Mathematics in Venezuela in the sixties by P. R Montgomery (1936-2013)


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La visión del profesor Montgomery de la Universidad de Kansas que estuvo como profesor en la Universidad de Oriente (UDO) a fines de los años 1960. Point of view of professor Montgomery from University of Kansas about the state of mathematics in Venezuela in the late 1960's. Montgomery was a visiting professor at University of Oriente (UDO).

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Mathematics in Venezuela in the sixties by P. R Montgomery (1936-2013)

  1. 1. University Mathematics in Venezuela Author(s): P. R. Montgomery Source: The American Mathematical Monthly, Vol. 75, No. 6 (Jun. - Jul., 1968), pp. 664-667 Published by: Mathematical Association of America Stable URL: . Accessed: 27/07/2013 09:44 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact . Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. This content downloaded from on Sat, 27 Jul 2013 09:44:11 AM All use subject to JSTOR Terms and Conditions
  2. 2. 664 MATHEMATICAL EDUCATION NOTES [June-July 7. David Page,Do somethingaboutestimationlUpdatingMathematics,2 (April1960). 8. M. L. Stein,S. M. Ulam and M. B. Wells,A visualdisplayofsomepropertiesofthe distributionofprimes,thisMONTHLY, 71 (1964)515-520. 9. ScientificAmerican,March1964,Cover(andinthatissue,MartinGardner,'Mathematical Games,"120-128). 10. StephenI. Brown,'Prime','Elementary,'and'Fundamental'Comparisons,ThePentagon, 26 (1967)95-105. UNIVERSITY MATHEMATICS IN VENEZUELA P. R. MONTGOMERY, UniversityofKansas 1. Introduction.The writerhas just returnedfroma two-yearstay at the UniversityofOriente(UDO) in Cumana, Venezuela, whichis a new university dedicated to educational experimentationand innovation. My stay was par- tiallysupportedby the Ford Foundation, in a cooperativescience project be- tweenUDO and the Universityof Kansas. In addition to mynormalduties of teachingand servingas advisorto the mathematicsdepartmentofUDO, I bad opportunitiesto make several shortvisitsto otheracademic institutionsin the country.These visitsweregenerallyof an "official"natureand hence may not have been as fruitfulas desired.This report,like the visits,is rathershortand cursory,but, like the visits,I hope it accuratelyreflectsthe generalpatternof teachingmathematicsat thepost-highschoollevelinVenezuela. There are fourdifferenttypes of such institutionsin Venezuela. The most common are the autonomous universitiessuch as Central (UC), Zulia, Los Andes,and Carabobo. At thesame level are the two institutions,UDO and the ExperimentalCenterforHigher Education, whichare operated directlyunder the M4inistryofEducation (ME) througha rectorand universitycouncil. (ME recentlyannounced plans to startanotheruniversitysimilarto UDO in Cara- cas.) The ME also operates two normalcollegesor Pedagogical Institutes(IP and IPE) forthe purposeoftraininghighschool teachers.These Institutesare not consideredequivalent to universitiesand the degreestheyofferare not ac- cepted as universitydegrees.Finally,thereare severalsmall privateuniversities ofwhichAndresBello Catholic Universityis themostimportant.These private universitiesdo not play a verylargerolein theeducational system.The schools vary in size fromtens of thousands of students at UC to a few hundredat IPE. At all ofthese places, mostof the mathematicstauglhtis of the algebra- calculus variety. 2. Students. Most of the universitystudents come fromfive-yearschools or liceos. However, there are two other types of secondary schools: normal schools, which prepare primaryteachers,and technical schools, which teach such diverse subjects as commercialfishing,agronomy,and industrial arts. These studentsare not accepted into any ofthe autonomousschools and may onlyenterUDO or the IP's. The mathematicalcontentofthe secondary(and primary)programsis very old-fashionedand does not contain any of the newerapproaches. It is tightly This content downloaded from on Sat, 27 Jul 2013 09:44:11 AM All use subject to JSTOR Terms and Conditions
  3. 3. 1968] MATHEMATICAL EDUCATION NOTES 665 controlledby ME and has not had any revisionsformanyyears,althoughthere is presentlyat work a committeeof the Ministrystudyingpossible changes. Algebra and Geometryare combined throughoutthe curriculumratherthan beingdivided intoseparate yearsas is thecase in theU.S. In thepast, teachers have tended to omit or treatvery lightlycertainsubjects,geometrybeingthe principalvictim. Because of the rigidityof the curriculum,then, the student would never be exposed to this material.No morethan thirtyper cent ofthe people teachingmathematicsin the liceoshave had any advanced (i.e., college) trainingin mathematics;the remainderbeinghighschool graduatesonly. Because of the nature of the highschool testingprogram,the studentstry to memorizeeverything,even entirepages. In generaltheyhave extremedif- ficultyapplying known material to new problems. Their studying is made furtherdifficultby various other factors,such as classes for25-30 hours per week; studyingusually done in theircrowdedpensionesor understreetlights; a sizable numberof studentslack textbooks.As would be expected, the failure rate is quite highwiththelargestcontributingfactorbeingpoorbackground. 3. Faculty. In general,the facultyoftheseinstitutionsis composedof part- time instructorswho teach forvaryingnumbersof hoursper week, and have sole controloverthecourses.Two exceptionsto this patternare UDO and IPE, who employ, almost exclusively,full-timeprofessors.This part-timefaculty consistsofA.B.-level mathematicsmajors or engineers.However, bothUC and UDO have a high percentageof professorswith graduate trainingin mathe- matics.The facultieshave a quite highratio ofnon-Venezuelansand thisratio is higheramong those professorswith advanced training.Rather substantial effortsare being made to overcome this dependence on foreignprofessorsby scholarshipprogramswhich have been initiatedin several of the universities. Most ofthesefellowshipsare forstudyin theUnited States, but otherstudents are sentto France, Germany,Italy and Puerto Rico. The average teachingload is difficultto compute with the part-timehelp, but at UDO, forexample,mostprofessorsteach about twelvehoursper week. (This representstwo courses,sincemostcoursesmeetsixhoursweekly.) There is very little professional interchange between the universities (sometimesnot even withinthe university).The only scientificsociety with open membershipis the Venezuelan Society forthe Advancementof Science, which has a small and not very active mathematicssection. At least two of themathematicsdepartmentspublishquarterliesalong thelinesofMathematics Magazine whichare directedat highschool studentsand teachers. 4. Classes. The size oftheclasses varies fromthelargesections(70-100) of theintroductorycoursesto quitesmall sections(2-10) fortheadvanced mathe- matics majors. As stated above, most meet six hoursper week. This time in- cludes some hours of "practice" which correspondroughlyto laboratory in chemistryor physics.Very littlehomeworkis assigned and littlestudyingis done outsideofclass exceptforpreparationforexams.The practiceclasses are, This content downloaded from on Sat, 27 Jul 2013 09:44:11 AM All use subject to JSTOR Terms and Conditions
  4. 4. 666 MATHEMATICAL EDUCATION NOTES [June-July in theory,intendedto helpovercomethislack ofoutsidework.There are widely differinguses of thesehours.Some professorsuse themstrictlyfordrill,others incorporatethe drill in the lectures,spreading it over several class periods, and otherprofessorsignorethe drillsaltogether. Discipline in classes and exams is far differentfromwhat is normallyen- counteredin the U.S., the studentsbeingless attentiveand less orderlyin the classroom.The full-timeprofessorsare much stricter,however,than the part- timefacultyand thosewithhighertrainingtend to be even moreso. Texts are not as widespread as in the United States. Many courses have an officialtext,but not all studentsbuy it. Many professorsalso use and sell theirown notes. Good up-to-datebooks in Spanish are rareand expensive. 5. Programs.There are two differenttypesof mathematicsdegreesoffered. The most popular is a four-yearprogramforprospectivehighschool teachers and is the only programavailable at the IP's. Several of the universitiesalso offerthisprogram.The other,whichis consideredto be a mathematicsmajor, is a five-yearprogramand is offeredat mostuniversities.The five-yearprogram is not quite the equivalent of a Master's degree althougha thesisis usuallya partialrequirement. Both programsare considerablyheavier than the correspondingprograms in theU.S. This is caused, in part,by thelack ofgraduateschoolsin Venezuela and thelack ofopportunityforfurthertrainingelsewhere.At UDO, forexample, studentstake more than one hundredcreditsin mathematicsin the five-year program.At the same time,muchofthe learningand teachingis superficial. Usually the studentsdo not have a choice of courses. Instead, the depart- ment offersa pensum,which is a listingof the courses a student must take duringeach semesterat the university.He thenpasses the semesteras a unit, not the individual courses,although if a previous course is a prerequisite,it musthave been passed. (There are severalways ofpassing a failedcourse,not all of which involve retakingit.) Some of the pensumsare quite modernand include such subjects as differentialgeometryand algebraic topology. Others are veryoutdated and includesuchcoursesas sphericaltrigonometryand theory of equations. Most of the courses a mathematicsstudent takes are mathe- matics; very little work is taken outside the department.Minors, or double majors, are unknown.UDO has adopted a creditsystemsimilarto that used in the U.S. and although still influencedby the pensums,it promisesto give moreflexibilityand adaptability to both the departmentand the students. In a given semester,under these pensums,a student takes an average of 18-20 credits.The actual numberof class hoursis somewhathigherdue to the practice hours.This representsabout six courses and may include one or two courses outside mathematics,in, forexample, English, Humanities, or Social Science. 6. Libraries. With the exceptionsof the librariesat UC and UDO, these facilitiesin Venezuela are almostnon-existent.About all that is available to a This content downloaded from on Sat, 27 Jul 2013 09:44:11 AM All use subject to JSTOR Terms and Conditions
  5. 5. 19681 MATHEMATICAL EDUCATION NOTES 667 professoror a studentare those books he buys himself,(professorseven have to purchase their personal texts) and these are only available throughthe Caracas bookstores.UC's libraryis moreestablishedand has a bettercollection ofjournals. UDO's libraryhas a widerchoice oI books at all levels,due chiefly to thefactthatthemathematicsdepartmentreceived$10,000 in 1966 fromthe InternationalDevelopment Bank to purchase books and journals. 7. Conclusion. Judged overall, mathematicsis not very popular in Vene- zuela. More attentionis graduallybeingpaid to it,and severaloftheuniversities are tryingto build betterdepartmentsthat are beginningto shed the service role whichhas been theirprimaryfunction.Still thereare littleprospectsfor a "market"outsideoftheuniversities. The mostimmediateneed at the presentis qualifiedteachersat all levels- primary,secondary,and university.For the universities,the scarcityof texts is of crucial importance,and in the secondaryschools a curriculumrevisionis longoverdue.As ineverything,theproblemsofthedepartmentsand universities have a special character,sometimesreferredto as "Latin," whichmakes them appear almost unresolvable.The Universityof Oriente,with its experimental natureand approach,could be a strongmodernizingfactorifits influencecould be spread to theotheruniversities. TEACHING GROUP THEORY TO COLLEGE FRESHMEN G. E. CARUSO,Lea College on Lake Chapeau, Minnesota An experimentalstudywas conductedat Nassau CommunityCollegeduring 1965 and 1966 to determinethe relativeeffectivenessof the abstract approach and the concreteapproach of teachingthe theoryof groups,rings,and fields to freshmen.In the abstract approach, with student participationkept to a minimum,rigorousdefinitionsweregivenfirst,followedbyexamplesillustrating the propertiesofthesystems.Proofsoften theoremson groups wverethenpre- senteddeductively.In the concreteapproach, specificexamplesof the systems were presentedfirst.The studentswere then encouragedto participatein dis- cussions aimed at discoveringgeneralizationsfromnthese concrete examples. The proofsoftheoremsfollowedparticularexamplesand problemsexemplifyillg each theorem. The basic hypotheseswere: 1. The experimental(abstract) groupwillshow a higherachievementin learningthetheoryofgroups,rings,and fieldsthan the control(concrete)group.2. The experimentalgroupwillshowa higherachieve- mentin delayed recall of thesesystemsthan the controlgroup. Two sequences oflessonplans weredeveloped in detail,one fortheabstract approach and the other forthe concreteapproaclh.Eaclh consisted of eleven 50-minutelessons and extendedover a four-weekperiod. Fromna pilot study precedingthe experimentan item analysis was performedon the resultsof a preliminarytest. Two tests were constructedfromthese results: one to be This content downloaded from on Sat, 27 Jul 2013 09:44:11 AM All use subject to JSTOR Terms and Conditions