Teaching Mathematics to Non-Mathematics Majors Through Applications

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http://dx.doi.org/10.1080/10511970601182772

PRIMUS, XVIII(5): 411–428, 2008Copyright ! Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970601182772

Teaching Mathematics to Non-MathematicsMajors Through Applications

Sergei Abramovich and Arcadii Z. Grinshpan

Abstract: This article focuses on the important role of applications in teachingmathematics to students with career paths other than mathematics. These includethe fields as diverse as education, engineering, business, and life sciences. Particularattention is given to instructional computing as a means for concept development inmathematics education courses and the role of interdisciplinary projects in teachingupper level calculus. Various problems and models that emphasize the geometric rootsof mathematics, its connections to other disciplines, and relevance to the outsideworld have been presented as meaningful tools to stimulate students’ intellectualcuriosity and develop their self-confidence in the subject matter. Didactical signifi-cance of famous results is discussed.

Keywords: Applications, instructional computing, teacher education, non-mathematicsmajors, interdisciplinary projects, upper level calculus, discrete mathematics.

1. INTRODUCTION

A great number of university students who make their professions other thanmathematics do not have a favorable perception of their mathematicalexperience and are unaware of the importance of mathematics in theirprofessional and everyday life. This is true for soon-to-be professionals inthe fields as diverse as education, engineering, business, and life sciences.Mathematics that is important to life varies across the spectrum of conceptsand topics. One informative example is the classical triangle inequality — thesum of any two sides of a triangle is greater than the third side — and itsnumerous generalizations. The triangle inequality, d(x, y) ! d(x, z) + d(z, y)for any three points x, y, z in a metric space (X, d), and the one for somespecial functions [10, 23] are worth mentioning in this regard. Indeed, thesegeneralizations of the classical inequality illustrate one of the most profound

Address correspondence to Sergei Abramovich, Department of Curriculum &Instruction, State University of New York at Potsdam, 44 Pierrepont AvenuePotsdam, NY 13676-2294, USA. E-mail: [email protected]

Downloaded At: 19:10 19 January 2010

notions of mathematics epistemology — the creation of an increasing numberof general concepts on higher levels of abstraction that preserves the unity ofmathematics [9].

Consider mathematics teacher education. Though elementary pre-tea-chers understand how to build a triangle out of three straws — a classroomactivity recommended by New York State Learning Standards [34] — manyare unaware that from the perspective of measurement, the lengths of thestraws are numerically related to each other, forming an inequality. Inparticular, this knowledge is important while purchasing the right amountof material at the lumberyard or cutting a long wooden stick into three piecesin preparing a triangular frame. Of course, compared to many real-lifeproblems requiring both geometry and enumerative combinatorics, such atriangle problem is quite simple.

It is interesting to note that having elementary pre-teachers discover thetriangle inequality through an appropriate pedagogy of guided discovery notonly brings about an applied flavor into their study of mathematics, but it has areal epistemological significance also. Indeed, it shows the emergence ofmutual relationships among different areas of mathematics. More precisely,it shows one-directional rudiments of the application of arithmetic and combi-natorics to geometry. At this elementary level one can see how counting skillscan be applied to geometry — a branch of mathematics concerned with thestudy of shape and space. However, when counting skills have gone throughmultiple stages of generalizations, one discovers that knowledge about shapeand space accumulated over time can inform generalized abstract analyticskills. In other words, geometry becomes an appropriate tool of applicationto analysis. Therefore, while one-directional rudiments of relationships thatexist among different areas of mathematics can be recognized early in mathe-matics as an intellectual endeavor, a mutual relationship among those areasoccurs at a later stage of the development of mathematical knowledge.

As many mathematical concepts studied at the tertiary level can betraced back to their precursors at a lower level, problematic situationsassociated with higher concepts can be rendered simply to accommodateapplication-oriented teaching of pre-college mathematics. By the same token,slight variation of many problematic situations associated with lower con-cepts can give birth to higher concepts. Appreciation of the dynamic structureof mathematics epistemology can, in particular, become a basis for a pro-ductive collaboration between mathematics and education departments at auniversity. It is through such collaboration that the whole K—16 mathe-matics curriculum could become connected and oriented to applications.This, in turn, has the potential to provide a structure for an integratedmathematics education so that a public appreciation of the subject matterwould become a reality.

This paper suggests that a unified pedagogy for non-mathematicsmajors structured by the interplay of pure and applied ideas [14, 22, 24]

412 Abramovich and Grinshpan


has great potential to improve mathematics preparation across the board.One of the main features of this pedagogy is that instructor/student rela-tionships within appropriately designed coursework originate on theexpert/novice plane and then quickly move to the master/apprenticeplane [3]. Since problematic situations that commonly come to lightwhen teaching mathematics through applications may be new to bothparties, the master/apprentice metaphor is used here in a sense that astudent is given an opportunity to learn mathematical ideas through activeparticipation in cognitive apprenticeship under the guidance of a courseinstructor.

In what follows the authors share experiences in teaching mathematics toeducation, engineering, business, and life sciences students. Particular atten-tion will be given to the utilization of computing as a means of problemsolving and concept development in mathematics education courses [3, 5, 7]and the role of interdisciplinary projects in teaching upper level calculus [22].What unites the authors’ experiences is the way mathematics has been taught— by addressing the needs of society — be it the construction of computa-tional learning environments to help local schools with technology integra-tion or solving mathematical problems at the request of local businesses anduniversity researchers. Finally, the authors argue that teaching mathematicswith attention to big ideas is conducive to creating informed entries intopivotal points of mathematical development that, in some cases, inspires thestudents to use mathematics as an appropriate research tool in their specificprofessional fields.

2. TEACHING MATHEMATICS TO PRE-SERVICE TEACHERS

Many educational researchers have expressed a perspective on acquiringknowledge with a focus on applications as a basis for reform in schoolmathematics instruction. For example, building on Dewey’s [17] notion of‘‘reflective inquiry’’ — a problem-solving method that blurs the distinctionbetween knowing and doing — Hiebert, Carpenter, Fennema, Fuson, Human,Murray, Oliver, and Wearne [25] argued that students at the pre-collegelevel, starting from the elementary grades, should be allowed to problematizemathematics content and in doing so to acquire knowledge through applyingit. This kind of intellectual milieu cannot be realized in the classroom unlesspre-service teachers (hereafter referenced as teachers) are offered some kindof experience in mathematical modeling and applications as part of theirstudies [6, 11, 14, 15, 28]. One way to provide the teachers with thatexperience is through teaching a technology-rich course on the constructionof computational learning environments they can utilize during their practi-cum or student teaching. The following two illustrations reflect such anapproach implemented by the first author.

Teaching Math to Non-Math Majors 413


2.1. Illustration 1: Greatest Common Divisor as a Problem-Solving Tool

Problem-solving activities and concepts presented in this illustration emergedfrom the construction of an electronic manipulative environment for explor-ing percentage problems in the elementary classroom. The environment inquestion utilizes a spreadsheet’s capacity to be used as a manipulative with ahot link to numeric notation. In such environment a student is presented witha rectangular grid that contains less than 100 cells. On this grid a number ofcells are shaded (Figure 1) so that the shaded part constitutes a whole numberpercentage of the entire grid. A task for a student (at the upper elementarylevel) is to evaluate what percent of the grid is shaded and enter thispercentage number into the cell of the spreadsheet designated as an answerbox. If the student’s answer is incorrect, a computer-generated messagesuggests to continue the task on an otherwise hidden, identical adjacentnon-shaded grid. The objective of this new task is to give a student anopportunity to use an incorrect answer as a thinking device and, in doingso, to shade a region on the adjacent grid (the spreadsheet is programmed insuch a way as to enable an interactive evaluation of the shaded part innumeric form) which does correspond to this answer. That is, if the rightanswer is p% and a student’s answer is q% (q 6" p), the task is to shade q% ofthe adjacent grid and, by seeing the self-created representation of q%, toreconsider the original evaluation.

Technically, however, the idea of turning a negative evaluation into agenerator of new meanings cannot be implemented in this environmentwithout restrictions because not every whole number percentage can berepresented by a shaded part of a grid with less than 100 cells. For example,if a student evaluates the shaded part of a 50-cell grid (Figure 1) as 25%, thetask of shading 25% of an adjacent 50-cell grid would be impossible. In sucha way, the tension between a non-authoritative pedagogy and the semioticstructure of the task has led to the following problematic situation: Theenvironment must somehow restrict a student’s possible guessing.

Figure 1. What percent of the grid is shaded?



Because the teachers’ contribution to pedagogy of the environment wasstrongly encouraged, a way around this difficulty was found collaboratively.After a discussion the following idea was accepted: One has to create anenvironment in which a student is offered a choice of selecting answers froma set of percentage numbers including the correct answer (a so-called multi-ple choice problem). The main question to be answered was: Given arectangular grid, what is the total number of choices available on this grid?In attempts to find an answer to this question the following content-boundedproblem emerged.

On an n-cell grid, 0 , n ! 100, k cells are shaded. In how many wayscan one shade k cells on an n-cell grid, 0 , n ! 100, in a whole numberpercentage provided that shape and location of the cells is not important? Inother words, given a positive integer n, in how many ways can one choose apositive integer k ! n such that 100k/n is a positive integer also?

The problem essentially was formulated by the teachers themselvesand the role of the instructor was to create an inquiring environment andprovide help in the transition from the stage of informal inquiry to thatof its formal linguistic representation. Such an environment can bedeveloped by using a spreadsheet in which computational capacity todeal with algebraic expressions depending on two variables makes itpossible to explore the quotient 100k/n numerically; that is, to computeits values for every pair of whole numbers k and n within the specifiedrange.

Figure 2 shows a fragment of a template with percentage numbers(100k/n) generated in the k-range from 1 to 100 and n-range from 1 to 100.

Figure 2. Spreadsheet solution to Problem 1.



To clarify, note that the number 25 in cell J4 means that two (cell B4) is 25%of eight (cell J2); blank cell J5 indicates that three (cell B5) is not a wholenumber percentage of eight (cell J2); the number four in cell J1 means thatthere exist four ways of shading an eight-cell grid in a whole numberpercentage. Numerical evidence of whole number percentages provided byspreadsheet modeling when attended with an attitude of reflective inquirycalled for interpretation of the visual information. This interpretationincluded:

1. creating a problematic situation by asking a meaningful question;2. conjecturing and debating an answer to this question, and3. providing a formal demonstration (proving) of the conjecture.

The following note by a pre-service teacher found in a class assessmentsummarises a computer-enhanced discussion of the above problem andpoints out its solution: ‘‘The most unexpected discovery [we made in theclass] was that the number of cells that result in whole number percentagesis equal to the greatest common divisor between the original number and100.’’ It is the computational environment that mediated the inductivediscovery of the solution by the teachers. In turn, the availability of aplausible conjecture enabled the teachers to grasp the starting point of adeductive argument and accomplish the demonstrative phase of the solutionwith relative ease.

2.2. Illustration 2: Geometrization of Hidden Properties of Fractionsin Context

The above use of greatest common divisor as a problem-solving toolrepresents an example of application of counting skills to geometry. Thepresent illustration will show how, by the same token, geometry can informcounting skills. Furthermore, it will demonstrate how geometrization ofarithmetic can be superior to computing as an inquiry into a real-lifesituation becomes more complex. As was mentioned above, this illustrationreflects the first author’s work with teachers in developing technology-richcurriculum materials at the request of a local school. The Principles andStandards for School Mathematics [33] stress the importance of students inprimary grades discussing the concept of probability and apply it in realisticproblem solving situations. To this end, a group of teachers had come upwith a spreadsheet-based environment for the intuitive evaluation of prob-abilities (chances) of events that have a direct appeal to the curiosity ofyoung children. The environment integrated M&M mathematics (i.e.,hands-on use of the multicoloured candies as mediators of mathematicalthinking) and the graphing facility of spreadsheets. In particular, third-grade



students were expected to use a spreadsheet in exploring the following proble-matic situation [4].

Billy wants to eat red M&Ms only. There are four red plain out of 10total in one bag and 10 red peanut out of 20 total in another bag. Billy ateone red M&M from each bag. How many more red peanut M&Ms does Billyhave to eat in order to give Mary more chances to get a red plain M&M thana red peanut M&M?

For curriculum materials to provide adequate assessment tools, one hasto come with a number of similar situations that can be offered to youngchildren to explore. With this in mind, the teachers were advised that on aformal, decontextualized level the above situation concerns the property of aproper fraction to decrease as both its numerator and denominator decreaseby the same number. In algebraic terms, the situation can be formulated asfollows. Let a, b, c, d be positive integers such that a , b, c , d, a/b , c/d –find the smallest positive integer n such that a#1

b#1 >c#nd#n. The introduction of

parameter n allows for the variation of difficulty of the situation; for exam-ple: Is it possible to find a quadruple (a, b, c, d) for which n = 1? A purecomputational approach to this question resulted in the following two bags:one with six red plain out of 32 total and another with four red peanut out of20 total. This finding enabled the teachers to see that although originally, thechances may be in favor of red peanut M&M, Billy can reverse the chancesafter eating just one red candy from each bag.

Consequently, this led the teachers to inquire: Is it possible to find twobags for which the chances could be reversed two, three, four, etc., times?It turns out that this inquiry is not easy to answer, even computationally.However, recourse to geometry can immediately clarify the situation and,in addition, provide an algorithm for generating different types of bags.Indeed, the chances for each bag can be associated with the functionsf(x) = (a – x)/(b – x) and g(x) = (c – x)/(d – x) which decrease mono-tonously as x grows large. Thus, generally speaking, in order for Billy toreverse the chances in favor of red plain M&M after eating n candies fromeach bag (n , a, n , c), the graphs of f(x) and g(x) should intersect on theinterval (n-1, n). This leads to the inequality n# 1 < ad#bc

a$d#b#c < n fromwhich the appropriate quadruples of integers can be chosen. Finally,because the two graphs can intersect at one point only, the chances cannotbe reversed more than one time.

Note that this investigation, stemming from an elementary situation, isappropriate to be discussed within a mathematics course for secondary pre-teachers. In doing so, many concepts and techniques important for the studyof calculus can be introduced in context, among them: fractional-linearfunctions, hyperbolas, asymptotes, inverse variation, monotonic behaviour,inequality solving, graphing, and generalizing. Through creating links amongdifferent parts of the mathematics curriculum, teachers can see K–12 curri-culum as an integrated whole.



3. A PROJECT OPTION IN CALCULUS COURSES

A focus on applications in mathematics teacher education gives futureteachers one very important experience (exemplifying mathematical ideasin the ways which are usable) to be imparted to their own students. Thenone can recognize at the pre-college level that mathematics knowledgestems from the need to resolve real-life situations of different degree ofcomplexity. The Curriculum Principle put forth by the National Council ofTeachers of Mathematics [33] includes the notion that all students at thislevel should be offered experiences ‘‘to see that mathematics has powerfuluses in modeling and predicting real-world phenomena’’ (p. 16). Thisemphasis on applications goes beyond the pre-college level. Indeed, mathe-matics has been greatly developing and penetrating all the spheres of life,making collegiate mathematics education a necessary yet controversialelement of modern culture.

Mathematics language is abstract. Thus traditionally, university mathe-matics for non-mathematics majors is taught by distancing it from realitywith no connections to students’ professional interests. In this setting, quite anumber of soon-to-be-professionals do not see the importance of mathe-matics in their prospective fields. Furthermore, abstractness in teachingoften results in the problem of communication. As Maull and Berry [31]noted in connection with teaching engineering mathematics, there may bediscordance between terminology and ideas used by a lecturer-mathematicianand their interpretation by the students. As a result of being too theoretical,mathematics education at the university level becomes ineffective: non-mathematics majors study the subject matter ‘‘because they have to.’’ Analternative approach to mathematics education is based on the well-knownnotion of ‘‘learning by doing’’ [8, 17, 24, 35], which makes possible ameaningful interplay of pure and applied ideas. This approach has greatpotential to bring about experiential learning into calculus — the basic coursein collegiate mathematics curriculum.

Specific alternative pedagogies of teaching undergraduate mathe-matics to students majoring in science, engineering and business havebeen suggested by a number of authors. These include teaching calculusthrough assigning research projects to students based on the extension ofcurriculum materials found in traditional textbooks [12] or teachingmathematics through modeling difficult problems that can not be solvedusing traditional procedures and strategies [19, 32]. In the context oftechnology-enabled pedagogy one can refer to the ideas dealing withteaching computer-based service mathematics through the design ofcomputational environments aimed at communicating mathematicalideas [36] or with downplaying traditional mathematical formalism byemphasising numerical methods as a bridge between mathematics andengineering [26].



Yet another alternative is to teach undergraduate mathematics by enga-ging students in activities that address mathematical problems emerging invarious domains outside of mathematics [27, 37]. Through illustrations ofthis section, long-term efforts to incorporate this kind of alternative pedagogyinto the teaching of undergraduate mathematics by introducing interdisci-plinary projects (stemming from business/industry and university research)into calculus courses for engineering, business, and life sciences students atthe University of South Florida (USF) will be shared. While some veryspecific problems and models that are far beyond those discussed in thecontext of teacher education will be highlighted, all illustrations in thispaper are permeated by the master/apprentice metaphor of mathematicslearning.

It has been more than two decades since the Committee on theUndergraduate Program in Mathematics recommended that ‘‘[s]tudentsshould have an opportunity to undertake real world mathematical modelingprojects, either as term projects in an operation research course as indepen-dent study, or as an internship in industry’’ [13, p. 13]. Its most recent reportrecommends that ‘‘programs for students preparing to enter the nonacademicworkforce should include . . . project[s] involving contemporary applicationsof mathematics’’ [14, p. 96] and suggests that ‘‘exposing students in mathe-matics courses to discipline-specific contexts for various mathematical topicshas a positive effect on their ability to transfer knowledge between courses’’[14, p. 36]. Thus, in accord with these recommendations, introducing inter-disciplinary and industrial elements into the teaching of calculus throughpromoting the idea of the project option for non-mathematics majors is ameaningful pedagogical alternative to traditional instruction. This makes itpossible for a fair part of mathematics education to address the needs ofuniversity (non-mathematical) research and community businesses. In doingso, another element of collaboration can be brought to bear: namely, colla-boration between mathematicians and other researchers oriented towardscontemporary science/business problems and turned into joint supervisionof non-mathematics majors. More about such collaboration and projectssupervised by the second author at USF can be found in [22, 30].

The project option is a substitute for the final exam in upper levelcalculus. It appeals to those students who are already involved in businessor research: company employees, research assistants, honor students, andinterns. Also, some students’ performance as an apprentice to a mathe-matics professor is much more superior to their self-struggle with a time-limited test. In fact, two students may work on a project supervised by amathematics professor and company expert/university researcher. Thus, asan educational approach, the project option emphasizes the value ofwell-structured productivity of social learning and downplays the role ofcontinual testing as a traditional vehicle of knowledge assessment inmathematics [29].



3.1. Illustration 3: Useful Integrals — from ‘‘bar myth’’ to Wetland Area

An interesting project based on the application of calculus to the restaurantbusiness was carried out by an engineering freshman who also worked asbartender. There has been a tradition of pouring 2 oz. of liquor into a standardsnifter glass by laying the glass on its side and filling it with liquor until liquidreaches the rim of the glass (Figure 3). This trick that bartenders called ‘‘barmyth,’’ allowed them to avoid using jiggers. The student challenged this traditionwith the help of mathematics. He first measured the height (level) h of thetraditionally filled glass, then bent a piece of copper wire along the glass’ surfaceand traced the wire onto graph paper. In such a way, an experimental curve wascreated. On this curve, several points were selected and plugged into a curve-fitting program to obtain the equation y = f(x) of the glass’ profile curve.

The final step was to find volume V of the filled glass by using the

formula V " !Rh

o

f 2%x&dx, where h was determined by averaging five similar

measurements. It turned out that the traditional way of pouring ‘‘2 oz. ofliquor’’ was, in fact, 0.375 oz short (the cost for this amount of liquor wasabout $1). As a result of this project, the manager of the restaurant involvedhas made using jiggers on snifter glasses mandatory and described thestudent’s finding as ‘‘an invaluable gift’’ to the restaurant business.

Another example related to the use of integral calculus in applications isthe project dealing with finding area of a restored wetland proposed by anenvironmental restoration manager. Mathematically, the project concerns the

analysis and calculation of area type integralsR

C

r2d", where C is a Wetland’s

boundary, r and " are suitable polar coordinates. A student contributor, major-ing in environmental science, effectively dealt with the calculation of suchcontour integrals as an alternative to the traditional use of simple polygons(e.g., parallelograms, rectangles, triangles) in approximating areas. Beingextremely useful in applications, contour integrals are not usually includedinto the calculus curricula for non-mathematics majors. At the same time, these

Figure 3. Challenging ‘‘bar myth.’’



integrals can be introduced as a natural extension of the case connected to thearea of disk when the contour C is a circle centered at the origin.

3.2. Illustration 4: Special Functions Connection

Some real-life projects carried out by engineering students are deeply rooted intheoretical mathematics. The following project, dealing with the problem oflighting roadways, comes near the theory of special functions. A constructioncompany that offered this project used to employ fixture characteristics and lamplumens in combination with construction site plans. This approach of defininglight levels gives estimates at best, which are sufficient for normal applicationsthough. However, some businesses, like hospitals and banks where night shiftsare customary, demand exact lighting levels from the company, especially inspecific areas of both horizontal and vertical surfaces. For such levels to be found,one has to analyze the illumination of a patch of a roadway lit by an ‘‘infinitely’’long set of lights spaced at regular intervals and having equal heights.

An engineering senior who demonstrated excellent engineering intuition inresolving a non-trivial mathematical situation carried out the project. Essentially,the situation was reduced to the analysis of the following infinite series

X1

n"#1

h3

%%n# a&g&2 $ h2h i3=2

where (see Figure 4) h is the light height, g is the distance between lights (thenatural restriction is 0, h, g), and a determines the piece of roadway (mod g)currently under consideration (0 ! a, 1, a = 0 at the current light and a = 1 at

Figure 4. Defining the light level of a roadway.



the next light). The series represents the total illumination of the consideredpatch of roadway. The illumination is high if a = 0 and it is low if a = 1/2.

From a mathematical perspective, problems involving representationsthrough this kind of series are challenging even for experts in the theory ofspecial functions. To this end, note that one can compare the above serieswith the values of the generalized z-function (see, e.g., [10]); namely, z(3,a)and z(3,-a). Nonetheless, the student managed to avoid mathematical diffi-culties associated with the complexity of the series by using a computer fornumerical analysis of its partial sums (with g = 150 feet and h = 40 feet) fordifferent values of parameter a.

3.3. Illustration 5: Business Analysis

Of particular interest are real life problems aimed at market, investment,sales and profit analysis. These concern business students, though not onlythem. Some material on probability and statistics included in the USFengineering and life sciences calculus curricula allows engineering and lifesciences students to analyse both sides of the medal: professional and finan-cial. It results in combining engineering and scientific ideas with financialissues in the students’ projects. A project, proposed by a medical companydirector and conducted by a student with diversified interests in business,engineering, medicine, and law, illustrates the point.

A parent company of a number of medical clinics distributed throughoutthe United States is engaged in the expansion of its clinics both nationallyand internationally. The analysis of potential markets has been necessary as aguide to this expansion. This study explores a variety of characteristics ofindividuals suffering from a chronic disease, which is a fast growing healthproblem in the United States. Mathematical analysis of multiple samplepopulations, which involves density function modeling, is developed as afactor in the determination of market location. Some normally distributedrandom variables and the Gaussian function play a key role in approximatingdensity functions. Also, the study takes a look at the company data collectionand operating procedures. Some recommendations are made regarding pos-sible changes in collecting, structuring, and analyzing the demographic data.

4. ON THE INTERPLAY BETWEEN PROJECT-BASED CALCULUSCURRICULUM AND TEACHER EDUCATION

The above-mentioned real-life projects could be rendered simply to embarkon numerous mathematical activities for teachers (and their students alike).For example, in the context of integration, related topics may include findingthe area of a circle (consider a circular wetland) and volumes of basic solids



(consider a cylindrical glass). Also, such basic mathematical activities assorting and counting can be interpreted as the rudiments of business analysisdealing with the determination of market location based on exploring variouscharacteristics of a population sample.

Furthermore, discrete concepts are extremely effective in solving real-world problems. Difference equations are worth noting in this regard.Examples that are appropriate for elementary and middle school teachersinclude the sum and product of n consecutive counting numbers starting fromone as counting tools serving diverse contexts [1]. Fibonacci numbers can beintroduced to secondary mathematics teachers through the ‘‘knowing isdoing’’ [8, p. 189] pedagogy as solutions to a second order differenceequation describing the growth of the population of rabbits breeding inideal circumstances. More complicated phenomena, in particular those aris-ing in biology and medicine [18], are appropriate for inclusion in upper levelcalculus courses with a project-based component. A project associated withthe application of difference equations to Alzheimer’s research at USF isdescribed in [22].

Connecting real-world research projects for engineering, business, andlife sciences students to mathematical activities for prospective teachersallows for collaboration between mathematics and education faculty. Thiscollaboration enables the development of integrated mathematics curriculafor different majors. What is more important, such connectivity of applica-tion-oriented mathematical topics has the potential to introduce precursors ofhigher concepts at a lower level.

5. FAMOUS RESULTS AS USEFUL DIDACTICAL TOOLS

Real-life applications of mathematics provide a great deal of stimulation forvarious kinds of research in the subject matter field, involving professionalmathematicians and students of different majors alike. This is not to say thatapplied mathematics is the only meaningful source of the development ofmathematical thought. Indeed, there are many problems within mathematicsitself that can be used to motivate and keep motivating those who seek togain full appreciation of mathematics as a fundamental science. Some ofthese problems can be recommended for inclusion in the mathematics curri-culum for non-mathematics majors. The authors’ experience indicates thatfamous theorems and conjectures with origins in both pure and appliedmathematics have the potential to trigger the imagination and thought processof those whose minds are open to challenge, and thus can be utilizedappropriately as useful didactical tools.

For example, the statements and historical details of such problems asFermat’s Last Theorem proved by Andrew Wiles [40] and the Bieberbachconjecture proved by Louis de Branges [16] (see also [20, 39]) may be



included into some basic mathematics courses for non-mathematics majors.Proofs of these famous results not only require more than elementary means;they are also quite complex. However, as Stewart noted, ‘‘the fact that proofis important for the professional mathematician does not imply that theteaching of mathematics to a given audience must be limited to ideaswhose proofs are accessible to that audience’’ [38, p. 187].

Fermat’s Last Theorem states that the equation xn + yn = zn has no non-zero integer solutions for x, y and z when n . 2. In particular, this theoremcan be introduced to future teachers of mathematics as a way of answeringthe question: Is it possible to extend the interpretation of Pythagorean triplesas partitioning a square into the sum of two squares to include similarrepresentations for higher powers? As detailed elsewhere [2], the use of aspreadsheet with secondary pre-teachers makes possible a way of visualizingFermat’s Last Theorem by ‘‘modeling’’ non-existing solutions to the aboveDiophantine equation for n . 2 in much the same way as for n = 2. Likewise,it is quite possible that with the help of technology or through other means anatural bridge between the statement of Fermat’s Last Theorem and somegeometric properties of modular elliptic curves in Wiles’ proof will beaccessible to future mathematics students.

The Bieberbach conjecture states that for each n = 2, 3, . . . and eachanalytic function f(z) = z + a2z

2 + a3z3 + . . . that is one-to-one in the unit disk

D = {z: j z j, 1}, the inequality j an j! n holds. All extremal cases are coveredby the rotations #K%#z&, jlj = 1, of the Koebe function K(z) = z/(1-z)2 =z + 2z2 + 3z3 + ' ' ' .

This result with its stunning record alone (see, e.g., [21]) can inspirestudents’ interest in learning such important mathematical concepts as one-to-one functions, power series, convergence, and Taylor coefficients which,in particular, are appropriate to be discussed with engineering majors. In anengineering calculus class, one such discussion resulted in an interestingobservation: a rotation of the Koebe function is the extremal function forsome characteristics in the maximum power transfer theory. This observationled to a project proposed by a professor of electrical engineering at USF andwas carried out by an engineering major under the supervision of the secondauthor. The geometric roots of the Bieberbach conjecture are worthmentioning here as well. For example, its proof for n = 2 is based onpresenting a plane set area as a contour integral and thus it is accessible tonon-mathematics majors enrolled in an upper level calculus course.

6. CONCLUSION

It appears that the effectiveness of mathematics education across grades canbe greatly enhanced by connecting students’ learning to the needs of thesociety. Second, changes in K–16 mathematics curricula should be made



from top to bottom enabling certain elements of higher concepts to bemeaningfully introduced at a lower level. Third, mathematics (both contentand methods) courses for teachers should develop an appreciation of mathe-matics as an exploratory science and provide grade-appropriate experienceswith significant mathematical concepts that penetrate the subject matter frombottom to top. Fourth, some famous results with origins in both pure andapplied mathematics can be utilized appropriately as useful didactical tools.Finally, calculus as the basic collegiate mathematics course should haveflexible curricula that permit the inclusion of topics with the true flavor ofcurrent mathematical applications.

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BIOGRAPHICAL SKETCHES

Sergei Abramovich is a Professor of mathematics education in the School ofEducation and Professional Studies, State University of New York atPotsdam. He was born in St. Petersburg (then Leningrad), Russia, did allhis studies at the University there and earned a Ph.D. in differential equationsand mathematical physics. After a number of years of teaching and doingresearch in pure and applied mathematics, he became involved in the math-ematical education of gifted students at the secondary level. In the early1990s, he moved to the United States to become a visiting professor at theUniversity of Georgia. He also had a visiting appointment at the Universityof Illinois at Chicago prior to coming to Potsdam. His current researchinterests span across K–16 mathematics education with particular emphasison the appropriate use of technology in the development of concepts, bothelementary and advanced.



Arcadii Z. Grinshpan (Ph.D., 1973) was born and educated in St. Petersburg(Leningrad), Russia. Since his student years at the St. Petersburg (Leningrad)University he has been active in complex analysis, inequalities, mathematicalmodeling, and numerical analysis. His academic and industrial experienceenabled him to look for an answer to the problems facing collegiate mathe-matics education for non-mathematics majors. He is the Director of IndustrialMathematics and Head Coordinator of the Mathematics Umbrella Group atthe University of South Florida.



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Teaching Mathematics to Non-Mathematics Majors Through Applications