Mathematics Reform: Looking for Insights fromNineteenth Century Events

Paola SztajnIndiana University

In the nineteenth century. Warren Colburn defendedunderstanding as the avenue to learning arithmeticand questioned the memorization method in use since the seventeenth century. Colburn’s work -wasappreciatedby educators in the common school era, and his book is still considered an important one inthe history ofmathematics education. Many criticisms ofColburn’s ideas, however, emerged during histime, and teaching for understanding never fully reached nineteenth century mathematics classrooms.This episode in the history ofmathematics education raises questions about the success of contemporaryattempts to reform school mathematics.

An ongoing debate in school mathematics is theone between understanding and skills, meaning anddrill, or, in current terms, conceptual and proceduralknowledge. Different instructional programs proposedthroughout history have promoted diverging ideasabout teachingmathematics. At distinct points intime,prevailing views about mathematics education haveemphasized understanding before drill or vice-versa.Diametrically opposed views, nonetheless, have coex-isted in an enduring dispute.

Definitions of drill and skills have remained quiteconstant over the years. Explanations for understand-ing, on the contrary, have certainly changed. Currentperspectives are influenced by information processingideas about internal representations and mental struc-tures: people look fordescriptions ofthe processes thatoccurinthemind inorderto findunderstanding (Hiebert& Carpenter, 1992). In the past, however, understand’ing was a simpler term that carried a commonly agreedmeaning�It was an insight experience.

Mental discipline was the leading theory in educa-tion by the end of the nineteenth century. Brains, thetheory claimed, need to be exercised like muscles. Inthe United States, nineteenth century education issynonymous with mental discipline. In mathematicseducation, mental discipline is synonymous with drilland learning by rote.

At the beginning ofthe nineteenth century, mentaldiscipline was growing in America. At that time.however, a big debate arose between teaching math-ematics with understanding and teaching throughmemorization. This debate was launched by WarrenColbum, a young Harvard graduate. His book FirstLessons in Arithmetic on the Plan ofPestalozzi, withSome Improvements (1821) defended understandingand discovery as the avenue to learning arithmetic.

Colbum questioned the study of arithmetic based

on memory, rules, and catechism. He believed thatsimple arithmetical calculations understood by thepupil were among the best exercises the mind and thatstudents would understand arithmetic if they wereallowed to discover its rules. Colbum’s ideas ofunderstanding and discoverywerenot dissociatedfrommental discipline, the emerging theory of his time.Rather, he emphasized the importance ofmental disci-pline, later expanded and formalized during that cen-tury (Bidwell & Clason. 1970).

This article identifies the presence of an advocatefor understanding in mathematics education under theheading ofmental discipline. Colbum’s work and itsrepercussion in the teaching and learning ofarithmeticduring the first half of the 1800s is analyzed. Tohistorically situate Colbum’s work, a broad perspec-tive of American education during that period is dis-cussed. Then, the work ofPestalozzi and the impor-tance of arithmetic in his work are examined. Thisarticle also looks at Colbum’swork and its meaning fornineteenth century mathematics educators. The re-searcher explores critiques to the works ofPestalozziand Colbum that emerged during their time. and offersa hypothesis concerning the failure of teaching forunderstanding during the common school, mental-discipline era. Inconclusion, questions thatcanhelpusbetter direct our current movement towards under-standing in school mathematics are suggested.

Education in Nineteenth Century America

Education innortheasternandmidwesternAmericaduring the first decades ofthe nineteenth century has tobe understood as part of the social, economic, andpolitical movements thatwere going on at the time. Bythe 1800s the United States was already larger thanmost European countries. Improved means for ex-

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changing information and merchandise were then im-portant forthe country to stay united. The constructionof roads and turnpikes, railroads, and canals, allowedthe transportation ofpeople and goods at faster speedand lower prices than ever before.

As the market society expanded, common schoolreformers took the lead advocating free educationdedicated to moral development Education shouldcreate a national morality� a promise of good workhabits, reduction ofcrime, and protection ofproperty.By the 1820s, urban free schools hadbegun anirrevers-ible transition into publicly funded systems. Increas-ingly, Americans began to share the vision that freeschooling was essential to unify the population andproduce republican citizens (Kaestle, 1983).

The 1830s and the 1840s were the decades ofcommon school movements. Common school crusad-ers advocated state controlled schools teaching thesame body of knowledge to all children regardless ofsocial background. Children should be taught a com-monsocial and political ideologyto constitute a nation,decreasing conflicts and social problems (Spring, 1990.p. 74). Schooling expanded throughout the countryand by the 1850s most white American children in thenorthern states had access to free education.

Economic changes in America during the earlynineteenth century transformed education. At thesametime, commerce, trade, and finance also modifiedthe importance given to learning basic arithmetic.Because applications such as public statistics appeared,and commercewasno longerrestricted to small groups,arithmeticgainedpractical andpopularinterest Ameri-can sensitivity and attention to quantified materialsincreased because of the "changes that were makingnumbers an integral part oflife" (Cohen, 1982. p. 117).A greater number ofpeople needed more mathematicsthan ever before. By the time free schools began toreach most American children, it was accepted that allchildren needed to learn the rudiments of arithmetic.Mental discipline had also grown in America andarithmetic was increasingly perceived as a way toimprove the mind. The emergence ofcommon schoolsprovided the necessary vehicle for arithmetic instruc-tion: The teaching and learning of arithmetic could beused forpractical as well as moralpurposes throughoutthe United States.

Arithmetic and the Work ofPestalozzi

During the common school era. Johann HeinrichPestalozzi had an important role in the schooling ofyoung Americans. Many common school reformers

embraced Pestalozzi’s ideas because they saw in hiswritings thehumane instructionalmethod theyneeded.based on maternal love and cultivation of the moralfaculty (Spring. 1990). Pestalozzi "sought to establishmoral order and social harmony through education"(Binder, 1974, p. 24) and his ideas about "internalizeddiscipline through propermotivation," that challengedtraditional pedagogical practices, are listed by Kaestle(1983, p. 67) among the significantly new develop-ments in moral discipline in the nineteenth century.

FormanyAmerican educators, Pestalozzi *s ideaswere the necessary defense to teach for understanding.These educators believed that too much stress was putonmemorization and if, instead ofmemorizing, under-standing would be reached, memory would "take careof itself (Pestalozzi’s School. 1827, p. 68). Theyargued that there was "enough in the intuitive under-standing of every child to accomplish the completegrowth and maturity of its faculties, if its reason beproperly trained and nourished" (p. 68).

Pestalozzi proposed that all knowledge shouldbegin with observation. An object should be broughtbefore a child’s consciousness and while observing it,the child should perceive its unity, its form, and give ita name. In this way, the basic elements ofknowledgeforPestalozzi werenumber, form, and word. Numbershad such prominence in Pestalozzi’s learning theorythat he wrote specifically about instruction in arith-metic: Numbersgive "infallible results" and are"neverthe germ of errors." Since number instruction leads"toward the purpose of all instruction� intelligentideas� [it] must be honored as the most important ofall the departments . . . [and] must also be pursueduniversally" ("Selections." 1859, p. 698).

Pestalozzi "made the utmost efforts to bring arith-metic before the intuition of the child" ("Selections,"1859, p. 698). For him, arithmetic arose from thecombination and separation of units. He believedfigures are abbreviations of computation and shouldnot weaken the comprehension of arithmetic in themind. All progress in arithmetic should be based in anunderstanding of the material relations. He believedthat if understanding was not present "the very firstmeans ofattaining intelligentideas wouldbedegradatedto memory and imagination, and thus made powerlessfor its real object" (p. 699).

Colburn’s Reform: The principle of analyticinduction

Reform in arithmetic instruction in the UnitedStates began in 1821 with the publication ofColburn’s

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First Lessons in Arithmetic on the Plan ofPestalozzi,mth Some Improvements (Cajori. 1925). With thisbook. Colbum criticized the instructional approachtraditionally used in teaching arithmetic. He defendedreasoning instead ofmemorizing inmathematics class-rooms, and advocated discovery methods in the teach-ing and learning of arithmetic. Colbum believed theusual teaching method reversed the natural learningprocess because it required pupils to learn generalprinciples before they understood the particular ideasthat formed each principle.

Through his work, Colburn popularizedPestalozzi’s plan and "rendered it capable of beingapplied by the humblest mediocrity" (Edson, 1856. p.306). Like Pestalozzi. Colbum claimed that all stu-dents should leam from observation of the real worldsurrounding them. Therefore, he proposed that arith-metic teaching should start with the observation ofsensible objects and with practical examples. ForColbum, examples with abstract numbers were "ofvery little use, until the learner had discovered theprinciples from practical examples. They are moredifficult inthemselves. forthelearnerdoes not see theiruse and therefore does not so readily understand thequestion" (Colburn, 1970, p. 16).

At the beginning of the 1800s, there were threeAmericanbooksonarithmetic: Pike’s(1779).DabolTs(1800). and Adam’s (1801). In these books, arithmeticwas a set of mathematical rules to be memorized.Students were given large abstract numbers withouthaving any concrete experience. Also. they werepresented the four basic operations and had to performthe operations with meaningless figures. In such asituation, Colburn believed the most that studentscould learn was how to mechanically perform incom-prehensible operations and translate figures into words.Furthermore. Colburn argued that this was uselessbecause students developed no concepts of the effectproduced by the operations (Colburn. 1825, p. 19).

ForColburn, students should beled through appro-priate questions to the discovery of the rules thatgovern arithmetic. Students should first be allowed todevelop their own methods of solving problems.Through specially chosen examples purposefully ar-ranged in a sequence of increasing difficulty, studentsshould continually improve their solution methods.Arithmetic rules would then emerge naturally from thestudents’ strategies. In summary, a learner should gofrom the particular to the general and mathematicalreasoning should proceed upon the principle of ana-lytic induction (Colburn. 1825. p. 16).

In the years that followed Colburn’s first book, his

plan was referred to as the inductive method. Inteaching by induction, students were not "directed tolearn a rule, but to draw inferences from known data"(Arithmetic. 1826, p. 22). Whenever a new principleshould be taught, it was introduced "notby announcingit in the shape of an abstract proposition, but throughthe medium of a problem involving this principle"(Mathematics, 1827, p. 46). In this method, studentswere left to themselves to answer questions and dis-cover the laws of arithmetic.

The inductive method was also referred to as ananalytic, natural, or new method. The debate betweenunderstanding and drill in the teaching and learning ofarithmetic was then represented by analytic versussynthetic, natural versus common, or the new versusthe old method. By the analytic method, for example.instruction should begin "with examples, and at lengtharrive at a rule" (Carter. 1827, p. 30). The syntheticmethod, on the other hand, concisely stated rules andfollowed them by applications.

In the natural method, the relations among num-bers were not "established by prescription. They existin the very nature ofthings; and every child maybe ledto discover them, by the powers of his own mind"(May. 1829, p. 221). This method opposed the com-mon method where learners were first introduced to"general rules, abstract principles, and tedious defini-tions" and expected to commit them "faithfully bymemory" (p. 220).

Finally, the new method was the one in whichinstruction began with practical example and smallnumbers so that any student could "easily reason uponthem" (Arithmetic, 1837. p. 12). The student was also"thrown immediately upon his own resources and ...compelled to exert his own power" (p. 12). In the oldmethod, the "learner was presented with a rule whichtold how to perform certain operations on figures,...[and] no reason was given" (p. 12).

Although Colbum was proposing an innovation athis time, he did not question the emerging theory ofmental discipline. He believed that few exercisescould strengthen the mind "so much as arithmeticalcalculations, if the examples are made sufficientlysimple to be understood by the pupil" (Colburn, 1825/1970. p. 16). An original feature in Colburn’s plan. tohelp discipline the mind. was that most calculationsshould beperformed mentally. Before childrenlearnedto write the figures they should be able to understandthe questions and solvethem mentally. Once a solutionwas obtained, Colbum believed children should beable to explain how they proceeded in order to achieveit Therefore, in arithmetic classrooms, teachers should

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ask not only for answers but for procedures carried outin the mind as well.

This arithmetic in the mind, called mental arith-metic, was highly celebrated after the publication ofColburn’s First Lessons. The idea became popular ina marketeconomy where mental calculation was a plus(Cohen, 1982, p. 136). Colbum’s book was alsolargely accepted among educators who realized mostofthe books they used were "ofthe same sort that wereinuse and prevailed whenyoung children were thoughtquite incapable of learning arithmetic" (North Ameri-can Review, 1822,pp.381-383).

Before the publication of Colbum’s book, it wassaid that a student could not "take pleasure in dwellingso long on what he does not in the least understand"(Carter, 1827.p.31). Colbum’s new instructional planwas thus appreciated because children did not have torely on memory to answer a question, and their mindswere brought to the constant work ofinvestigation anddiscovery. Aftersome educators read Colbum’s book.they believed it could do "infinite good" to arithmetic(Mathematics, 1827, p. 44). Therefore, this book"seemed to have revolutionized the mode of teachingelementary mathematics in the schools of New En-gland" (Edson, 1856. p. 297).

Reform Failure: The Emerging Critiques

By the 1850s mathematics teaching and learningwas already back to "the *good old-fashioned way* ofmemorizing" (Cohen. 1982. p. 138). Mental arith-metic, forexample, proposed by Colbum to strengthenunderstanding, was transformed over the years into aninitial drilling procedure: Students were to answerquestions quickly without having to write down theircalculation. In the classrooms, after learning to count.students practiced addition, for example, "at first verythoroughly inthe head and thenon slate: they write andrecite additiontables, countingthem each time" (Arith-metic, 1842. p. 48).

Between the 1830s and the 1850s. critiques toColbum’s ideas grew among educators. As early as1834, an article at the American Annals of Educationand Instruction claimed that parents were "expressingadesirethattheirchildrenmightleam Arithmetic in thegood old way of rules and examples" (Notices, 1834,p. 148). This same periodical had previously advo-cated the use of the inductive method of teaching.However, after reporting an emerging pressure to re-turn to the synthetic method, some writers agreed thatthe old way might actually be "often necessary forthose who are to enter the counting house" (p. 148).

In 1837 another critique ofPestalozzi came out inthe American Annals of Education and Instruction.His methods were said to prepare children who were"well qualified for mathematical and abstract reason-ing, but not prepared to apply it to the business ofcommon life" (Pestalozzian, 1837, p. 12). Studentswho learned underPestalozzi’s principles did not havegood "stores ofknowledge" (p. 12) for immediate use.Their minds had become so "accustomed to receivingknowledge divided into its most simple elements andsmallest portions, that [they were] not prepared toembrace complicated ideas, or to make those rapidstrides in investigation and conclusion" (p. 12).

Compared to the synthetic teaching way, totaladherence to the analytic method started to be consid-ered "not less defective" (Notices, 1838. p. 287). Thenew method was no longer considered appropriate forelementary educationbecauseuneducatedminds"mustbe taught at the outset more dogmatically" (Treatises,1839. p. 265). Supporters ofthe inductivemethod werealso criticized forconsidering "more its intrinsic scien-tific beauty, than its actual adaptedness to the purposesof instruction, [and therefore, there was] an evidenttendency to a return to the old [teaching] mode" (p.265).

Recommendations emerged for pursuing arith-metic in the classrooms "in the old-fashioned manner,except that [teachers should see] more the importanceof encouraging each pupil to study everything outhimself (Extract, 1839, p. 77). From counting, calcu-lating, and memory(three possible modes ofaddingupa column of figures), the last was one more timebelieved to be "undoubtedly the best. If you oncecommit the addition table thoroughly, you have it fixedfor life" (Arithmetic, 1842, p. 52). Finally, it was onceagain accepted that, despite the risk of "being consid-ered old fashioned,... rules should be recited by eachmemberofthe class, thenpracticed for sometime�themore the better" (F. C. B.. 1855. p. 210).

What happened to Pestalozzi’s and Colburn’s ad-mirers between the 1830s and the 1850s? Manyarticles published in educational journals at the timechanged from praising their ideas to strongly criticiz-ing them. After a few decades ofimportant advocatesfor understanding in the teaching and learning ofmathematics,thependulum swungbackto drill. Withinthe framework ofmental discipline, arithmetic instruc-tion returned to rote memorization, and today drill isthe term mainly used to characterize nineteenth cen-tury mathematics education in America.

One can only hypothesize why this occured. Onepossibility is that teaching for understanding was dis-

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cussed injournals and universities, but it did not reachAmericanclassrooms. Students ofthe early nineteenthcenturycommonschools, therefore, did notexperienceColbum’s instructional plan. Reactions to Colbum’sideas may have emerged from educators in academiawho had different positions concerning how childrenleam or from some participating teachers accustomedto the memorization method who voiced their opin-ions. In this case, the debate between understandingand memory never left academic circles and educa-tionaljournals, and memorizationneverleft nineteenthcentury arithmetic classrooms.

There is little historical evidence about what hap-pened inside the classrooms and what teachers did(Cuban, 1993, p. 2). and therefore it is hard to knowhow many teachers actually taught for understandingat the beginning of the nineteenth century. However.figures about textbooks in use during the first half ofthe nineteenth century give some data on teachers*utilizationofColbum’smethod. Colbum’s bookneverovertook DabolTs place at schools. In the state ofNewYork, for example, during the time when Colbum’sFirst Lessons was praised in journals, the increase inthe number of users of the book did not challenge theposition occupied by Daboll. The number ofbooks inuse in 1827 and in 1831, for example, changed asfollows: DabolTs.from349to469; Adams’, from 91to 102; Pike’s, from 80 to 46; and Colbum’s, from 1to 26 (School books. 1832, p. 378).

In the state of Connecticut, Colbum also did notpass Daboll. In 1839,127 school societiesused Daboll’sbook, the most used school book, and only 60 usedColbum’s (Barnard, 1839, p. 192). The next year.Daboll came in second with 98 books in use whileColbumhad 57 (Barnard, 1840, p. 224). Smith’s book.published in 1827. combining the inductive methodwith a more rigid and traditional practice, went fromsecond in 1839 to first in 1840. In his 1841 report, theSecretary ofthe Board ofCommissioners in Connecti-cut noticed that, in the classrooms, the old instructionalmethod was found more frequently than teaching forunderstanding, and arithmetic instruction in schoolsstill was "not commenced with a due understanding ofthe first principles" (Barnard, 1841, p. 250).

Epilogue

Themathematics educationcommunity to needs torealize that the understanding-drill pendulum for theteaching and learning ofmathematics does notswing ina historical vacuum. During the early 1800s, with thegrowth of cities and market economy, education in

America had to provide citizens the moral foundationand practical knowledge needed for the healthy devel-opment of a young nation. Educating all children wasnecessary, and the very first rudiments of a schoolsystem similartothepresentoneemerged. Pestalozzi’stheories fit this historical moment, providing the sup-port required to promote moral education for all chil-dren.

Arithmetic education was also embedded in thosesocial and educational movements. Developingnumeracyamong alargerportion ofthe population wasanew taskeducators had to face. Focusing attention onColbum and his book illuminates an attempt to dis-seminate new beliefs about how all children couldleam arithmetic. In a period of social change, whennew teaching approaches were sought, Colbum de-fended teaching through understanding.

Although Colbum’s First Lessons "is creditedwith being the most popular arithmetic text ever pub-lished" (Bidwell & Clason, 1970, p. 13). it was neverfully implemented in American classrooms. His ideaswere debated among educators but were probably nottranslated into regular practice. As common schoolsdeveloped throughout the country and an increasingnumberofchildren were educated, Colbum’s plan wasmodified. Memory and drill incorporated ideas such asmental arithmetic, created to makeleamingmoremean-ingful, and dominated the teaching and learning ofarithmetic even after Colbum’s effort.

What makes this nineteenth century episode im-portant for the 1990s? Recognizing the "return" oflearningby rote after abig crusade in favorofmeaning-ful instruction helps us organize our current campaignfor understanding. Parallels between Colbum’s planand today’s struggle broaden perspectives on how toimplement understanding in the teaching and learningof school mathematics. There are similarities betweenthe 1800s’ industrial era and the current beginning ofwhat is being called"an information society" (NationalCouncil ofTeachers ofMathematics. [NCTM], 1989,p. 3). There are also analogies between the commonschool attempt to educate all children and the currenteffort to promote numeracy for all.

As the community of mathematics educators de-fends teaching for understanding, Colbum’s episoderaises questions to ponder. Why did the educationalpendulum swung back to learning by rote? Why wasColbum’s book not used in classrooms? If the debatebetween understanding and memory did not leave theuniversity arena, it did not reach practitioners in class-rooms. Itmaybe, forexample, thatmostteachers nevertried out Colbum’s ideas because they were not even

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aware of their existence. In that case, are educatorsdoing a better job of communicating with practicingteachers? All possible explanations for the reasonswhytheearlynineteenthcenturyeffortto teach throughunderstanding did notmanage to overcome two centu-ries of rote learning can help educators learn fromhistory and avoid repeating the same mistakes.

References1

Arithmetic. (1826). The Teacher’s Guide andParent’s Assistant, 7(2), 21-25.

Arithmetic. (1837). Common School Advocates,7(2). 11-13.

Arithmetic. (1842). Connecticut Common SchoolJournal, 4(5), 48-52.

Barnard, H. (1839). First annual report of thesecretary of the Board of commissioners of commonschools in Connecticut. Appendix 5: Different kindsof books in use in the different school societies. Con-necticut Common School Journal, 7(13), 191-192.

Barnard, H. (1840). Second annual report of thesecretary of the Board of commissioners of commonschools in Connecticut. Appendix 2: Abstracts ofreturns respecting the books in use or recommended tobe used by school visitors. Connecticut CommonSchool Journal, 2(13). 224.

Barnard, H. (1841). Third annual report of thesecretary of the Board of commissioners of commonschools in Connecticut. Connecticut Common SchoolJournal, 5(16). 242-284.

Bidwell. J. K., & dason, R. G. (Eds.) (1970).Readings in the history of mathematics education^Washington, DC: National Council of Teachers ofMathematics.

Binder, F. M. (1974). The age ofcommon school,1803-1865. New York: John Wiley & Sons.

Cajori, F. (1925). A history of elementary math-ematics with hints onmethods of teaching. New York:The Macmillan Company.

Carter. (1827). Improvement in arithmetical in-struction. American Journal ofEducation, 2(1), 30-37.

Cohen, P. C. (1982). A calculating people: Thespread of numeracy in early America. Chicago: TheUniversity of Chicago Press.

Colburn, W. J. (1825). First lesson in arithmetic onthe plan ofPestalozzi with some improvements. In J.K. Bidwell & R. G. Clason (Eds.) (1970). Readings inthe history of mathematics education. Washington,DC: National Council of Teachers of Mathematics.(Reprint from the book)

Cuban, L. (1993). How teachers taught: Con-stancy and change in American classrooms, 1880-1990 (2nded.). New Yoric: Teachers College Press.

Edson,T.(1856). Educationalbiography: WarrenColburn. American Journal of Education, 2.294-316.Extract from confessions of a school master (1839).Connecticut Common School Journal, 2(5), 76-77.

F.C.B.(1855). Arithmetic: Teaching rules. Con-necticut Common School Journal, 10,208-210.

Hiebert, J. & Carpenter, T. P. (1992). Learning andteaching with understanding. In D. A. Grouws (Ed.),Handbook of research on mathematics teaching andlearning. New York: Macmillan Publishing Com-pany.

Kaestle, C. F. (1983). Pillars ofthe republic. NewYork: Hill and Wang.

Mathematics. (1827). Teacher’s Guide andParent’s Assistant, 7(3), 44-46.

May, S. J. (1829). Errors in common education.American Journal of Education, 4(3), 213-225.

National Council of Teachers of Mathematics.(1989). Curriculumandevaluationstandardsfor schoolmathematics. Reston, VA: Author.

Notices. (1834). American Annals of Educationand Instruction, 4. 148.

Notices of books. (1838). American Annals ofEducation, 5.287-288.

Pestalozzian system ofeducation. (1837). Ameri-can Annals of Education and Instruction, 7, 5-14.

Pestalozzi’s school. (1827). Teacher’s guide andParent’s Assistant, 7(5). 68-69.

Review of an arithmetic on the plan ofPestalozziwith some improvements. (1822). North AmericaReview, 381-383.

Selections from publications byPestalozzi. (1859).American Journal of Education, 7,519-722.

School books in the United States. (1832). Ameri-can Annals ofEducation and Instruction, 2, 371-384.

Spring, J. (1990). The American school: 1642-1990 (2nd ed.). New York: Longman.

Treatises on algebra. (1839). American Annals ofEducation, 9. 263-268.

Footnote: ^ere are two different journals titledAmerican JournalofEducation. The firstone waspublishedin the 1820s and later became the American Annuals ofEducation’, the second onewaspublishedbyHenryBamard,in the 1850s. They areboth listed with the same name in thereferences and can be distinguished by their dates.

Author Note: The author is supported by a scholarshipfrom Conselho Nacional de Desenvolvimento Cientifico eTecnol6gico, Brazilian government

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Editors Note: The author could not be reached to provide known address was Education 3002, Indiana University,completereferences, and itdidnotseem worthwhile to delay Bloomington, IN 47405.publication while trying to locate her. Paola Sztajn’s last

/ /^-The Further Adventures of a Molecular Biologist

Professor Bear Was Quite Disconcerted to Discover That Bruno, Joey, and FredWere Enrolled in Modern Abstract Algebra 621.

He Had Never Really EnjoyedTeaching Sour Groups.

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