Humanistic Mathematics Network Journal

Issue 7 Article 9

4-1-1992

Mathematics as a Humanistic DisciplineElena Anne MarchisottoCalifornia State University, Northridge

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Recommended CitationMarchisotto, Elena Anne (1992) "Mathematics as a Humanistic Discipline," Humanistic Mathematics Network Journal: Iss. 7, Article 9.Available at: http://scholarship.claremont.edu/hmnj/vol1/iss7/9

MATHEMATICS AS A HUMANISTIC DISCIPLINE

ElenaAnne MarchisottoProfessorofMQ1hem1ltics

California State UniversityJ8Il NordhoffStreet. Nonhridge

It is said that the mono which adorned thedoors of Plato's Academy advised. "Let no oneunversed in mathematics enter here." For theGreeks, the study of mathematics furnished thefinest training field for the mind. It occupied anesteemed place in the curriculum of Plato'sAcademy. No person was considerededucated ifhe did not know mathematics.

Mathematics hasnotretainedsuch adominantposi tion in modem education. However the in­creasing importance of the discipline in thetwentieth century because of the computer andrelated teehnologieshas generatedrenewedinterestin its study. Today's academies universities andcolleges - are requiring the study of mathematicsfor more students than ever before . In 1983. thelargest public university system in the country ­The California stateUniversitySystem -establisheda mathematics course as a graduation requirementfor all students at any of its nineteen campuses.Liberal ans colleges throughout the United Statesare reinstating mathematics requirements for allmajors. Mathematics plays a central role in thecurriculum of most universities and collegesthroughout the world today.

With the exception ofspecializeduniv~rsitieslike MIT and CalTech, Mathematics Departmentsnationwide are considered service departments.offering the majority oftheir courses to students infields other than mathematics. Such depanmentsas Engineering, Computer Science, Business, thePhysical Sciences, and the Social Sciences includea COre of mathematics courses which are essentialto meet the mathematical needs of their majors.

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A conference was held at Williams College inWilliamstown. Maryland, in 1982, to evaluate themathematical needs of students in other disciplinesand the implementation of curricula to meet thoseneeds. Papers were presented by mathematicians andmathematics educators identifying requirements forspecific field s: Isaac Greber for Engineering, WilliamScherlisandMary ShawforComputerScience.StanleyZionts for Business, Jack Lockhead for the PhysicalSciences, and Robert Norman the Social Sciences.

In Engineering, the required mathematics corerepresents topics that are regarded as fundamentalmathematics as well as those topics which students areexpected to know in order to solve engineeringproblems in other courses. They include the under­standing of limits, functions, complex variables, in­tegral equations, and the calculus of variations. Thecentral analytical tool of the engineer is the ability toderive, solve, and understand differential equatio ns.Other mathematics courses which are becoming moreimportan t to engineering majors with the adventof thecomputer are probability and statistics, Boo lean al­gebra, and numerical methods.

The requisites for Computer Science are many.The modes of thought which characterize mathemat­ics are crucial to prospective computer scientists.Probably the most important contribution mathemat­ics makes is teaching these students how to reasonabstractly and problem solve.

Until the 1950' s the role of mathematics inBusiness Programs was minimal . About that time , thediscipline cal led Management Science was incorpo­rated into business schools, and more and more

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quantitative techniques were employed in businessclasses. Mathematical applications to managementproblems are abundant. Mode ls that include linearprogramming and computer simulation are widelyused in industry. Students enrolled in BusinessPrograms need a mathematics background whichincludes algebra, beginning differential and integralcalculus. matrix algebra, linear programming, andsimulation, all with applications in the field.

Mathematics is the language of the PhysicalSciences and has traditionally been at the center ofall programs in those fields. Calculus is essential tothe study of the physical sciences. What students inthose disciplines need most from their mathematicspreparation is a basic understanding ofvariables andfunctions and how to express them in mathemat icallanguage.

Mathematicspreparation for the Social Sciencesis somewhat different from that for prospectivephysical science. mathematics, or engineering ma­jors. In fact, unlike these disciplines, there is nogenerally accepted body of mathematics that everysocial science student is expected to know. Yetsocial scientists, when questioned, win indicate awide range of mathematics that they find useful intheir field s. These include probability, manipulativealgebra, computing, statistics, calculus and differ­ential equations, combinatorics.Hnear algebra, setsand relations.

In each of the above-me nt ioned fields the va­lidity of prerequisite courses in mathematics is un­questioned to provide the appropriate backgroundfor students to pursue subsequent coursework intheir disciplines.The service functionofmathematicsis clearly defined and well justified .

However, teaching mathematics as a servicecourse has some built-in liabilities. The experienceof teaching mathematics to students for whom thesubject isnot theirmajorfield is often less satisfactorythan teaching mathematics majors. Non-mathemat­ics majors approach the subject from a perspective

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that is different from those for whom mathematics istheir major field. Non-majors are more interested inwhat mathematics can do (within the limited focuspeculiar to their discipline) than what mathematicsis. They generally study mathematics only becauseit is useful in preparation for their fields, and thissometimes impairs their enthusiasm for the subject.

Is utility sufficient as a motivation for learn­ing? Some mathematicians (like c.F. Gauss who hasbeen credited with boasting that pure mathematics isuseless) will ask if utility is even a necessary con­ditionfor study.Others (like Philip Davis and ReubenHersh) will question if utility can be measured:

. .. the meanin gs of the expression"mathematical utility" embrace aes thetic,philosophical , historic, psychologica l,commercial, scientific, technological , andmathematical elements. Even this does notinclude all possible meanings... One candistinguish between utility within the fielditself and utility to other fields. Even withthese subdivisions, the notion of utility isexceedingly slippery (1981, p. 80).

Measuring the mathematical utility of a courseis often not a realistic task. However, even if onedetermines that what is presented in a mathematicsclass is useful for some purpose. the learnin g ex­perience is incomplete if utility is the only foc us. Afixed goal of learning a specified syllabus may bepursued, and perhaps attained . But if at no time theissues of how and why the goal is important arediscussed in the classroom , the students get no

.perspective on the mathematics being taught. Theyfocus only on the mechanics of the discipline. Theyhave no real interest in concepts, in learning whatmathematics is in addition to what it does . Forstudents in Engineering, Computer Science, Busi­ness, and the Physical and Social Sciences, prereq­uisite mathematics coursework should serve, illaddition to itsutilitarian goal ofprep aration forw(Jli:in the field , to provide them with some context ofhow the mathematics they find useful has come to

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be, how it relates to disciplines other than theirs, andhow it affects their lives.

Thesegoalsbecomeevenmoreimponantwhenmathematics course requirements are extended tostudents in non-math related fields: the humanities,the arts, etc. As Lynn Anhur Steen indicates in theJanuary, 1986 issue of Focus, the newsletter of theMathematical Association ofAmerica: "Forstudentsintheartsandhumaniti~mathematicsisaninvisible

culture - feared, avoided, and consequentlymisunderstood.n These students see no utility inlearning mathematics. The biggest challenge inteaching liberal ans students is enabling them torecognize mathematics as a creative, human en­deavor, as a "humanistic discipline".

Professors are frequently unprepared to meetsuch a challenge. From experience, they know thatmathematics is an ever-changing process whichpermeates our personal as well as professional lives,but they often don't know how to teach that [Q

students. Although they recognize the intrinsic valueof mathematics, wonhy of study as a major field,necessary in providing the essentials for other dis­ciplines, appropriate as a component of liberaleducation, they have not been trained to conveythese ideas to students in the classroom. Courseswhich emphasize the humanistic aspects of math­ematics are not pan of the traditional curriculum fora Ph.D. in mathematics. Most mathematiciansteaching in colleges and universities have nevereven studied the pedagogy of their discipline. Theyleachmathematics as they do mathematics.They areexcellent mathematicians, but sometimes they losesightof the need to present their subject in its humancontext

This is not true of the mathematics communityin general. Mathematicians like Morris Kline, PaulHalmos, Lynn Anhur Steen, Anneli Lax, AlvinWhite, RJ . Moore, Reuben Hersh and Philip Davisarejust some ofmany who have written extensivelyabout teaching mathematics humanistically. The

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journals regularly include articles with this perspec­rive. For example, the American MathematicalMonthly (April, 1982) featured an article in thesection entitled "The Teaching of Mathematics"edited by Mary and Roben Wardrop which placesmathematics squarely in the center of human de­velopment:

Mathematics has played a central role in thedevelopment of modem civilization. It hasbeen essential not only to the growth ofscience and technology, but has had profoundeffects on philosophy and other forms ofthought as well (Page 270).

Mathematics has a place in history. It is pan ofthe human experience. There are few who will denythat this historical perspective affords all studentsgood reason to learn mathematics. Yet for a varietyofreasonsthisaspectofmathematicsisoftcnignoredin the teaching of the discipline. Students, in hugelecture halls facing blackboards strewn with Greeksymbols, often get no evidence that mathematics isa growing, living, creative process, changing withtime. Presentation of topics to comply with crowdedsyllabi and thick, heavy textbooks cluttered withformulas, theorems and proofs, most times reinforcethe students' belief that mathematics is a disciplineexisting outside of time and human activity. Thisview distorts mathematics and discourages interestin the subject. As Davis and Hersh point out inDescanes Dream:

A detemporalized mathematics cannot tellus what mathematics is, why mathematics istrue, why it is beautiful, how it comes to be,or why anybody should care a fig about it.Butifoneplaoes mathematicssquarely withinhuman time and experience, it becomes awarm and rich source of possible meaningsand actions. Its ultimate mystery is neverdispelled, yet it is exhibited as one of theprimary creations of the human intellect(1986, p. 20 I).

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When students see mathematics as a humanendeavor. they have more than utilitarian reasons tounderstand why they should learn it. At the annualmeetingofthe Mathematical Association of Americain San Antonio. Texas. in January. 1987. Anneli Laxof the Courant Institute so indicated: "Humans re­tain what they learn best if they can put new materialinto a human context. e.g. co nnect it to past expe­rience. future aspiration s, previously acquired mentalstructures."

Mathematics instructors need to help studentsrecognize mathematics as a human process. integra lto life, not fragm ented from it. Pan of this involvesa fostering and understanding of mathematics inrelation to otherdisciplines. According toJean Piaget,all learning is interdisciplinary. This basic fact hasoften been ignored in the teaching of mathematics.For too long students have studied mathematics as aclos ed unchanging subject in isolation from otherdisciplines.

At the San Antonio meeting, Lax spoke aboutthe effect ofa fragmented approach to mathematics:

The isolation of mathematics from the rest oflife causes further fragmentation of mathematicsitself in to arithmetic, algebra, geometry, etc., andthese subdisciplines get cut up into sections ormodulesorskills to bemastered, tested and forgotten.

Attempts to free mathematics from this isola­tion and fix it fmnly as an integral pan of the humanexperience arecurrentIy being pursued in universitiesthroughout the country, The NEXA program at SanFrancisco State University, for example, uses aninterdisciplinary approach to instruction which in­volves faculty from different departments team­teaching mathematics classes. There are concertedeffort s nationwide ro iruegra te real-life applica tionsinto mathematics instruction. There is renewed in­terest in teachin g what George Polya described as" the most characteristically human activity ...problem solving" (Polya, p. ix).

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But perhaps the most significant effort to re­vitalize the teaching of mathematics is being un­dertaken today by a network of mathematicians andmathematics educators devoted to what isdescribedas " the humanistic dimensions of mathematics".Alvin White of Harvey Mudd College hosted aconference in Claremont. California, in March 1986to b)' to develop a concrete definition for thephrase"mathematics as a humanistic discipline" andto discuss appropriate pedagogical goals for teach­ing relative to that definition . Participants in theconference assigned a variety of different, butideologically complementary meanings tothe phrasewhich gave rise to a series ofeducational objectives.In a monograph discussing the conference, Whitewrote:

The conceptof"mathematics asa humanisticdiscipline" is not as well defined as ageometric series or a triangle. but it is moreevocati ve. Man y mathematicians who haveheard the phrase are not troubled by the lackof a succinct definition. but are excited bythe richness of the fruitfulness that theyanticipate...the concept, even if iJJ defined.challenges traditional ways of reaching andlearn ing mathematics at all levels.

The thirty-six conferenceparticipants identifiedtwo pedagogical goals which result from definingmathematics as a humanistic discipline:

I) teaching mathematics humanistically - alteringthe nature of the teaching and learningenvironment, and

2) teaching humanisti c mathematics - reconstructingthe curriculumand the discipline ofmathematicsitself.

The first goal , teaching mathematics human­istically, seeks a student-centered classroom, plac­ing the student in the position of inquirer rather rhanpassive leamer. It encourages the development ofllcommunity of learners with both profe ssor and

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students learning from each other. To teach math­ematics humanistically. the professormust recognizethe emotional climate of the learning environmentand provide appropriate accessforsrudents reengagein participatory learning within that environment

The second goal, teaching humanistic math­ematics, strives"to integrate the humanistic elementsof science with the mathematics" (White, 1976,p.245) in the classroom. Itadvocatescurricula whichrelates mathematical discoveries tothehuman beingswho made them. It encourages the exploration oftherelation of mathematics to other disciplines, and tothe culture in which it is embedded. To teach hu­mani stic mathematics, the professor must allow thestudents to experience the curri culum within thecontext of its past, present. and potential impactupon the world in which they live.

With the dual themes of teaching mathematicshumanistically and teaching humanistic mathematicsin mind , the conferees outl ined founeen desirableObjectives for the classroom. Among them are toprovide students with

an appreciation of the fundamentalinterrelationships of all knowing.

a curriculum which relates mathematics tootherareas, such as science, technology, humanities,and ethical issues, including personal ethics.

an understanding of the human dimensions thatmotivate d iscovery, such as competition,cooperation, the urge for holistic pictures incontrast to pieces.

The thread which unifies all the pedagogicalgoals outl ined at the conference is perhaps bestexpressed by Alvin White in his article BeyondBehavioral Objectives:

. . . our guidelines and teaching objective sshould not have as their major target orfocusthe mastery of fact s and techniques. Rather

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the facts and techniquesshould be the skeletalframework which suppons our objective ofimbuing our students with the spirit ofmathematicsanda senseofexcitememaooutthe historical development and the creativeprocess. The concepts and relationships ofmathematics should be presented as thebuilding blocks of this magnificent edificecreated by the human imagination (White,1975.p.850). .

No mathematician would deny these goals inteaching . Not only are they not controversial, theyare not even new ideas. Early in the twentieth cen­tury, curriculum interest groups called "humanists"led by Charles William Eliot and William TorreyHams recognized mathematics as "an importantpart ofour Western cultural heritage" (Stanic, 1986,p. 192). Mathematicseducators, predominantly at ornear the University of Chicago, advocated breakingdown the barriers between mathematics and otherdisciplines and between the various areas withinmathematics. They promoted the teaching ofmathemaricswithinthecontextofhumanexperience.

Certainly, those who have read Morris Kline,who have studied Piaget, who have focused on theteaching of mathematics as well as the doing ofmathematics. are most likely pursuing these objec­tives in their classrooms. But what is obscuringthose goals for others?

In Mathematics Tomorrow, Peter Hilton out­lines those forces which he believes undermineeffective mathematics education. He includes thequest for instant satisfaction (ignoring the fact thateducation is a slow, gradual, cumulative processwith rewards that are largely long-term); the viewthat the prime purpose of education is to guaranteea high material standard of living (affirming theutilitarian goal of learning); and the mechanicalapproach to teaching elementarymathematics whichemphasizes rote calculationand memorydependence(compounded by the fact that many teachers are notskillful in mathematics, do not enjoy the subject

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themselves. do not feel comfortable with it. andconvey these attitudes to the students). He is par­ticularly critical of standardized tests :

These tests. superimpose a degree of artificial­ityon that which is already present in the curriculum.They force students to answer artificial questionsunder artificial circumstances; they impose severeand artificial time constraints; they encourage thefalse view that mathematics can be separated outinto tiny water-tight compartments; they teach theperverted doctrine that mathematical problems havea single right answer and that all other answers areequally wrong; they fail completely to lake accountof mathematical process. concentrating exclusivelyon the "answer" (1981 , p. 7 9) .

In this quote. Peter Hilton focu ses on howstandardized test s distort the teaching curriculumand promote those very practices which discouragelearning: failure to make mathematics real for thestudent. fragmentation of the discipline. disregardfor mathem atics as a creative process.

But standardized tests are onl y a symptom oftraditi onal ineffective approaches toward teach ingmathematics. Even when the inappropriateness ofstandard pedagogy is exposed. significant changedoes not occur. What is the reason for such inertia inthe sys tem, for what Hilton calls "the remarkablestability of those practices which mili tate againsteffect ive mathematic edu cation" (1981, p. 79 )1According to the title of an Opinion articl e in theJan uary 21. 1987. Chronicle of Higher Edu cation"Mathematics Te achers Are Too Lazy to ChangeTheir Ways ; as a Result Teaching is Stagnant." Inthat art icle, James T. Sandefur states :

Mathematics. like other disciplines, isevolving. but the teaching ofmath is stagnant.Instead of developing new courses, asprofessorsdo inotherdisciplines. weconrinueto teach the same topi cs that were taught tous. . . There has been little ou tside pressurefor a change in the way we teach math. For

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one thing, non-mathematicians rarely knowenough math to be able to make concreteproposals. Alsomanyofthem are intimidatedby mathematicians who tell them that changeis impossible and suggest that such a proposalonly shows their ignorance. As a result,mathematicians have been able to dictatewhat is covered in math courses and to resistany pressure for change. (Sandefur, p. 44).

There is some truth in what Sandeful is saying.More than laziness. however. the culprit is fear. In anarticle in the Edu cation journal, Alvin White dis­cu sses the rea sons why he found it difficult todeviate from conventiona l approaches to teachingCalculus when he was considering experimentingwith a humanistic alternative: " In addition to havingto contend with the skeptic ism of my traditionalistscolleagues, I also had to deal with my own selfdoubts" (White, 1974, p. 132) . It is as Carl Rogersind icates in Freedom to Learn:

J believe that all teachers and ed ucatorsprefer to facil itate . .. (a] meaningful type oflearning. . . Yet in the vast majority of ourschools. at all educational levels, we arelocked into a traditional and conventionalapproach which mak es significant learningimprobable if not impossible.. . It is notbecause of any inner depravity that educatorsfollow such a self defeating system. It isquite literally because they do not know anyfeasible alternative" (Rogers, 1969, p. 5).

Mathematics professors mu st not o nly identifythe problems in traditional mathematics education .but they must prov ide alternatives - practical wayscourses can be taught, different hypotheses uponwhi ch instruction can be built. new goal s for whichteachers and students can strive. They must alsocreate a climate receptive to such alternatives. If thepresent network of humanistic mathematicians andmathematics educators which is assembling can dothi s, they will succeed in ways that their earlytwentieth century co unterpans did not. The eV1·

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dence to date ispromising. Thenew "humanists" areseeking to provide concrete alternatives. They aregenerating interest throughout the country in teachingmathematics humanistically.Theyare attempting toillustratewhyandhowmathematicsshouldhetaughtto all students.

Once mathematics is recognized as panof thehuman experience. the approach to its teaching willseekto demonstrate howmathematics has evolved.how it works in ourpresent environment. and whatare its possibilities for our future. It will focus onwhat is beautiful, creative. and imaginative inmathematics. stimulating students towant to discovermore about the subject, and itsrelationship to otherdisciplines. very few persons choose mathematicsas acareer. Mathematicsc1assescan and mustengagethe non-mathematicians without losing the pro­spective majors. George Polya claims that to bejustified, mathematics courses must conform totwoprinciples:

First, each student should be able to derivesome profit from his study irrespective offuture occupation.Second, such students ashave someaptitudefor mathematics should be attracted toit andnot get disgusted with it by ill-advisedteaching (p.122).

Lynn Anhur Steen describes mathematics asan "enabling force" (Steen, 1986, p.l). Mathematicsprofessors must help students understand math­ematics initshistorical perspective, with itscreativepresentand imaginative future.It is these humanisticaspects of mathematics which will provide the ap­proach to its teaching and the reasons for its study.

REFERENCES

Bringuier, Jean-Claude. (1980). Conversations withJean Piaget, Chicago: TheUniversityofChicagoPress.

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Davis, Philip. (1985). What Do 1Know? A Study ofMathematical Self-Awareness. The CollegeMathematics Journal, Vol. 16, No. 1,22-40

Davis, Philip and Hersh, Reuhen, (1986). DescartesDream, Chicago: Harcourt Brace Jovanovich.

Davis, Philip and Hersh, Reuben, (1981). TheMathematical Experience, Cambridge:Birkhauser.

Hilton, Peter J. (1981). Avoiding Math Avoidance.Mathematics Tomorrow, (Eel. Lynn ArthurSteen), New York: Springer-Verlag.

Lax, Anneli. (1987). Mathematics as a HumanisticDiscipline. (Paper presented at the annualmeeting of the Mathematics Association ofAmerica inSan Antonio. Texason January 23.1987).

Polya, George. ( 1981 ). Mathematical Discovery,New York: John Wiley & Sons.

Rogers, Carl. (1969). Freedom to Learn, Columbus,Ohio: Charles E. Merrill Publishing Co.

Sandefur, James T. ( 1987) . Mathematics TeachersAre Too Lazy to Change Their Ways; as aResult,Teaching is Stagnant. ChronicleofHigherEducation, January 21,1987,44 - 45.

Stanic, George M.A. (1986). The Growing Crisis inMathematics Education in the Early TwentiethCentury. Journal for Research in MathematicsEducation, Vol. 17, No.3, 190 - 205.

Steen, Lynn Arthur, (1986). Restoring Scholarshipto Collegiate Mathematics. Focus, Vol . 6 .No.!,1-7.

Wardrop, Mary and Wardrop, Roben (Eds.) . (1982)Minimal Mathematical Competencies ForCollege Graduates. TheAmerican MathematicalMonthly, Vol. 89, Number 4,266 - 272 .

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White, Alvin.( 1987). Mathematics asaHumanisticDiscipline. (to be published. in theInternationalEncyclopedia of Education, PergamonPublishing Co.

While, Alvin. (1976). A New Paradigm for themathematics classroom. International Journalof Mathematical Education in Science andTechnology, Vol . 7, No .2, 243 - 246.

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White, Alvin M. (1975). Beyond BehavioralObjectives. American Mathematical Monthly,Vol. 82, No.8, Octoher 1985, 849 -851.

While, Alvin M. (1974). Humanistic Mathematics:An Experiment. Education. Vol. 95, No.2, 128- 131.

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