDept. of MathematicsManhattan CollegeRiverdale, NY [email protected]

Dina YagodichDept. of MathematicsFrederick Community CollegeFrederick, MD [email protected]

Holly ZulloDept. of MathematicsWestminster CollegeSalt Lake City, UT [email protected]

Brian WinkelDirector, SIMIODECornwall, NY 1258 [email protected]

Abstract

We cite broad support for using modeling in mathematics course-work in undergraduate studies and address the possibilities for doingso in the undergraduate differential equations course.Wedescribeone current support community, SIMIODE, that isded-

icated to using modeling in differential equations courses.Further, we give specific examples of use and narratives by several

facultywhohave engaged in amodelingapproach to teachdifferentialequations.

The UMAP Journal 37 (4) (2016) 381–406. c�Copyright 2016 by COMAP, Inc. All rights reserved.Permission to make digital or hard copies of part or all of this work for personal or classroom useis granted without fee provided that copies are not made or distributed for profit or commercialadvantage and that copies bear this notice. Abstracting with credit is permitted, but copyrightsfor components of this work owned by others than COMAPmust be honored. To copy otherwise,to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP.


Modeling in the Classroom andDifferential EquationsThe recent COMAP/SIAM report, GAIMME—Guidelines for Assessment

and Instruction in Mathematical Modeling Education [8], encourages us to domodeling throughout themathematics curriculum,with rich examples andpractical guidance.Eric Mazur, the distinguished Harvard University physics professor,

leads physics colleagues in doing and encouraging active and engaginglearning by having students build things, discover principles, and modelreality to learn physics. Mazur has been advocating this for years withbroad coverage in public and academic literature [1].The 2015 CUPM Curriculum Guide [9] recommends,There are major applications involving differential equations in allareas of science and engineering, and so many of these should beincluded in the ODE course to show students the relevance and im-portance of this topic.

The report also does a reality check by admitting that many topics in tradi-tional differential equations course have been de-emphasized while use ofmodeling and technology has increased.In theMarch2014 issueofSIAMNews, DavidBressoud, formerPresident

of theMathematicalAssociationofAmerica,makes a similar call to increasethe relevance ofmathematics courses aswe prepare our students to face thechallenges of tomorrow [4, p. 7].Thus, there is an imperative to motivate and enrich the study of mathe-

matics by using real-world examples from other disciplines. We see differ-ential equations as an ideal and natural course for achieving this directive.Our students are asking for relevance and real-world engagement of

the mathematics that they are learning. Modeling can motivate studentinterest and pique curiosity about the many applications of mathematics.It is a good time to be teaching mathematics, for its usefulness is becomingclearer and mathematics is becoming more accessible through the use ofsimulations, data, animation, and technology.

Modeling, Inductive Learning, CuriosityModeling stimulates interest and interest can sustain a student. Paul J.

Silvia, a social psychologist, has determined thatWhen interested, students persist longer at learning tasks, spendmoretime studying, read more deeply, remember more of what they read,and get better grades in their classes.. . .

Modeling-First 383

In the case of interest, people are ‘dealingwith’ an unexpected andcomplex event—they are trying to understand it. In short, if peopleappraise an event as new and as comprehensible, then theywill find itinteresting.. . . finding somethingunderstandable is the hinge betweeninterest and confusion. . . . [24, p. 58]Using modeling to teach mathematics is a consummate inductive ap-

proach. In a seminal study on inductive teaching and learning, in thepremier engineering education journal, the American Society for Engineer-ing Education’s (ASEE) Journal of Engineering Education, Michael Prince andRichard Felder assert thatInductive teachingand learning serve as anumbrella term that encom-passes a range of instructional methods, including inquiry learning,problem-based learning, project-based learning, case-based learning,discovery learning, and just-in-time teaching. [23, p. 1]

The authors conclude thatinductivemethods are consistently found to be at least equal to, and ingeneralmore effective than, traditional deductivemethods for achiev-ing a broad rangeof learningoutcomes. [23, p. 1]

One of the authors, in a closing essay for PRISM, the magazine of ASEE,says,Another well-entrenched tenet of traditional instruction is the notionthat students must first master the underlying principles and theoriesof a discipline before being asked to solve substantiveproblems in thatdiscipline.An analysis of the literature (rendered in [23]) suggests that there

are sometimes good reasons to “teach backwards” by introducing stu-dents to complex and realistic problems before exposing them to therelevant theory and equations. [14, p. 55]Colleagues around the academic community believe in inductivemeth-

ods that are in touch with the real world, and they are using this approach.In a paper for teachers of American history, Lendol Calder forcefully sup-ports our approach, by quoting the distinguished professor of history,Charles G. Sellers (UC Berkeley):The notion that students must first be given facts and then at somedistant time in the future will “think” about them is both a cover-upand a perversion of pedagogy. . . . One does not collect facts he doesnot need, hang on to them, and then stumble across the propitiousmoments to use them. One is first perplexed by a problem and thenmakes use of the facts to achieve a solution. [6]Modelinguses an inductive approach. It alsouses andneeds intuitionas

a call for action and an attempt at a first step. Modeling can foster intuition


and bring it to a higher level in students through attempts at buildingindividual models. Henri Poincare (brought to our attention in [29]) said,The principal aim ofmathematical education is to develop certain fac-ulties of the mind, and among these intuition is not the least precious.It is through it that the mathematical world remains in touch with thereal world. [21, p. 128]It is by logic that we prove, but by intuition that we discover.

[21, p. 129]Keith Stroyan speaks to the very positive role of student inquiry and sum-mary through projects:It is easy to get sidetracked by algebra or trig skills and boil the coursedown to template exercises. That ends up reinforcing students’ im-pression that math doesn’t solve real problems.Projects canhave aprofound impact on students because they ‘take

intellectual ownership’ of the problem. Most of my students can de-scribe their projects a decade after they took the course. [28, p. 1123]We have found that students who engage in projects and “struggles”

with tough modeling situations remember the experience positively. Theultimatemodeling experience for undergraduates is theMathematicalCon-test in Modeling (MCM)tmand the Interdisciplinary Contest in Modeling(ICM)tm, in which students work in teams of three for four days to pro-duce a mathematical model for a significant problem [19]. Students whoparticipate in MCM/ICM say this experience was the best undergraduateactivity theyhad, handsdown! What does this say about our current courseofferings?

Some Specifics on the Modeling-First ApproachWhat does amodeling approach in a differential equations course really

look like? We offer an outline.• Students are presented with a question: Perhaps some objective is tobe achieved or a phenomenon might be understood. Ideally, there is astakeholder who has interest in the answer.

• Students are then either given data or run experiments to collect theirown data.

• In groups or as a class, students develop a mathematical model, leadingto differential equations.

• Students are motivated to solve the differential equations because theywant to answer the originating question. Techniques of analytic solu-tion, numerical solution, and/or use of technology become relevant andimportant.

Modeling-First 385

• Solution in hand, students reflect on their answer. Does it make sense?Does the solution capture the key phenomenon driving the physical sit-uation being modeled? Does the model need to be modified or iterated?What is the long-term behavior of the system? Was it possible to answerthe stakeholder’s question?

• Reflection can then be extended by encouraging students to generalizewith more questions. Will the solution always behave like this? Whatcan be said about similar systems? Does a small change in the parame-ters cause a change in the output? What happens if the initial conditionschange? Is there/what is the “tipping” point? What other useful infor-mation can be provided to the stakeholder?In the remainder of this article we offer material, experiences, and rele-

vance for teaching the differential equations in a modeling context.

Building a Community of Support at SIMIODEIn response to calls for modeling in mathematics courses, we propose

a serious paradigm shift to a modeling-based approach for teaching differentialequations. The differential equations course is a natural place for modeling,both for motivating and applying the mathematics under study.SIMIODE (Systemic Initiative for Modeling Investigations and Oppor-

tunities with Differential Equations), for teachers and students at www.simiode.org [25], is a rich environment devoted to the teaching of differ-ential equations using modeling and technology up front and throughoutthe learningprocess. SIMIODEoffers a community of support inwhich col-leagues can explore, communicate, collaborate, publish, teach, contribute,develop teaching materials, and form an archive of resources.At SIMIODE, we are building a complete environment for teachers and

learners with communication, with groups within and among campuses,projects for students and teachers, reviews, models, data, and videos (theselast are at [26]). In SIMIODE, colleagues can join a group, form a group ofinterest or purpose, begin a discussion, and then collaborate and commu-nicate with others, build projects, and offer reviews of materials.UsingSIMIODEYouTubevideos, students can collectdataonTorricelli’s

Law andmodel it with a first-principle physics approach for building a dif-ferential equation. At SIMIODE [25], one can see material [41] associatedwith this Torricelli’s Law video in the Modeling Scenario section of Re-sources.SIMIODE has a manuscript management system [16] that handles ma-

terials submitted to SIMIODE. Double-blind, peer-reviewed publicationsare offered online for all to use.With SIMIODE, we are declaring that it is time to commit to inductive

approaches in differential equations instruction using modeling and tech-nology and to encourage students to explore and build their own models


using differential equations.We offer narratives of three faculty colleagueswho are embracing teach-

ing differential equations using modeling.

Differential Equations at WestminsterCollegeWe discuss a 300-level differential equations course that Holly Zullo

taught at Westminster College (Salt Lake City, Utah) in Spring 2016.

BackgroundNo programs on our campus require differential equations, although

physics students are strongly encouraged to take it andmathematicsmajorsmay take it as a mathematics elective. As the teacher of this purely electivecourse, I could choose topics as I saw fit. I started the semester with 17students, and13 successfully completed the course. Theother four studentsstopped attending at various points in the semester, mainly due to personalissues (some having to do with the 8 A.M. class period). Of the 13 whocompleted the course, about half weremathematicsmajors, with the othersscattered between physics, 3-2 engineering, chemistry, and environmentalscience.This was a 4-credit course that met for two 110-minute sessions each

week, for a total of 29 class meetings, including the final. I used a fairlystandard print text [11]. I usedExcel andMATLAB, alongwithdfield.jarand pplane.jar [22]. Most students had very little experience with Excelor MATLAB, although the physics majors were proficient in Mathematicaand a few students knew some Java programming.Course grades were based on the following components:

• WebWork: 15% of final grade– 24 assignments, primarily computational

• Written homework, labs, projects: 50% of final grade– 2 written homework assignments using problems from textbook– 11 labs (more on these later)– 1 mini-project (more on this later)– work on labs and project in pairs

• Exams: 35% of final grade– midterm and final, both given in-class and weighted equally– involved some modeling, but largely computational.

Modeling-First 387

ModelingI have a vested interest in working modeling into every possible class

component, because every year I advise students in theMathematical Con-test inModeling (MCM)[19], andmyclassesare about theonlymathematicscourseswhere studentswill seemodeling. Since only twoofmydifferentialequations students were seniors, I now have 11 students who are ready tocompete in the MCM next year! Competition aside, modeling is a naturalfit with differential equations. So I strove to work modeling in whereverpossible. I wanted students to see many models as we discussed topics inclass, to explore models more deeply through labs, and to tackle a modelof their choosing for a project.

Modeling Woven Through ClassMany times, I would introduce a new type of differential equation to

students by asking them a modeling question. This began on day 1, whenafter a brief discussion of what constitutes a differential equation and asolution, I posed the following question [18] to students:The amount of chemical in a lake is decreasing at a rate of 30% peryear. If p(t) is the total amount of the chemical in the lake as a functionof time t (in years), which differential equation models this situation?1. p0(t) = �302. p0(t) = �0.303. p0(t) = p� 304. p0(t) = �0.3p5. p0(t) = 0.7p

I followed that with a question about Newton’s Law of Cooling andunits, and then we hit on second-order differential equations with the fol-lowing question:A branch sways back and forth with position f(t). Studying its mo-tion you find that its acceleration is proportional to its position, sothat when it is 8 cm to the right, it will accelerate to the left at a rateof 2 cm/s2. Which differential equation describes the motion of thebranch?Later in the course, I introduced systems by having students develop

a simple compartment model for a medication moving from the gastro-intestinal tract into the bloodstream. Students then further explored sys-tems through a fun series of questions [18] on the Lanchester combatmodel,adapted from [2]. On another day, we discussed SIR models, using part of[27].


All of thesemodels engage students andallow themtodiscuss themean-ing behind each piece of a differential equation, thus increasing students’understanding of their rationale, the nuances, and pieces of a differentialequation. Most of my students loved all of these models, although one stu-dent did complain about the combat model, since she didn’t like the ideaof losing her own troops or of increasing their fighting power to better killother troops. This concern aside, using models to motivate new materialwas, not surprisingly, a huge success.

Modeling in LabsModeling was worked into the course wherever possible, but the most

in-depth work occurred in the labs. Four of the labs explicitly involvedmodeling, while the others helped to develop analytical skills important tomodeling and to understanding differential equations:• qualitative analysis of graphs,• understanding of numerical algorithms,• analysis of long-term behavior of solutions to differential equations,• and so on.Students worked in pairs (or a group of three, if needed) on all of the labs.Early on I used a SIMIODE Modeling Scenario on the sublimation of

dry ice [39], with the Excel Solver approach for parameter estimation. Formany students, this was their first exposure to the idea of trying multiplemodels to see which fit the best. This activity probably would have bene-fited from my spending a little more time on the chemistry and geometryof this situation, since I was a bit surprised by my students’ general lackof familiarity with basic reaction orders. I wound up suggesting plausiblevalues for r in the model

m0(t) = �km(t)r

form(t), the remaining mass of dry ice. At this stage in the course, it waschallenging enough for most groups to pick a couple of values for r, solvethe resulting differential equation, and then put the solution into Excel tofind the best fit value of k. One group, however, did manage to use Solverto optimize over both k and r. I devoted an hour of class time to this lab,and students finished it at home.The followingweek, I had students do a similar lab (based on [7]) where

they used two different methods to estimate the parameters needed to fita logistic model to population data for Yellowstone National Park’s bisonpopulation. We again used Excel’s Solver to minimize the sum of squarederrors in order to estimate parameters in the model.In another lab, students startedwith a simple exponential growthmodel

for a hypothetical zebra mussel population. They learned about Euler’s

Modeling-First 389

method by comparing their numerical solutions with various step sizes tothe analytical solution, all thewhilediscussing the eventual size of the zebramussel population. They also experimentedwith two different eradicationplans, using Euler’s method to solve the resulting ordinary differentialequations. This lab was also done with Excel, allowing students to seeclearly the iterative process.Several of the next labs had students using MATLAB to solve ordinary

differential equations, and dfield.jar and pplane.jar [22] to providegraphical analyses; but the equations in these labs were not presented incontext. While I would like to modify these labs to include real scenariosand resulting equations, they still provided students an opportunity todevelop their analysis and writing skills.By week 10, students were ready for the SIMIODE Modeling Scenario

on the drug ibuprofen [45]. In this lab, students develop and compare fourmodels for how ibuprofen moves through the body. Rather than havingstudents test the three sets of given parameters for each model, I had themdo that for just the first model but then find their own best parametersfor each later model. Through their earlier work with Excel, students hadlearned exactly what it means to find parameters to minimize the sum ofsquared errors. Wanting a bit more sophistication at this point, in this labI showed them how to use MATLAB’s optimization toolbox to do this.While using that is much more of a “black box” approach, students wereconceptually ready for that jump, which allowed them to use MATLAB tosolve ordinary differential equations that they were writing. While I hadonly devoted one hour of class time to most of the earlier labs, this one wasmore complex, and we spent our entire two-hour class period on it; thenstudents finished their work at home.

Modeling in ProjectsEach of the labs was completed within a week’s time, including usually

an hour of class time and whatever other time was needed at home. I hadintended tohave studentsworkona longer,more intenseproject thatwouldtake themmore than aweek to complete; but class time ran short, as didmytime to develop such a project. Instead, I settled for a one-weekproject rightbefore the final exam. Studentsworked in pairs, and the project culminatedin an oral presentation to the class, with no other written submission.Students were offered several options for the project, including any of

several SIMIODE activities, further study of the SIR model that we hadlooked at earlier, or playing with ideas of chaos through any of severalexplorations outlined in our textbook. Several groups chose to study chaos,two chose SIMIODE activities (one chose [38] on LSD and one chose [15],an activity centered on numerical modeling of a pendulum). By this point,studentswere able towork quite independently on theirmodeling projects.


ChallengesMost students in our programhave very little, if any, experiencewriting

even short technical papers of the type that I expected for these modelingactivities. Despite what I thought were clear instructions, at first manystudents didn’t include their graphs or didn’t record the actual parametervalues they found. Students also struggled to discuss strengths and weak-nesses of their models. In particular, when fitting a model to data, theyreally had to learn how to discuss where the model fit well, what charac-teristics of the data it captured, and what it did not capture. They werevery comfortable comparing values for the sum of the squared errors (SSE)and selecting amodel based solely on that numerical comparison, but theirqualitative analysis skills were much slower to develop. Develop they did,though, and therewerenoticeable improvementsby the endof the semester.Other challenges arose simply from having enough engaging activities,

both large and small, for students to work on, and also finding appropri-ate technology support. SIMIODE has a wide range of activities that aresuitable for what I called my labs. The technology platform for many ofthese activities, though, is Mathematica, which is too expensive for manyschools. It would be helpful to see more platforms employed, includingopen-source options, and that is a goal of SIMIODE. Also, there is a needformore very short or very long activities, such aswould be useful for 20-30minutes in class or for an extended project, respectively. The MathQuestClassroomVoting questions [18] provide some help with in-class activities,but much more is needed.

RewardsOverall, teaching differential equations with modeling was a huge suc-

cess. I enjoyed it, the students enjoyed it and learned, and my physicscolleague was very pleased with what his students were learning. As anapplied mathematician, it is very important to me that the mathematics Iteach is put into context, and there is no better way to do that than throughmodeling. The students were excited about the applications and aboutlearning how to use differential equations for modeling. One student whostruggled mightily through the proofs in my discrete mathematics coursein the fall rediscovered his joy in this course and would frequently com-ment, with a huge grin on his face, “I love modeling!” Another studentwrote on the end-of-semester course evaluations, “labs were the bane ofmy existence but i [sic] learned a lot from them!” Another highlight was amorning late in the semester when one of the physics majors said, “Peter[physics professor] really loves that you’re making us do all these labs!”This was confirmedwhen I later ran into the same physics professor and hecongratulated me on doing such a great job with the differential equationsclass, saying he loved what the students were learning with their labs, and

Modeling-First 391

he was so glad they were working with MATLAB.

Conclusions and Future WorkThis course played an important role in these students’ development,

especially sincemost of their other courses focus on very traditional, proof-based mathematics. As part of the modeling experience, students alsolearned to write short technical papers and gained basic proficiency withExcel andMATLAB. The next time I teach differential equations I intend toincorporate even more modeling. There is room for me to include realisticscenarios and equations in many of the labs that currently just exploreequations out of context. I can also make time for a bigger project, so thatstudents have an opportunity to explore a model (or several models) ingreater depth. As SIMIODE continues to develop, it will be even easier toselect resources to support these goals.

Differential Equations at ManhattanCollegeWediscuss a 200-level differential equations course taught by Rosemary

Farley at Manhattan College (Riverdale, New York) in Spring 2016.

BackgroundThis course is required of every student in the School of Engineering

and some students in the School of Science. As a required course, there isa syllabus with topics that have to be covered in preparation for a commoncumulative final exam. In particular, all the traditional methods of solvingdifferential equations by hand must be covered.I taught two of the seven sections and 56 of the 192 students who took

the course. All but one of my students were in the School of Engineering.Every student passed the course, but 4 received a grade of D (a grade thatis not acceptable for this required course).This was a 3-credit course that met for two 75-minute sessions each

week. One of those weekly sessions was in a computer lab. The differ-ential equations committee chose the textbook [3] together with the onlinehomework systemWiley Plus [30]. For the Spring 2016 semester, the com-mittee added systemsof first-order differential equations to the curriculum.Linear algebra is not a prerequisite for the course.Course grades were calculated as follows. There were three in-class ex-

aminations, totaling 60%, Maple labs were worth 15%, and a cumulativecommon final examination was worth 25%. Questions requiring mathe-matical modeling appeared on all examinations.


The MotivationMy research colleague Patrice Tiffany and I attended the MAA-PREP

Workshop on SIMIODE at Carroll College in Helena, Montana in the sum-mer of 2015. We left convinced that a modeling-first approach should beused in our differential equations classes atManhattanCollege. We formedan interesting and symbiotic partnership. I have taught the course for 20years, while she has never taught it. Unfortunately for me, she was onsabbatical in the Spring semester when the differential equations coursesare primarily taught. Fortunately for me, she was my sounding board forthe labs that I created in her absence. I benefited from her guidance, asshe asked questions and expressed concerns about important issues thatescaped my detection. I note her contribution, first to give credit where itis due and second to stress that beginning something like this is so mucheasier if one can work with a colleague.

Getting StartedWhen thinking about using a modeling-first approach, I worried about

completing the syllabus. At the beginning of the semester, the differentialequations committee created a common final exam that had one questioninvolvingmodeling. That question had four options, with students pickingone. The rest of the test looked like most traditional differential equationsfinal exams:• first-order differential equations,• first-order systems of two differential equations,• second-order homogeneous and non-homogeneous differential equa-tions, and

• Laplace transforms.I wanted to commit to the modeling-first approach, but I also needed tomake sure thatmy studentswoulddowell on thefinal. So Imade importantdecisions from the start.Since I was in a computer lab on Tuesdays and in a traditional class-

room on Fridays, I decided to do modeling-first activities every Tuesday.Occasionally, I used about 25 minutes of the 75-minute class for a regularlecture. However, I committed 50minutes of eachweek—that is, 1/3 ofmyallotted weekly class time—to modeling-first activities.I realized that I had to be realistic about covering the material. I did this

in two ways:• First, I believe that the best decision I madewas to use themodeling-firstapproach on required topics (more on this later).

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• Second, I supplemented lectures with videos on the material covered.This was not a flipped classroom. I covered all the material in class andgave more examples and explanations on videos. The creation of thesevideos was time-consuming, and I do know that there is a great dealof material available online already. However, I wanted my students toknow that I was really committed to the modeling-first approach. Manyadmitted that they placed a higher value on the videos because I madethem. The response to the videos was overwhelmingly positive. In fact,I have never had as much gratitude expressed for my work.I also decided to use the computer algebra system Maple. Some of

my students had used Maple in their calculus classes and some had not; Iassumed nothing about Maple knowledge. The code is easily learned (andeasily forgotten when it is not being used regularly). In fact, with so manymenus in Maple now, Maple is getting easier to use; the learning curve isnot steep. I have found that there is little difference between the studentswho have usedMaple before and thosewhohave not. I also believe that thetechnological tool chosen is not important. However, it is important thatdifferential equations students use technology. Themost recent report fromthe Differential Equations group of the Committee on the UndergraduateProgram in Mathematics [9] states that the ordinary differential equationscourse “is easily the course in the introductory undergraduatemathematicscurriculum in which the use of technology is most essential.”

Using Technology in the LabIt is imperative in every lab setting that mathematics take center stage.

From the beginning, I de-emphasize code by explaining thatwhen studentsknow what they want to do, I will provide the code for them to do it.There is one rule that helps a great deal: No student is allowed to ask

for help with any code until the student has written down what they wantthe code to do. Some students will say, “I don’t know what to type.” Typ-ically what they really mean is “I don’t know what to do.” It took abouttwo weeks, but finally all students realized that asking for code was notgoing to help them actually create a differential equation. Rather, they hadto interpret the words given about how something like a disease spreadsand come up with a differential equation that made sense. Once they un-derstood this, their impatiencewithMaple diminished; and their questionschanged, becoming centered on the mathematics in their problem.

Modeling-First: Classes in the LabOn the first day of class, the students were given them&m Immigration

andDeath scenario [37] from SIMIODE. This introduction to themodeling-first approach was quite successful. Students followed the directions care-


fully and arrived at the difference equation quickly. They used the rsolvecommand in Maple to find a closed form solution and graphed it. All sortsof interesting analyses happened. However, I was disappointedwhen theystruggled to see the emerging calculus ideas as we went from a differenceequation to a differential equation. The material was important enough tohave a full class discussion about this issue. While it took time, it was avery fruitful and important discussion.The remaining Tuesday modeling classes followed the same pattern.

We did no more data collection. Rather, problems would be posed to thestudents, and their task was to formulate them as initial value problemsand solve them in any way they could.Let me reiterate that the best decision I made was to use the modeling-

first approach on the required topics. This made me relax about coveringmaterial in the syllabus. Before we did any method of solving differentialequations by hand, the students had some kind of modeling-first activityin the lab. Let me explain further.The first time that students saw Newton’s Law of Cooling was in a

lab where this law was stated in words. The students were then given aproblem and were told that they could use Newton’s Law of Cooling tosolve it. They had to decide what variables were essential and what to callthem. They had to translate the law into a differential equation, identifyinitial conditions and important facts, and answer the question. Since theyhad not learned how to solve any first-order differential equations yet, theyused Maple.This method of modeling-first continued throughout the course. The

first time that they saw a salt concentration problem with one tank, or aspread-of-disease problem, was in the lab. With no lecture beforehand ex-plaining how to proceed, studentswere asked to solve these problems. As Iwalked around the room, answering questions, asking questions, and giv-ingadvice, students started to speakmathematically. Theynaturally startedto work together, agree on variable names, and discuss what was impor-tant. They began to discusswhether or not their answersmade sense. Theyfound problems such as salt concentration problems relatively easy whenthey realized that the units involved could be used to derive the differentialequations. They struggled with the spread of disease in a boarding schoolproblem [20] and had to be guided toward creating a differential equationthat included the product of those infected and those not infected. Regard-less of how hard they found the questions, they were all engaged. Theredefinitely were leaders and followers; but because they each had to handin a write-up of the solutions, each student made sure to clarify any mis-understandings. For the labs involving first-order differential equations, Idrew on portions of the SIMIODE scenarios [40; 42; 43].Systems of differential equations were started with a modeling-first ap-

proach by considering a predator–prey model. Students were told to as-sume certain facts about how both the prey and predator populationswere

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changing. Then they had to find a system of two differential equationsand initial conditions. We talked about the fact that they were really beingasked to translate the given assumptions into mathematical equations, andthat a far more difficult problem would have been to come up with thoseassumptionsbasedonobserved information. Then the studentswere askedto open a Maple worksheet that created a system of differential equationsthat modeled a predator-prey scenario. In this worksheet, the system ofdifferential equations was not solved. Rather, both the direction field anda particular solution for certain initial conditions were plotted. Then theywere asked to discuss what this animation revealed in terms of predatorsand prey. It was interesting to listen as groups talked aboutwhatwas beingrevealed on the plot. The students were speaking about how one popula-tiondependedon the other. Itwas also great to realize that thesedifferentialequations were coming alive as a result of this exercise.The students were then given a two-tank salt concentration problem.

Realize that the first system that they had ever seen was the predator-preysystem a few minutes before! It was simply fascinating, and a bit unnerv-ing, to watch as students tried to cram every piece of information into onedifferential equation. The reaction was one of relief as they realized thatthey needed two differential equations! This lab was great. Students sawsystems of differential equations from a modeling-first perspective. Theyanalyzed their solutions and answered all the subsequent questions fromseveral different perspectives. For labs involving systems of first-order dif-ferential equations, consider the SIMIODE scenario on salt concentrations[47].To use a modeling-first approach in solving second-order differential

equations, we tackled the spring-mass problem. My goal was to have stu-dents derive the second-order differential equation that describes such asystem. I gave a great deal of guidance, asking specific questions aboutthe forces acting on such a system and about Newton’s Second Law of Mo-tion. It turned out to be such a positive experience. The students were allengaged and involved in figuring out how to think about such a system.They were immediately talking about units and about what made sensephysically. Ultimately, they arrived at a second order initial value problemof the form

my00(t) + by0(t) + ky(t) = 0, y(0) = y0, y0(0) = v0.

They understood that b is a damping constant and k is a spring constant.However, something interesting happened when they solved subse-

quent word problems. Many of the students derived the equation eachtime, rather than just applying the formula for the spring-mass system.Doing that showed me that they really understood the material. I didsuggest to them, however, that on a timed test they might not derive itagain. More than one student said that they liked their way because itinvolved understanding. This lab was a great success. For the labs in-


volving second-order differential equations, consider SIMIODE scenarioson spring-mass systems and on an introduction to second-order differentialequations [44; 46].

Reflections and ConclusionsI was far too ambitious on my first labs. By trying to do too much in

a lab setting, I actually accomplished less. Everything was better when Ishortened the labs. I worried too much about time management because Idid not realize that the modeling-first approach had unforeseen benefits.For example, when we talked about autonomous differential equations

and I asked whether we had seen such equations before, there was a re-sounding “Yes.” Most students not only knew that the boarding-schoolproblem was such an equation but they could describe the graph and itsasymptotic behavior. When we did a Newton’s Law of Cooling problemin lecture, and I asked what information was given, students immediatelycalled out the correct differential equation and the correct initial conditions.There was little hesitation and their answerswere perfect. So we picked uptime in lecture because this material could be covered quickly.It was interesting that most of my students approached the modeling

problems on tests and the final exam in a way I had not seen previously.In a typical modeling problem, most students wrote down the differentialequation, equations for every condition given, and an equation for whattheywere asked tofind—before solvinganything. Thiswasadirect result ofusing a computer algebra system to solve differential equations. Having allof the informationwritten down before solving anything kept the studentson track.Would I do this again? Absolutely. I willmake some changes. I will take

time to check the SIMIODE website [25] frequently because new scenariosare being added all the time. I had just finishedworking onmyown spring-mass lab in the Spring when I found a new one that I could have used. Weneed not reinvent the wheel: We can use what others have shared.In that vein, I am also so happy that I kept a blog about SIMIODE [12].

It is rather shocking to realize that what I did last semester is already a blurto me. Reading my own blog about my successes, failures, and surprisesalong the way clarified so many of the ideas reflected in these past fewpages. In addition to my blog, every lab I did and all the solutions are alsoavailable through the SIMIODEwebsite [13]. I wholeheartedly suggest thatyou take the time to write your own blog and share it with the communityof learners in SIMIODE. It is quite a learning experience. It has taken mejust this one semester to become committed to the modeling-first method.This method has definitely helped both me and my students change theway we think about differential equations and make us all better problemsolvers.Given all the modeling my classes did, it was not surprising (but it

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was gratifying) that my students had the highest scores on the modelingproblems on the common final. The actual overall average on this questionfor all seven sections was 6.8 out of a possible 10. My students’ averagewas 7.6. In fact, my students did very well on the cumulative final as awhole, having an average of 82%. That is good news. I really am anxiousto teach differential equations again in the Spring of 2017, modifying theapproach I used this past semester, having my research colleague back oncampus, and benefiting fromanother severalmonths of the ever-expandingSIMIODE library of scenarios.

Differential Equations at FrederickCommunity CollegeThe comments here pertain to a 200-level 3-credit differential equations

course taught by Dina Yagodich at Frederick Community College (FCC)(Frederick, Maryland) over a number of semesters, running from Spring2014 through Summer 2016.

BackgroundTeachingupper-levelmathematics courses suchasdifferential equations

is challenging at a community college. Our students transfer to a varietyof colleges and universities after completing their coursework with us, soour task is to ensure that our students are ready for their junior- and senior-level courses. After a survey of our top-10 transfer schools, certain topicsand skills were deemed critical for differential equations. These includeLaplace transforms and asmuchMATLAB instruction as possible. Startingin Spring 2016, a 1-creditMATLAB coursewas added as a pre-/co-requisitefor differential equations; it includes basic matrix operation skills to assistinstruction with systems of differential equations.A breakdownof the classes’make-up is given inTable 1. Many students

who are aiming for engineering use the General Studies major for moreflexibility. A new major option, STEM: Sci, Tech, Eng, and Math, wasadded in Spring 2015.This was a 3-credit course that met for 15 weeks (including the final

exam), with either two 75-minute sessions each week or one 150-minutesession. Since most transfer colleges require a grade of C or better, un-successful students include students who withdrew, audited the course, orearned an F or a D (see Table 2). Student grades are based on 60% exams(midterm/final), 20% weekly homework (pencil and paper), 10% weeklyquizzes, and 10% on modeling and MATLAB projects.In Spring 2014 through Spring 2015, the textbook [51] was used. How-

ever, in Fall 2015, the switch was made to [17], an open education resource


Table 1.Student interests.

Semester Students Interests

Spring 2014 32 (2 sections) 13 Eng, 12 Gen Stud, 1 Chem, 1 Math, 5 HS studentsFall 2014 9 (1 section) 7 Eng, 2 Gen StudSpring 2015 33 (2 sections) 17 STEM, 10 Gen Stud, 1 CS, 1 Math, 4 HS studentsFall 2015 13 (1 section) 10 STEM, 2 Gen Stud, 1 CSSpring 2016 10 (1 section) 4 STEM, 2 Math, 1 CS, 1 Gen Stud, 2 HS studentsSummer 2017 6 (1 section, online) 1 STEM, 5 Gen Stud (students attending a 4-year univ.)

Table 2.Grades.

Semester Students Number successful

Spring 2014 32 (2 sections) 23Fall 2014 9 (1 section) 8Spring 2015 33 (2 sections) 30Fall 2015 13 (1 section) 9Spring 2016 10 (1 section) 7Summer 2017 6 (1 section, online) 5

(OER) textbook, paired with Schaum’s Outline notes [5]. Excel and MAT-LAB were used as software support.

MotivationBrianWinkelhadpresented informationabout teachingdifferential equa-

tions with modeling at the 2013 Joint AMS–MAA meeting in Baltimore.His presentation included a “sure-fire” activity using m&m candies [37]to model population decay. I tried it. It worked! Since I was fairly newto teaching differential equations, I was ready to change the format of theclass from how it had been taught by colleagues in the past. Addition-ally, my background is in electrical engineering and applied mathematics,so a modeling approach seemed very appropriate for the class. However,rather than completely change the course, I opted to add modeling slowly,semester by semester, to see how it fit into the course.

Initial SemestersIn each of the first three semesters, I included only one or twomodeling

activities, including one the first day of class. I used the m&m modelingactivity [37], with some minor adjustments to fit the activity into about30 minutes. The first experiment models population decay without anyimmigration, and studentswere able to confidently predict the steady-state

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solution. The class evaluated the number of generations required by eachof the groups to reach a population of zero, andwe had a fruitful discussionabout the statistics behind experiments. Studentswere able, with guidance,to create a difference equation and then move to a differential equation. Atthe time, MATLAB was not required for the course, so I just demonstratedusing the software to find the solution.The second half of the m&m modeling activity involves immigration,

and no student was able to predict what the steady-state value would be.Watching students continue to repeat experiment until the population set-tled to the number that they had predicted (which it never did!) was veryenlightening. Once the solution was found using MATLAB, students wereable to come up with reasoning to explain the actual steady-state solution.This development led directly into a very gooddiscussion about themodel-ing cycle: taking data, making a model, testing the model against the data,and then revising the model.A memorable quote from a student evaluation from that semester an-

swered the question “What class assignment or activity did you find to bethe most useful?” with the following:Believe it or not, the m&m activity on the first day of class really stood out tome. It helped to show the real-world applications of differential equations andI thought it was amazing that we could build an equation using real-worlddata.Even if no other modeling is done the rest of the semester, starting a

coursewithamodelingactivity suchas thispaintsa clearpictureof the typesof problems that differential equations can be used to solve, demonstratesthe modeling cycle, and introduces numerical solution methods such asMATLAB.

Later Semesters and Online ExperienceBy Fall 2016, I added more modeling problems. One that stood out

was not based on data; rather, it asked the students to calculate how muchadditional anesthesia to administer to a dog undergoing surgery after theoriginal dosage begins to wear off [48]. The problem took students bysurprise; they did not realize that they had already learned the tools tosolve this problem. Working in groups, students were able to formulatea differential equation to model the situation. At this point, analyticaltechniques to solve this type of equation had been taught, so students wereable to solve it without computer assistance. A student commented in acourse evaluation that the activity that was the most useful was “The dogdrug, it was fun and it was a real life example.”The SIMIODE community allows for continual renewal in themodeling

classroom. I made a change to the m&m modeling scenario based on aconversation at the 2016 Joint AMS-MAA meeting in Seattle, Washington


with another member of SIMIODE. When immigration was added to themodeling problem, I gave each student a different number of m&m’s atthe beginning of the experiment. Some groups had a number greater thanthe steady-state solution, others had a number less than the steady-statesolution, and one group was given the exact number of the steady-statesolution. Students were truly surprised when all groups settled to thesame number. This was a great way to bring the class together as a wholeand understand how population modeling works.In Summer 2016, the class was offered online for the first time. Because

of the repeated success with the m&mmodeling in the classroom, I createdan assignment for students to do on their own following the proceduresfrom the face-to-face class. They discussed the results in a discussion boardand I then posted a video [50] explaining the modeling project. In futureonline sections, I hope to integrate more of the material available at theSIMIODEWebsite as well as host a class discussion board there.

Conclusions and Future PlansStudent comments from Fall 2015 helped keep me focused on continu-

ing to usemoremodeling in the classroom. Answering the question “Whatspecific recommendations do you have to improve the course?,” two stu-dents commented “Maybe do theory after we do examples of the problem first”and “I suggestmore focus on example/model-based learning.” Andanother com-ment that answered “What class assignment or activity did you find to bethe least useful?” was negative only because more modeling was desired:“At first I enjoyed the modeling problems, but I don’t feel like we did enough, orwent through enough of them to really benefit from it.”Over this sequence of semesters, themovewasmade from an expensive

textbook to open education resources, in addition to the move towardsmodeling-first. I initially settled on one OER book for the theory, pairedwith Schaum’s Outline notes [5] for a larger homework set. I will startbranching tomore thanoneOERbookaswell as [10] tomeet theneedsof thecourse. Since there is no textbook currently published teaching differentialequations as modeling-first, using a variety of textbooks and resources willallow for tailoring the material. Making a textbook change, however, tookmy focus off more fully integrating modeling throughout the course.My Spring 2017 schedule will be very similar to my Fall 2016, and we

have a five-week break between Fall and Spring semesters. I am going touse that time to completely convert my class to a modeling-first approach.In particular, I will be using more MATLAB, to give students the abilityto solve modeling problems that students cannot yet analytically solve.My biggest challenge will be the Laplace transform topics, but systems ofdifferential equations can easily be reworked using a modeling focus.In Fall 2016, two of my students from Spring 2014 stopped by to say

hello. After chatting about their experiences at their respective four-year

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universities, I asked them if they remembered them&m candy experiment.One said, “Oh, the populationmodeling experiment” and the second com-mented that it was a great way to start the course. There are not many firstdays thatmy students remember, but these two students still remember thatfirst day, two-and-a-half years later! This reinforces my belief that hookingstudents early into the “why” of differential equations sets the tone for theentire semester and beyond.

Conclusion and ReflectionsOvermanyyears and inmany types of institutions (small liberal arts col-

leges, research universities, technology institutes, and military academies)Brian Winkel has found that applications and modeling motivates and en-gages students. He has written about these experiences and documentedthem in many articles, for example [31; 32; 33; 34; 35; 36; 49].Showing students that mathematics matters in the real world through

modeling opportunities offers motivation and generates curiosity in theapplication and in themathematics. Curiosity is good and serves as amoti-vational force [24]. Such activities motivate students to learn mathematics,and seeingmathematics in context helps students recall and transfer math-ematical knowledge to other situations and disciplines. Most important,the confidence that comes with building and testing a mathematical modelfar outweighs algorithmic instruction and examples and exercises devoidof any reality.The narratives offered here by colleagues illustrate the rich possibili-

ties in doing modeling in a differential equations course, and the authorsknow such experiences are well received by students. We encourage read-ers to bring context to the mathematics of differential equations by doingmodeling in the course in ways that are appropriate for individual settings.More andmore avenues and resources for usingmodeling in differential

equations courses are available through the broader literature and throughthe resources at SIMIODE. SIMIODE makes it easier to incorporate mod-eling, while the collegial support found in the SIMIODE community per-mits sharing of teacher experiences and materials. The time to move to amodeling-first approach in teaching differential equations is never betterthan now.

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About the AuthorsRosemaryFarley isProfessorofMathematicsatMan-

hattan College. She has more than 20 years of experi-ence introducing technology both to mathematics stu-dentsat all levelsand tomathematicseducatorsatwork-shops and national conferences. She has used tech-nology to generate research questions for presentationsby students at undergraduatemathematics conferences.She has contributed to PRIMUS and is a member ofSIMIODE and MAA.

Dina Yagodich started her career in electrical engi-neering. After taking a few years off from working inher field to stay at home with her kids, Dina startedteaching part-time at night to keep her resume active.After one or two semesters, she was hooked on teach-ing and has never looked back. Completing a secondmaster’s, in Applied Mathematics, gives Dina a solidfoundationboth inmathematicsaswell as engineeringand an appreciation of how important it is to tie mathematics instructionto the “why” and not just the “how.” Dina currently teaches full-time atFrederick Community College in the Mathematics Dept.

Holly Zullo has been teachingmathematics at var-ious small colleges for 21 years. She has always incor-porated modeling into her classes wherever possible,and she has worked with hundreds of students in theMathematical Contest in Modeling.


Brian Winkel began his teaching career in a liberalarts college after a Ph.D. inNoetherian ring theory. Hemigrated quickly to interests in applications of math-ematics, which was fostered by a mid-career stint inengineering institutes and then fosteredby teaching inmilitary academies. Along the way he founded andedited Cryptologia, a journal devoted to all aspects ofcryptology, and PRIMUS—Problems, Resources, and Is-sues in Mathematics Undergraduate Studies. Currentlyhe serves as the Director of SIMIODE, a 501(c)3 non-profit organization.

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Fish Mixing (Teacher Version)Eric SullivanDept. of Mathematics, Computer Science, and EngineeringCarroll CollegeHelena, MT [email protected]

Elizabeth CarlsonDept. of MathematicsUniversityof NebraskaLincoln, NE [email protected]

Abstract: This activity gives students a chance to build a differential or differenceequation model similar to standard textbook mixing problems.The activity usestangible objects (“fish”) and a student-designed restocking and fishing plan for alake. The mixture is of two species of fish; one is the only species currently in thelake and the other is introduced via restocking. Students generate data via a briefsimulation and then conjecture forms of a differential or difference equationmodel.

Keywords: mixing, simulation, fish, fishing, modelingTags: differential equations, first order, linear, restocking

StatementA lake in northernMontana is dominated byArctic graylingfish (hence-

forth “species A”), but the Dept. of Fish, Wildlife, and Parks is planning aslow introduction of bull trout (“species B”). The lake is popular with fish-ers, who remove both species from the lake regularly.The Dept. of Fish, Wildlife, and Parks has carefully estimated the num-

ber of fish taken by sportfishing eachweek and has decided to keep the fishpopulation as constant as possible by replacing the fish lost by equal num-bers of Arctic grayling and bull trout. For example, if there areN = 50 fishin the lake at the beginning of the week and fishing removesM = 10 fish

The UMAP Journal 37 (4) (2016) 407–416. c©Copyright 2016 by COMAP, Inc. All rights reserved.Permission to make digital or hard copies of part or all of this work for personal or classroom useis granted without fee provided that copies are not made or distributed for profit or commercialadvantage and that copies bear this notice. Abstracting with credit is permitted, but copyrightsfor components of this work owned by others than COMAPmust be honored. To copy otherwise,to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP.


that week, then the lake will be restocked with 5Arctic grayling and 5 bulltrout. Hence, the population of the lake will remain atN = 50 fish, assum-ing that no new fish are hatched. Both fish species swim freely throughoutthe lake, and both are targeted by similar bait used by fishers.

Inyour lake, youwilluseN = andM = .

In summary:

• The week starts withN = fish.

• The fish swim freely around the lake.• M = fish are removed from the lake at randomduringthe week.

• M = fish are restocked at the end of the week.

M/2 = of those fish are Arctic grayling, and

M/2 = of those fish are bull trout.

1. Conjecture:

(a) What do you think will happen to the populations of species A andB over a long period of time?

(b) Is it possible that species A will be eliminated from the lake with therestocking plan? Explain.

2. Simulate:

(a) Use pennies to represent yourN fish anddecidewith your partner(s)which coin face represents which species (e.g., heads up is speciesA). Start your lake with 100% species A.

(b) Decide with your partner(s) how to simulate the swimming of fish,the fishers, and the Dept. of Fish, Wildlife, and Parks’ restockingplan. Simulate roughly 15 weeks of the fish population representingspecies A and B with coins. Be sure to let the fish swim thoroughlyaround the lake and keep track of the proportions of species A andB, in a form like that shown on the next page.

3. Model:

(a) Propose a verbal model for the rate of change of species B in the lake:

rate at which species B changes = +

(b) State explicitly any assumptions that you are using in your verbalmodel.

Fish Mixing 409

Week # Number in population Proportion of populationspecies A species B species A species B

0 1 012

3

4...

......

......

(c) Introducemathematical notation for your proposedmodel andwriteyour verbal model mathematically. Be sure to include any necessarycondition(s):

model:

condition(s):

4. Analyze:

(a) According to your model, what is the long-term effect on the fishpopulation in the lake? Use your model to justify your answer alge-braically and graphically.

(b) Solve your mathematical model (either numerically or analytically)and compare with your data.

(c) (Extension) Suppose now that the Dept. of Fish, Wildlife, and Parksdoes not attempt to keep the population in the lake constant. Thatis, suppose that fishing reduces the population byM1 fish eachweekand theDept. of Fish,Wildlife, and Parks restocksM2 fish eachweek.Fully explore this scenario.

Comments (for Instructors Only)This scenario ismeant to be a lead-in to classicmixing problems via a bi-

ological situation, but the instructor should keep inmind that this is not theonly model that could arise from this scenario. These instructor commentsfor this modeling scenario are organized into the following categories:

• an analytic solution to the typical first-order linear differential equationthat arises from this scenario;

• an analytic solution to the typical first-order linear difference equationthat arises from this scenario;


• practical considerations for running the simulation in the classroom;• suggestions for guiding students through the modeling process; and• a presentation of three sample student data sets.

Analytic Solution to the Differential Equation Model

If b(t) is the proportion of species B in the lake’s fish population at time t(weeks), then the intended (and likely the simplest) mixingmodel could be

b′(t) = −αb(t) +12· M

Nwith b(0) = 0, (1)

where α is a rate parameter ([α] =1/time), [M ] = fish/week, [N ] = fish,and the factor of 1/2 represents the fraction of the restock that is species B.Equation (1) is a linear first-order differential equation that lends itself toseparation of variables, integrating factors, or themethod of undeterminedcoefficients. Using any of these methods, the solution to (1) is

b(t) =M

2αN

(1 − e−αt

). (2)

Note that a(t) (the proportion of species A) can be modeled as a(t) =1 − b(t) since these are the only two species under consideration and thepopulation remains constant. The value of α is the percentage of the pop-ulation that is replaced at any given time.To provide a concrete example, consider the case where N = 50 and

M = 10. In this case, we have α = 10/50 = 0.20; and the proportion ofspecies B populating the lake is given (after some simplification) by

b(t) =12

(1 − e−0.2t

). (3)

The equilibrium solution for (1) occurs when b′(t) = 0. Hence, the equi-librium is b(t) = 1/2. This can also be seen in equation (3) by consideringthat limt→∞ b(t) = 1/2. Furthermore, we can see the equilibriumof b = 0.5graphically by using a slope field (see Figure 1).

Analytic Solution to the Difference Equation Model

If students choose to model this scenario with a difference equation, theanalogous equation to (1) is

bn+1 = bn − αbn +12· M

Nwith b0 = 0, (4)

where bn is the proportion of the lake populated by species B at the end ofweek n. Using the method of undetermined coefficients and conjecturing

Fish Mixing 411

0 5 10 15 200

0.2

0.4

0.6

0.8

1

time (t)

proportionofpop

ulation

b′(t) = −αb(t) + M2N

Figure 1. Slope field for b′ = −αb + M2N

with several solutions shown.

that the analytic solution takes the form bn = C1(1 − α)n + C2, we arriveat the solution

bn =M

2αN

(1 − (1 − α)n

). (5)

IfN = 50 andM = 10, then α = 10/50 = 0.20 and the solution to (4) is

bn =12

(1 − 0.8n) . (6)

Similar to the differential equation model, the difference equation modelpredicts an equilibrium of beq = 1/2.Solutions (3) and (6) are given in Figure 2 to show that the continuous

differential equation and the discrete difference equation give qualitativelysimilar predictions. Notice, though, that the difference equation predictsa larger proportion of population for species B when the curvature of thesolution is greatest.

Practical Considerations for the Simulation

We use coins to model the fish. When mixing the “fish,” students mayinadvertently flip coins. Calling the students’ attention to this fact at leastencourages them to beware of this possible issue. The experiment could bedone equally well with m&m’s, two-sided poker chips, or any two easily-identifiable objects of the same size.Students will come upwithmany creativeways to remove fish from the

lake. The methods of fishing, restocking, and swimming are purposefullyleft vague in the statementof theproblemtogive the students someroomfor


0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

time

proportionofspeciesB

Difference and Differential Equation Solutions

b(t)

a(t)bnan

Figure 2. Solutions to equations (3) and (6) with N = 50 and M = 10(α = 0.20). Note the slight differences in curvature between the discreteand continuous models.

creativity. Anicediscussioncomes fromaskingstudentswhy it is importantto let the fish “swim” and how this affects the choice of fishing technique.Some groups will inevitably realize that if the mixing is thorough enough,then thefishing can takeplace fromanywhere in the lakewithout additionalbias. Allowing the students to discuss this fact leans on their potentialknowledge of randomization and random sampling from statistics. Someexamples of student techniques are:

• Place several different coins in the lake that act as boats. Assume thatthe boats all draw only exactly their allotted number of fish.

• Thoroughly mix the fish and then have the fishers draw the M closestfish from one “shore.” Given thorough mixing this method does notintroduce any bias.

• Drop an object (such as a pen or a needle) into the lake and draw the fishthat are touching the point of the object.

• Use coins or chips with different colors to represent the different species,place them in a jar, mix them thoroughly, and “fish” blindly from the topof the mixture.

Finally, the choices ofM andN determine how long the simulationwilltake during class. If N = 50, then it may be best to use M ≈ 10. Thischoice should show approximate convergence to an equilibriumwithin 15iterations (see Figure 2). We have run the experimentwithM = 4 andwithM = 6, but the convergence is not obvious in a timelymanner and leads tosome confusion with the students.

Fish Mixing 413

Suggestions for Guidance with the Modeling Process

Since this modeling scenario can lead to a first-order linear differentialequation, we expect that it will likely be done with students early in asemester. In our experience, students are more likely to begin modelingthe scenario as a difference equation earlier in the course, since modelingdiscrete change is often more comfortable.If students are having trouble with the verbal and mathematical model,

then we find that directing them to a partial model such as

Bnew − Bold = +

gives the students a firmer starting point, where B is the total number ofspecies B fish in the lake. We intend for students to give a model such as

Bnew − Bold = (B removed via fishing) + (B added via restocking) .

Given that this is truly a discrete scenario (where time ismeasuredweekly),it may be sufficient to allow the students to stickwith the discrete differenceequation; but the formulation can also be used to build the differentialequation by taking dB/dt ≈ Bnew −Bold. The extent of the emphasis on thediscretemodel vs. the continuousmodel is at thediscretionof the instructor.When the studentsbeginbuilding their verbal andmathematicalmodels

theyoftendon’thaveafirmgrasp that thegrowthratemightbeproportionalto the amount in the current population. After giving the students severalminutes to struggle with the model, we allow them to write their ideas andpartial models on the board for everyone to see and critique. Three samplestudent models are as follows:

Bnew − Bold = (some function of B) + M/2; (7a)ΔB = some linear function of B; (7b)dB

dt= inversely proportional to time. (7c)

After further exploration, the students who generated equation (7a) ex-plained that they expected the data to be approximated by the model

Bnew − Bold = −α(B) · B + M/2,

explicitly stating that the constant of proportionality depends on the con-centration of species B. This lends itself to several families of possible mod-els. In particular, one groupdecided to try andfit the data to the differentialequation model

dB

dt= −αBr + C,

where α, r, and C were unknown parameters to be fit with software (e.g.,Excel’s Solver). Unfortunately, this type of model can potentially lead to


an equilibrium value that contradicts the students’ conjectures about theequilibrium.The students who generated equation (7b) observed that when plotting

ΔB vs.B, the resultwas approximately linear for their data. This suggestedto them that the scenario could be modeled by the difference equation

ΔB = −αB + C.

Upon further investigation, they realized that a linear regression of ΔBon B could be used to estimate α and C without regard to the context ofthe problem. The danger of this approach is that the inherently randomnature of the simulation will not yield data that allow an easy tie from theestimated parameters to the context of the problem. In other words, if α isestimated as α ≈ 0.17 instead of α = 0.20 in Figure 2, then students mayfail to make the connection between the value of α and the ratioM/N . Asecond danger to this approach is that the small data sets may not lendthemselves to observing a linear relationship in theΔB vs. B data.The students who generated equation (7c) indicated that their data and

intuition suggested that the change is inversely proportional to time. Un-fortunately, this approach leads to a logarithmic model for B(t) with nosteady-state solution.Students initially seem to prefer working with the total population in-

steadof theproportionof the population. As theproblemevolves, itmaybebeneficial todiscuss the advantages tousing total populationvs. populationproportions. Whenworkingwith proportions, the actual value of the initialpopulation isn’t relevant, since it cancels out of the analytic solution, forboth the difference and differential equations. Furthermore, the parameterα should approximate the growth rate of the species B population; but ifwe are dealing with total populations, the value of αwill be divided by thesize of the population, making the physical meaning of the constant moredifficult to see and understand. Even with these advantages, the tangiblenature ofworkingwith totalpopulations is potentially farmore comfortablefor students and should be encouraged until the model is nearly complete.

Sample Student Data

Three sample student data sets are given in Table 2with correspondingplots in Figures 3–5. These data sets were built usingN = 50 total fish andM = 6 fish replaced at each iteration. Each figure shows solutions to boththe differential equation (1) and difference equation (4)with α = 0.12.The value of α is given by the physical problem, but this is not the

value of the parameter that fits in the least-squares sense. That value canbe found using technology (e.g., Excel’s Solver); but the reader should becautioned that if the students don’t use a physically meaningful value forthe parameter, then the equilibrium point can be affected and the ultimatemeaning of the model may lose its impact.

Fish Mixing 415

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

time (weeks)

Proportionofpop

ulation

an Data Set #1

a(t) Differential Equation Solutionan Difference Equation Solution

bn Data Set #1

b(t) Differential Equation Solutionbn Difference Equation Solution

Figure 3. Data Set #1 with N = 50 and M = 6, along with the analytic solutions to thedifferential and difference equations ((1) and (4)) with α = 0.12.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

time (weeks)

Proportionofpop

ulation

an Data Set #2

a(t) Differential Equation Solutionbn Data Set #2

an Difference Equation Solution



0 2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

time (weeks)

Proportionofpop

ulation

an Data Set #3

a(t) Differential Equation Solutionan Difference Equation Solution

bn Data Set #3




Table 2.

Three sample student data sets withN = 50 total fish andM = 6 fish replaced at each iteration.Each data set shows the proportion of the population inhabited by the given species.

Set #1 Set #2 Set #3Week (n) an bn an bn an bn

0 1 0 1 0 1 01 0.94 0.06 0.96 0.04 0.94 0.062 0.88 0.12 0.92 0.08 0.88 0.123 0.82 0.18 0.88 0.12 0.84 0.164 0.8 0.20 0.84 0.16 0.82 0.185 0.78 0.22 0.80 0.20 0.78 0.226 0.74 0.26 0.76 0.24 0.74 0.267 0.72 0.28 0.74 0.26 0.68 0.328 0.74 0.26 0.72 0.28 0.68 0.329 0.72 0.28 0.70 0.30 0.64 0.3610 0.70 0.30 0.68 0.32 0.66 0.3411 0.68 0.32 0.64 0.3612 0.68 0.32 0.64 0.3613 0.64 0.36 0.64 0.3614 0.62 0.38 0.60 0.4015 0.60 0.40 0.60 0.4016 0.62 0.3817 0.62 0.3818 0.60 0.4019 0.58 0.4220 0.56 0.4421 0.58 0.42

Last,we suggestusingM > 6 in future implementations, sinceFigures3and 4 do not show a clear convergence to an equilibrium point in a shortamount of time.

Final NoteThis modeling scenario was created during the July 2015 SIMIODE De-

velopers Conference at Carroll College in Helena, MT. Eric Sullivan is anAssistant Professor of Mathematics at Carroll, and Elizabeth Carlson was asenior mathematics major during the 2015–16 school year.

Reviews 417

ReviewsCasazza, Peter, Steven G. Krantz, and Randi D. Ruten (eds). 2016. I, Math-

ematician II; ix+208 pp, $39 + shipping (print), $25 (download), freedownload for COMAP members. ISBN 1-933223-99-5, 1-933223-98-7.Bedford, MA: COMAP.

The book is a collection of 21 articles, divided into three sections:

• Who are mathematicians? (8 items),• On becoming a mathematician (6 items), and• Why I became a mathematician (7 items).With one exception, all are published for the first time. This is a secondvolume of such essays, succeeding Casazza et al. [2015].The editors say that the articles in thefirst two sections “will be of partic-

ular interest to budding mathematicians, budding math teachers, buddingmath communicators, and in turn their teachers.” They say that “manyarticles do not fit squarely into either category,” which is the case.The last category is not tight, either. Geraldine Taiani devotes one 18-

line paragraph to her mathematical development, the rest of her four-pagearticle being devoted to explaining how to determine in what years thebirthday of someone born on the 13th of a month will fall on a Friday. Thisis not to say that the material is not clear, interesting, and entertaining—because it is.Much the same is true of the other contributions. The book is best

viewed, I think, as a collection of pieces by authors writing about prettymuch whatever they wanted to within what were probably relaxed guide-lines given by the editors.The results are worth reading no matter what category you put them

in. For example, Christian Wenzel has a fine piece on Immanuel Kant’sviews on mathematics and aesthetics, and their intersection. I don’t knowwhere else it could have appeared, and it’s good that it’s available to awideaudience.RobertStrichartz’s“Bochner’s ‘formulism’ illustrated”contains interest-

ingmaterial on Fourier series andwavelets, andmuch else as well. There’sa proof that there are irrational numbers a and b such that ab is rational that,if it wasn’t new to me, I had forgotten:

TheUMAP Journal 37 (4) (2016) 417–420. c©Copyright 2016 byCOMAP, Inc. All rights reserved.Permission to make digital or hard copies of part or all of this work for personal or classroom useis granted without fee provided that copies are not made or distributed for profit or commercialadvantage and that copies bear this notice. Abstracting with credit is permitted, but copyrightsfor components of this work owned by others than COMAPmust be honored. To copy otherwise,to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP.


Let a = b =√

2.

• If ab =√

2√

2is rational, then we are done.

• If√2√

2is irrational, raise it to the irrational power

√2 to get(√

2√

2)√

2

= (√

2)√

2·√2 = (√

2)2 = 2;

so with a =√

2√

2and b =

√2, we are again done.

We have no idea which example is right, but we know that one is. Neat!Dean Simonton, in “Are pure mathematicians the lyric poets of sci-

ence?,”givesanexplanationof theobservationthat, on theaverage,winnersof Nobel prizes live longer than those who are nominated but not chosen.I had thought that the failures of the non-winners ate on them and drovethem to earlier graves, but not (necessarily) so. Winners are often chosenafter having been nominated several times; so, on the average, they areolder than those who were only nominated and thus, on the average, havea longer future expected lifetime. Reasonable!H.O. Pollak’s “What is applied mathematics?,” though a bit unfocused,

is full of good material. Marco Abate’s “Mathematical memories” is anutterly charming brief memoir. And so it goes throughout the collection:there is something for everyone in its widely-varied contents.

References

Casazza, Peter, Steven G. Krantz, and Randi D. Ruten (eds). 2015. I, Math-ematician. Washington, DC: Mathematical Association of America.

Underwood Dudley, Prof. Emeritus, Depauw University, Greencastle, IN;[email protected] .

Consortium for Mathematics and Its Applications (COMAP) and the Soci-ety for Industrial and Applied Mathematics (SIAM). 2016. GAIMME:Guidelines for Assessment and Instruction in Mathematical Modeling Ed-ucation. Bedford, MA: COMAP, and Philadelphia, PA: SIAM; 216 pp,$20 (P). ISBN978-1-611974-43-0. Freedownloadathttp://www.comap.com/free/GAIMME, http://www.siam.org/reports/gaimme.php.

First, a disclaimer: I was present for some of the initial discussions thatwould eventually lead to the writing of this report. However, I was notinvolved in any aspect of its creation. I must commend the writing teamand editors for taking an initiative that began as free-ranging discussionsand transforming it into a vehicle for a concrete, informative, and usefulguide to the implementationofmodeling throughout theK-16mathematicscurriculum.

Reviews 419

As stated in its introduction, this resource is not a curriculum, set oflesson plans, or even precise prescriptions of how to conduct a modelingcourse. What it does provide is clear guidance onwhat does andwhat doesnot constitute modeling, combinedwith practical advice on howmodelingexperiences can be constructed, facilitated, and assessed. It does take a verystrong position that modeling—if it truly is modeling—is necessarily open-ended and messy, emphasizing the journey rather than the final product.I am reminded of my review in this journal [Bressoud 2014] of Shiflet

and Shiflet’s Introduction to Computational Science: Modeling and Simulationfor the Sciences. Shiflet and Shiflet provide a traditional modeling curricu-lum that explores and explains different types of models. While such acourse supports opportunities for modeling, it is not an example of whatthis report advocates. Instead, this report calls for true modeling experi-ences that require problems where neither the information that is needednor the techniques to be employed are clear. The intention is to preparestudents for the kinds of uses of mathematical knowledge that they mighthope to encounter in the world outside the classroom.This is high-stakes teaching. The payoff is enormouswhen students ex-

perience the thrill of coming to a satisfying answer to a perplexing problemand see how the mathematical tools that they have learned can be appliedto meaningful situations. But it is not without risks. The instructor mustnegotiate trade-offs between accessibility of the problems and depth of themathematics to be employed. And this is teaching that is time-consumingfor both instructor and student.The point of this report is to provide guidance to help with these trade-

offs. It tackles the difficult issues of how to prepare students for modelingexperiences,when andhow to provide support and scaffolding, andhow toassess student efforts. There are beefy chapters on what modeling can andshould look like at different levels: one for grades K-8, a second devotedto modeling in high school, and a third on modeling in the undergraduatecurriculum. There are appendicesof resources andexamples. And there is adelightful chapter inwhich each of thewriters comments onunderstandingthe nature and importance of modeling. Particularly striking is HenryPollak’s clarification of the vision of modeling that permeates this report:

Every application ofmathematics usesmathematics to understand, orto evaluate, or to predict something in the part of the world outsideof mathematics. What distinguishes modeling from other forms ofapplications of mathematics are (1) explicit attention at the beginningof the process of getting from the problemoutside ofmathematics to itsmathematical formulation, and (2) an explicit reconciliation betweenthe mathematics and the real-world situation at the end. Throughoutthemodelingprocess, consideration is given toboth theexternalworldand the mathematics, and the results have to be both mathematicallycorrect and reasonable in a real-world context. [Pollak 2003, 649]


For myself, the only times that I have incorporated real modeling intoany of my courses was in Macalester’s course in quantitative reasoning.The results were mixed, and the mathematical content of the papers thatwere written by students was minimal. This report provides no simplesolutions, but I would have benefited enormously from its advice.

References

Bressoud, David. 2014. Review of Shiflet and Shiflet [2014]. The UMAPJournal of Undergraduate Mathematics and Its Applications 35 (4): 354–356.

Pollak,HenryO. 2003. Ahistory of the teachingofmodeling. InAHistory ofSchoolMathematics, edited byGeorgeM.A. Stanic and JeremyKilpatrick,647–669. Reston, VA: National Council of Teachers of Mathematics.

Shiflet, Angela B., andGeorge B. Shiflet. 2014. Introduction to ComputationalScience: Modeling and Simulation for the Sciences. 2nd ed. Princeton, NJ:Princeton University Press. 219

David M. Bressoud, Department of Mathematics, Statistics, and Computer Sci-ence, Macalester College, 1600 Grand Ave., St. Paul, MN 55105; [email protected] .

Annual Index 421

Annual Index for Vol. 37, 2016Author IndexAcknowledgments. 37(4): 424.

Annual Index. 37(4): 421–423.

Arney, Chris. Cyber modeling: Full of challenges. 37(2): 93–97.

, and Tina Hartley. Results of the 2016 Interdisciplinary Contest in Modeling.37(2): 99–120.

, and Yulia Tyshchuk. Judges’ Commentary: Refugee immigration policies.37(2): 215–225.

Arney, Kristin Rachelle C. DeCoste, Kasie Farlow, andAshwani Vasishth. Judges’ Com-mentary: Water scarcity. 37(2): 179–193.

Author Index. 37(4): 421–422.

Atwood, Bruce, and Sijia Liang. When will a soda can balance? 37(1): 9–18.

Bian, Fuping, Jessica Libertini, and Robert Ulman. Judges’ Commentary: Spread ofnews through the ages. 37(2): 145–154.

Campbell, Michael, and Joseph Hanna. The optimal can: An uncanny approach. 37(1):43–63.

Campbell, Paul J. Acknowledgments. 37(4): 376.

. Coding for all? 37(4): 333–338.

. STEM the tide? 37(1): 1–7.

Carlson, Elizabeth. See Sullivan, Eric.

DeCoste, Rachelle C. See Arney, Kristin.

Deitsch, Jordan. See Hurst, Matthew.

Driscoll, Patrick J. Results of the 2016MathematicalContest inModeling. 37(3): 237–250.

Electronic copies. 37(4): 423.

Elgersma, Michael. See Wagon, Stan.

Errata. 37(4): 423.

Farley, Rosemary, Dina Yagodich, Holly Zullo, and Brian Winkel. Modeling-first ap-proach to teaching differential equations. 37(4): 381–406.

Farlow, Kasie. See Arney, Kristin.

Flamino, James, AlexNorman, andMadisonWyatt. Characterizing information impor-tance and its spread. 37(2): 121–144.

Garfunkel, Solomon A. Publisher’s Editorial: Announcing the Doug Faires Award.37(3): 233–236.

Gordon, SheldonP. Theflavor of amodeling-based college algebra / precalculus course.37(1): 65–82.

Gross, Julia, Clayton Sanford, and Geoffrey Kocks. Projectedwater needs and interven-tion strategies in India. 37(2): 155–178.

Guide for Authors. 37(1): 90–92.

Hanna, Joseph. See Campbell, Michael.


Hattle, Anna, Katherine Shulin Yang, and Sicheng Zeng. Modeling the Syrian refugeecrisis with agents and systems. 37(2): 195–213.

Hurst, Matthew, Jordan Deitsch, and Nathan Yeo. Beneath the surface: Thermal-fluidanalysis of a hot bath. 37(3): 251–276.

Kocks, Geoffrey. See Gross, Julia.Li, Yulei. See Yang, Hui.Liang, Sijia. See Atwood, Bruce.Libertini, Jessica. See Bian, Fuping.Media Contest. 37(3): 209.Norman, Alex. See Flamino, James.Olson, Gary, and Daniel Teague. Teaching modeling and advising a team. 37(2): 227–

232.Oliveras, Katie. See Olwell, David H.Olwell, David H., Carol Overdeep, and Katie Oliveras. Judges’ Commentary: The

Goodgrant Challenge papers. 37(3): 325–331.Overdeep, Carol. See Olwell, David H.Ren, Jingze, Haonan Run, and KaiWang. An educational fundingmechanism based on

data insight. 37(3): 305–324.Reviews. 37(1): 83–89; 37(4): 417–420.Reviews Index. 37(4): 423.Roberts, Catherine A. Judges’ Commentary: The space junk papers. 37(3): 301–304.Run, Haonan. See Ren, Jingze.Sanford, Clayton. See Gross, Julia.Shannon, Kathleen M. Judges’ Commentary: Hot bath problem. 37(3): 277–281.Singmaster, David. The utility of recreational mathematics. 37(4): 339–380.Statement of ownership, management, and circulation. 37(3): after 332.Sullivan, Eric, and Elizabeth Carlson. Fish mixing (teacher version). 37(4): 407–416.Teague, Daniel. See Olson, Gary.Tyshchuk, Yulia. See Arney, Chris.Ulman, Robert. See Bian, Fuping.Vasishth, Ashwani. See Arney, Kristin.Wagon, Stan, and Michael Elgersma. A stable cup of coffee. 37(1): 19–41.Wang, Kai. See Ren, Jingze.Wang, Zhaoqi. See Yang, Hui.Winkel, Brian. See Farley, Rosemary.Wyatt, Madison. See Flamino, James.Yagodich, Dina. See Farley, Rosemary.Yang, Hui, Yulei Li, and Zhaoqi Wang. Will we survive the space junk? 37(3): 283–300.Yang, Katherine Shulin. See Hattle, Anna.Yeo, Nathan. See Hurst, Matthew.Zeng, Sicheng. See Hattle, Anna.Zullo, Holly. See Farley, Rosemary.

Annual Index 423

Reviews Index(Names of authors are in plain type; names of reviewers are in bold.)Aldous, Joan M., and Robin J. Wilson. Graphs and Their Applications: An Introductory

Approach. James M. Cargal. 37(1): 86–89.Benjamin, Arthur, Gary Chartrand, and Ping Zhang. The Fascinating World of Graph

Theory. James M. Cargal. 37(1): 86–89.Casazza, Peter, Steven G. Krantz, and Randi D. Ruten (eds). I, Mathematician II. Un-

derwood Dudley. 37(4): 417–418.Consortium for Mathematics and Its Applications (COMAP) and the Society for In-

dustrial and Applied Mathematics (SIAM). GAIMME: Guidelines for Assessmentand Instruction in Mathematical Modeling Education. David M. Bressoud. 37(4):418–420.

Even, Shimon, andGuyEven (eds.). GraphAlgorithms. JamesM.Cargal. 37(1): 86–89.Jungnickel, Dieter. Graphs, Networks and Algorithms. James M. Cargal. 37(1): 86–89.Fortnow, Lance. The Golden Ticket: P, NP, and the Search for the Impossible. James M.

Cargal. 37(1): 83–85.Nickerson, Raymond S.Mathematical Reasoning: Patterns, Problems, Conjectures, and

Proofs. James M. Cargal. 37(1): 85–86.Reitner, EdnaE., andClaytonMatthewJohnson. Limits of Computation: An Introduction

to the Undecidable and the Intractable. James M. Cargal. 37(1): 83–85.

ErrataVol. 18, No. 2 (1997)“Running a faster race,” by Deborah P. Levinson

p. 113: The last expression in the solution to Exercise 7 should read

F

k2

(kT + e−kT − 1

).

Vol. 36, No. 2 (2015)“Results of the 2015 Interdisciplinary Contest in Modeling, by Chris Arney, and Amy

Krakowka Richmond”p. 110: http://www.comap.com/undergraduate/contests/mcm/contests/2014/

results/2014_ICM_Results.pdf

�−→http://www.comap.com/undergraduate/contests/mcm/contests/2015/

results/

Vol. 37, No. 1 (2016)spine of the issue: Vol 37.1 2015 �−→ Vol 37.1 2016

Electronic CopiesElectronic copies of the Journal and its articles are available to COMAP members.

An author of a contribution to any issue of the The UMAP Journal from Vol. 1 onward,may obtain from the editor an electronic copy in PDF format, for personal or classroomuse or for posting at the author’s Website.


AcknowledgmentsI am very grateful to the associate editors, whose names appear on the masthead.

Not only do they do the bulk of work of evaluating manuscripts, but they also solicitnew works and encourage and guide potential authors.

I am also indebted to the additional individuals listed below who have reviewedmanuscriptsduring thepast year. Their careful evaluationand judgmenthaveenhancedthe quality of the articles andModules that have appeared in the Journal. (Reviewers ofsome papers considered for or published in Vol. 37 of the Journal were acknowledgedalready in Vol. 36, No. 4.)

The Journal offers many opportunities for participation in COMAP’s work.

• To contribute an article, UMAP Module, or Minimodule, or referee manuscripts—please contact me;

• to review books, software, films, or exhibits—please contact the Reviews Editor;• to write a self-contained expository essay about an area of mathematics, howevergrand or small—an era, a concept, a theorem, an idea, a term—please contact the OnJargon Editor;

• to contributean InterdisciplinaryLivelyApplicationsProject (ILAP)Module—pleasecontact the ILAP Editor;

• to encourage and stimulate colleagues to prepare and submit suitable material—please contact me about being nominated to join the Editorial Board.

Contact information for the associate editors in charge of Reviews, On Jargon, andILAPs, as well as my own information, are on the masthead of every issue.

Finally, the associate editors and Iwould like to thank the Journal’s authors, withoutwhom none of this would be possible, and its readers, whose benefit and enjoyment arethe culmination of our enterprise.

Paul J. Campbell, Editor

Chris Arney, U.S. Military AcademyJeffrey A. Graham, Susquehanna CollegePaul A. Isihara, Wheaton College (Illinois)James Walsh, Oberlin College

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