History of Math

Humanistic Mathematics Network Journal #266

ABSTRACTThe main goal of this essay is to discuss, informally,an intuitive approach to the history of mathematicsas an academic discipline. The initial point of depar-ture includes the analysis of some traditional defini-tions of the concept of 'history' taken from standarddictionaries. This concise dissection attempts to sug-gest the complexity of the discipline.

KEY WORDS: HISTORY, HISTORY OF MATHEMAT-ICS, HISTORIOGRAPHY, METHODOLOGY, CHRO-NOLOGY.

The term 'history' is familiar to almost everyone, andmost people believe they know its meaning intuitively.The lay person sometimes thinks of history in termsof dates, names, places and of colorful anecdotes oninteresting characters. History provides a record ofwhere someone lived and what he did. In short, his-tory is the repository of the past. But what exactly isthe history of mathematics as an academic discipline?

The events that took place yesterday are now part ofhistory. Photographs are history: they reflect the waywe once were. History is studied in elementary school,especially that of student's native countries. Teachers,who often seem old enough to have taken part in someof the historical events, teach -over and over again-anecdotes, names, places, dates, and so on. Historicalmovies, TV programs and books are popular. Themedia affects the way people understand history as adiscipline. Unfortunately, sometimes the public'sknowledge of important historical issues is derivedfrom the popular media (especially movies), and notfrom professional sources. Thus, their understandingof these events can be distorted.

Unlike the word mathematics, the term history is usedon a daily basis by the news media. Some reportersmay believe that history is actually being made whenthey type or read the news. On occasions, reportersand anchormen have trouble attempting to disassoci-

ate themselves from the event they are reporting on.

Of course, in most cases, the term history is misusedby professional reporters. Often, for example, TVsports commentators discuss player's individualrecords while narrating a baseball game. These statis-tics include the player's batting average, number oftimes at bat in the game, number of stolen bases, etc.Sometimes, the commentators also narrates theplayer's background. They mention the college theplayer attended; where he played in the minorleagues, his previous professional teams and so on.The commentator may also discuss some of theplayer's qualities as a human being (e.g., generosity,sportsmanship). Then, the sportscaster may try to ex-plain why the player was motivated to become a pro-fessional. After the commentator has finished withthese statistics and anecdotes, he often says: '... and,ladies and gentlemen, ... the rest is history'.

This popular expression 'and, ladies and gentlemen,... the rest is history'- suggests that once the informa-tion or anecdote has been revealed to the audience,the rest of the account is common knowledge and re-quires no further explanation. Or, perhaps, the phrase'... the rest is history' is synonymous with '... the rest isunimportant' or '... the rest is well-known'. In fact, theterm 'history' may suggest that something is not ofcurrent concern or lacks importance (e.g., My youthis history) [2, p. 614].

History, however, is much more than long lists of data.Think back to your student days. Did your teacherdemand knowledge that was more substantial andprofound that mere factual information contained ina list of data, especially on a test? Sometimes theyasked essay questions which required some explana-tion of historical events. Consider an often asked ques-tion on why XV century navigators attempted to finda new route to India. Students, apparently, fail to dis-tinguish between the factual aspects of historical dataand the interpretation of historical evidence. In gen-

History of Mathematics, an Intuitive ApproachDr. Alejandro R. Garciadiego

Dept. of MathematicsFaculty of Sciences

Universidad Nacional Autonoma de Mexico

Humanistic Mathematics Network Journal #26 7

eral, young students do not understand that the words'history' and 'chronology' are not synonymous.

The word history may be defined as "[ ... ] a narrativeof events; [ ... ]" [2, p. 614]. This definition seems toindicate that there is indeed a subtle difference be-tween the terms chronology and history. History de-velops a narrative and, therefore, it is not simply a listof dates and names arranged sequentially. History,viewed from this perspective, is familiar to us. His-torical works (e.g., a movie, a book, a play) containmore than just a simple list of names and dates. Inmany cases, the presentation (or form) of the materialis as important as the raw material (or historical data)itself. For example, the movie industry attempts toproduce and sell good films to entertain an audience,not necessarily to reproduce good history. For this,producers, directors and writers pay special attentionto the presentation of the narrative. In most occasions,they even add external elements to produce a moreinteresting movie or a more attractive story. In fact,almost any historical movie or TV show contains adisclaimer asserting that some characters, situationsand dialogues were added for dramatic purposes.Similarly, a scientist -being not a professional histo-rian- may judge a historical textbook by its literaryand entertainment qualities and not necessarily by itshistorical accuracy and objectivity. Some classical textsused for many years by the mathematical communitymay provide excellent examples [in particular, items3 and 7 of the references].

Before attempting to refine the definition of history, itis important to understand that a historical scholar-ship does not necessarily arise from a description ofpast events -analogously, doing mathematics involvesmuch more than dealing with numbers. Some people,other than historians, are constantly interested in thepast. Take, for instance, a private investigator. He maybe trying to solve a homicide case. His task is to re-construct how a murder took place. Depending on thecase, the private eye will have to answer certain ques-tions. Some may deal with 'historical' factual inquir-ies: for example, attempting to answer when, whereand how the event took place. Other questions mayinvolve non-factual issues; for example, the state ofmind of the killer at the moment of the crime. Is theprivate eye an historian?

He is trying to reconstruct the past, presumably the

goal of the historian. No doubt there are some simi-larities between the two professions. Every privatedetective must be acquainted with some of the meth-odological techniques used by historians. He mayhave visited a newspaper archive and read old items.Perhaps, he may have studied autobiographicalsources (e.g., diaries, address books, old photographs,unpublished correspondence, etc.) to determine theactivities of the person he is investigating. On the otherhand, the historian may enjoy the detective work ofhis profession. A wonderful example is remarkablyillustrated by Reid's attempts to determine where EricT. Bell spent his childhood [12]. Furthermore, it is al-ways an intellectual challenge to find a difficult source,or to devise new ways of understanding or interpret-ing historical data, or to prove a point in a more effec-tive way.

Nevertheless, private investigators do not always at-tempt to reconstruct the past. They are usually inves-tigating events that occurred in the immediate past,and have not yet formed a conclusion. His investiga-tion will lead to a conclusion (e.g., a court verdict),and will likely be influenced by the actions of the de-tective. Furthermore, his goal is not to reconstruct thepast in itself, but to use some information about theevent for other purposes. For example, some detec-tives/reporters have written books (or reports) onpolice investigations attempting to reconstruct theevents associated with crimes committed in the past.One of these books [18], more than three hundredpages long, attempts to reconstruct the last day in thelife of Marilyn Monroe. Some authors, on occasions,criticize the original investigation, presenting newevidence that may reveal the real reason that some-body died (e.g., Marilyn Monroe and JFK, amongmany others [see, for example: 15, 16 and 17]).

Historians may not only find certain aspects of adetective's methodology questionable (including theprivate investigator's use of sources, inferences, con-jectures, goals and extrapolations), but may also criti-cize his/her lack of objectivity. It is extremely diffi-cult for detectives not to become personally involvedin the events surrounding the case. Detectives usu-ally receive an economic reward for their activities(except perhaps for the fictional character Mike Ham-mer who seems to get always involved for friendshipand personal reasons) and have an obligation to getresults for their clients, not necessarily to find the his-


torical truth. On the contrary, history, mathematics andmost other academic disciplines, demand the highestpossible level of rigor and objectivity. For example,Frege criticized Cantor's definitions of the concept ofset [menge] and cardinal number because Cantor re-lied on each individual's mental capacity to abstractcertain properties. That sort of definition was too sub-jective, depending on the personal perspective of eachparticular mathematician. In the same way, the closerthe historian is to the person, event or idea that he/she is studying, the more difficult it is for him/her topresent an objective reconstruction of the event. Butlet us attempt, once again, to define the term 'history'.

Some biographical treatises include, as a guide, a chro-nology listing major events and corresponding datesin a person's live. As dis-cussed earlier, a chronologi-cal treatment of events is notnecessarily the same as a his-torical investigation. Fur-thermore, some researchersmay begin an investigationintending to conduct a his-torical study, but may notproduce results that meetcontemporary professional standards. For example,consider the term 'class'. A historian may ask who firstconceived this term, and when and where the con-cept was first used. He might examine an old refer-ence to find out whether the term class was alreadyin use. If it was, then he could look up an even earlierreference and eventually find out who first used theterm. It is very possible that the researcher's final re-port will simply be a chronology itself, naming theperson who initially used the term, and how its mean-ing may have changed over time. But this chronologi-cal narrative may not explain why the concept wasformulated in the first place or modified over time.

A historian may argue that a chronological descrip-tion of events does not necessarily fulfill the criteriaof rigor established by the professional communityof historians. In fact, it is quite easy to attempt to ex-plain facts retrospectively. A mathematician may ar-gue along similar lines. To the general public, any textusing numbers may be thought to involve mathemat-ics. Mathematicians, however, would regard this assimplistic; in fact, most professional books or articlesin mathematics contain no numbers at all, except forthe page numbers.

Sometimes, the nature of a historical discipline maybe partially understood by asking why some peoplestudy it. Some people want to clarify who discovereda particular idea. Hubbert Kennedy ingeniously pro-posed 'Boyer's law', in part because of the many occa-sions in which credit for a concept has been given tothe wrong person. Boyer's law states that "mathemati-cal formulas and theorems are usually not named af-ter their original discoverers" [8, p. 67]. Boyer men-tions [4], between the XVII and the XIX centuries, atleast thirty cases, of mathematical results that havenot been credited to the appropriate person.

Sometimes, influenced by political ideology, histori-ans are affected by nationalist sentiment, attributing(or attempting to attribute) discoveries (previously at-

tributed to western math-ematicians) to their owncountrymen [13]. There isanother reason why peoplestudy history: To under-stand the present by study-ing the past. Some profes-sional mathematicians,sooner or later, decide toinvestigate the origins of

the concepts they use today.

At present, less emphasis is given to describing eventsin the lives of the great men of mathematics than tolisting discoveries in the various branches of math-ematics. Historians of mathematics place greater em-phasis on the development of ideas. They attempt tofind the key concepts that influenced the developmentand evolution of their discipline. In particular, theytry to find the key questions that played the role ofAriadne's thread conducting mathematical research.So, some historians attempt to reconstruct how math-ematical ideas originated, evolved and learn how theyinfluenced other ideas. Historians might also focuson the development of a particular school of thoughtand its 'philosophical' program. Others might attendthe social influences that affected the thought of math-ematicians or the relationship between themathematician's work and the society in which helived. Others might be interested on the social condi-tions surrounding the academic community or, on thecontrary, on the effects of mathematical ideas on soci-ety. Some historians analyze the way that institutionsand governments implement scientific policy. Othersstudy the effects of economic factors such as produc-

❝...history lacks the absolute character of math-ematical reasoning. Within mathematics, once thebasic premises are settled, there is no space foralternative results.


tion and consumption on science. Indeed, historiansclaim that ideas do not occur in a vacuum. Knowl-edge is a human creation, not a preexisting entity,closely associated with specific individuals, institu-tions and schools of thought.

No doubt, some people are tempted to write embel-lished accounts of events for posterity, includingsports, knowledge (in this case, mathematics), poli-tics, and, most importantly, war. Most likely, histori-ans belonging to opposite ideological camps wouldpresent radically different accounts of the same event.For example, a Spanish soldier and an Aztec Indianwould describe the conquest of Mexico in the XVI cen-tury from very different perspectives. The same situ-ation occurs in the history of the American Civil War,as in the description of almost any human event. Thetriumphant narrative of a Yankee might differ fromthe pessimistic account of the Southerner. They maydiffer on the origins of the conflict, and the impor-tance and consequences of some events. Here restsone of the major methodological differences betweenmathematics and history. History provides room foralternative interpretations. In fact, if different personshave witnessed an event, there may well be as manydifferent accounts, as beautifully illustrated by the lateAkira Kurozawa's Rashomon [Film. B/W. 1950], inwhich four different characters, all apparently equallytrustful and credible, narrate four conflicting interpre-tations of the same event. The nature of history pro-vides room for diverse reconstructions of the past, asreflected by the diversity of published historical ma-terial. These reconstructions will vary in their degreesof plausibility and consistency in terms of both chro-nological and technical characteristics of the discipline.Thus, history lacks the absolute character of math-ematical reasoning. Within mathematics, once thebasic premises are settled, there is no space for alter-native results. For example, if within Euclidean ge-ometry we have shown that the sum of the innerangles of any triangle is equal to the sum of two rightangles [Euclid, 1-32], then there is no room for anypossible alternative, thanks to the principle of ex-cluded middle.

The possible existence of alternate interpretations mayexplain why, on historical disciplines, many histori-cal monographs are published on the same topic. Forexample, Drake [5] and Koyré & [10] have presenteddifferent historical accounts on whether Galileo con-ducted empirical experiments. Later, other historians

commented on the works of Drake and Koyré, dis-cussing each author's viewpoints [19]. Still later, someother historians, using additional sources, may dis-cuss the works of the earlier critics. And so on. Buthistorians are not satisfied by merely presenting analternative view of the events in question; more im-portantly, they want to convince their readers thattheir own conceptualization is better than earlier oralternative accounts. This last point is extremely im-portant.

Indeed, in most cases, historians are not satisfied bymerely narrating past events. For example, Ferreirés[6] convincingly argues that some of Dedekind's ideasunderlie Cantor's theory of transfinite numbers. Like-wise, Rodriguez-Consuegra consistently disputes [14]that there is a general philosophical method inspiredby Russell's philosophy of mathematics (which hedeveloped at the turn of the century), in spite ofRussell's apparently endless intellectual evolution. Incurrent thinking, an ordered (or unordered) collectionof information on a particular topic does not consti-tute a historical study. As May has already argued:"history arises when chronology is selected, organized,related and explained." [11, p. 28 (my emphasis)].

History is also defined as "a branch of knowledge thatrecords and analyzes past events" [2, 614]. The keyword here is 'analyzes'. Contemporary historians donot record past events in a passive manner, but evalu-ate the past from a critical perspective. "If error andignorance," as Adler and van Doren say, "did not cir-cumscribe truth and knowledge, we should not haveto be critical" [1, p. 166]. By critical, at this level, onesimply means a position that evaluates and judges.Certainly, every historical treatise is based on a biasedframework affected by the author's ideology and back-ground. Historians cannot disassociate themselvesfrom previous knowledge and be completely impar-tial. When an author has a critical attitude, however,he/she does not intentionally supports a political orideological position.

The reader should not misunderstand the meaningof the word judge. The historian does not make ethi-cal or moral assertions, nor is he/she concerned aboutthe empirical truth of a concept. His goal is to deter-mine why a particular novel idea, theory or interpre-tation was accepted over other conceptualizationscurrent at the time, or immediately beforehand. Inorder to do so, the historian is required to ask appro-


priate questions: What problem was the mathemati-cian attempting to solve? What conceptual tools wereavailable? What comprises a 'rigorous' solution to theproblem according to the standards of the time? Inthe case of the history of mathematics, Wilder has as-serted that: "we don't possess, and probably will neverpossess, any standard of proof that is independent ofthe time, the thing to be proved, or the person or schoolof thought using it" [20, p. 319].

Historical understanding does not exist independentlyof other kinds of knowledge. Firstly, it is highly de-pendent on the intellectual background and histori-cal assumptions of the practitioner. Secondly, it is ab-stracted from a totality surrounding it. The historiancannot encompass this totality and necessarily omitspossible associations with other intellectual disci-plines; very frequently he/she omits sociological fac-tors. The historian understands that scientific thoughtis profoundly influenced by science, arts, technology,philosophy and theology, among other areas. How-ever, the relationship of mathematics to these otherdisciplines may be obscured by current thinkingwhich emphasizes the independence of mathemati-cal thought. The task of the historian is to transcendthe constraints of the present and reveal these influ-ences.

There is no single approach to studying historicalquestions. Historical research is strongly affected bypersonal values, background, interests, the surround-ing environment and the characteristics of the time inquestion. The only way to formulate interesting his-torical questions is to expand the knowledge of thepast. The more one knows, the more one would liketo know. It is obviously necessary to read the classicsand the works of great historians. While doing this,keep in mind that, there was less of a rigid distinctionbetween mathematics, the sciences, philosophy andothers branches of knowledge up to the turn of thepresent century than there is today. Most of the intel-lectuals of the past were competent in all these disci-plines. In order to understand the scientific ideas ofan important historical figure, it is necessary to un-derstand the person's theological and/or philosophi-cal thought. Descartes, for example, made importantcontributions to philosophy, mathematics, physics,music and medicine, and, perhaps, to other disci-plines. If the historian of mathematics wishes to un-derstand Descartes' contributions to geometry, he/she

needs to study his writings in other subject areas.

To conclude, consider the following analogy to his-torical research. Suppose that you are attempting toput together a one thousand piece puzzle. Most peoplebegin with the following strategy: they separate outborder pieces with a straightedge to form the frameof the puzzle. Then, if possible, they try to separateout pieces with the same color, to build small sectionsof the puzzle. Finally, with the help of the picture onthe top of the box, they attempt to assemble all of thesmaller sections together, to produce the whole pic-ture. Similarly, historians are trying to reconstruct thepast by presenting a picture, story or narrative to theiraudience. Unlike a puzzle, however, historians do notknow the shape, size or the number of pieces of thepuzzle! To make matters more complicated, the his-torian does not even know the overall 'image' of theevents he/she is trying to reconstruct, and thereforehas no framework (like the pieces that form the bor-der) to begin developing the story. Moreover, the his-torian knows that he will never possess all pieces ofthe puzzle (except in trivial cases). So, at some point,he will have to conjecture (or imagine) the potentialshape, size and design of some of his pieces. But tomake the situation even more difficult, others mayvisualize the image(s) he wishes to present in a differ-ent way. Furthermore, even if they have the same gen-eral perspective, another colleague may visualize anindividual piece or sections of the puzzle in a differ-ent way. The reconstruction of the past is never final,unique or totally accurate; but a plausible and logicalsynthesis of the available evidence.

Most importantly, after a historian has developed aplausible interpretation of the evidence, he has thedifficult task of proving the soundness of his formu-lation, just as a mathematician must prove a new re-sult. It is not acceptable just to assert a new point ofview. Knorr [9], in his monograph discussing the his-tory of some pre- Euclidean concepts, provides a con-vincing elegant argument for the specific date for thediscovery of incommensurable quantities and a gen-eral goal for the entire corpus of Euclid's Elements.Then, he goes on to argue why this is a plausible case.

We have considered several important and profounddifferences in the methodologies used in mathemat-ics and historical research. One of the most importantdifferences is apparent when the scholar begins a new


project. The mathematician is usually aware of thefuture outcome of his/her research (e.g., a proof ofthe well-ordering theorem, a proof of Fermat's lasttheorem, etc.). It is usually extremely difficult to reachthe goal (sometimes it has taken more than 300 years!),but, at least, the goal is known. On the contrary, thehistorian does not usually know what conclusions he/she will make. He does not yet know the key ques-tions, concepts and results. Most importantly, the his-torian is unaware of the driving factors that influenceda mathematician’s career. To some extent, it may bevery simple to list and describe the publications of acertain individual. But it might be extremely difficultto learn what influences affected the mathematician’sideas and contributions. On some occasions, afterreading dozens of articles or hundreds of manuscriptunpublished folios, which may or may not be relatedto each other, the historian might find himself in com-plete darkness, without understanding the pivotalideas behind the mathematicians's concepts and pub-lications. As May pointed out, despite much hardwork and effort in conducting his/her research(searching, finding, organizing and selecting), the his-torian may not be able to explain what underlay amathematician's thinking.

REFERENCES1. Mortimer Adler and Charles van Doren. 1972. How to read a

book. New York: Simon & Schuster.2. American Heritage Dictionary, The. 1985. Boston: Houghton

Mifflin Company. (2nd college ed).3. Eric T. Bell. 1937. Men of Mathematics. New York: Simon

and Schuster.4. Carl B. Boyer. 1968. A history of mathematics. New York:

John Wiley & Sons, Inc.

5. Stillman Drake. 1978. Galileo at work. His scientific biogra-phy. Chicago: Chicago University Press.

6. José Ferreirés. 1999. Labyrinth of thought. A history of settheory and its role in modern mathematics. Basel: Birkhäuser.

7. Leopold Infeld. 1948. Whom the Gods Love. New York:Whittlesey House.

8. Hubbert Kennedy. 1972. "Who discovered Boyer's law?"American Mathematical Monthly 79: 66-77.

9. Wilbur Knorr. 1975. The evolution of the Euclidean elements.Dordrecht.

10. Alexandre Koyré. 1966. Etudes galileennes. Paris: Hermann.1966.

11. Kenneth O. May. 1974. Bibliography and research manual ofthe history of mathematics. Toronto: University of TorontoPress.

12. Constance Reid. 1993. The search for E T Bell, also knownas John Taine. Washington, DC: Mathematical Associationof America.

13. K. Ribnikov. 1987. Historia de las Matematicas. Moscow: Mir.[Translated into Spanish by Concepción Valdés from the origi-nal Russian edition 1974].

14. Francisco Rodriguez Consuegra. 1991. The mathematicalphilosophy of Bertrand Russell. Basel: Birkhäuser.

15. Anthony Scaduto. 1976. Who Killed Marilyn? New York:Woodhill.

16. Milo A. Speriglio. 1983. Marilyn Monroe: Murder Cover-Up.New York: Seville.

17. Anthony Summers. 1980. Conspiracy. London: VictorGollancz.

18. ——. 1985. Goddess: the Secret Lives of Marilyn Monroe.New York: Macmillan.

19. Pierre Thuillier. 1983. "Galileo et l'expérimentation." La Re-cherche 14 (# 143, Avril). Pp. 584-597.

20. Raymond L. Wilder. 1944. "The nature of mathematical proof'.The American Mathematical Monthly 51: 309-323.


INTRODUCTIONWhat kind of contribution can education make insupplementing what takes place in the political arenaso that future generations will be less prone to definetheir personal and national self- interest primarily inadversarial and military terms?

This article is based upon the assumption that educa-tion can make a difference. How much of a differenceit can make and how long we will have to persist be-fore we achieve results that are more than negligibleis an issue of great concern. But we do need to makeat least modest beginnings if there is any hope thatwe will not blow each other up in a worldwide effortto eradicate differences. As a beginning, we need tothink of education in all areas from a more humanis-tic perspective.

My particular interest is with the subjects of math-ematics, technology, and the natural sciences. In con-temporary primary and secondary education they are,for the most part, presented as devoid of emotional,cultural, and humane values.

There are many reasons for concern over this amoral,ahistoric, and isolated conception of the scientific dis-ciplines. Mathematics in particular is usually per-ceived by the public, and has generally been presentedby teachers, as an area of knowledge characterizedalmost exclusively by facts and truths. What can bechanged in the education of mathematics teachers thatwill enable them to participate in a more humanisticcurriculum that acknowledges diversity?

Something of particular concern to this author is thatthis kind of narrow approach that does not refer tobasic human values tends to ignore differences in stu-dents' religious and cultural backgrounds, as well asthe traditional and cultural quality of the mathemati-cal experience. Thus, the approach does not strive to-wards better understanding among students from

different backgrounds, nor towards having them seeka common ground. Not only is there a total lack ofconnection with emotional and humane values thatmight enable students to appreciate their own andeach others culture, but by ignoring the lengthy birthpangs of the evolution of ideas, this warped view ofthe scientific disciplines inaccurately portrays the na-ture of mathematical discovery itself.

Though in general, students are more diverse than wetend to acknowledge, I am concerned in particularwith the differences among the students whom I teach,who come from two very different cultures. I teach insouthern Israel, at Kaye College, a teachers' collegethat attracts both Jewish and Bedouin students. KayeCollege was established in the 1950s as a teachers'seminary, and in 1982 was transformed into a college.Since the early 1970s, it has offered teacher trainingfor all teaching tracks, including the Bedouin sectortrack. Efforts to attract Bedouin began three years ago,when the college designed a "Couples Project" courseto address curricula that acknowledges the differentcultures.

While formal education has been part of Jewish tradi-tion for centuries, the nomadic Bedouin in this regionfirst began sending their children to educational set-tings in 1948, when the State of Israel was established.Today, the majority of Israel's Bedouin have settled inpermanent cities, towns, or villages and work in agri-culture, construction, and education, among otherfields, and the number of Bedouin students in highereducation is constantly increasing. The Israeli govern-ment looks to education as means of Bedouins' inte-grating into Israeli society (Mofet, 2001; Ben-David,Y., 2001).

The students from both cultures come together in acourse I have designed entitled "History of Mathemat-ics and its Interlacing in Teaching Mathematics". Theyknow very little about the development of mathemat-

Humanizing Mathematics: The Humanistic Impression in theCourse for Mathematics Teaching

Ada KatsapKaye College of Education

Beersheva, Israel


ics, let alone the contributions each of their cultureshas made to this development.

There is a need to address the question of what can bechanged not only because of the specific cultural di-versity I face with my own students, but because ofcultural diversity throughout much of the world. Bychanging the basic principles of, and setting new goalsfor mathematics teaching, we can extricate ourselvesfrom the discipline's silence on these vital matters.Teachers must be equipped with strategies for con-veying mathematical knowledge that transcend skillsfor structuring logical foundations of scientificthought. Pupils must not only assimilate a concentra-tion of logical phrases and ways of thought that un-derpin mathematical knowledge, but they must learnto do so through a process that reflects, at least to someextent, the historical ways by which humankind ar-rived at such knowledge.

HISTORICAL AND HUMAN PERSPECTIVESThe mathematical text, with its abundant symbols,poses numerous difficulties - even impossibilities - formany studying such a seemingly bland subject. Forthis reason, teaching ways of making the mathemati-cal text come alive is of the utmost importance. Oneidea that has gained momentum in the past few de-cades is teaching mathematics through history.

Many researchers acknowledge the importance of his-tory of mathematics in mathematics teaching andlearning. Garner (1996), who reviewed the profes-sional literature on the issue, concluded that "the studyof history is essential for those who would attempt toteach mathematics." In a previous article, Garner(1995) claims that "students may be brought to a moremeaningful understanding" of mathematics topics and"gain insight into how pure mathematics feeds appliedmathematics in [which] seems abstract." According toHarakbi (1994), a "retrospective look at the historicaldevelopment of mathematics allows the teacher torefresh and deepen both the understanding of a spe-cific topic and didactic ways of presenting it. Engag-ing in the history of mathematics requires adoptionof different points of view, including exposure to theobstacles and prohibitions mathematicians encoun-tered on their way to glory. Therefore, experience inthis area can be a factor in reducing anxiety aboutmath" (Harakbi, 1994).

But history has much more to offer than merely re-vealing the steps through which an idea evolved -which itself can be viewed as yet another dry exposi-tion to be absorbed. Educators often assume that thereis no connection between the mathematical experienceand emotional concomitants. In fact, emotional com-ponents are lacking at every turn in the developmentof mathematics and of personal and historical effortsto understand it. To remedy this situation, the studyof an historical development of mathematics shouldnot only reveal the evolution of ideas, but also inves-tigate the emotional arena within which these ideasdeveloped.

In a letter to his son, when the latter was on the vergeof creating non-Euclidean geometry, Bolyai's fathermade a telling remark: "For God's sake, please give itup. Fear it no less than the sensual passion, because ittoo may take up all your time and deprive you of yourhealth, peace of mind, and happiness in life" (Boyer,1968). Yet Sizer (1984) wrote, "Learning is human ac-tivity, and depends absolutely on human idiosyn-crasy." The latter statement is based on the idea thatmost prospective teachers do not begin their math-ematical studies with human idiosyncrasy. Rather, itis obvious that the opposite is true. We must create anatmosphere and conditions in the classroom that areconducive to understanding, and teaching studentsto tolerate, what is different not only between twoindividuals from different cultures, but also betweenindividuals who are apparently highly similar in say,two people from the same family.

The purpose of this paper is to show how a course in"History of Mathematics and its Interlacing in Teach-ing Mathematics" provides an opportunity for all in-volved to develop introspection about the nature ofmathematics, while at the same time refreshing andstrengthening the participants' espoused and non-es-poused humanistic values. The paper examines howthe history of mathematics can aid in the teaching ofmathematics to students, in the following ways:

• Acknowledging the idiosyncratic world views ofthe mathematicians who developed the ideas

• Seeking an arena within which we can arrive at abetter understanding of the emotional compo-nents in the development of mathematics


• Celebrating cultural diversity

• Intensifying a humanistic world view.

THE COURSE DESCRIPTIONA course such as the one we are discussing would tra-ditionally be called "History of Mathematics," and thiswas the case at Kaye. However, some four years ago,the name of the course was changed to "History ofMathematics and its Interlacing in Teaching Math-ematics," with the aim of emphasizing the humanis-tic aspect of mathematics teaching/learning. Alongwith the new course title came a new syllabus, whichwas constructed both to improve teachers' attitudestowards teaching mathematics and to expose them todiverse ways of learning mathematics.

The course participants are from two sectors: in-ser-vice teachers, ranging from novice to veteran, cur-rently enrolled in a B.Ed program, and preservicemathematics teachers in their third year of collegestudies. Both groups include Jewish and Bedouin stu-dents. This diversity creates a rich opportunity fordevising cooperative learning strategies in the class-room.

A discussion of the syllabus is held every year, at thebeginning of the course; the discussion focuses onsyllabus goals, content expectations, and terms. Thediscussion intensifies when course participants areinformed that they will be: a) preparing teaching unitsduring the school year; b) participating with eachother in presentations; c) working in pairs or in three-somes; and d) becoming acquainted with the unit theywill be preparing, which is to be included in a vol-ume that is a compilation of their collective work overthe year.

What usually occurs at this point is that the students,who have never met before, look each other over andopenly or covertly seek out a partner. In this first les-son, the instructor does not dictate whom the studentsshould choose. Later, should anyone seek advice, theinstructor may make a suggestion. As for pairing upBedouin and Jewish students, I direct the students tothink crossculturally, aiming for at least some mixedpairs, in accordance with the integrated tone of KayeCollege. It should be noted that although thepreservice teachers are already accustomed to work-ing in pairs during their studies, the in-service teach-

ers may not be. Another issue is how to combine his-tory content with mathematics content in the lessonset - that is, how to combine the science of symbolswith the humanities when that science ostensibly ne-gates any involvement with the humanities.

The first semester is devoted to familiarizing studentswith the chronological development of mathematics,reading and understanding texts from the history ofmathematics, and attempting to make these texts moreaccessible to and appropriate for pupils by turningthem into an integral part of the mathematics lesson.Most of the participants already have some knowl-edge of the better-known mathematicians and theirdiscoveries.

I present the students with examples lesson plans com-piled either by myself or by students of this course inprevious years. These lesson plans give rise to intense,critical discussion, sometimes positive and sometimesnegative. Often, comments such as "We'll do it differ-ently" are heard; these comments indicate the teach-ers' caring and their desire to create a better product.

One lesson set, for example, is "Area: What is it?" Thisset has three sections, each focusing on a differentsphere of the mathematics of area: geometry, geomet-ric algebra, and algebra. Here, historical material iscombined with mathematical material, and discus-sions of important personages in mathematics are setagainst solving mathematical problems (Katsap, 2000).Another example is "The Golden Section," which com-bines knowledge of ancient Greece, architecture, art,the works of Leonardo Da Vinci, and the beauty ofnature. These elements touch on much of mathemat-ics as well as on many humanistic areas in whichmathematics plays an integral part. This first semes-ter focuses on "classroom cohesiveness" and work insmall groups. The students continually present theproducts of their small-group work to the class; later,these products will include units prepared by the stu-dents. This develops the students' openness and theirdesire to help other group members and to obtain helpfrom them, and also fosters healthy competition andan orientation towards achievement.

At the mid-point of the first semester, the studentsare given guidelines for preparing their units, as fol-low:


• The selected unit topic is to be integrated intomathematical areas studied in school

• There is no restriction on historical eras

• Relevant areas from the history of mathematicscan include: individuals who made a contributionto mathematics; mathematics problems; math-ematical discoveries; and any historical materialconnected with mathematics

• The study unit may: a) combine an historical math-ematical text with a mathematical topic relevantto teaching in the form of a complete lesson plan;or b) integrate a sequence of passages into lessonplans for the study of a particular chapter in math-ematics in the school.

The variety of options offered the students in choos-ing a study unit makes the choice easier for them, andgives them a greater sense of independence and self-confidence. At the same time, however, it means thatthey must take responsibility and demonstrate com-petence, criticism, and autonomy in their decision-making. All this may sound familiar to instructors ofteacher education courses, but again, combining ahumanistic subject such as history with a scientificsubject such as mathematics is somewhat unusual.After all, when have mathematics teachers had tostruggle with the question of whether to present anhistorical tale to the pupils and allow them to discovermathematical elements in it, or to teach them math-ematical laws (as is usually done), and then tell themabout the life and times of the individual who discov-ered the laws? Or, if an interesting historical discov-ery does not fit in with the mathematics curriculumbut the teacher finds it fascinating and wants topresent it to the pupils, how should it be combinedwith the regular mathematics lesson? In cases, theteacher (i.e. the students in my course) must take re-sponsibility and use his or her autonomy to make adecision.

Towards the end of the first semester, students choosepartners, and a timetable for the students' second-se-mester presentations of the study units is drawn up.Each student pair begins the process of selecting atopic to be investigated, prepared, and presented tothe class in a 45-minute presentation. Although thisprocess is already familiar to the students from their

previous studies, the innovation here is the mannerin which the students repeatedly consult with eachother before approaching the course instructor forhelp. Most of those who do consult with the instruc-tor need help in choosing and organizing the mate-rial collected in their search, or in finding written andelectronic information sources. The Internet plays atremendous role in knowledge sources, and learningto use it is an additional learning skill that this coursehelps develop. Most of the topics and content chosenby the students are accepted.

Some of the students "try out" their study units firston their own pupils in the classes they currently teach.They then can tell their fellow course participants howthe pupils responded. At the end of the unit presenta-tion, a discussion, usually quite lively, is held in thecourse class; usually both constructive and negativecriticism is offered. Often, there is a sense that the stu-dents say what they have to say for themselves morethan for others, and give themselves tips for the fu-ture.

Not surprisingly, none of the lessons presented by thestudents in the second semester take the form of a lec-ture. They all involve group class activity, whether itbe collaborative work, discussion, or drawing conclu-sions in groups and in full class discussions—provid-ing a welcome experience in changing teaching meth-ods and seeking diverse ways of improving teaching.

At the end of the second semester, after all work hasbeen presented in the class, two students are chosento compile the material on the computer and, togetherwith me, edit the volume. These students reported thatthey found it very rewarding to process the separatestudy units presented during the course (to be dis-cussed later) into a structured volume. The studentsalso participate in discussions to determine topic clas-sifications and to organize the study units accordingly.Each of the topics includes four to five units. Some ofthe titles that have come up in past courses are: Math-ematics in Judaism and Islam; From Primitive Count-ing to Probability; Numerals, Numbers, and Sets ofNumbers; Science, Art, and Craft; and Journey to theRoots of Geometry. The students claim that this workgives them a sense of contributing to the course's col-lective effort to promote advancement and innova-tion in mathematics teaching in the school. To date,four volumes have been published.


What is challenging in this course is the need for theinstructor to flow with the class, never knowing whattopics will be raised by the students during the year.This keeps the instructor on his or her toes, and stimu-lates the students' and the instructor's interest andcuriosity. Together, the instructor and the studentsform a learning community requiring constant com-munication and cooperation. Friere sees such a situa-tion as the ideal learning circumstances for liberatingeducation, in which "teacher and pupils form a com-munity of learners, and where both sides are essen-tial factors in the process of obtaining knowledge"(Friere & Schor, 1990).

UNIQUE TOPICSSome extremely interesting topics arose in this coursefollowing the students' extensive searching for knowl-edge sources. Before I compiled the course, I discov-ered the book Ayil Meshulash (Stature of the Triangle)(1960), written in the 19th century by a student of theJewish Torah scholar the Vilna Gaon (Genius fromVilna). It was based upon notes found after the Gaon'sdeath, that derived from his oral mathematics teach-ing. In this book, subtitled "On the wisdom of trianglesand geometry and some rules of qualities and alge-bra," author Shmuel Lukenik notes that the Gaon wasputting the notes and explanations in book form topreserve them for generations to come.

The Vilna Gaon, or Rabbi Eliyahu was a master ofTorah, Talmud, Jewish philosophy, Halacha (Jewishreligious law), and Kabbalah (Jewish mysticism). Thebulk of his written work concerns corrections andemendations of Talmudic texts, and interpretations ofthe Shulchan Aruch (code of Jewish law). In the aca-demic field, the Vilna Gaon wrote on geography (TheForm of the Earth) and grammar (Eliyahu Grammar).He was also interested in music, claiming that mostof the arguments of the Torah would be incompre-hensible without it (On-line Resources, 2001). Accord-ing to some sources, the Vilna Gaon is the author of"The Gaon's Theorem," a principle of the mathemat-ics of infinity (Feldman, A., 1999). Other sourcesclaimed that this theorem was called "Kramer's Theo-rem," Kramer being the Vilna Gaon's family name.Gerver (1993), however, stated that this suppositionwas unlikely, as the author of this article did notpresent proof for these two opinions.

After reading Ayil Meshulash, I had the idea of in-

cluding in the course Jewish and Islamic sources ofmathematics history and integrating them in math-ematics teaching. The result was a collection of mate-rials included in a chapter entitled "Mathematics inJudaism and Islam," unique to this course. Every year,a number of student pairs, both heterogeneous (Jew-ish and Bedouin) or homogeneous, work on this chap-ter.

Ayil Hameshulash was frequently chosen for investi-gation by these students, as it is both a mathematicalwork including mathematical explanations and geo-metrical definitions, theorems, and proofs as well asselected topics in algebra, and an historical document.Although the book was written 200 years ago, it stillcontains explanations that cannot be found today inany other mathematical work. The students in thecourse maintain that this book offers better explana-tions than those used today, and often say that theyfind it exciting to study such a venerable work. In oneof the lectures, the students compared the Vilna Gaon'spresentation of the right-angle triangle with that ofIsraeli mathematics teacher Benny Goren in his bookPlane Geometry - a text currently in wide use in Is-raeli schools. Most of the students in the class an-nounced that they would adopt the former methodin teaching this topic in their classes.

CULTURAL DIVERSITYNot surprisingly, homogeneous Jewish student pairsusually seek material on Jews who contributed tomathematics, as well as material found in the Torah,which is full of mathematics (the Jewish tradition ofinterest in studying mathematics comes from the To-rah).

Homogeneous Bedouin pairs of students usually seekmathematical material authored by Arabs. It is inter-esting to note that the mathematicians they chooseare often already familiar to them, but only as clericsor poets. As one of these is the Persian mathemati-cian, astronomer, philosopher, physician, and poetcommonly known as Omar Khayyam. Khayyam maybe best known for The Rubaiyat, but his bookMaqalatfi al-Jabr wa al-Muqabila is a masterpiece onalgebra and is of great importance in the developmentof algebra.

The Bedouin students' discovery of so many signifi-cant mathematicians gives them a different view of


mathematics, as well as greater pride in their people.In addition to their investigation of these individualsand their contributions to mathematics, the Bedouinstudents bring in Koranic writings, and all the stu-dents work on mathematical problems foundtherein—thus imbuing them with greater pride intheir Arab heritage.

Investigating the Jewish and Arab mathematiciansarouses a great deal of interest among the studentsparticipating in the course. As it is the first opportu-nity both the Bedouin and the Jewish students haveto interpret the original mathematical language of theVilna Gaon, in the ancient Rashi Hebrew of the time,as well as the mathematical language of the ArabicKoranic text. Thus, the students unwittingly acquirehumanist values, such as respect for the history andtradition of their own and other peoples.

REFLECTIONS ON HUMANISTIC EDUCATIONWhat does it mean for education to be humanistic?This course enabled the instructor to come up with anumber of tentative answers to this question. Brown(1996), who discussed humanistic mathematics edu-cation, claimed that first of all, there is a need to deepenthe understanding of the perception of humanism andhuman nature, and their lengthy history. After that,interpretations of education should be studied indepth, along with the development of the humanisticperception. Finally, mathematics itself should be re-assessed. In another article, Brown (1993) asks: "Howmight we use mathematics to convey knowledge andattitudes towards the world and about oneself thatwould be valuable in many non-mathematical con-texts?"

As of this point, I am currently investigating in morethan anecdotal way the implications of a course of thissort at Kaye College. Nevertheless, initial findingsfrom the pilot study show that most of the studentsgained a different view of mathematics. This was in-dicated by their considerable appreciation and esteemfor the value of the history of mathematics, and for itsinfluence on their view of mathematics and mathemat-ics teaching in the school. The students became moreopen to and more willing to accept knowledge accu-mulated through history that could be defined as ahumanistic value - i.e. appreciation of the evolutionof ideas of the many false starts that are part of thejourney.

Another humanistic value mentioned by the studentsin the pilot study was respect for different culturesand appreciation for other eras. This may sound ob-vious for history or literature classes, but it is rare withregard to mathematics classes. The pilot study alsofound that the course gave the participants a sense ofpower, and taught them to deal with new challenges.The covert competition that developed between thestudent pairs in presenting their material forced themto invest time and effort in a search for interesting,creative material and to find diverse ways of present-ing their topic—and also to keep trying. All this canbe classified as the humanist value of self-actualiza-tion.

One of the most important comments that came up inalmost every lesson, in one way or another, reflectedthe students' genuine desire to pass the torch to theirown pupils, and to make them aware of what theythemselves had learned and experienced. This wasevident among both the experienced teachers and thepreservice teachers, and was evidence of these teach-ers' caring and sensitivity towards their pupils. I of-ten heard the teachers say, "Let's send them [the pu-pils] to find the material themselves"; "Let's let thepupils prepare and teach a lesson, so they can dis-cover things on their own." This leads us to anotherhumanistic value: expressing humanity and respectfor others.

Finally, the students arrived at the material throughself-discovery. The experience of searching for thematerial and familiarity with and use of knowledgesources not previously accessible led to a humanistvalue that they expressed both orally and in writing:showing intellectual interest.

CONCLUSIONIncreasing teachers' awareness of the development ofmathematical knowledge can help them fulfill theirprofessional commitment, even narrowly construedas teaching pupils and instilling in them habits of sci-entific thought. Naturally, their pupils do not absorbonly the knowledge; they undergo a learning processthat reflects, in one way or another, the process expe-rienced by humankind in attaining this mathematicalknowledge.

Teachers' lack of knowledge - or their attitude of out-right dismissal - when faced with the question "Why?"


can make pupils passive, indifferent, or even averseto mathematics. Instead, teachers can encourage pu-pils to ask, "Why do I do this?" or "How can I explorethis?" Here, I do not refer to the knowledge of expertsin the history of mathematics, as that knowledge ispurely academic and has little to do with mathemat-ics education in the school. I refer only to relevant top-ics and areas of mathematics that enrich mathematicsteaching, thus enabling the mathematics teacher tofoster an atmosphere allowing mind and emotion totake flight together. Since the dawn of civilization,wise men and women have advanced mathematicalknowledge, and have also educated the younger gen-eration in "the qualities of the whole person." Muchof the teachings of Socrates, Plato, Aristotle, and manyothers are devoted to humanistic education. Amongthe principles of Platonic humanist perception, we findvalues such as "Worthless is the life of he who liveswithout investigation and a critical view of what isgood," alongside, "The full measure of a man is in hiswisdom and knowledge" (Aloni, 1998).

But for pupils today as well as for most teachers,Pythagoras's theorem is no more than c2 = a2 + b2.Knowing that this theorem has been in existence acrossthe world for over three thousand years, and how ithas been proven in many different ways, may excitethem and lead them to join the ranks of those who seethe beauty behind the rigid and ostensibly inimicallanguage of mathematics.Hersh (as cited inBrockmam, 2001) states:

Mathematics is neither physical nor mental,it's social. It's part of culture, it's part of his-tory, it's like law, like religion, like money, likeall those very real things which are real onlyas part of collective human consciousness.

The latter statement may be made more obvious while,in the marvelous book Even-Zifer the Doctor, Sissa(1960) tells the 12th-century story of the friendshipbetween a Jewish doctor from Cordoba and the Chris-tian royal scribe from Hungary, a friendship based onthe love shared by seekers of knowledge in the DarkAges and drawing on the human spirit. Both wereenamored of the enchanted science called mathemat-ics. Sissa writes, "Is it possible to integrate science intoa story? Is there not something amazing in the his-tory of science, as surprising and as fascinating as thehistory of nations?"

It is no wonder, then, that the students in the coursediscovered another world within a subject so familiarto them. As they explored the depths of history, thestudents learned of the hieroglyphics in the Rind andMoscow papyruses, mathematical discoveries fromancient Greece through the Middle Ages, stories ofcharity from the Koran, and astronomical calculationsfor the Jewish calendar. In addition, they looked atmodem mathematics and mathematicians such asDescartes, Euler, Gauss, and Lobadievsky.

It may be debatable whether there is a parallel betweenthe historical development of mathematics and theprocess of clarifying values expressed by studentsparticipating in the course. But the contact betweenthe two areas—mathematics and the history of math-ematics—as well as appropriate training methodsgave rise to positive humanistic energies and a newview of the subject that the students had chosen toteach. It is to be hoped that this learning added an-other aspect to their professional humanist develop-ment as teachers.

Today's focus on technological means cannot replacethe crucial human figure of the teacher, who is respon-sible for fostering pupils' curiosity about mathemat-ics—a curiosity rooted in human emotion. Teachersconstantly face the challenge of renewing educationand ways of learning in teaching mathematics in or-der, as Hook (1994) puts it, to shape teaching practicein our society "and [to] create new ways of knowingand different strategies for the sharing of knowledge."

ACKNOWLEDGMENTSThe author is grateful to Prof. Stephen I. Brown forhelpful comments on a previous draft of this article.

REFERENCESAloni, N. (1998). To be an individual: Ways in humanist educa-

tion. HaKibbutz HaMeuhad Publishers.Ben-David,Y.,(2001), The Bedouin in Israel. Israeli Foreign Minis-

try, Jewish Virtual Library, AICEBoyer, C.B. (1968), History of mathematics. John Wiley and Sons

Inc., BostonBrockman, J. (2001), What kind of thing is A number? A talk with

Reuben Hersh. EDGE Third Culture.Brown, S.I. (1993), Mathematics and Humanistic Themes: Sum

Considerations, Chapter 26 in Problem posing: Reflectionsand applications. Hillsdale, NJ: Hove and London: Edited byS.I. Brown and M. Walter, Lawrence Erlbaum Associates, pp.


249-278.Brown, S.I. (1996), Towards humanistic mathematics education,

in (first) international handbook of mathematics education,edited by Alan Bishop. The Netherlands: Dortrecht: KluwerPress, 1289-133 1.

Friere, P. & Schor, Y. (1990). The pedagogy of liberation. Tel Aviv(Hebrew).

Feldman, A.,(1999), Violence and the Gaon's principle. Jerusa-lem: Yad Yosef for the Enhancement of Torah and Judaism.Edited by A. Feldman, Vol. 2, No. 5, June 23.

Garner, M. (1995). Historical essays in calculus and pre-calculus:Teaching mathematics through history. Paper presented atInterface 1995, Atlanta, GA, October.

Garner, M. (1996). The importance of history in mathematics teach-ing and learning. Paper presented in October 1996

Gerver,M.(1993), Kramer vs. the GR"A. Mail. Jewish Mailing ListVol. 7 Number 54. 21 May.

Harakbi, A. (1994). Using the history of mathematics in training

teachers in school classrooms. Lecture at one-day confer-ence, "Increasing the Level of Mathematics Teaching in theElementary Schools"

Hooks, B. (1994). Teaching to transgress. New York: RoutledgeKatsap,A. (2000), History of Mathematics: "Area: What is it?"

Workshop at conference, edited by Mathematics Department.October 2000. Kaye College of Education

Mofet (2001), Focus on Colleges: Kaye Teachers College. P. 8.On-line resources, 2001, Torah from Dixie, Inc. Vilna Gaon.Vol. 7, Issue 35. September 8 2001.

Sissa, A. (1960), Even-Zifer the Doctor. Shaharit Books, Ma'arivPress.

Sizer, T. R. (1984). Horace's compromise: The dilemma of Ameri-can high schools. Boston, MA: Houghton Mifflin

Vilna Gaon (1960), Ayd Meshulash (Stature of the triangle): Onthe wisdom of the triangle and geometry and some rules onqualities and algebra. Jerusalem

Such a really remarkable discovery. I wanted your opinionon it. You know the formula m over naught equals infinity,m being any positive number? [m/0 = ∞]. Well, why notreduce the equation to a simpler form by multiplying bothsides by naught? In which case you have m equals infinitytimes naught [m = ∞ x 0]. That is to say, a positive numberis the product of zero and infinity. Doesn't that demon-strate the creation of the Universe by an infinite power outof nothing? Doesn't it?

Aldous Huxley, Point Counter Point(1928), Chapter XI

INTRODUCTIONWe are living in a mathematical age. Our lives, fromthe personal to the communal, from the communal tothe international, from the biological and physical tothe economic and even to the ethical, are increasinglymathematicized. Despite this, the average person haslittle necessity to deal with the mathematics on a con-scious level. Mathematics permeates our world, of-ten in "chipified" form. According to some theologies,God also permeates our world; God is its origin, itsultimate power, and its ultimate reason. Therefore itis appropriate to inquire what, if anything, is the per-

A Brief Look at Mathematics and TheologyPhilip J. Davis

Brown UniversityProvidence, RI

ceived relationship between mathematics and God;how, over the millennia, this perception has changed;and what are its consequences.

I begin with two stories. Recently, I spread the wordquite among my mathematical friends that I had beeninvited to lecture on mathematics and theology. Iwanted to get a reaction, perhaps even a suggestionor two.

One, a research mathematician, the chairman of hisdepartment, who, in his personal life would be con-sidered very devout in a traditional religious sense,told me that, "God could never get tenure in our de-partment."

Another friend, well versed in the history of math-ematics, told me, "The relation between God andmathematics simply doesn't interest me."

I think that these two reactions sum up fairly well theattitude of today's professional mathematicians.Though both God and mathematics are everywhere,mathematicians tend towards agnosticism; or, if reli-gion plays a role in their personal lives, it is kept in a


separate compartment and seems not to be a sourceof professional inspiration.

There is hardly a book that deals in depth with the4000 year history of the relationship between math-ematics and theology. There are numerous articles andbooks that deal with particular chapters of the story.Ivor Grattan-Guinness has written on mathematicsand ancient religions. Joan Richards has treated theinfluence of non-euclidean geometry in Victorian En-gland. Joseph Davis and others have treated the at-tempts at the reconciliation of science and religion byJewish scholars of the seventeenth century. But mosthistorians of mathematics in the past two centuries,under the influence of the Enlightenment and of posi-tivistic philosophies have avoided the topic like theplague.

This suppression has been an act of "intellectualcleansing" in the service of presenting mathematicsas a pure logical creation, "undefiled" by contact withhuman emotions or religious feelings. It parallels themany acts of iconoclastic destruction that have over-taken civilization at various times and places and isstill taking place. Why has it occurred? Numerousreasons have been suggested. Is it the Enlightenmentand positivistic philosophies?

But things are now changing. The separation of math-ematics and theology is now not nearly so rigid as ithas been since, e.g., Laplace's day. There is now a sub-stantial reversion in physics, biology, mathematics, etcto the older position. The material published runs fromwhat is very thoughtful and sincere to what might becalled "crazy." (And what is the test for what is andwhat is not "crazy"?).

Why? Is it part of the general perception that ratio-nalism has its limitations? The current generation findspositivistic philosophies lacking in social and emo-tional warmth and in transcendental values. It is nowtrying to reclaim those values with syntheses of God,the Bible, Apocalyptic visions, the Nicene Creed, Zero,Infinity, Gödel's Theorem, Quantum Theory, theOmega Point, the God Particle, Chaos, Higher Dimen-sions, Multiple Universes, Neo-Pythagoreanism,Theories of Everything, etc. etc. I find that most ofthis is bizarre. When it comes to specific statements,such as "God is a mathematician", I find the discus-sions both pro and con unconvincing, but I would not

say, as an older generation of positivists might havesaid, that the statement is meaningless.

The extent of the historic relationship between math-ematics and theology should not be underestimated.There is much that can be and has been said. Practi-cally every major theme of mathematics, its concepts,its methodology, its philosophy, have been linked insome way to theological concepts. Individual math-ematical features such as number, geometry, pattern,computation, axiomatization, logic, deduction, proof,existence, uniqueness, non-contradiction, zero, infin-ity, randomness, chaos, entropy, fractals, self-reference,catastrophe theory, description, modeling, prediction,have been wide open for theological questions andanswers.

As simple examples: is God constantly geometrizing?Does God have the power to make 2 + 2 other than 4?Does God predict or simply know?

The links between mathematics and theology are partof the history of mathematics and part of the math-ematical civilization into which we were born. Theyare part of applied mathematics. In recent years, theselinks have been extended to embrace theological re-lations to cognition, personhood, feminism, ethnicity.

The contributions of mathematics to theologicalthought have been substantial. The young John HenryNewman (1801-1890. Later: Cardinal) asserted that thestatements of mathematics were more firm than thoseof dogmatic theology. Hermann Cohen, philosopher(1842 - 1918) thought that mathematics was the basison which theology must be built. In many recent dis-cussions, as we shall see, mathematics takes objectivepriority over theology just as it did to CardinalNewman. On the other hand, one should remind one-self of the ascending hierarchical order in the days ofthe Scholastics (e.g., St. Thomas Aquinas, 1225-1274):mathematics, philosophy, metaphysics, with theologyat the apex.

In the other direction, the contributions of theologyand religious practice to mathematics were also sub-stantial - at least until around the 14th Century. Asexamples church and astronomical (secular) calendarsare mathematical arrangements and needed reconcili-ation. The Jewish philosopher and theologian MosesMaimonides (1135 - 1204) wrote a book entitled On


the Computation of the New Moon. Among Moslems,the determination of the qibla (the bearing from anyspot towards Mecca) was important and fostered thedevelopment of spherical trigonometry. These vari-ous demands led to improved techniques and theo-ries. Kim Plofker has only recently discussed the his-torical attempts to reconcile sacred and secular Indiancosmologies.

Astrology which very often had links to theology andreligious practice, demanded exact planetary posi-tions, and astrology stimulated and supported math-ematics for long periods of time and led to intellec-tual controversy. Astrology carries with it an implica-tion of rigid determinism and this came early intoconflict with the doctrine of free will. The conflict wasreconciled by asserting that though the stars at thetime of one's nativity control one's fate, God has thefinal say, so that prayer, repentance, sacrifice, etc.,undertaken as a free will impulse, can alter the astro-logical predictions. This is the message of ChristianAstrology, two books with the same title written cen-turies apart by Pierre Dailly (1350-1420) and by Will-iam Lilly (c. 1647).

With the discrediting of astrology as a predictive tech-nique (even as it remains a technique for individualsto shape their daily behavior), such contributions havecertainly been much less publicized or emphasizedin recent mathematical history than,, e.g., technologi-cal or military demands

On a much wider stage and at a deeper level, claimshave been made and descriptions have been given ofthe manner in which Christian theology entered intothe development of Western science. Here is the con-temporary view of Freeman Dyson:

Western science grew out of Christian theol-ogy. It is probably not an accident that mod-ern science grew explosively in Christian Eu-rope and left the rest of the world behind. Athousand years of theological disputes nur-tured the habit of analytical thinking that couldbe applied to the analysis of natural phenom-ena. On the other hand, the close historicalrelations between theology and science havecaused conflicts between science and Chris-tianity that do not exist between science andother religions.... The common root of mod-

ern science and Christian theology was Greekphilosophy.

The same claim might be asserted for mathematics,though perhaps with somewhat less strength.

A few western opinions over the ages, arranged moreor less chronologically, should give us the flavor, ifnot the details, of the relationship between mathemat-ics and theology. (See e.g., David King for Islamicwritings, and David Pingree and Kim Plofker for In-dian.)

However, while citing and quoting is a relatively easymatter, it is not easy to enter into the frame of mind ofthe authors quoted and of the civilizations of whichthey were part; how the particular way they expressedthemselves mathematically entered into the whole.Thus Plofker has written:

It is difficult to draw a clear and consistentpicture of the opinions of authors who rejectsome assumptions of sacred cosmology whileespousing others... To many scholars eager tovalidate the scientific achievements of medi-eval Indians according to modern criteria, thevery notion of their deferring to scripturalauthority [the Puranas] at all is something ofan embarassment.

To appreciate this, it helps to remember that the secu-larization and the disenchantment (i.e., disbelief inritual magic) of the world is a relatively recent eventwhich occurred in the late seventeenth century. Foran older discussion of this point, see W.E.H.Lecky.

To quote contemporary historian of mathematics IvorGrattan-Guinness:

Two deep and general points about ancientcultures are often underrated that people sawthemselves as part of nature, and mathemat-ics was central to life. These views stand incontrast to modern ones, in which nature isusually regarded as an external area for prob-lem-solving, and mathematicians are oftentreated as mysterious outcasts, removed frompolite intellectual life.

And David Berlinski, (contemporary, philosopher, and


science writer) writes:

As the twenty-first century commences, we arelargely unable to recapture the intensity ofconviction that for all of western history hasbeen associated with theological belief.

I now shall present numerous clips, mostly of olderauthors, organized according to certain mathemati-cal topics.

NUMBER

Perhaps the earliest mathematics/theology relation-ship is "number mysticism", the attribution of secretor mystic meanings of individual numbers and of theirinfluence on human lives. This is often called numer-ology and its practice was widespread in very ancienttimes. Odd numbers are male. Even numbers are fe-male. In Babylonia the numbers from I to 60 were as-sociated with a variety of gods, and these characteris-tics are just for starters. Since alphabetic letters wereused as numbers, the passage from numbers to ideasand vice versa was rich in possibilities.

"All is number," said Pythagoras (c. 550 BC), aroundwhom a considerable religious cult formed and whosecultic practices seemed to involve mathematics in asubstantial way. The historian of mathematics, CarlB. Boyer, wrote, "Never before or since has mathemat-ics played so large a role in life and religion as it didamong the Pythagoreans."

The words "or since" may be easily challenged with-out in the least denying the importance of mathemat-ics for the Pythagorean Brotherhood.

Some mathematical mysticism occurs in Plato'sTimaeus. There, Plato (c. 390 BC) takes the dodecahe-dron as a symbol for the whole Universe and says that:"God used it for the whole." For Plato, the world hasa soul and God speaks through mathematics.

Ideas of number mysticism spread from Pagan toChristian thought. The Revelations of John (c. end of1st Century) is full of numbers and of number mysti-cism. For example:

Here is the secret meaning of the seven starswhich you saw in my right hand and of theseven lamps of gold: the seven stars are the

angels of the seven churches, and the sevenlamps are the seven churches. (Rev. 1:20).

And then, there is the famous, oft quoted passage inRevelations 13:18:

...anyone who has intelligence may work outthe number of the Beast. The number repre-sents a man's name and the numerical valueof its letters is six hundred and sixty six.

Innumerable computations of the Second Coming, orof the Days of the Messiah have been carried out. Theidea that the end of the world can be computed is veryold.

The Apocalypse, foretold in Revelations, and said toprecede the Second Coming, has been and still is afavorite subject for mathematical speculation and pre-diction. The predictions are usually made along arith-metic lines and make use of some method of givingnumbers to the historic years. For the details of a com-putation of the date of the Apocalypse carried out byJohn Napier (15 50 - 1617), the creator of logarithmsand one of the leading mathematicians of his day, thereader is referred to the splendid book of KatharineFirth.

Religious authorities have often proscribed such com-putations. But such computations have never disap-peared and the desire to calculate the end of days ispresent in contemporary end-of-the-worldcosmologies based on current astrophysical knowl-edge as well as in tragic episodes of religious funda-mentalism.

Iamblichus (c. 250 - 330), a Neo-Platonist, in hisTheologoumena tes Arithmetikes (The Theology of Arith-metic) explains the divine aspect of each of the num-bers from one up to ten.

St. Augustine (354 - 430) asserted that the world wascreated in six days because six is a perfect number(i.e., a number equal to the sum of its divisors). Au-gustine also said: Numbers are the link between hu-mans and God. They are innate in our brains.

In the 12th Century, the Neoplatonist Thierry ofChartres opined: "The creation of number was the cre-ation of things."


The colorful mathematician and physician GeronimoCardano (1501 - 1576) cast a horoscope for Jesus andearned thereby the wrath of the hierarchy.

The humanistic Shakespeare, whose works displaylittle religious sentiment, has a line: "There is divinityin odd numbers." Was he perhaps picking up on theTrinity and the mystic number 3 ?

Blaise Pascal (1623 -1662), an early figure in the de-velopment of probability theory, "proves" the exist-ence of God by means of a wager:

God is or He is not. Let us weigh the gain andthe loss in selecting 'God is.' If you win, youwin all. If you lose, you lose nothing. There-fore bet unhesitatingly that He is. —Pensees.

"Pascal's Wager" has generated a vast literature of itsown.

Sir Isaac Newton, convert to (heretic) Arianism, al-chemist, theologian, (1642 - 1727), the "last of the ma-gicians" according to John Maynard Keynes, is so pre-eminent in mathematics and physics that the amountof material on his "nonscientific" writings—for longconsidered by historians of science to be an aberra-tion—is now substantial. See, e.g., James E. Force andRichard Popkin and also B.J.T Dobbs. Briefly, New-ton attempted to combine mathematics and astro-nomical science so as to prepare a revised chronologyof world history and thereby to understand the di-vine message. For example, we find in Newton's TheChronology of Ancient Kingdoms Amended:

Hesiod tells us that sixty says after the wintersolstice, the star Arcturus rose just at sunset:and thence it follows that Hesiod flourishedabout a hundred years after the death ofSolomon, or in the Generation or Age next af-ter the Trojan War, as Hesiod himself declares.

Also,

Newton saw his scientific work as evidenceof God's handiwork. He turned to religiousstudies later in life and considered it an inte-gral part of his thinking. Indeed, just as today'scosmologists are trying to find a 'Theory of

Everything' , Newton looked for a unificationof the sacred texts with his mathematico-physical theories. —Katz & Popkin.

In mathematician and clergyman John Craig's "Math-ematical Principles of Christian Theology" (1699),Craig calculated, based on an observed decline in be-lief and a passage in St. Luke, that the second comingwould occur before the year 3150. To a contemporarymathematician, Craig's reasoning is not unlike an ar-gument from exponential decay.

Expressions of number mysticism ebb and flow. Theyseem never to disappear entirely. Today, number mys-ticism is widespread. There are said to be lucky andunlucky numbers - an ancient idea. These selections,intended for personal use, are widely available inbooks and newspapers." Your number for the day is859." "In the year 1000 or 1666 or 2000 something goodor something bad will happen." The question ofwhether these kinds of assertions are "deeply be-lieved" is often irrelevant given the extent to whichits practice results in human actions.

Recently there were various to-dos about the new"Millennium", (including a billion-pound exhibitionin London) as though the year 2000 inherited mysticproperties from its digital structure. In the "Y2K flap",digital programming was indeed important, but theexcessive publicity and mild hysteria were hallmarksof a virulent attack of number mysticism.

The spirit of Pythagoras seems to have influenced thethought of a number of distinguished 20th Centuryphysicists. Arthur Eddington (1882-1944) and P.A.M.Dirac (1902-1984), for example, have searched forsimple whole number (i.e., integer) relationships be-tween the fundamental physical constants expressednondimensionally. Then, seemingly denying simplic-ity, Dirac wrote, "God is a mathematician of a veryhigh order. He used some very advanced mathemat-ics in constructing the Universe."

The number of amazing patterns that can be con-structed via simple arithmetic operations is endless,and to each pattern can be attributed mystic potencyor divine origin. Ivor Grattan-Guinness, who is also amusician and musicologist, in a section on "the powerof number" in his History of the Mathematical Sciences,gives instances involving Kepler, Newton, Freema-


sonry, and Bach, Mozart, and Beethoven. About W.A.Mozart (175 6-9 1), he says in part:

Mozart's opera The Magic Flute, written in 1791,to defend Masonic ideals against political at-tack, is crammed with numerology and somegematria. [Gematria = the identification of let-ters with numbers and used to arrive at in-sights].

But number, though operating within a theologicalcontext, is not always conceived within a mystic the-ology though we may now think otherwise. ThusLeibniz (1646-1716), "Cum Deus calculat etcogitationem exercet, fit mundus." (When God thinksthings through and calculates, the world is made.)

Today, we may omit "Deus" from this precept: via cal-culation we create everything from huge arches in St.Louis (which has a great spiritual quality), to designerdrugs or to the human genome map project. To somepeople, these numerical computations provide thelatest answer to the Biblical question, "What is manthat thou are mindful of him; the son of man that thoushouldst visit him?" (Ps. 8,4) without answering in theleast what the long-range effects of such computationswill be.

Thus, numbers. All of the instances cited, togetherwith those that follow in later sections, may be deemed"applied mathematics", for they apply mathematicsto human concerns and not to mathematics itself. Suchan expanded meaning would be in strong disagree-ment with the current usage of "applied mathemat-ics."

GEOMETRY; SPACE

We find in the Old Testament, Proverbs 8:27: "Hegirded the ocean with the horizon." The Hebrew wordfor gird is "chug." A mathematical compass is a"mechugah." Same root. God compasses the world.

The image of God as the one who wields the compasswas common. The Renaissance artists liked it anddrew it over and over. In Amos Funkenstein's splen-did Theology and the Scientific Imagination, you will findthat on his cover there is a mediaeval picture show-ing Christ measuring the world with a compass. Thecompass motif lasted well into the 18th century whenWilliam Blake (1757-1827, mystic artist and poet) pro-

duced a famous engraving that combined these ele-ments. Was this merely artistic metaphor, or was itstronger?

The world, therefore, was constructed geometrically.The classic statement is "God always geometrizes."

On a much more abstract level, Moses Maimonides(1135-1204, philosopher, theologian, and physician)denied the infinity of space. In this regard, he sidedwith Aristotle. On the other hand, Hasdai Crescas,poet and philosopher, (1340 - 1410) allowed it.

In Art: Dilrer (1471 - 1528), Michaelangelo (1475 -1564), and numerous other artists of the period, menwho were well versed in the mathematics of the day,looked for the divine formula that would give the pro-portions of the human body. The human body wasGod's creation and perfection must be found there.This perfection was thought to be expressible throughmathematical proportions.

Hermetic geometry (i.e., geometrical arrangementsthat were thought to embody occult or religious forces)abounded. Churches were constructed in the form ofthe cross. Secular architecture was not free of it: theCastel del Monte erected for Frederick IIHohenstaufen (1194 -1250), by Cistercian monks, dis-plays an intricate geometrical arrangement, a fusionof European and Arabic sensibilities, based on theoctagon and whose plan has been said to symbolizethe unity of the secular and the sacred.

In the Monas Hieroglyphica (heiros, Greek: sacred, su-pernatural) of John Dee (1564), mathematician, the firsttranslator of Euclid into English a man who was botha rationalist, an alchemist, and a crystal-ball gazer,delineates certain assemblages of figures that havepotency deriving from a mixture of their geometri-cal/astral/theological aspects.

Consider next the spiral. Much has been written aboutits symbolism: in mathematics, in astronomy, inbotany, in shells, and animal life, in art, architecture,decoration, in Jungian psychology, in mysticism, inreligion. To the famous Swiss mathematician JohannBernoulli (1667 - 1748) who created the mathemati-cally omnipresent Spiral of Bernoulli, its self-repro-ducing properties suggested it as a symbol of the Res-urrection, and he had its figure carved on his grave-


stone in Basle. Today, the double helix carries both abiological meaning as well as an intimation of humandestiny.

In my childhood, the circle persisted as a potent magicfigure in the playtime doggerel "Make a magic circleand sign it with a dot." The interested reader will findthousands of allusions to the phrase "magic circle" onthe Web. Magic ellipses or rectangles are less frequent.

The Buddhist mandalas which are objects of spiritualcontemplation, embody highly stylized geometricalarrangements. The amulets and talismans that areworn on the body, placed on walls, displayed in cars;the ankhs, the crosses, the hexagrams, the outlinedfish, the horseshoes, the triangular abracadabra ar-rangements and magical squares, the sigils (magicalsigns or images) of which whole dictionaries werecompiled in the 17th century, the hex signs placed onhouse exteriors, all point to geometry in the service ofreligious or quasi-religious practice.

There is a multitude of geometrical figures signs em-ployed in kabbalistic practices, each associated withstars, planets, metals, stones, spirits, demons, andwhose mode of production and use is specified rigor-ously. Wallis Budge, student of Near Eastern antiqui-ties wrote:

According to Cornelius Agrippa [physicianand magician, 1486 - 1535], it is necessary tobe careful when using a magical square as anamulet, that it is drawn when the sun or moonor the planet is exhibiting a benevolent aspect,for otherwise the amulet will bring misfortune

and calamity upon the wearer instead of pros-perity and happiness.

Let semanticists and semioticists explain the relation-ship between our geometrical symbols and our psy-ches for it lies deeper than simple designation (e.g.,crescent = Islam). The geometrical swastika, whichover the millennia and cultures has carried differentinterpretations, is now held in abhorrence by mostAmericans. The memory of World War II is certainlyat work here, but the geometry can go "abstract" andits meaning become detached from an original his-toric context.

Why has Salvador Dali (1904 - 1989) in his large paint-ing Corpus Hypercubus in the Metropolitan Museumin New York, placed a crucifixion against a represen-tation of a four dimensional cube? Art historian Mar-tin Kemp has commented:

Dali's painting does stand effectively for anage-old striving in art, theology, mathematics,and cosmology for access to those dimensionsthat lie beyond the visual and tactile scope ofthe finite spaces of up-and-down, left andright, and in-and-out that imprison our com-mon sense perceptions of the physical worldwe inhabit. The scientists' success in coloniz-ing the extra dimensions is defined mathemati-cally...

***To be continued...The remainder of this article, includ-ing bibliography and notes will be issued online inHMNJ #27.

Pat’s ProloguesIntroductions to the First Two Airings of Math Medley, A Radio Talk Show

Patricia Clark KenschaftDepartment of Mathematics and Computer Science

Montclair State UniversityUpper Montclair, New Jersey

Since May 1998 I’ve been hosting a weekly radio call-in talk show, “Math Medley.” It appears to be the firstof its kind, according to both the math and the radiograpevines. Each week I interview someone about“education, parenting, equity, and environmental is-sues with an underlying theme of mathematics,” to

quote my opening patter.

It’s been a joyous foray into show biz for a math pro-fessor. Everything is done by telephone—to and fromanywhere! Some of my guests are experienced radiohams, but many are appearing for the first time. All


have been fabulous people to enjoy for an hour. Theyhave included five past presidents of the NationalCouncil of Teachers of Mathematics (NCTM), the cur-rent president and president-elect, research mathema-ticians, K-12 classroom teachers, equity advocates andleaders in various environmental organizations. Therehave been shows on bailing out family finances, in-vesting thoughtfully, and environmentally aware con-sumerism—all important environmental issues. Youcan see a list of past and future Math Medley guestsand topics at www.csam.montclair.edu/~kenschaft.

Ten minutes into my first show a man called in to sayhe’d been reading a book called The Bell Curve, andhe wondered whether there was any hope in what Iwas trying to do. Five minutes later he thanked meand my guest and said he had learned a lot alreadyfrom the show! There was a hostile caller later thatshow who actually slammed up the phone at the theterrible suggestion that citizens of North Providencemight pay taxes to educate children in Prah....vi-dence(said with dripping disdain). Since then, all callershave been respectful, even when they disagree. Twoblind men have called in, a truck driver, many par-ents, a physician, and some math professors. Some-times people accept our invitation to solve math prob-lems on the air, and are willing to talk about solutionsfor five or ten minutes.

The first nine shows aired only in Providence, andMath Medley is now heard throughout Rhode Islandand in nearby Massachusetts at 990 AM from 11:00 tonoon. The show also can be heard in Arizona at 1100AM, currently from 8:00 AM to 9:00 AM. Arizona isthe only state that does not go onto daylight savingstime, so there is a discontinuity twice a year; I hopestrongly to keep the show at 11:00 AM Eastern Time.It can now also be heard live atwww.renaissanceradio.com.

Even more exciting, the shows are gradually being“archived” by webCT, so you can hear them any timeat www.webCT.com/math; then click on Math Med-ley on the upper right. As I write this, about sixty dif-ferent shows can be heard there, but it appears thatmany more will soon appear. It isn’t a completely pre-dictable process.

Each week I write an introduction to the show. Thefirst nine are somewhat representative, and they arereproduced below, slightly edited, to give the flavor

of Math Medley. Leads and support for publishingentire show transcripts would be appreciated. Sup-port for the shows (other than my teaching salary)would also be welcome; I hereby express gratitudefor the partial support from webCT, Texas Instru-ments, the American Mathematical Society, BurpeeSeeds, and Breadman Breadmaking Machines—somein the form of advertising. I would also appreciatearrangements to syndicate to other stations; the hoststation has been very cooperative. If anyone wouldlike tapes of the shows, I can provide them in limitedquantities, two shows for five dollars. I own the copy-right, so other stations are free to rebroadcast pastshows lifted from the web as long as they either donot make a profit or share the profits with me.

I am grateful for National American BroadcastingCompany for providing an opportunity for middleclass non-professionals in radio broadcasting. Theirtechnology has improved over the three years. Thefirst shows were the least technically satisfying, solater shows are becoming archived sooner. The onlyshow of the first nine now archived (as I write this)on webCT is Tom Banchoff’s. Personally, I think al-most all 165 shows thus far are worthy of botharchiving: and transcribing so people can quickly readand scan them, but obviously I’m biased.

The common opening for all shows is omitted in thefollowing transcripts. Alas, transcribing entire showstakes time and/or money, but the introductions canbe taken from my word processor. The topics andguests of the first nine are “Recent Changes inChildren’s Math Education” with Claire Pollard;“Changing Expectations of Teachers and Kids” withJohn Long; “Girls and Math” with Joan Countryman;“Changing Forms of Testing” with David Capaldi;“Ethnomathematics” with Gloria Gilmer; “NewTrends in Mathematical Research and Teaching” withTom Banchoff; “Gender Issues and Standardized Test-ing” with Cathy Kessel; “Environmental Mathemat-ics” with Barry Schiller; and “Hope” with TrinaPaulus.

RECENT CHANGES IN CHILDREN’S MATH EDUCATIONMAY 16, 1998, CLAIRE POLLARD

Today we will be considering “Recent Changes inChildren’s Math Education.” My guest will be ClairePollard, who is president of the Rhode Island Math-ematics Teachers Association. She also is a Resource


Teacher in the Providence public schools, where herjob is to help Providence teachers understand bothmath and how to teach it better.

I have two types of reasons for hosting a radio showto entice people into thinking about math. One is thatI enjoy it enormously, and it seems sad to me that morepeople don’t. I dream of a world where all people—well, almost all people—will enjoy math, in the sameway that most people enjoy singing or dancing. Littlechildren love math. At least, all the ones I meet in su-permarket lines and waiting rooms do. I wish that ourculture wouldn’t destroy that enthusiasm as often asit does. Your joy with math—like singing—doesn’ttake from my pleasure a whit. I hope that Math Med-ley can help many more people enjoy math. I lovemaking people happy when it doesn’t cost me any-thing.

My other reason for this show is less cheery. It’s that Ireally want there to be people on this earth a centuryfrom now. For that to happen, many people have toknow math. For example, we need statistics to under-stand complicated facts. Moreover, sharing limitedresources without violence requires that many peoplemust understand fractions and big numbers muchbetter than most adults do now.

I’m Dr. Pat Kenschaft, professor of mathematics atMontclair State University in New Jersey. My day jobis teaching college and graduate students pure math-ematics. Over the past decade I’ve won fourteen grantsfor helping elementary school teachers mathemati-cally because I believe that the day job of my guest isone of the most important in the world. I’ve taughthundreds of elementary school classes in the past de-cade, but she’s taught many more. Welcome to MathMedley, Claire Pollard.

CHANGING EXPECTATIONS OF TEACHERS AND KIDSMAY 23, 1998, JOHN LONG

My guest today will be Dr. John Long, Professor ofEducation at the University of Rhode Island and co-chair five years ago of the Rhode Island special legis-lative commission on math and science reform. Histopic will be, “The Changing Expectations of Teach-ers and Kids.” This affects all of us, because some daywe all will be at the mercy of today’s children. Nomatter how much money we have, no matter howloyal our own children are, and no matter how high

the walls we build around our home, younger adultsoutside our family will affect the quality of our wan-ing years. Therefore, their education will greatly in-fluence whether they make our elderyears pleasant—or unpleasant.

The recent Third International Mathematics and Sci-ence Study—called TIMSS by its friends—placedUnited States fourth graders somewhat above the in-ternational median, a welcome relief from the Firstand Second International Studies, when our country’sshowing was embarrassing. One reason for the im-provement has been the effort of about 200,000 of usaround the country in what is called the “math re-form movement.” Two hundred thousand is not asmall group—it’s roughly the population of Provi-dence. However, compared to the 50 million Ameri-can youngsters in kindergarten through twelfth grade,it’s not nearly enough. That’s one of us for 250 schoolchildren. Math reform needs many more people in-volved.

United States eighth graders still placed a little lowerthan median in the recent TIMSS study, and ourtwelfth graders came out at the bottom of the interna-tional ladder. There were only two countries belowus—Cyprus and the Union of South Africa.

The good news is that our youngsters do learn whenthey are taught. More good news is that when ourteachers are taught the subject matter and good waysof teaching it, their students learn. A recent study of900 districts in Texas found that teachers’ expertiseaccounted for about 40% of the variance in schoolchildren’s reading and math achievement. Indeed, thedifferences in black and white students were almostentirely explained by teachers qualifications and so-cioeconomic status. I am quoting from a blue-ribbonreport What Matters Most: Teaching for America’s Fu-ture led by two governors, one Republican and oneDemocrat. The report goes on to say that 28% of themath teachers in this country do not have even a mi-nor in math. It is no wonder that our youngsters donot do well on international exams.

My guest today is Dr. John Long, professor of educa-tion at the University of Rhode Island and one of thestate’s leaders in the math reform movement. Wel-come to Math Medley, Dr. John Long.


In 1992 the following problem was given to 8th gradestudents as a part of the National Assessment of Edu-cational Progress (NAEP)’s national assessment:

MARCY’S DOTSA pattern of dots is shown below. At each step, more dotsare added to the pattern. The number of dots added at eachstep is more than the number added in the previous step.The pattern continues infinitely.

(1st step) (2nd step) (3rd step)

• • • •• • • • • • •

• • • • • • • • •2 dots 6 dots 12 dots

Marcy has to determine the number of dots in the 20th step,but she does not want to draw all 20 pictures and thencount the dots.

Explain or show how she could do this and give the answerthat Marcy should get for the number of dots.

***The only answer accepted as “correct” was 420. Butthe quality of each explanation was graded as mini-mal, partial, or satisfactory or better.

Responses of 8th grade students in the national sample

Correct

No

response

Incorrect Minimal

16%

Partial Satisfactory

or better

63% 10% 6% 6%

This problem is a typical “IQ type” problem commonon NEAP tests. In order to solve it, a student needs

only 3rd grade mathematics. A student has to knowthat the number of dots in a rectangular array is theproduct of the number of rows and the number ofcolumns, and he/she has to be able to make one easymental multiplication. Finding the intended solutionrequires no ingenuity if a student has any experiencewith looking for numerical, and not visual, patterns.But providing a concise and clear explanatory write-up is difficult even for the best students.

The intended solution looks as follows:

1. Marcy should notice that the number of rows instep n = 1, 2 and 3, is n, and that the number ofcolumns is n + 1.

2. Therefore the number of dots is equal to n(n + 1),for n = 1, 2 and 3.

3. If this formula holds for other numbers, then forn = 20 the number of dots is 20 * 21 = 420. This isthe number of dots Marcy should get.

4. She should also check that the number of dots thatare added increases from one step to the next. Thatis easy, because the number of dots added in stepn is n(n + 1) – (n – 1)n = 2n.

But is there any reason to claim that 420 is the uniquecorrect solution? NO!

There are infinitely many patterns of dots that satisfythe conditions of the problem, and there is no over-whelming reason to claim that the one which seemsto one person the most obvious is the “correct” one.Clearly many 8th graders saw patterns that were dif-ferent from the one seen by the makers of the test.

Below are some possible solutions to the problem.(They are written as if they were students’ answers,but they were not.)

EXAMPLE 1There are many other solutions besides the obvious

Marcy’s Dots: A Problem on a National Test RevisitedPatricia Baggett

Department of Mathematical SciencesNew Mexico State University

MSC 3MB, Box 30001Las Cruces, NM 88003-8001

[email protected]

Andrzej EhrenfeuchtComputer Science Department

Box 430University of Colorado

Boulder, CO [email protected]


them!

We have to start with step 0, with no dots, and look atthe differences.

But 2 + 4 = 6, and this starts the Fibonacci pattern! Sothe next difference is 4 + 6 = 10, the next is 6 + 10 = 16,and so on. Then you have to add all these differencesup to step 20 to get the number of dots. I went onlyup to step 10 and I gave up. Not enough time. I wouldrather program a calculator to give me the answer.

CONCLUSIONIn order to make the solution to the original problemunique, one needs to add a few strong assumptions.For example, the number of dots, d(n), in step n isexpressed by a polynomial of second degree. The as-sumption about the degree of the polynomial isneeded because the polynomial

d(n) = n(n + 1) + (n – 1)(n – 2)(n – 3)

is a third degree polynomial, which is a solution tothe original problem.

Questions that are used on national and state testsshould be mathematically sound. Questions shouldbe testing mathematical knowledge that is expectedat a given grade level. Also, answers should be scoredobjectively and correctly. The problem of Marcy’s Dotsfails all three criteria. It is an example of rushing to-ward a solution, rather than thinking, what is a solu-tion to the stated problem (Buerk 2000).

It is hard to judge whether poor test questions areexceptions or if they are the norm, because test mak-ers protect themselves by keeping the contents of testssecret.

REFERENCESBuerk, D. (2000). What we say, What our students hear: A case

for active listening. Humanistic Mathematics Network Jour-

420, so I asked myself, what is the smallest possiblesolution?

The differences between the numbers of dots that areadded in each step must increase at least by 1. So wehave the following pattern if we make the differencesas small as possible.

Thus, the number of dots at the nth step is:

12 + 7 + 8 + … + (n + 2) + (n + 3)= 12 + (n + 10)(n – 3)/2.

So the smallest possible solution for the 20th step is267 dots.

EXAMPLE 2The question is about the number of dots, and notabout their pattern. So I decided to concentrate juston numbers. The ratios between the numbers of dotsare 3 and 2 (6/2 and 12/6), and that suggests an ex-ponential growth. However this cannot be “purely”exponential, because the ratios are not equal, so Ifiddled a little with formulas and found this one forthe number of dots, d(n), in step n:

d(n) = 2n +2(n – 1).

I used a scientific calculator to compute d(20) =1,048,614.

EXAMPLE 3The pattern can continue by repeating the ratios 3 and2.

Thus the number of dots in step 2n is 6n, and the num-ber of dots in step 2n + 1 is 2*6 n.

Therefore the number of dots in the 20th step is 6 10 =60,466,176. (I used a calculator.)

EXAMPLE 4We were learning about Fibonacci, so I started look-ing for Fibonacci numbers. And guess what? I found


nal #22. April, pp. 1-11.Kenny, P.A., Zawojewski, J.S., & Silver, E.A. (1998). Marcy’s Dot

Problem. Mathematics Teaching in the Middle School, 3(7),474-477.

National Assessment of Educational Progress (NAEP) (1992).

Mathematical Assessment.National Research Council (1999). Mathematics Education in the

Middle Grades. Washington, D.C.: National Academy Press.(Also available at http://www.nap.edu.)

Editor’s note: These are excerpts from a longer paper.

***It is easy to see that even in (ordinary) human life, and firstof all in every individual life from childhood up to matu-rity, the originally intuitive life which creates its originallyself-evident structures through activities on the basis ofsense-experience very quickly and in increasing measurefalls victim to the seduction of language. Greater and greatersegments of this life lapse into a kind of talking and readingthat is dominated purely by association; and often enough,in respect to the validities arrived at in this way, it is disap-pointed by subsequent experience.

Edmund Husserl, The Origin of Geometry

***• Sense statements may tend to be homeopathic to

the mathematical, and mathematical statementstend to be allopathic to the sense world.

• Mathematics (or geometry) opens betwixt infra-realization and super- nominalization, both ofwhich are programs.

• Mathematics is in the thinning ofprogrammaticness, as such a checking of pro-grams and unprogram-maticness.

• Mathematics leans on institutions of objectivity.

• Hollow mathematicians are at least correct.

AphorismsLee Goldstein

• Upon a people’s limited language, nonverbalmathematics was the first mathematics.

• I believe in nonverbal universals.

• We may read silence in dreams.

• Statements containing “there exists” could be thatpenultimate resort of the nonverbal.

• The nonverbal could hypothetically be as nomi-nal, not nominalist.

• Mathematics is nominalism’s self.

• A referential statement about mathematics mightbe as an unending hypothesis

• Science and the beginning of the world are notreferentism.

• The reference is the residue.

The basic idea of the above aphorisms is that a quick-ening and underlying programmatization of theunderstructure of things quickens the tendency towords and language and not the intuitive realismwhich precedes programs. This answers Husserl’squestion, then, and opens up a new field coincidentalor before grammar. As our grammar is hard-wired intoour brains, so is the programmetrical structure of theworld.


INTRODUCTION: A NEW ANSWER TO THE OLD QUESTIONIn this article, I address two works by two differentauthors in the May 1998 issue (#17) of HumanisticMathematics Network Journal. In “Real Data, Real Math,All Classes, No Kidding,” Martin Vern Bonsangue dis-cusses the famous question so often put to math teach-ers: “When am I ever gonna use this stuff?”Bonsangue confesses, “I gave shop-worn answers like,‘Math teaches you how to think, so it doesn’t matter,’or ‘Well, we can solve word problems with math,’neither of which I believed in” (17). His solution is tofind “real” examples (I explain the quotation marksin the following section), specifically data about earth-quakes and life expectancy, to use in his math classes.

A few pages later, Jeffry Bohl, in “Problems that Mat-ter: Teaching Mathematics as Critical Engagement,”deals with the question again. He creates a small tax-onomy of possible responses to the same age-old utili-tarian question: all answers are versions of “tomor-row” (you will need the information for an upcomingchapter, course, etc.), “jobs” (you will need the infor-mation to land a good career), “general mentalstrength” (you will need the information to think bet-ter), or “tests” (you will need the information to passthe test, quiz, next assessment).

I would like to propose another category of responsethat stems from what I believe is the general principleat work behind Bonsangue’s search for “real” ex-amples. Specifically, the response to “When am Igonna need this?” involves recognizing that the ques-tion, though designed to be teleological, cannot,strictly speaking, be answered. As those of us whoare a little more experienced and worldly than ourstudents well know, you never know when you will needto know something.

“REAL” DATA: WHAT IS IT AND WHY DO WE LIKE IT?One problem with mathematics textbooks is their lackof real—perhaps we should use “accurate” or “con-temporary”—data. Bonsangue’s math problems in-volve real data: earthquakes, a subject of intrinsic in-

terest for his Californian audience; and life spans ofpre-civil war women of different races, a relativelypeculiar subject involving history, detective work, andheritage. However, on the face of it, neither topicwould necessarily treat a student’s question of “Whenam I going to need to know this?” By introducing datacollected from “the field,” as Bonsangue does, we ex-pect relevance and a “hook” that will capture students’imaginations. In my experience, our expectations aresatisfied.

Somehow, more traditional problems that also use realdata are hardly so enticing. For example, “A bus de-parts at noon from Newport News (where I teach)averaging 42 mph. Another bus departs an hour andfifteen minutes later averaging 60 mph. If I am goingto Washington, D.C., 200 miles away, which bus getsme there first?” This problem may actually be basedon extant bus schedules; furthermore, D.C. is a won-derful city to visit and so knowing this informationhas some value. The difference is, perhaps, that thesedata look like a “standard” word problem, whereasBonsangue’s data are unorthodox, deriving from veryunmathematical sources.

Indeed, math problems that seem to come from non-mathematical problems are often what we mean by“real data,” and they are often very interesting to stu-dents. Even though knowing the “Distribution of ‘Felt’West Coast Earthquakes January 1, 1990 through July11, 1996” (Bonsangue 19) is not a good answer to“When am I going need to know this?”, it has a goodchance of engaging students because it smacks of re-ality. It uses information that somehow creates or dem-onstrates a nexus between life, math, and nature. Inother words, Bonsangue’s problems are truly human-istic mathematics—they genuinely engage many ar-eas of human experience and knowledge at once—and this is what excites students. Math problems areoften most interesting when they combine the math-ematics with other areas of learning.

WHEN DO YOU NEED TO KNOW ANYTHING?

When is a Math Problem Really “Real”?Dr. Michael E. Goldberg

Hampton Roads [email protected]


I begin with a digression: years ago, I was one of twofinalists for a job teaching physics and earth science.During the second interview, I learned that my com-petition had more experience and expertise than I did,but she had one weakness: she was tentative aboutdriving a school van for an after-school program. TheGoldberg family car for 20 years was a huge, full-sizedvan (this was the pre-minivan era), so I was happy tooffer my services as driver. I got the job. Never in mywildest dreams did I imagine, going into that secondinterview, that my competence driving vans wouldbe critical to my teaching career. This experience con-firmed one of life’s great lessons, one of the most criti-cal I have to offer to anyone, young or old: a funda-mental aspect of life is that you never know what youneed to know until you need to know it.

Intuitive understanding of this principle may well bewhy working with real data almost always seems somuch more interesting than traditional math prob-lems. Even if the purpose of the knowledge is not clearnow, a problem-solver is gaining real, applicableknowledge that may be of utility another day. HereBohl’s “tomorrow” answer applies. Bohl believes, “weneed to teach mathematics through the mathematiza-tion of real, socially relevant situations” (29). It maysound curmudgeonly, but I find “socially relevant” aproblematic phrase: it too often means, “socially rel-evant in the eyes of a teacher and not a student.” WhenI taught composition, I frequently covered “sociallyrelevant” material and found I unintentionally putstudents in combative, uncomfortable situations.When I did not try to introduce the material, it cameto light in more natural and potent ways. Trying to besocially relevant can be problematic.

However, I agree with the spirit of the advice: we areteaching not only math, but life skills (or, at least, cul-tural skills). Herein lies the connection betweenBonsangue’s “real data” and Bohl’s “general mentalstrength.” Social relevance is in the eye of the be-holder, but if students understand they are learningabout their world, not some abstract theoretical prin-ciple, they are more likely to file it away for use an-other day. If students feel that you have broadenedtheir knowledge base, as opposed to broadening theirmathematical knowledge base, they will often feel morefulfilled. This is not to say, “don’t teach math.” Infact, my point is that a general, humanistic knowl-edge base should be integrated with mathematics. The

most exciting mathematical problems are the ones thatseem to come from elsewhere besides math books.

According to A Handbook of Literature, by Holman andHarmon, “humanism suggests a devotion to thosestudies supposed to promote human culture most ef-fectively—in particular, those dealing with life,thought, language, and literature of ancient Greeceand Rome” (233). It may be (unfortunately) difficultto find students eager to digest information about an-cient Greece and Rome, but problems engaging otheraspects of humanism tend to excite students greatly.It is, after all, far rarer that an English teacher getsasked, “When am I going to need to know this?”

Though the humanities have historically struggled forvalidation in America, I have found few students ask-ing why they needed to read Hamlet. The presump-tion among students is that Shakespeare, by virtue ofhis importance to literature, history, philosophy, etc.,is automatically worthwhile. Problems engaging morethan one area of human thought (like the onesBonsangue has constructed) fit the bill. They teachmathematics, but they also stimulate other interests,effectively displacing the “when do I need to knowthis” question. I, too, prefer mathematical problemsthat in some way ask students to engage other disci-plines (art, history, literature, film, etc.), ideally a fewat once. I also believe that the “social relevance” willoften be contained within such problems automati-cally, by virtue of their complexity.

To summarize this section, let us review why the busproblem described above is relatively drab to my stu-dents. (I should point out that it would not be dull toall students. The trip to Washington, D.C. is, for some,a rewarding concept to entertain.) There do not ap-pear to be many dimensions or applications to theproblem. We need to go to a specific place and themath helps us make a better decision about how toget there. The “better decision” is better only insofaras it saves us about half an hour. The beauty of takingreal data is that these data are not math problems first:they are problems dealing with life and thought, prob-lems dealing with humanism. It thus turns out—andthis is surprising and exciting to many novices—thatmathematics is part of the pursuit of humanistic (notjust “human”) knowledge. A friend in college wasfond of pointing out that math departments are notin “arts and sciences” divisions not because math is a


science, but because it is an art. A problem that can beput in a student’s “I may need to know this some-day” file is an important, artistic problem.

AN EXAMPLE: GREAT MATH MOMENTS IN LITERATUREAs should be quite obvious, I am a humanist at heart.I have a doctoral degree in English literature; wrote adissertation on James Joyce (himself no minor human-ist); and spent ten years teaching composition, film,and English literature. (Thus, I can claim with someauthority that students do not question the teachingof Hamlet.) I also have a strong background in mathand physics, but for a decade I taught very little math,save my “calculus tip of the day,” when I taught awriting course geared toward college engineeringmajors. For years, people asked me about my diversebackground. Would I ever need to know all that math?Why was it so interesting to me anyway (I was anEnglish major)? I come by my devotion tointerdisciplinarity honestly, and I believe that it is pre-

cisely the search for links between different modes ofthought and different knowledge bases that best fur-thers learning.

Having mused about the intrinsic needs of the young(and old?) mind to encounter more than just math, itis time to turn to my experience and my suggestionsfor teaching humanistic mathematics. I will keep thissection short; this Journal contains a wealth of ideasregularly, and I only want to add enough to connecttheory with practice.

I am now a high school mathematics teacher and Idelight in adapting non-mathematical situations intomy classroom. I enjoy asking students to look at rela-tively obvious “real life” problems (some examples:altering recipes, renting cars, predicting landfall of ahurricane, buying concert tickets, carbon-datingbones) as well as those that, in talking aboutBonsangue’s “life expectancy” project, I call “relativelypeculiar.” These are particularly enjoyable in my cal-culus class, where I used a problem called “the per-fect glass of chocolate milk” to explain saturation (andderive a formula), and used derivatives to calculatehow far a professional wrestler jumped by knowinghis “hang time.” My point is this: both of these arefairly simple problems, but they are taken from dailyexperience and thus, judging by my students’ re-sponses, evoke great interest.

Unique to my classroom is a series of posters that Icreated (along with some friends and a graphic art-ist), called “Great Math Moments in Literature.” Witha few notable exceptions, I generally did not take ex-amples from science fiction. Though I enjoy sciencefiction very much, to sample it would be to take the“obvious” choices (kind of like using the bus prob-lem). The best posters are those that use examples fromwork seemingly unmathematical in nature. There isroom for Madeleine L’Engle, Thomas Pynchon, LewisCarroll, and Arthur C. Clarke, but what pleases methe most are the posters that come from less scientificor mathematical sources.

My favorite example is a poster I use while discuss-ing irrational numbers with students in my Algebra Iclasses. We should never forget how apt the term de-noting numbers like π or really is: irrational num-bers do not make sense to the novice. Only experience,it seems, eventually allows us to get a handle on these

2


quantities that we can write as symbols, point to on anumber line, but never know the exact value of. Whenthe class arrives at this head-spinning concept, I pointto a poster that contains a quotation from the Bible:“Then [Solomon] made the molten sea; it was round,ten cubits from brim to brim, and five cubits high. Aline of thirty cubits would encircle it completely.” Theposter can (and does!) occasion a variety of wonder-ful discussions, including ones of metaphor (“themolten sea” is a poetic term for a gigantic chalice) orof the nature of units (a “cubit” is about 18 inches, thedistance from the elbow to the fingertip). Generally,though, I prefer to point out the biblical approxima-tion of π to 3 (circumference of 30 divided by diam-eter of 10). The class can begin drawing and measur-ing circles, research the circumference/diameter rela-tionship other ways (on the Internet?), or merely en-gage in a lively discussion about how accurately wecan measure anything. Whatever discussion/exerciseensues, by the time we are done, the students are sat-isfied that they have gained knowledge of philoso-phy, theology, geometry, and, by the way, Algebra I.The concept of irrational numbers becomes a part ofa much bigger, more important discussion.

I close with an imagined example, using anotherposter in my classroom. The poster cites a laterSherlock Holmes story (“The Final Problem”), inwhich the great detective explains Moriarty’s evil ge-nius. It seems,

[Moriarty’s] career has been an extraordinaryone. He is a man of good birth and mathemati-cal faculty. At the age of twenty-one he wrotea treatise upon the Binomial Theorem whichhas had a European vogue. But the man hadhereditary tendencies of the most diabolicalkind.

This would be an enjoyable starting point for a dis-cussion of the Binomial Theorem (given a great dealof time or, perhaps, a graduate-level classroom, itmight even be fun to let students do some researchand take a crack at writing Moriarty’s treatise). Itseems to me also a good place to investigate the na-ture of proofs and theorems. I say this because moststudents accept math as “truth.” How, they mightwonder, could someone write a “treatise” on an in-controvertible truth? How could such a paper enjoy“a vogue”? Isn’t math either true or false? The point,

then, would be to use a seemingly innocuous quota-tion as an introduction into the complexities and un-knowns of mathematics, thence an introduction intothe complexities and unknowns of our world.

Sherlock Holmes’ genius lies in two abilities: he is re-markably observant and he has a huge knowledgebase. Just to take one example, in “The Five OrangePips,” he deduces that the Lone Star must be a boatfrom America because he knows it borrows the nick-name for Texas. This is impressive knowledge for aproper English gentleman in 1892, for whom know-ing state nicknames would appear to be utterly use-less information. It is utterly useless, until Holmesneeds it one day to solve this case; armed with thisinformation, the case is solved quickly and effectively.Holmes is therefore one of the great, if not the great-est of all exemplars of the “you do not know whatyou need to know until you need to know it” prin-ciple. Citing him as a model is attractive: he uses the


seemingly “mathematical” science of deduction tohelp with the most human of issues, including bro-ken relationships, jealousy, greed, and evil. If mathproblems were like Sherlock Holmes stories, studentswould be endlessly engaged, entertained, and edu-cated. Diversity, interdisciplinarity, and real data makemath problems that much more like Sherlock Holmesstories.

CONCLUSIONI am not suggesting we teach literature in math classes.I am suggesting we break down the barriers betweendisciplines and I have chosen Sherlock Holmes as ametaphor. In my own teaching I try (though, of course,I do not always succeed) to apply the framework Ihave outlined here. Any good teacher—not only onewho handles math—must have his or her own an-swers to “When am I going to need to know this?”However, the best answer is not always a time, date,or place. The best answer is that no one quite knows.We may, in part, teach math so that students can beready for tomorrow, do well in jobs, improve theircritical thinking, and pass the next test. In fact, I havesome support for all of these reasons. However, we

also have an imperative to teach humanism, tomathematize the world in ways such that our studentsare ready for whatever life offers them. After all, in“real life,” people get jobs teaching physics just be-cause they know how to drive a van.

REFERENCESThanks to Dr. Andrew Poe, who helped scout out both “Great

Math Moments in Literature” used in this piece.Bohl, Jeffrey. “Problems That Matter: Teaching Mathematics and

Critical Engagement.” Humanistic Mathematics Network Jour-nal 17 (May 1998). 23-31.

Bonsangue, Martin Vern. “Real Data, Real Math, All Classes, NoKidding.” Humanistic Mathematics Network Journal 17 (May1998). 17-22.

Doyle, Arthur Conan. “The Final Problem.” The Complete SherlockHolmes. Volume I. Doubleday: Garden City, 1930. 469-480.

- - - . “The Five Orange Pips.” The Complete Sherlock Holmes.Volume I. Doubleday: Garden City, 1930. 217-229.

Holman, C. Hugh, and Harmon, William. A Handbook to Litera-ture. 6th Ed. New York: MacMillan, 1986.

The Holy Bible. Revised Standard Version. Collins’ Clear-TypePress: New York and Glasgow, 1952.

Book Review: The Teaching Gapby James W. Stigler and James Hiebert

Review Part II: Contrasting U.S. and Japanese Beliefs aboutMathematics Teaching

Michael L. BrownSimmons College

Boston, MA [email protected]

The Teaching Gap. James W. Stigler and James Hiebert.The Free Press (Div. Simon & Schuster Inc.): New York,1999. ISBN 0-684-85274-8

This is the second of three articles that together forman in-depth book review of The Teaching Gap. Aftera brief summary of the first article1, we explore thecontrasting cultural beliefs that support mathematicsteaching in the U.S. and Japan, and in doing so, findseveral surprises that are relevant to college teaching.

SUMMARY OF THE FIRST ARTICLEAs our first book review article discusses, The Teach-

ing Gap addresses critical questions about how math-ematics teaching is actually done in the U.S., Japan,and Germany, based on uniquely valuable data: thefirst videotaped national random samples, in eachcountry, of eighth-grade mathematics classroom les-sons. Remarkably, teaching varied greatly from oneculture to the next, and comparatively little withineach culture, giving an empirical foundation to thepivotal claim in the book that "teaching is a culturalactivity."2 The authors claim, and we are persuaded,that their findings of cultural differences go far be-yond the eighth grade. Indeed, we believe they havemuch of importance in common with what we en-


The Teaching Gap. James W. Stigler and James Hiebert.The Free Press (Div. Simon & Schuster Inc.): New York,1999. ISBN 0-684-85274-8

This is the second of three articles that together forman in-depth book review of The Teaching Gap. Aftera brief summary of the first article1, we explore thecontrasting cultural beliefs that support mathematicsteaching in the U.S. and Japan, and in doing so, findseveral surprises that are relevant to college teaching.

SUMMARY OF THE FIRST ARTICLEAs our first book review article discusses, The Teach-ing Gap addresses critical questions about how math-ematics teaching is actually done in the U.S., Japan,and Germany, based on uniquely valuable data: thefirst videotaped national random samples, in eachcountry, of eighth-grade mathematics classroom les-sons. Remarkably, teaching varied greatly from oneculture to the next, and comparatively little withineach culture, giving an empirical foundation to thepivotal claim in the book that "teaching is a culturalactivity."2 The authors claim, and we are persuaded,that their findings of cultural differences go far be-yond the eighth grade. Indeed, we believe they havemuch of importance in common with what we en-counter at the college level, as we explore in what fol-lows.

For the perennial and largely unsuccessful efforts toreform American education, the authors accordinglyoffer a deep and elegant explanation: in these efforts,"we have been acting as if teaching is a nonculturalactivity." The Japanese system for improving teach-ing, on the other hand, "is built on the idea that teach-ing is a complex, cultural activity."

The U.S. and Japanese typical lesson patterns were

Book Review: The Teaching Gapby James W. Stigler and James Hiebert

Review Part II: Contrasting U.S. and Japanese Beliefs aboutMathematics Teaching

Michael L. BrownSimmons College

Boston, MA [email protected]

starkly contrasting. While they both began with a re-view of previous work, in the U.S. this led into "pre-senting a few sample problems and demonstratinghow to solve them," with students then practicingsolving similar problems, followed by checking andcorrecting some of this practice. In Japan, the initialreview led into presenting a new problem, which stu-dents then worked on trying to solve. Then the class dis-cussed and compared the students' various methods,along with any the teacher showed. The teacher fin-ished by highlighting the principal points.

Thus, after a new problem is presented, in Japan stu-dents develop solution procedures, "without first"being shown "how to solve the problem," while in theU.S. a solution procedure is "almost always" shownto students first. Only then are they required to workproblems using it. Mathematics students in the U.S.are thus found to be occupied mainly with masteringunconnected skills by repetitive practice, while theirJapanese counterparts devote time in comparableamounts to, on the one hand, solving problems thatgenuinely challenge them and having concept-focuseddiscussions, and on the other hand, skill practice.

As we emphasized in the previous article, this is inour view a crucial contrast, and one which, as we ex-amine here, seems to have much bearing on U.S. col-lege mathematics teaching.

CULTURAL SCRIPTS AND BELIEFS ABOUT TEACHINGTo explain why a nation's mathematics lessons fol-low these distinctive and, in the case of the U.S. andJapan, contrasting, patterns, the authors introduce thenotion that these lessons were created and carried outby teachers who, as members of a culture, "share thesame scripts" [italics ours]. A script is "a mental ver-sion of...teaching patterns" such as the U.S. and Japa-


nese patterns whose contrasts we have summarizedabove. In this way The Teaching Gap develops its claimthat how one teaches is primarily determined by one'sculture, rather than being an instinctive gift or ac-quired in "college teacher-training programs."

And the script, in turn, "begins forming early.... Aschildren move through twelve years and more ofschool, they form scripts for teaching." So students(as well as teachers) come to the classroom after yearsof being socialized into a pattern of expectations thatare mutually aligned in consonant ways. This is, Ithink, why the authors' generalizations have so muchpower beyond the eighth-grade context from whichthey were derived—these findings are statementsabout deeper patterns of cultural participation. Inparticular, as we remarked in the previous article, weat the college level may find it worthwhile to considerthe extent to which our teaching also fits the U.S. pat-tern and contrasts with the Japanese pattern, as sum-marized above.

Teaching is therefore a complex system that, like othercultural activities, "evolve[s] over long periods of timein ways that are consistent with the stable web of be-liefs and assumptions that are part of the culture."

Thus we see that the authors' fundamental logic is:beliefs give rise to and sustain cultural scripts (whichcharacterize a culture's system of teaching), and thesystem in turn determines the various manifest fea-tures of teaching that we can observe, say, in videodata.

And what are these beliefs about? A nation's teachingscript seems to depend on a few central beliefs about(1) what mathematics is, (2) the way students learnabout it, and (3) the teacher's function during the les-son. The authors have inferred a set of beliefs for Japa-nese and U.S. teaching in these three areas based notonly on teachers' answers to questionnaires but di-rectly on how they behave.

CULTURAL BELIEFS ABOUT WHAT MATHEMATICS ISIn regard to what mathematics is, the U.S. pattern fits"the belief that school mathematics is a set of proce-dures...for solving problems." Sixty-one percent ofAmerican teachers, "asked what 'main thing' theywanted students to learn from the lesson,...describedskills.... They wanted the students to be able to per-

form a procedure, solve a particular kind of problem,and so on."

It is also riveting and thought-provoking to learn that"[m]any U.S. teachers also seem to believe that" thisview of mathematics ("learning terms and practicingskills") is "not very exciting. We have watched themtrying to jazz up the lesson and increase students' in-terest in nonmathematical ways: by being entertain-ing, by interrupting the lesson to talk about otherthings...or by setting the mathematics problem in areal-life or intriguing context.... Teachers act as if stu-dent interest will be generated only by diversionsoutside of mathematics." While we may question theinclusion of creating a "real-life...context" in the list,the thrust of the description hits home, I think, sur-prisingly forcefully: I suspect it is a rare college math-ematics educator indeed who has not taken it forgranted, to one degree or another, that these extrinsicactivities in the classroom are a necessary concomi-tant of winning the prize of students' enthusiasm forthe mathematics. While the authors' description mayat first be disturbing, on reflection it is encouraging,in that it gives us a new context in which to view thesepulls away from the mathematics in our classrooms,and thereby a hope that they are not ineluctable.

These "diversions" are compromises, and thus to someextent compromising: they are moves away fromwhere we, including those of us in the college class-room, feel we ought to be. What is hopeful in this for-mulation is that the blame is placed neither on us noron our students, but on our system.

Contrasting beliefs about what mathematics is seemto underlie Japanese teaching. "Teachers act as if math-ematics is a set of relationships between concepts,facts, and procedures. ...On the same questionnaire[as the one given to American teachers], 73% of Japa-nese teachers said that the main thing they wantedtheir students to learn from the lesson was to thinkabout things in a new way, such as to see new rela-tionships between mathematical ideas."

Moreover, in stunning contrast to the U.S. teachers'beliefs about what mathematics is, "Japanese teach-ers also act as if mathematics is inherently interestingand students will be interested in exploring it by de-veloping new methods for solving problems. Theyseem less concerned about motivating the topics in


nonmathematical ways.

CULTURAL BELIEFS ABOUT HOW MATHEMATICS SHOULD BELEARNEDBased on this U.S. belief about mathematics as largelya set of procedures, a natural belief about how math-ematics should be learned then follows: "incremen-tally, piece by piece...practicing [procedures] manytimes, with later exercises being slightly more diffi-cult than earlier ones.” Again, does this not seem sofamiliar as to be taken as virtually the only way to dothings?

The authors connect "this view of skill learning" andits "long history in the United States" to behavioristpsychology, B.F. Skinner, and related work. Unfortu-nately this fertile connection is relegated to a merefootnote3 and seems very worthy of amplification.

A further American belief about how mathematics islearned also follows: "Practice should be relativelyerror-free, with high levels of success at each point.Confusion and frustration, in this traditional Ameri-can view, should be minimized; they are signs thatearlier material was not mastered." The authors offerthe example of a lesson in adding fractions, sayingthat these beliefs would lead to a presentation se-quence with "like denominators,...then...simple frac-tions with unlike denominators,...warn[ing] about thecommon error of adding the denominators (to mini-mize this error), and later...more difficult" unlike de-nominators. But (at a more advanced level of course)do we not act out of the same beliefs much of the timein our college teaching, and construct our lessons ac-cordingly, seeking to maximize success rates all alongthe way, and minimize frustration?

The Japanese belief about how mathematics is to belearned involves "first struggling to solve" problems,then discussing how to find solutions, and then hav-ing various methods compared and related. "Frustra-tion and confusion are taken to be a natural part ofthe process, because each person must struggle witha situation or problem first in order to make sense ofthe information he or she hears later. Constructingconnections between methods and problems isthought to require time to explore and invent, to makemistakes, to reflect, and to receive the needed infor-mation at an appropriate time." They quote a Japa-nese teachers' manual that, in the lesson on adding

fractions, advises allowing students to make the mostcommon error, i.e., to add the denominators, then toreflect on the "inconsistencies" they will find thereby.The teacher should start with, "for example, _ + _,"then compare the various ways that students comeup with to solve this problem, such as adding denomi-nators to get 2/6. Thus, the Japanese believe that"struggling and making mistakes and then seeing whythey are mistakes" are necessary to learning.

CULTURAL BELIEFS ABOUT THE TEACHER'SRESPONSIBILITIESThe last set of beliefs concerns what teachers in eachcountry regard themselves as responsible for in theclassroom.

American teachers seem to think they should parti-tion work into units that are doable for most of theclass, telling students all that they need to know inorder to do the work, and then giving them lots ofdrill. Note carefully, however, that telling studentswhat they need to know to do the work typically re-duces to showing them how to do problems just likethe practice problems. "Confusion and frustration" arebelieved to be intrinsically bad, evidence that teach-ers have fallen short in some way, and, upon theiroccurrence, the teacher will rush to give whatever helpis required to put students back on the right path. U.S.teachers thus "try hard to reduce confusion by pre-senting full information about how to solve problems."Again, I would surmise that we can see ourselves, atleast some of the time, in this description, and indeedthat we perhaps might not even have considered ittoo plausible that there was any alternative to doingwhat is described here.

Also, that we can see our choices in teaching as muchas we do in these descriptions is indirect confirma-tion of the authors' thesis that these choices are cul-turally conditioned.

A natural consequence of the foregoing beliefs in theU.S.—because students must pay continuous closeattention to their teacher solving model problems inorder to be able to carry out the same solution meth-ods on their own—is that "U.S. teachers also take re-sponsibility for keeping students engaged and attend-ing." In particular, as one illustrative consequence, adetail that is nevertheless emblematic is that the U.S.teacher typically prefers the overhead projector rather


than the blackboard, because the projector is betterable to focus attention. U.S. teachers have other ployswhose goal is to keep the attention of students fromwandering: "They pump up students' interest by in-creasing the pace of the activities, by praising studentsfor their work and behavior, by the cuteness or real-lifeness of tasks, and by their own power of persua-sion through their enthusiasm, humor, and 'coolness."'Notice that this is a whole category of teacher behav-ior in the U.S. that is distinct from the injection of ex-trinsic diversions as discussed above, but which I thinkcan often share some of the same dubious or compro-mising qualities.

Taking "responsibility for keeping students engagedand attending" is so fundamental to the way we dothings in American classrooms that, here again, onemight well have taken for granted that it could not beany other way. Tying this to an emphasis on proce-dural learning is encouraging, I think, because it sug-gests that the necessity for such measures is mutable.

Surprisingly, teachers in Japan "apparently believethey are responsible for different aspects of classroomactivity" from their U.S. counterparts. The Japanesebeliefs about what the proper role for the teacher isare at the heart of what college teachers can most ben-efit from in The Teaching Gap, since it is this third set ofbeliefs that goes directly to what Japanese teachersactually do in the classroom. We accordinglyquote the authors' description at some length:

... They often choose a challenging problem tobegin the lesson, and they help students un-derstand and represent the problem so theycan begin working on a solution. While stu-dents are working, the teachers monitor theirsolution methods so they can organize the fol-low-up discussion when students share solu-tions. They also encourage students to keepstruggling in the face of difficulty, sometimesoffering hints to support students' progress.Rarely would teachers show students how tosolve the problem midway through the lesson.

Japanese teachers lead class discussions, ask-ing questions about the solution methods pre-sented, pointing out important features of stu-dents' methods, and presenting methodsthemselves. Because they seem to believe that

learning mathematics means constructing re-lationships between facts, procedures, andideas, they try to create a visual record of thesedifferent methods as the lesson proceeds. Ap-parently, it is not as important for students toattend at each moment of the lesson as it is forthem to be able to go back and think againabout earlier events, and to see connectionsbetween the different parts of the lesson.

The authors then say, picking up on an example citedabove, that, "Now we understand why Japanese teach-ers prefer the chalkboard to the overhead projector.Indeed, now we see, in a deeper way, why they can-not use the projector."

This priority that they "go back and think again...andsee connections," and the consequent depotentiationof the need for continuous attention, seem to addressat once two prominent issues in our college teaching:respectively, how to encourage the making of higher-order, more abstract connections, and how to lessenthose diversionary pulls away from the mathemati-cal material itself to extrinsic matters, as discussedabove, and instead allow the inherent qualities of themathematics to be what keep students attending. In-deed, this priority seems to be, in the context of courseof the whole Japanese system, an explicit remedy forthe latter problem.

A COROLLARY BELIEF ABOUT VARIATIONS IN STUDENTS'PERFORMANCEA further fascinating and striking consequence of theJapanese script, closely related to the three sets of be-liefs that we have just considered, is a positive valua-tion of what is often viewed as a chronic barrier tobetter results in American classrooms, certainly in-cluding those in the colleges—namely, differing lev-els of performance and ability in the classroom. (Infact, the statistical distribution of such levels is notonly often widely spread out but, very likely muchmore often in the colleges than in the public schools,indeed actually bimodal or even multimodal, due tothe implementation of two or more distinct tiers ofadmissions policies at many colleges. Thismultimodality typically makes this barrier even moreawkward to surmount.)

Remarkably, the Japanese see such individual differ-ences "as a resource for both students and


teachers...because they produce a range of ideas andsolution methods that provide the material for stu-dents' discussion and reflection." The larger the class,the more assured the teacher can be that a satisfac-tory range and an assortment of types of responseswill be produced, hence the more reliably planned thelesson can be! Moreover, this range also gives teach-ers the means to address the differing levels of per-formance and ability among students. The Japanesehave in fact quantified and systematized this ap-proach: "Japanese teachers have ready access to in-formation of the form 'When presented with problemA, 60% of students will use Strategy One, 20% Strat-egy Two, 15% Strategy Three, and 5% some other strat-egy."'

What a very different and more constructive "take"on the problem of disparate performance levels theJapanese script seems to allow.

CLOSINGIn this second book review on The Teaching Gap, wehave given detailed emphasis to the contrasting setsof teachers' beliefs in the U.S. and Japan because thesefindings seem exceptionally relevant to the practiceof mathematics teaching at the college level. In thethird and concluding article, we focus on ideas relatedto improving mathematics teaching in the U.S., andon ways to carry these explorations onward, both inaction and in reflection. There as well, the focus is onwhat we find most worthy of being better knownamong mathematics educators at the college level.

REFERENCES1 Brown, Michael L., "Book Review: The Teaching Gap, by

James W. Stigler and James Hiebert. Review Part 1: Math-ematics Teaching as a Cultural Activity," Humanistic Math-ematics Network Journal, 24 (May 2001).

2 All unattributed quotes in this article are, like this quote, fromThe Teaching Gap.

3 This is Footnote 3 on Page 187 of The Teaching Gap.


Growney, Joanne. My Dance is Mathematics.Bloomsburg, PA: JoAnne Growney, 2001, 30 pages,$5.00. May be ordered from the author.

My Dance is Mathematics is a set of twenty-four poemsexpressing appreciation, understanding, and love ofmathematics from a variety of different perspectives.The relationship to mathematics may be seen in thecontent of the poem, the structure, or both. Poems areoften humorous. I enjoy reading them so much that Ihave given the booklet as a gift many times. Readersof Mathematics Magazine, The College MathematicsJournal, The American Mathematical Monthly, and theHumanistic Mathematical Network Journal have seensamples from this collection.

The content of a poem in "My Dance is Mathematics"may be related to mathematical subject matter, or theassociated pedagogy; it may be about a mathemati-cian, or a mathematician's life. For instance, everymathematician would understand and enjoy the por-trayal of a familiar experience in the poem, "Misun-derstanding."

Ah, you are a mathematician,they say with admirationor scorn.

Then, they say,I could use youto balance my checkbook.

I think about checkbooks.Once in a whileI balance mine,just like sometimesI dust high shelves

One of my favorites is called "A Mathematician'sNightmare." On the surface, it seems to be about de-cision-making in pricing and shopping, but it is anexcellent depiction for a student or lay reader of theCollatz Conjecture, a famous unsolved problem. (A

Review of JoAnne Growney’s “My Dance is Mathematics”Sally Lipsey

Brooklyn CollegeNew York, NY

footnote gives the explanation.) Another favorite is abeautiful and poignant poem about Emmy Noether.It is called "My Dance is Mathematics," for which theentire collection is named. In this poem, JoAnneGrowney asks "If a woman's dance / is mathematics,/ must she dance alone?"

In "December and June," the poet uses vibrant imagesto compare winter and summer scenes and feelings.The structure of the poem is based on prime factor-ization in a delightful way. Here is the first stanza:

coldwinds howl

geese go southnights long tea steepstemperatures fall low

ponds freeze snowmen growtoboggans slide down hillsidessun hides ice coats spring waitswood-fires flame snowballs fly

winds howl groundhogs hibernate

Poems with very different content, "Changing Colors"(about universal emotions) and "Counting" (about thenumber system) are based on the same poetic struc-ture: the numbers of syllables in consecutive lines areconsecutive positive integers. One of the poems,"ABC," ftmny and clever, is also a bit Eke an exercise,expressing a mathematical idea by using each letterof the alphabet in turn. We enjoy the fun and may beinspired to try one ourselves.

JoAnne Growney's poetic talent, insight, and humorprovide repeated pleasure.


SUMMARYThis paper describes my efforts to incorporate prob-lem-solving portfolios into my liberal arts mathemat-ics course. I begin with a description of the compo-nents of the portfolios and the factors I consider inevaluating them. I then address some of the moresignificant obstacles I have encountered as well aswhat I consider to be among the major benefits. Aselection from one student’s portfolio is appended.

***My use of portfolios in my liberal arts mathematicscourse (Math 102, “The Nature of Mathematics”) grewout of my desire to expand on my traditional and of-ten too-narrow use of homework, tests, and not muchelse to formally assess my students’ work. My officeadjoins those of some of my colleagues in the EnglishDepartment, and I have had the opportunity to ob-serve their successes with the use of the portfolio asan assessment tool. The liberal arts mathematicscourse has an emphasis on a variety of problems suit-able for thoughtful investigation and comes with theadded benefit of typically being populated by students

with particular talents in writing and the arts. As such,it seemed to be a course that was conducive to pilot-ing portfolios.

The portfolio, as I use it, is made up of three majorparts. One is the problems themselves. Prior to thestart of the quarter, I select between twenty and thirtyproblems from my list of homework problems anddesignate them as being eligible for inclusion in thestudents’ portfolios. I select problems that will requiresome investigation and analysis on the part of the stu-dent; the straightforward computational problems arenot included—unless there is something special aboutthem that an insightful approach might reveal. Asthe quarter progresses and I make further assign-ments, I will also mark some of them as eligible forinclusion. Selected test problems can be included aswell. Each student selects five of the eligible prob-lems to include in her portfolio. I suggest that stu-dents will want to rewrite their solutions to make themlook more presentable; many students do this even incases where it might not have been necessary. Foreach of the five problems, the student then writes apage or two about the process she used to solve theproblem, the techniques she attempted to use, whatworked, what didn’t work, and what she learned fromboth the successful and the unsuccessful attempts. (Iusually refer to word counts rather than page lengthsas guidelines, in order to eliminate problems withfonts, margins, and the like. I give a guideline of 250– 500 words for these papers, but I also generally tellstudents that the paper needs to be as long as it needsto be—no more, and no less. A well-written paperthat is a little shorter is better than a poor paper ofgreater length.)

The second part—which almost always appears at thebeginning of the completed portfolio—is a reflectiveintroduction, a 500 – 1,000 word piece that ties theproblems together. I point out that, in most cases, therewill be some common thread running through the

Portfolio Assessment in Liberal Arts MathematicsMike Kenyon

Yakima Valley Community College, Grandview Campus500 West Main Street

Grandview, WA [email protected]

An excerpt from The Magic of Math followsthe conclusion of this article.


problems each student chooses, and I ask the studentsto identify those commonalities and reflect on whatthis body of work shows about their progress overthe course of the quarter. One student, for example,opted to showcase the problems she felt most directlyapplied to “the real world.” Many students, though,have been successful with this even (or especially)when their problems do not, at least superficially, ap-pear to have much of anything in common. In fact,some will go out of their way to choose problems thatseem to have nothing whatsoever to do with eachother. Those students are then able to point out some-thing deeper about the problems and their solutionsto them that shows how they have developed theirability to investigate complex problems. In somecases, students have simply chosen their favorite prob-lems. One student explained her selections with thesewords: “With the wide range of subjects covered inthis class, some of them of realistic use in daily lifeand some of them fairly abstract, I found the varietyof problems coupled with their difficulty was forcingme to use critical thinking skills. More importantlythan that I found myself enjoying this activity …Maybe just like everything else in life problem solv-

had for me personally … Although this particularcourse has come to an end, I do not believe I havearrived at the end of my mathematical journey.”[Babcock] Another student explained, “In the past, Ihave always considered math an intriguing subject,though not terribly exciting. However, that opinionhas changed dramatically. I have discovered, throughthe topics addressed in this course, that the subject ofmath possesses many amazing, if not ‘magical’qualities…Overall, each piece in this portfolio dem-onstrates not only my skills as a student, but also myabilities to learn new information and apply it to dif-ferent circumstances.” [Juris] Having read those com-ments and others like them, as well as the portfoliosthat contain them, I am led to believe that compilingthe portfolios has afforded the students the opportu-nity to look back on their work and see the individualproblems in the larger context of the entire quarter.That, I think, makes the students more likely to real-ize what they have done and to reflect on what it maymean.

The final part of the portfolio is a 500 – 750 word re-view of the book Fermat’s Enigma by Simon Singh. Weread the book over the course of the quarter and dis-cuss it chapter-by-chapter in class, and also watch theNOVA video The Proof. I point out that this is, afterall, a book about a problem (albeit a very difficult andfamous one) and its eventual solution and as suchseems entirely appropriate in a problem-solving port-folio. Where possible, I encourage students to includereferences to their book reviews in their reflective in-troductions, but I note that this isn’t always possible,and if it doesn’t seem to fit with the rest of the mate-rial in the introduction, they shouldn’t try to force it.This component could, of course, easily be removedin a course that did not have outside reading.

Each of the five problem write-ups counts for 10% ofthe portfolio grade, as does the book review. The re-flective introduction counts for the remaining 40%. Iassess each paper in seven areas:

1 The essay should have a clear focus on a specificmain idea.

2 The essay should be relevant to the reader.3 The essay should show that the student has made

an effort to be aware of herself as a problem solver.4 The essay should include specific examples (from

the attempts (successful or otherwise) to solve the

ing was more about the journey than the destination,so I chose the problems in this portfolio based on howinteresting the journey was.” [Lyczewski]

It is in the reflective introduction that the students aremost likely to look back on the course as a whole andfully appreciate all that they have done. As one thirty-something student wrote, “After an absence of 17years from mathematics … even though I did well inMath 95 [Intermediate Algebra] and developed someunderstanding of algebraic concepts, I continued tohave serious reservations about taking another mathclass. I still failed to see the relevance mathematics


problem, from the reading, from class discussions,and the like) in support of the points being made.

5 These examples should be analyzed, not merelystated.

6 The essay should be well-organized.7 Mechanics, usage, grammar, and the like should

be correct.

The student earns a score of 4, 3, 2, or 1 (excellent,good, fair, or poor, corresponding roughly to gradesof A, B, C, or D, respectively) in each area on eachpaper. (The exception is that I find it is usually notappropriate to look at the student’s awareness of her-self as a problem solver in her book review.) The scoreson each paper are averaged to give an overall scorefor that paper, and I add the scores for each paper(counting the reflective introduction four times) to geta score for the portfolio out of 40 possible points.Merging this with the rest of the student’s work forthe quarter (homework, tests, etc.) turns out to be alittle challenging, because a grade of 3 out of 4 is a Bon the portfolio, but 75% is not a B on other assign-ments. To compensate for this, I end up with a per-cent score for the portfolio (but 30 out of 40 is not 75%in this scale), and I have the rather absurd situation ofrecording a portfolio grade to the nearest hundredthof a percent. It works well enough, though. (The gorydetails are available upon request.)

I gave students very little direction on this project thefirst time I assigned it. I gave them a two-page hand-out describing the guidelines and criteria during thefirst week of class and occasional reminders as thequarter went on. We spent little if any class time onit, though I did make it clear that I was entirely will-ing to read over rough drafts and offer my feedback.(Several students each quarter take advantage of this.)I remain uncertain as to whether I had a stroke of ge-nius in this minimalist approach or just got uncom-monly lucky, but I was simply delighted with the qual-ity of the work I received. It was clear that the stu-dents had taken the project seriously and were deter-mined to showcase their best work.

Since then, I have asked students who have writtenexceptionally good portfolios for permission to copythem and share them with my colleagues and futurestudents, so I now have samples to share with myclasses. I have also expanded (and, I think, improvedon) my explanatory handouts and discussed what I’m

looking for in more detail. This includes devising ahandout that explains in a sentence or two what char-acteristics will earn a paper a score of 1, 2, 3, or 4 ineach of the seven areas, all of which seems to makefor a much better-conceived project.

(I have never, incidentally, been turned down when Iasked a student for permission to copy his or her port-folio for future use. One student, in fact, was delightedby the request, saying that no teacher—and certainlynot a math teacher—had ever wanted to keep a copyof her work before!)

There are, of course, challenges involved in under-taking such an endeavor. Not least of these is the timeinvolved in evaluating them. Portfolios frequently runin excess of twenty pages; a length of thirty pages isnot all that rare. I find that I require at least 45 to 60minutes (often more) to read and comment on eachportfolio. (I never write comments directly on thestudent’s paper. All of my comments are on separatepages.) I simply cannot read more than four or five ina single sitting; at that point, my brain turns to jelly,and it’s futile for me to try to go on. Couple this withthe rest of the time pressures that come at the end ofthe quarter, and it’s a non-trivial exercise. (Don’t ex-pect any sympathy from the students, either!) To ac-commodate this, I generally make the portfolios dueeither at the start of the last week of classes or the endof the week before. I don’t give extensions, or, moreprecisely, the truly extraordinary conditions thatwould warrant me giving an extension have not yetarisen.

The nature of this writing is such that plagiarism israther unlikely to be an issue, but it is still good tokeep an eye on any overwhelming similarities thatmight appear between students’ work. And one stu-dent submitted a problem that hadn’t even been as-signed (much less listed as eligible for inclusion) thatquarter. My first clue came when she wrote that itwas the fourth problem on our third test, which onlyhad three problems. In fact, she had lifted “her” workalmost word-for-word from one of the sample port-folios I had provided from previous classes. Fortu-nately, such cases of academic dishonesty are both rareand relatively simple to identify.

There is no grade of F represented in the 1-through-4grading scheme, and at first that bothered me at some


fundamental level. What, I wondered, would I do ifand when I received a truly atrocious paper? As itturns out, though, I have never, in the three classeswith whom I’ve done this, received a complete port-folio that merited a grade of F. There have been caseswhere students have decided that three or four prob-lems instead of five would be sufficient (it’s not), orthat the book review isn’t a necessary component (itis), or that it’s all right to turn in a portfolio three dayslate (“Late portfolios will not be accepted” is not, inmy view, subject to multiple interpretations).

There is no provision in the grading for mathematicalaccuracy. I concluded that since the student canchoose any five problems she wants from a list thatends up including close to forty options, she is obvi-ously going to choose ones that she knows she hasright. This has almost always been an accurate as-sumption. In the few cases where it has not been, theaccompanying paper has (perhaps predictably) beenso poor as to more than penalize the student for themathematical errors.

By and large, I have been thoroughly impressed bythe quality of the work I have seen in these portfolios.Students have taken the project quite seriously, donea great deal of careful introspection and analysis, andcomposed excellent bodies of work reflecting theirprogress.

There is, of course, some initial (and sometimes last-ing) resistance to the idea of writing in a mathematicscourse. But that resistance often gives way to a cer-tain sense of welcoming the opportunity to use othertalents to showcase their mathematical progress, es-pecially in a course targeted to the liberal arts student.In several cases, I have seen students whose abilitiesdo not appear to be reflected in their test scores doingsignificantly better when they are given an opportu-nity to use another assessment instrument. Better yet,

the converse does not hold: I have not yet had a stu-dent who was doing well otherwise but submitted apoor portfolio.

An unexpected benefit to this project has been thenumber of students who have gone on to take addi-tional mathematics courses. We generally assume thatthe liberal arts mathematics course will be the lastmathematics course a student takes, and it does notserve as a prerequisite for any further courses. Butsince I have begun using portfolios, I have had a sur-prisingly large number of students elect to take col-lege algebra in subsequent quarters. (Both liberal artsmath and college algebra have intermediate algebraas their prerequisite.) Whether this is a result in anyway of the students’ experiences with the portfolioproject is not at all clear, of course, and I haven’t ana-lyzed those students’ performances in college alge-bra. But the very fact that they opt to take it is en-couraging.

There does not appear to be any obvious reason thisconcept would not work just as well in other math-ematics courses, but while I have thought about do-ing so, I have not extended my use of portfolios intomy other courses. I expect that minor (or even major,in some cases) modifications would be in order, de-pending on the specific course in question.

I am entirely willing to share samples of my students’work (the best ones, of course!) as well as the currentincarnations of my handouts, rubrics, and feedbacksheets. Feel free to contact me and let me know whatyou want to see.

I am grateful to my colleagues in the English Depart-ment for their advice and the resources they have pro-vided. John Delbridge and Tracy Case Arostegui en-couraged me to go forth with the idea when I men-tioned it. Sandy Schroeder provided me with read-ing material as I began, and Dodie Forrest’s rubric wasan excellent model for my own. Carolyn Calhoon-Dillahunt has always made herself available to re-spond to my numerous questions, and has also beenvital to the crafting of this essay.

WORKS CITEDBabcock, Joddi-Jay, A Mathematical Journey, Winter Quarter 2001.Juris, Lacie, The Magic of Math, Winter Quarter 2001.Lyczewski, Cherie, Cherie’s Enigma, Winter Quarter 2000.


I have discovered that there are many “magical” quali-ties involved in numbers and their arrangements. Atthe beginning of this course I was given a very in-triguing puzzle which has been passed down throughthe ages of mathematics. It is called the Magic Square.The goal is to create a square of equal proportion (3x3,4x4, etc.) made up of numbers whose sums, from anydirection, equal the same number vertically, horizon-tally, and diagonally. I was allowed to make it as largeor small as I desired. Wishing to be different from therest of the class, I took up the challenge of creating a5x5 magic square.

Logic and strategy have never been among my strongpoints. In fact, in ordinary circumstances I will dowhatever it takes to avoid them at all cost. However,I had only to look at the now-monumental squareawaiting its 25 numbers to see that random guessingwas not going to be even a remote possibility withinmy time frame. I decided that a little research wasnecessary in order for me to better understand thecreation of the magic square. After a search of theInternet, I was able to find a website containing doz-ens of different examples. My main “strategy” at thispoint was to pick out as many common patterns be-tween the examples as I could which might give me aclue in how to construct my own. Surprisinglyenough, it didn’t take me too long to discover a com-mon similarity in many of the 5x5 squares. Most ofthe numbers seemed to wrap around the square as ifit were bent into a spherical shape like a pillar. Withthis observation in mind, I attempted my own square.

For no particular reason, I decided to place my firstnumber (1) in the bottom row, middle column. Num-ber 2 then went in the top row, second column fromthe right. I wanted to make the numbers wrap aroundthe square while angling downward diagonally. Since2 could not be placed lower than 1, I went back to thetop and moved over one column to the right. Thefigures 3, 4, and 5 continued in the spiraling pattern.When I reached 6, I could no longer go down because

number 1 already occupied the space. After a greatdeal of pondering, I found it was possible to move 6to the space directly above 5 and then continue onwith the pattern. I utilized this move whenever an-other number blocked a needed space. Finally, afterthree hours, I triumphantly placed the last number(25) in the top row, middle column and the puzzlewas complete! Every row, column, and diagonaladded up to exactly 65.

Overall, I have learned more during the completionof this assignment than I ever expected. I have dis-covered the challenge and fascination of mathemati-cal puzzles and felt the thrill in finding the solution. Ihave also discovered the great importance of logic,patterns, and observations. It is hard to even imaginethe possibilities of solving a puzzle of this size by ran-dom chance. This project stretched my critical think-ing skills by forcing me to study the puzzle and cre-ate a logical strategy to solve it. Once I had discov-ered the pattern, the entire mystery came to a victori-ous conclusion in mere seconds. I have finally un-locked the “magic” in the box!

“Magic in a Box”Excerpted with permission from

The Magic of Mathby Lacie Juris

Winter Quarter 2001


ABSTRACTThis paper describes a divisibility rule for any primenumber as an engaging problem solving activity forpreservice secondary school mathematics teachers.

INTRODUCTIONMy students, preservice secondary school mathemat-ics teachers holding majors or minors in mathematicsor science, were raised to believe that there were some"neat" divisibility rules for numbers like 2, 3, 5, 9, 10,100, some considering the last digit or digits and someconsidering the sum of the digits. They have also heardof some "weird" and totally unuseful divisibility rulesfor 7, 11 and maybe even 13. Usually, the former areintroduced, and at times even proved, in junior-highschool. The latter are mentioned briefly without aproof, or omitted altogether. In the AS (After Sput-nik) era of growing dependence on calculating ma-chines, who could possibly be interested in divisibil-ity rules?

The curiosity of one student generated an interestinginvestigation, that I wish to present here. This studentdiscovered, in fact found on the internet, a divisibil-ity rule for 7, and wondered why it worked. I blessedher curiosity and suggested that the class work on it.The results went far beyond our original intentions: adivisibility rule for any prime number has been de-rived and proved. More than the mathematical exer-cise, I wish to share the exciting mathematical inves-tigation and experimentation in which the studentsengaged.

I will present the results as a problem solving activitythat started with collecting data through observationand incorporated several rounds of implementing a"What if Not?" strategy (Brown & Walter, 1990). I willpresent the results as students' engagement in gener-alizing and specializing (Mason, 1985), and will con-

Divisibility: A Problem Solving ApproachThrough Generalizing and Specializing.

Rina ZazkisFaculty of Education

Simon Fraser UniversityBurnaby, B.C. V5A 1S6 Canada

[email protected], 1998

clude with a brief discussion on the relevance of suchan activity and several ideas for possible extensions.

PROLOGUEConsider the divisibility rule for 3: A number is divis-ible by 3 if and only if the sum of its digits is divisibleby 3. Let's prove it for a 4 digit number.

Consider an expanded notation of a 4 digit numberwritten with digits a, b, c and d from left to right.

1000a + 100b + 10c + d == (999a + a) + (99b + b) + (9c + c) + d applying asso-ciativity and commutativity of addition= (999a + 99b + 9c) + (a+b+c+d)

The first addend in this sum (999a + 99b + 9c) is al-ways divisible by 3. The second addend (a+b+c+d) isthe sum of the digits. Therefore the number is divis-ible by 3 if and only if the sum of its digits is divisibleby 3.

Even though this proof refers to a four digit number,it gives a general idea how the proof can be extendedto a number with n-digits. The strategy used in thisproof is representing a number as a sum of two ad-dends. The divisibility of one component is obvious.The divisibility of the second component determinesthe divisibility of the number. A similar strategy willbe applied in the following proofs.

DIVISIBILITY BY 7 - INTRODUCING THE ALGORITHM.Divisibility of a number by 7 can be determined us-ing the following recursive algorithm:

1. Multiply the last digit of the number by 2.2. Subtract the product in (1) from the number ob-

tained by deleting the last digit of the originalnumber.


3. Continue steps 1 and 2 until the divisibility of thenumber obtained in (2) by 7 is "obvious". The origi-nal number is divisible by 7 if and only if the num-ber obtained in step (2) is divisible by 7.

Examples of implementation:(a) Is 86415 divisible by 7?86415 8641 - (5x2) = 86318631 863 - (1x2) = 861861 --- > 86 - (lx2)= 8484 --- > 8 - (4x2) = 0Yes, 0 is divisible by 7 therefore 86415 is divisible by 7

(b) Is 380247 divisible by 7?380247 --- > 38024 - (7x2) = 3801038010 --- > 3801 - (0x2) = 38013801 --- > 380 - (1x2) = 378378 --- > 37 - (8x2) = 2121 is divisible by 7 and therefore 380247 is divisibleby 7. (We could continue one step further to get a zero).

(c) Is 380245 divisible by 7?380245 --- > 38024 - (5x2) = 3801438014 --- > 3801 - (4x2) = 37933793 --- > 379 - (3x2) = 373373 --> 37 - (3x2) = 3131 is not divisible by 7 and therefore 380247 is not di-visible by 7.

When examples similar to the above were presentedin class, the immediate response for many studentswas a desire to try it out, to carry out the algorithm onnumbers of their choice and verify divisibility with acalculator. This generated a large body of evidence tosuggest that the algorithm "works". This also gener-ated two related questions:

• Why does this work?• Why does this work for 7?

The first question is drawn by the desire to under-stand the algorithm and to prove that it determinesdivisibility by 7 for any natural (or integer) number.The second question is drawn by the desire to deter-mine the special place of the number 7 in the algo-rithm. Specializing on 7 in turn invites generalization:Does it work for 7 only? Will the algorithm work foranother number? For which numbers will it work?How can the algorithm be modified to work for an-other number?

WHAT IF NOT 7?While experimenting with other numbers, a lucky trialby one student prompted a conjecture, that exactlythe same algorithm can be applied to determine di-visibility by 3. This conjecture has been supported byseveral examples, however, no other number wasfound for which the above algorithm can be appliedto determine divisibility. To encourage further inves-tigation I suggested the following variation.

DIVISIBILITY BY 19 - VARYING THE ALGORITHMDivisibility of a number by 19 can be determined bythe following algorithm:

1. Multiply the last digit of the number by 2.2. Add the product in (1) to the number obtained by

deleting the last digit of the original number.3. Continue steps 1 and 2 till the divisibility of num-

ber obtained in (2) by 19 is "obvious". The originalnumber is divisible by 19 if and only if the num-ber obtained in step (2) is divisible by 19.

Examples of implementation(a) Is 15276 divisible by 19?15276 1527 + (6x2) = 15391539 153 + (9x2) = 171171 17 + (lx2) = 1919 is divisible by 19 and therefore 15276 is divisibleby 19.

(b) Is 12312 divisible by 19?12312 --- > 1231 + (2x2) = 12351235 --- > 123 + (5x2) = 133133 --- > 13 + (3x2) = 1919 is divisible by 19 and therefore 12312 is divisibleby 19.

For convenience of reference in further discussion, weshall name this algorithm a trimming algorithm.

WHY-QUESTIONS TO WONDERExperimenting with the two variations of the trim-ming algorithm presented above there are (at least)two questions that arise: (1) Why is the last digit mul-tiplied by 2? (2) Why does the algorithm involve sub-traction in case of 7 and addition in case of 19?


DIVISIBILITY BY 17 - ANOTHER VARIATIONA different variation on the rimming algorithm canbe used to determine divisibility by 17. In this casewe multiple the last digit by 5 and subtract the prod-uct from the "trimmed" number.

Examples:(a) Is 82654 divisible by 17?82654 8265 - (4x5) = 82458245 824 - (5x5) = 799799 --- > 79 - (9x5) = 34 we may stop here or con-tinue one step further34 --- > 3 - (4x5) = -17Conclusion: 82654 is divisible by 17.

(b) Is 17456 divisible by 17?17456 --- > 1745 - (6x5) = 17151715 171 - (5x5) = 146146 14 - (6x5) = -16Conclusion: 17456 is not divisible by 17.

REPHRASING THE WHY-QUESTIONSThe similarities among the three algorithms are obvi-ous. However, the last variation suggests rewordingof the first question:

(1) How is the multiplier of the last digit of the num-ber determined? (Why was it 2 in case of 19 and 7and 5 in case of 17?)

The second question remains basically the same:

(2) Why does the algorithm involve addition in somecases and subtraction is others?

DIVISIBILITY BY 7 - A SPECIFIC "GENERIC" PROOFAfter experimenting with a variety of examples thestudents became convinced that the algorithms doindeed represent a divisibility rule. However, theywere still seen as some magic tricks. The interest inwhy (they work) took over from the initial excitementof how they work.

Let us prove the divisibility algorithm for 7.

Consider any natural number n. If N is the numberobtained from n by deleting the last digit a, we canalways represent n as 1ON+a (Example: 3456 = 10 x345 + 6) We are interested in connecting our originalnumber n and the number obtained by the algorithm,

namely, N-2a. In fact, we would like to prove that n isdivisible by 7 if and only if N-2a is divisible by 7.

Applying simple arithmetic we get:10N + a 10 (N - 2a) + 20a+ a

10 (N - 2a) + 21aThe last addend (21a) is divisible by 7 for any digit a.Therefore n is divisible by 7 if and only if N-2a is di-visible by 7. Now we can treat the "new" number (N-2a) as the number for which divisibility by 7 has to beestablished using the same method.

WHAT IF NOT 7?What if divisibility by a prime number p is in ques-tion? Separate proofs, similar to the above, can bedeveloped for a variety of numbers. Inviting studentsto develop these proofs and discuss similarities amongthem may help in generalizing to attain an algorithmwhich determines divisibility of a number by anyprime p.

WHEN P=17For example, to determine divisibility by 17 we lookedfor a number of the form 10k± 1 divisible by 17. Wefound 5 1. Therefore k=5. This is the number used inthe trimming algorithm to establish divisibility for 17.Since 51 has the form 10k + 1, the number obtained inthe algorithm should be of the form N - ka, thereforethe algorithm involves a subtraction of a product ofthe last digit by 5.

WHEN P=31What is divisibility rule for 31? 31 itself differs by 1from the closest multiple of 10. Therefore k=3 and thealgorithm involves subtraction.

Example:Is 4185 divisible by 31?4185 418 - (5x3) = 403403 40 - (3x3) = 31Conclusion: Indeed, 4185 is divisible by 31.

WHEN P=13What is the divisibility rule for 13? We find 39 as amultiple of 13 that differs by 1 from a multiple closestof 10. Therefore k--4 and the algorithms involves ad-dition.

Example:Is 4173 divisible by 13?


4173 --- > 417 + (3x4) = 429

429 42 + (9x4) = 7878 7 + (8x4) = 39Conclusion: 4173 is divisible by 39.

EXISTENCE PROOFWe believe that so far the why questions (1) and (2)raised earlier have been answered. Now it is time towonder whether it is possible to find an appropriatetrimming algorithm to determine divisibility by anyprime.

The mathematical answer is no. However, the "hu-man" answer is - almost. Such an algorithm can bedetermined for all the primes except 2 and 5. (How-ever, since 2 and 5 have well known divisibility rules,we will focus on the other primes.)

The existence of a trimming algorithm for p dependson the existence of a multiple of p which is larger by 1or smaller by 1 than a multiple of 10. It is obvious thatsuch a multiple does not exist for 5 and 2 which arefactors of 10. Let us prove its existence for all otherprimes.

Formally, let's prove that for any prime p, p ≠ 2, 5,there exist natural numbers k and m such that |mp -10k| =1

Let us consider the last digit x of p. The possibilitiesare 1,3,7 or 9, since this digit cannot be even or 5. If x= 1 or x = 9 the prime itself differs by 1 from the clos-est multiple of 10. In this case m = 1 and k is deter-mined accordingly. If x = 3, let m = 3, then the lastdigit of mp is 9 and the number mp is smaller thanthe closest multiple of 10 by 1. If x = 7, let m = 3, thenthe last digit of mp is 1 and the number mp is biggerthan the closest multiple of 10 by 1. Therefore if x=3orx=7, then m=3 and k is determined accordingly.

In summary, for any prime p, p ≠ 2, 5, it is possible todetermine a divisibility rule based on a trimming al-gorithm.

NUMBER THEORY CONNECTIONIn a Number Theory text (e.g. Long, 1987, p. 98) thefollowing can be found as an exercise:

(a) If p is a prime and (p, 10) = 1, prove that thereexist integers k and y such that yp = 10k+1.

(b) Let n = 10a + b. If p is a prime with (p, 10) = 1prove that p/n if and only if p/(a–kb), where k isdetermined in (a). However, without a concreteexperience the relationship between this exerciseand divisibility rules may not be apparent to manystudents.

FOR FURTHER INVESTIGATIONIn Polya's tradition, the fourth step in problem solv-ing is "looking back" (Polya, 1988). This involvessearching for alternative solutions or solution paths,generalizing solutions and exploring situations towhich the problem or the method of solution can beapplied. We have presented one level of looking backat the problem of divisibility by 7 by exploring thedivisibility algorithm for any prime. Further, "look-ing back" at the general divisibility rule, we can ex-tend our investigation by asking several "what if not"questions.

• What if not primes? Can a similar algorithm beused or modified to determine divisibility by acomposite number? What properties of primeswere used in our proof? For what composite num-bers can the algorithm be applied or modified?

• In all the above examples we have chosen thesmallest multiple of p that was bigger by I orsmaller by I than a multiple of 10. However, inour proof there was no reference to the choice ofthe smallest k. So, what if not the smallest? Willthe algorithm still work? In other words, is thealgorithm, for which existence is proved above,unique?

• Which familiar divisibility rules can be seen asspecial cases of the general algorithm?

A COMMENT ON USEFULNESSThere is a common trend in mathematics educationto focus on "applicable" mathematics, on mathemat-ics that is related to "real life situations". From thisperspective, there is a danger of labeling divisibilityrules as "unuseful".

I believe that usefulness, together with mathematicalbeauty, is in the eye of the beholder. For me, a prob-lem that attracts students' interest and curiosity, thatgenerates an engaging investigation, that invites stu-


dents to make conjectures and test conjecture - is mostuseful. I believe that an engaging mathematical in-vestigation is useful for all learners, and the excite-ment of mathematical investigation is especially use-ful for individuals planning for a teaching career. Ihope that my students feel the same.

BIBLIOGRAPHYBrown, S. and Walter, M. (1990). The Art of Problem Posing, Sec-

ond Edition (Hillsdale, NJ: Lawrence Erlbaum.Long, C. T. (1987). Elementary Introduction to Number Theory,

Third Edition. London: Prentice Hall.Mason, J. (1985). Thinking Mathematically. Wokingham, England:

Addison Wesley.Polya, G. (1988). How to Solve it, Second Edition. Princeton, NJ:

Princeton University Press.

One day, a boy named Data was playing with his dogVariable. Data lived in a magical world called “Num-ber Land.” Everything revolved around numbers.People were named after math terms. They used mathto label their houses. All of their jobs revolved aroundnumbers. Their whole language was made of mathterms.

“Son, come and finish your math chores,” said hismom, Mrs. Sequence. The Sequence family liked tokeep their numbers or equations in an ordered list.

“Aww, okay Mom,” he answered.

“Use like terms to organize your equation,” com-manded his mom. The word equation translates tobedroom in Number Land.

“Okay,” replied Data. His equation looked like this: -7+n-8n+2=. So he combined like terms and got 2n+2=.

“You’re almost done, but you need to know the valueof n. In this equation, n is equal to 1,” she said.

“Okay, Mom. Then my room is 4,” he replied.

“Good job, honey. That’s correct. Your equation is niceand organized,” exclaimed Mrs. Sequence.

“Here puppy. Come on Variable, let’s play with myfriend Identity,” said Data. “It’s the funniest thing thatall his Variables are exactly the same as my Variable.“Can Operation come over too?” asked Data.

“I guess,” said Mrs. Sequence.

“My friend Operation sure knows how to do adding,subtracting, multiplying, and dividing very well.When I have friends over, I never invite Operationand Inverse Operation,” Data explained. “Inverse, aswe call him for short, always excludes Operation fromeverything.”

“That’s smart, honey; you sure know a lot of infor-mation,” exclaimed Mrs. Sequence. So he called Iden-tity and Operation and asked them to come over. Theyarrived five minutes later.

“Come on in you guys. I’ll be right there but first Iwant to check the mail.” So Data went to the mailboxand looked through it and found a letter. He broughtit inside and opened it. “Oh. No. It’s an exponent let-ter,” he explained as he walked into the house.

“What does it say you have to do?” asked Operation.

“It says that I have to rewrite this letter 34 times or goto Division Crest to see what’s at the top,” Data yelled.So the three friends set off to Division Crest. “I won-der what’s up there,” said Data as they began to climbthe mountain. “Probably the same thing that’s on thetop of every mountain, a peak.” So they hiked all theway to the top. At the very top, they found a beauti-ful gleaming stone.

“Oh, my goodness, it’s an emerald!”

“We’re rich!” exclaimed Data.

“According to my calculations, you will now be 5,000.3times richer,” said Operation.

Equation StoryWhitney Perret

A short story by a student in Margaret Schafer’s 7th grade class at Hale Middle School, Los Angeles


Jeru is a student in the Starseed and Urantian Schoolsof Melchizedek and a minister of Aquarian ConceptsCommunity in Sedona, Arizona U.S.A.

***“Until we can understand the assumptions in whichwe are drenched we cannot know ourselves”

~ Adrienne Rich

Has science, in its earnest endeavor to free itself fromthe shackles of oppressive medieval thought, unwit-tingly shackled itself from a truer perception of real-ity? Has science, in its haste to distinguish itself fromsuperstition and free itself from restrictive religiousthinking, embraced postulates it would not have, hadthere not been a justifiably strong reaction againstmedieval religious mores?

The URANTIA Book offers this discernment on the sub-ject:

The mother of modern secularism was the totalitarian me-dieval Christian church. Secularism had its inception as arising protest against the almost complete domination ofWestern civilization by the institutionalized Christianchurch. (p. 2081 - §2)

One example, in my view, of the scientificcommunity’s illogical embrace of a postulate is thegenerally accepted theory that early life formed as theresult of the spontaneous coming together of aminoacids to form proteins. As you may know, this theorycame about as a result of the following hypothesis andexperiment summarized below:

In 1952, Harold Urey, a Nobel Prize winner, then of theUniversity of Chicago, suggested that the first living cellmay have come into existence as the result of a lightningflash searing its way through a smoggy primeval atmosphere

composed of hydrogen, ammonia, water vapor and meth-ane. Not that the lightning could have alchemized a livingcell at a single stroke; but it might, Dr. Urey proposed, havecombined the gases into a number of different amino acids,and these, in turn, might have combined into proteins, andthese, in their turn, might have combined themselves intothe first living cell.

In 1955, only three years later, one of Dr. Urey’s students,Stanley Miller, mixed the four suggested ingredients in abottle, discharged an electric spark through them for a week,and discovered on analyzing the result that he had indeedbrought about the formation of a number of different aminoacids. (Martin Cecil, On Eagles Wings p.39)

What is of interest here is that the scientific commu-nity as a whole has, apparently, accepted this findingas a valid hypothesis regarding the origin of life de-spite the statistical remoteness of this possibility. Howremote? Consider the following analysis regarding thechance amalgamation of proteins from amino acidsbelow:

In 1964 Malcolm Dixon and Edwin Webb, on page 667 oftheir standard reference work Enzymes, point out ….. that-depending on the laws of chance arrangement alone—inorder to get the needed amino acids close enough to form agiven protein molecule there would be required a total vol-ume of amino acid solution equal to 10 to the power of 50times the volume of the earth.

But here we are dealing with the chance origin of avery simple protein. What are the odds in favor ofthe formation of a larger protein molecule such ashemoglobin?

S.W. Fox and J.F. Foster have worked this out for us intheir Introduction to Protein Chemistry, page 279. Theyhave shown that only after the necessary amino acids had

Does a mathematical/scientific world-view lead to a clearer or moredistorted view of reality-purposive musings inspired from readings in

The URANTIA Book, The Cosmic Family, Volume I, and elsewhere.Jeru, Spring 2002

PO box 4305Sedona, AZ 86340

[email protected]


come together to form random protein molecules by the pro-cess described above, and only after these protein moleculeshad been formed in such a quantity that they filled a vol-ume 10 to the power of 512 times our entire known uni-verse …could we reasonably expect that just one hemoglo-bin molecule might form itself by luck alone. (Martin Cecil,On Eagles Wings p.39-40)

Clearly from the standpoint of statistical analysis theidea of life forming spontaneously is absurd. Yet, thisidea is pervasively held throughout the scientific com-munity. I suggest we have accepted this incongruousnotion rather than submit ourselves to the remotestpossibility of returning to the horrendously oppres-sive conditions of medieval times. That is to say, thereis, I believe, an unspoken fear that should the idea ofa personal god be accepted in mainstream scientificthought that this would then lead inexorably to a re-turn of the oppressive mores of medieval times wherescientists would find themselves beholden to and per-secuted by the church, as were Copernicus andGalileo. Thus, I submit, the animus within the scien-tific community towards considering the existence ofa personal god has more to do with fear and humanprejudice than with honest scientific analysis.

What has thus apparently evolved over recent centu-ries is the development of an existent paradigm of agodless science.

As we know, paradigms are sets of rules (filters ifyou will) for viewing the world. Regarding the clas-sic, The Structure of Scientific Revolutions, by ThomasKuhn, Ronald C. Tobey in A Beginner’s Guide to Re-search in the History of Science offers these insights:

Kuhn distinguished between two kinds of science - normalscience and crisis (or revolutionary) science. Normal sci-ence is science pursued by a community of scientists whoshare a paradigm. Revolutionary science is not. A para-digm is a consensus among a community of practicing sci-entists about certain concrete solutions—called “exem-plars”—to central problems of their field. Their consensusis based on commitment to the paradigm. The commitmentis derived from their training and their values; it is not theresult of critical testing of the paradigm. Normal science isintellectually isolated from “outside” influences, includ-ing the paradigms of other scientific fields and nonscien-tific events and values. Commitment to their paradigm givesa powerful “normality” to the paradigm, enabling scien-

tists to disregard phenomena that appear to contradict it-”anomalies.” (http://www.horuspublications.com/guide/cm106.html)

One may wonder then, to what extent the scientificcommunity is willing to promote this paradigm of agodless science. Is the scientific community’s invest-ment in a godless science so deeply entrenched that itcollectively disregards possibilities to the contrary?Consider Werner Heisenberg’s comments on his ownuncertainty theory. (The uncertainty theory simplystated is that, regarding sub-atomic particles, it is im-possible to know with certainty both the momentumand position of a particle at the same time, the greaterthe certainty of one quantity the less the certainty ofthe other, in contrast, this is not the case with larger(Newtonian) size objects, such as billiard balls wherethe position and momentum can be known with cer-tainty and at the same time.)

In view of the intimate connection between the statisticalcharacter of the quantum theory and the imprecision of allperception, it may be suggested that behind the statisticaluniverse of perception there lies hidden a “real” world ruledby causality. Such speculation seems to us—and this westress with emphasis—useless and meaningless. For phys-ics has to confine itself to the formal description of the rela-tions among perceptions. (W. Heisenberg, Zeitschrift fÜrPhysik, 43 {1927} p.197)

What is this ‘hidden “real” world ruled bycausality’?…apparently Heisenberg was unwilling toconsider it. Why? Perhaps it was because an honestexamination of this phenomenon could lead one toconceive of a personal god present amongst the par-ticles. A possibility apparently at odds with prevail-ing scientific thought then and now.

In the realm of sub-atomic particles, the observer in-deed has a cause-and-effect impact upon the observed.How could this be unless there was indeed a causal-ity connected to the presence of the observer? Statedotherwise, there exists a relationship between the hu-man observer and the physical matter beingobserved….a relationship. Here then is a clue thatthe universe is not static but that in fact our actionshave a discernible affect upon it.

Because our actions are intimately connected to ourthoughts and attitudes, we may thus expand the ma-


trix of reality being considered to include the attitudeand thought-life of the observer as well as the dis-cernible matter being observed. From this point ofview, reality becomes more fluid then perhaps we havepreviously conceived. Accepting for the moment anomnipresent personal creator we can also envisagethe presence of a divine or cosmologic vibration pat-tern whose very presence is revealed to us in directmeasure of our spirituality.

This phenomenon is hinted at in The Cosmic Family,Volume I where our relation to cosmologic vibrationpattern is revealed.

As you incorporate patterns of thinking within yourself,these energy patterns create messages within your physi-cal body that either respond to a cosmologic vibration pat-tern within the divine mind or to confusion, non-divinepattern, disharmony and self-assertion. (p. 155)

From this vantage point the relation of observer tothe observed may be expanded from merely a con-sideration of the study of sub-atomic particles to one’srelation to spirituality generally.

Again drawing from The URANTIA Book:

Moral convictions based on spiritual enlightenment androoted in human experience are just as real and certain asmathematical deductions based on physical observations,but on another and higher level. (p.2077 - §8)

This higher level apparently functions with remark-ably elastic properties as is again revealed in The Cos-mic Family, Volume I:

As you become honest, the gift of honesty is given. As youbecome patient, the gift of patience is given. As you be-come giving, the gift of things are given to you. As youseek wisdom over pride, wisdom is given. (p. 119)

The above discussion, in and of itself, I doubt willconvince many materialistically minded thinkers toembrace the reality of a living personal god but it is,nonetheless, worth considering; albeit many will likelyyield to the temptation of embracing a mechanisticview of reality rather then to consider the presence ofan intelligent creator behind the scenes.

Unfortunately, this mechanistic view taken to its logi-

cal conclusion:

…reduces man to a soulless automaton and constitutes himmerely an arithmetical symbol finding a helpless place inthe mathematical formula of an unromantic and mechanis-tic universe. (The URANTIA Book p.2077 - §4)

Challenging the mechanistic view-point generally TheURANTIA Book points out:

The inconsistency of the modern mechanist is: If this weremerely a material universe and man only a machine, sucha man would be wholly unable to recognize himself as sucha machine, and likewise would such a machine-man bewholly unconscious of the fact of the existence of such amaterial universe. (p.2078 - §6)

Clearly, however, not all scientists have embraced thisparadigm of a godless science. Many have, in fact,contributed to what has become known as the DesignArgument. (Stated simply the Design Argument pro-motes the idea that because there is so much evidencefor design in nature, both biologically and cosmically,that this therefore can be taken as evidence of the ex-istence of a designer.) Chief proponents of this viewinclude Isaac Newton who in his addendum to thePrincipia book three (the General Scholium) reasoned:

The planets and comets will constantly pursue their revo-lutions in orbits given in kind and position, according tothe laws above explained; but though these bodies may, in-deed, continue in their orbits by mere laws of gravity, theycould by no means have at first derived the regular positionof the orbits themselves from those laws.

Another proponent of this Design Argument wasWilliam Paley, author of Evidences of the Existence andAttributes of the Deity collected from the Appearances ofNature. Frederick Ferre in his classic essay Design Ar-gument summarizes Paley’s work in this way:

In that work Paley argued explicitly for the presence of in-telligently designed features in nature. The marks of de-sign, he said, are what we observe in contrasting a watchwith a stone. The stone, for all we can tell, might just have“happened”; but the watch is clearly put together out ofparts that work together in an arrangement that is essen-tial to their function, and the function of the whole has adiscernible and beneficial use. Wherever we find such aconstellation of characteristics, Paley said, we must admit


that we are in the presence of “contrivance” and designand since in our experience the only known source of suchcontrivance is the intelligence of some designer, we are en-titled—obliged—to infer an intelligent designer some-where behind anything possessing the above men-tioned marks of design. (Design Argument, FrederickFerre, Dictionary of the History of Ideas, Vol. I, p. 674)

Apparently then, we have arrived at two competingworld views; a mechanistic godless universe versusan omnipresent intelligent designer, with the humanscientist cast adrift somewhere in between.

Again, The URANTIA Book offers these insights:

The universe is not like the laws, mechanisms, and the uni-formities which the scientist discovers, and which he comesto regard as science, but rather like the curious, thinking,choosing, creative, combining, and discriminating scien-tist who thus observes universe phenomena and classifiesthe mathematical facts inherent in the mechanistic phasesof the material side of creation. Neither is the universe likethe art of the artist, but rather like the striving, dreaming,aspiring, and advancing artist who seeks to transcend theworld of material things in an effort to achieve a spiritualgoal. (p.2080 - §7)

Thus it appears it is the pursuit of science, rather thanthe science itself, which may offer us the most mean-ingful approach to reality.

Along this line of reasoning Cardinal NicholasCusanus of the fifteenth century observed:

Mathematics induces the mind to withdraw somewhat fromphysical immediacy into the sphere of reflective meanings,thus preparing for our further move toward God’s invisiblereality. (Idea of God 1400-1800, James Collins, Dictio-nary of the History of Ideas, Vol. II p. 346)

If one will allow the supposition that mathematics isthe product of the human mind then the field, takenas a whole, reflects the breadth, width and height ofhumanity’s ideational habitat; offering at the sametime an expansive as well as conditioned conceptualframework.

In this regard The URANTIA Book offers this insight:

While the domain of mathematics is beset with qualitative

limitations, it does provide the finite mind with a concep-tual basis of contemplating infinity. There is no quantita-tive limitation to numbers, even in the comprehension ofthe finite mind. No matter how large the number conceived,you can always envisage one more being added. And also,you can comprehend that that is short of infinity, for nomatter how many times you repeat this addition to num-ber, still always one more can be added. (p.1294 - §11)

Returning to our original proposition: Does a math-ematical/scientific world-view lead to a clearer ormore distorted view of reality? The URANTIA Bookagain offer us these clarifying insights:

Mathematics, material science, is indispensable to the in-telligent discussion of the material aspects of the universe,but such knowledge is not necessarily a part of the higherrealization of truth or of the personal appreciation of spiri-tual realities. Not only in the realms of life but even in theworld of physical energy, the sum of two or more things isvery often something more than, or something differentfrom, the predictable additive consequences of such unions.The entire science of mathematics, the whole domain of phi-losophy, the highest physics or chemistry, could not predictor know that the union of two gaseous hydrogen atoms withone gaseous oxygen atom would result in a new and quali-tatively superadditive substance—liquid water. The under-standing knowledge of this one physiochemical phenom-enon should have prevented the development of materialis-tic philosophy and mechanistic cosmology. (p.141 - §4)

In conclusion, on the subject of mathematical reason-ing, the universe, reality and whether or not there isan omnipresent personal god, perhaps the most stimu-lating close to this essay would be the query, so elo-quently posited in The URANTIA Book, “…whencecomes all this vast universe of mathematics without aMaster Mathematician?” (p.2077 - §4).

REFERENCESA Beginner’s Guide to Research in the History of Science, Ronald

C. Tobey, Horus Publications, Berkeley, California.Dictionary of History of Ideas, Philip. P. Wiener, editor, Scribner,

New York.The Cosmic Family, Volume I, by Gabriel of Sedona, The Starseed

& Urantian Schools of Melchizedek Publishing, Sedona, Ari-zona.

On Eagles’s Wings, Lord Martin Cecil, Two Continents Publish-ing Group, London, England.

The URANTIA Book, URANTIA Foundation, Chicago, Illinois,1955.


Emerson said (and I’m sorry for not having theexact quote), that every institution is the workof a man. We can think about that on two lev-

els; Alvin kept the torch burning for the rest of us tobe sure, but the shadow of the man is also the “hu-man” in Humanistic Mathematics.

The past decades have seen mathematics in unimag-inable transition. The WWII years may have been aperiod which experienced an admirable collaborationof thinker, but its legacy, the 50’s, was a period inwhich you had to take sides: “applied” or “pure.” Ei-ther way you were stuck.

In the 60’s, pure math was filtering into textbooks tocreate what might be (in your mind) “the only decentCalculus book” or something entirely unreadable.One-third of the female freshman class at Brown/Pembroke in those Sputnik years declared math astheir incoming major; but non-standard analysis killedmost of them off. Often a student’s text was theteacher’s notes, and that minimal, purple-printed testasked you to prove the Fundamental Theorem of Cal-culus and solve some problems. (Perhaps your texthad problems, but there weren’t many! )

In the 70’s, during the Vietnam War we saw the vagueemergence of the computer and simultaneously dis-trust began to brew about putting pure mathemati-cians into the position of guardians of engineeringresearch centers. I remember teaching at Washingtonand Lee in those days, trying my hardest to persuademy class that the computer would be useful! (We wereusing it to add three floating point numbers and ifyour punch cards didn’t spill all over the floor in tran-sit to the computer room, you found Fortran couldn’tdo the problem accurately.)

The 80’s were a period good for women and minori-ties. Cultural concerns, including issues of teachingflourished. Note, 1986 marks the inception of the Hu-manistic Math Network.

The 90’s! What can one say? They went by in a micro-second, raising issues of the calculator, the computerand the web, but equally, a brand new interest in teach-ing (e.g. calculus reform). Teaching can be the “tar-baby” of mathematics, and we have to temper ourinterest in being converts to a religion-of-one with theneeds of unthinking, financially driven governingbodies.

How then will we, in the future, hang onto the spiritof humanism, collaboration, and gentleness that learn-ers from previous decades enjoyed and shared? Myown feelings are quite optimistic. We are again ben-efiting from the blending of the applied and the pure,and we have a greater awareness of student-teacherrelations and issues of assessment with which we mustalways focus on the need to possibly remodel our-selves.

To create this last issue of the “hand-held” HMNJ, Iessentially asked Alvin to step aside as I asked for con-tributors from the journal’s past. Touching thesepeople enriched me. It reminded me of all that wasbest about mathematics; the people I contacted werepeople whose books or articles I may have read, insome comfortable reading corner—books “about”mathematics—how we do it and what we mean by itand why we do it. But there are many warm and won-derful members in my immediate mathematical com-munity.

Web generation we now may be, and the terms of theweb will be broader, more scattered, faster and yetmore time-consuming. But we can exploit the webversion of HMNJ by tuning in, to create a new com-munity which retains that understanding of the “hu-man” for which our institution of mathematics ismerely the shadow. Thanks, Alvin, for keeping us re-membering!

Sandra KeithMathematics, St. Cloud State UnivMN 56301

Special Section:On the publication of the 26th issue of

the Humanistic Mathematics Network Journal


Alvin White and the HMNJ

When I first met Al White at the winter mathematics meeting in San Antonio in 1987, Ialready knew his name. We both had links

to Stefan Bergman. Alvin did his doctoral disserta-tion under Bergman at Stanford, and I, a few yearspreviously, had been a graduate student takingBergman’s courses (at Harvard) and later post doc’dfor him. Al and I both had high regard for Bergmanas a human being, which was not always the case inthe mathematical community. This was an initial ba-sis for our friendship. The second basis and more sig-nificant one, was our mutual concern for the human-istic aspects of mathematics, however you care to de-fine that term.

Al saw the need for a periodical devoted to promot-ing the humanistic aspects of mathematics. In thosedays he was one of the few voices crying in a wilder-ness of unreflective accomplishment. He told meabout the HMNJ and his plans for it. I was fooled byhis soft-spoken and occasionally bumbling manner.But Al carried through with his plans.

Things don’t just happen; they don’t just sprout likemushrooms after a penetrating rain. When one looksclosely, there is always an individual who conceives,then creates and nurtures what has been created. Inthe case of the HMNJ, that person is Alvin White, andover the years the journal has become the leadingoutlet disseminating articles on the humanistic aspectsof mathematics. I congratulate Al for piloting theHMNJ with skill for so many years and I hope he willbe able to guide it through its present crisis.

Today when, on the one hand, society is ever increas-ingly mathematized, and when, on the other hand,mathematics offers many an escape from the difficultdilemmas of civilization, the necessity of relatingmathematics to the lives of those who create it andthose who are affected by it is increasingly important.Hence the need for reflecting on and publicizing suchreflections. Looking over several past issues of theHMNJ, I observe that it contains a wide variety ofsubjects. There are, among others, didactics at all lev-els of instruction; biography; linguistics; heuristics;methodology; history and futurology; esthetic; poetry;the spiritual element; women’s issues;ethnomathematical issues, book reviews.

What I, personally, would like to see in future issuesof the HMNJ is much more discussion of how math-ematics has entered our daily lives and what its ef-fects have been for good, for bad, or for neither. Asexamples, (some of which have indeed been discussedin past issues),product striping, ATM’s, votingschemes, the impossibility of counting (census resultsare a matter of litigation), other parts of mathematicsthat are litigious (e.g., what math can be copyright),use of mathematical statistics as court evidence, therole of mathematics in social and economic decisionmaking, gambling schemes, the role of mathematicsin “green” issues, the role of mathematics in defense.

Thus, referring to the last topic, there is to be in Au-gust of this year a conference in Sweden discussingthe role of mathematics in war. I would think that anumber of papers presented at this conference wouldbe appropriate material for reprinting in the HMNJ.What I would caution against is the HMNJ becominga journal that deals more and more with classroomissues. There are many journals that concern them-selves with pedagogical techniques and curricularquestions. There are indeed serious problems in theclassroom that need consideration, but despite theinadequacies in the teaching and learning of math-ematics, the fact is that the mathematization of soci-ety is going forward at an increasing pace, often set inposition by the fiat of a techno-competent elite, andwhich require constant reexamination and discus-sion. The HMNJ should be characterized by its con-cern for the problems that mathematizations create.

Philip J. DavisDivision of Applied MathematicsBrown UniversityProvidence, RI 02912


Reflections on the Founder

For me, the Humanistic Mathematics Networkis inextricably linked with Alvin White, itsfounder and editor. In this special edition of the

Journal, it is probably appropriate to describe what Iknow of Al, a man whose life and career embody theprinciples that the many readers of the newsletter findso compelling.

My first meeting with Al White has become a carica-ture in my mind; the remembered emotion of theevents has exaggerated the details. This is probablynot what happened, but it is what I remember hap-pening.

I came to Harvey Mudd College for one semester toteach a few courses. Al had written me a very kind e-mail to welcome me, and so I decided to stop by hisoffice and introduce myself. He was seated with a stu-dent, both of them at a desk far across the vast roomfrom where I stood in the doorway. I stuck my headin the room; I excused the interruption to say myname; I said that I would come back later.

But Al jumped up, papers flying from him the waypigeons fly from a cat. He clambered past piles uponpiles of books to grasp my hand in both of his. Wel-come, he said, and I am so glad to meet you. He askedme about an article I had written about mathematicalwriting, and gazed into my eyes earnestly while heprofessed his own passion for teaching. For severalminutes he explained his philosophy on the impor-tance of teaching well, of communicating well, ofkeeping the next generation always in mind. Students,he said, students come first. Always. I remember look-ing past Al at the student still sitting on the chair bythe desk, on the other side of piles of books, on the farside of a vast room. Al followed my gaze back andlaughed at himself. It is so good to have you here, hesaid to me, and he clambered back to his student.

My deepest obligation to Al is for a panel discussionthat he organized. I was still a young, untenured math-ematician, but Al invited me to be one of the panel-ists. On one side of me sat Leonard Gilman, the au-thor of “Writing Mathematics Well”, and whom I hadidolized from afar. On the other side sat JoAnneGrowney, a mathematician/poet whose books stilltravel back and forth between my shelves and my lap.

I remember Leonard Gilman playing the piano to de-scribe mathematics in music, and I remember thatJoAnne Growney read poems of her own and of oth-ers. (I don’t really remember what I did at all, butmaybe you do: there were 500 people or so in the au-dience). It was a wonderful evening, and a wonder-ful opportunity, and I am deeply grateful to Al forincluding me.

Alvin White has made his career at a small school(about 600 students) that is probably the only LiberalArts/engineering School in the nation. While youmight think that this combination would naturallyengender a newsletter like “Humanistic Mathemat-ics”, in fact I found during my semester there that thestudents (and even many of the faculty) were invari-ably practical-minded, with little room for fluff. Giventhe choice between truth and beauty, they wouldchoose truth.

And yet Alvin White began a journal-and a nationalmovement-that combines mathematics with poetry,that says that people come first, that defines math-ematics as a human endeavor. Given the choice be-tween truth and beauty, Al does not choose. Instead,like Plato and Keats, Alvin White says there is nochoice:

Truth is beauty, and beauty is truth: that is all ye know,or need know, on earth.

Thank you, Al!

Annalisa CrannellDepartment of MathematicsFranklin & Marshall CollegeLancaster PA 17604


On the Twenty-Sixth Edition ofHMNJ

I am indeed saddened by the news that HMNJ inits hard copy form will be coming to an end. I’mgrateful that EXXON has supported it as it has.

But I don’t think on-line journals are nearly as useful,as they are hard to read and often text and diagramsget changed on the screen as far as layout is concerned.Anyway, online journals are much harder to keep trackof. With HMNJ, I often recall the color of the issueeven if not the date, so can find it again more easily.Also, I can browse through it by my nightstand—something I can’t do if it is only on the computer!

Nevertheless, while hoping that the HMNJ continuesto be a fruitful place to visit online, something mustbe said about what it has been til now. Namely, it hasbeen a unique place for publishing diverse kinds ofarticles. For example, where else could long, valuable,and thoughtful articles like, “Will you Still be Teach-ing in the Twenty-First Century” (#23), “Tilings in Artand Science” (#12), and the timeless, “The ClassroomEncounter” be found?

Also, the book reviews have been very thoughtful.Thinking of this as the last issue, I now wish I hadshown my appreciation to Alvin more directly, andcertainly I never thought of writing to EXXON to tellthem what a valuable contribution they made.

So if nothing else, I do want, in this last issue, to thankAlvin for his vision, which he created this journal andfor all the hard work he put into it. I think he and thejournal have given encouragement to a lot of peoplewhose work and teaching will have greatly benefit-ted from it. His great chapter, “Teaching Mathematicsas if Students Mattered,” in the book “Teaching as ifStudents Mattered” (1985), should be read by olderfolk again, or used to introduce younger ones whohave never heard of this book, to ideas now consid-ered innovative, published over 15 years ago. I thinkthis chapter is even more relevant now than then be-cause of the onslaught of testing and the “TELL thefacts” type of teaching.

AN ALPHABET FOR THE TWENTY-SIXTH EDITION OF HMNJ

Alvin with his ideas of the need to teachBetter by encouraging

Concepts andDialoguesEncouraged us all to learn aboutFascinating subject matter withGood ideas.Humanistically Hearing (not just listening) and pro-

vidingInspiration and Intuition tempered withJudgment and always withKindness, he showed usLearning thatMatters, that seemsNatural, thatOpens students’ eyes toPosing of problems andQuestion-asking, together withReaching toSolve those problems byThinking in new ways—Usefully, and in aVariety of ways, demonstrating the importance ofWriting.eXxon supported this journal over theYears so as to provide us with theZest to continue our teaching so others, too, will love

math.

Good luck to the HMNJ of the future!

Marion WalterMath DepartmentUniversity of OregonEugene, OR 97403

Being an editor of a periodical is a tedious, thankless job. So many details to worry about. Somany chances to do or not do something that

somebody will complain about. When readers are sat-isfied, they thank the author, not the editor. When theyare dissatisfied, they blame the editor, not the peoplewho could have contributed articles but didn’t getaround to it.

And then, what about the courageous, heroic indi-vidual who actually dares to found a new publica-tion, and that newsletter miraculously makes it, en-dures, finds an angel and an audience! Do we remem-ber him or her, when years later we benefit from hisor her creative and daring impulse?


So, I say that what Alvin has done is awesome. Now,what about the down side? HMNJ—(Alvin White,actually) accepts what comes. Some is more original,some less. Some is less profound, some is more. Theliterary-scientific quality of each issue has to be vari-able. But it is authentic! HMNL is a an outlet, a ve-hicle for the large inchoate mass of math teachers whowant to humanize mathematics. As such, it is incom-parable and irreplaceable.

Thanks, Alvin.

Reuben HershUniversity of New MexicoAlburquerque, NM

Humanistic EducationalMathematics

My first memory of the notion of (if not theterm) humanistic mathematics is of thePasadena meeting of mathematics educa-

tors in March of 1986. Alvin White assembled quite afew impressive people, including some of my heroesof the time and later. We had a great time not reach-ing consensus about what humanistic mathematicswas.

Fortunately, the lack of consensus didn’t deter Alvinfrom forming the network and newsletter. Without aneed for consensus on the definition of the term, mem-bers seem to fall into at least one of two camps: em-bracing mathematics as a human activity (usuallymeaning as part of humanities, having strong connec-tions to literature and history and the fine arts), orteaching mathematics with attention to how humansthink and feel and learn.

My own interest is in teaching mathematics as a meansof educating humans. I use “educating” in the senseof encouraging development in intellectual, ethical,and identity domains. I’m especially interested in thematuring of peoples’ conceptions of knowledge, asdescribed by Perry (1970) from believing that everystatement is either right or wrong to seeing truth asrelative to context. As Perry found, this cognitive de-velopment is intertwined with ethical and identitydevelopment. Letting students in on the ambiguity,warmth, imperfections, and elegance of this field thatis commonly seen as precise and austere and perfect

can help shock them out of the comfort of a black andwhite world.

So I’m interested not so much in mathematics educa-tion-i.e., in helping students come to attain mathemati-cal skills or to understand mathematical ideas or evento “think like mathematicians’~-but in how mathemat-ics can be taught to enhance education. Back in the1970’s, Steve Brown coined the term “educationalmathematics” for that interest-a term included in thetitle of the Institute I direct. That Steve continues toshare this interest is evident in his most recent book(Brown, 2001).

Educational mathematics depends on humanisticmathematics in a variety of ways. To use mathemat-ics as a tool for education, we need to be aware ofhow humans think and feel and learn. We can use linkswith humanities astorce to encourage students to growand change. But the goals are somewhat different.And, while focused on these goals, the major issuesof mathematics education-reform curricula, assess-ment, etc.-are of interest only peripherally. I’ve writ-ten about these issues in various places, and I sup-port myself by working with them, but what I con-sider my most enthusiastic publications have beenabout using mathematics to encourage developmenton the Perry Scheme (e.g., Copes, 1982, 1993).

What about the future of humanistic mathematics edu-cation and educational mathematics? To the extentthat, as Postman & Weingartner (1969) claim, teach-ing is a subversive activity, many teachers will con-tinue to teach humanistic mathematics by teachinghumanely. A few will communicate the aspects of thefield that make mathematics like the humanities.

As for educational mathematics, I suspect that we arenot much further ahead than we were 20 years ago.NCTM’s influential publications (1989, 2000) ignorethis role of mathematics. The reform movement re-fers to areas outside mathematics only as a source ofapplications or as a means to help students under-stand mathematics better. The political battles areabout what to teach of and about mathematics: pri-marily, as facts and procedures or as a set of tools forproblem-solving. The justification of mathematics asa means to the end of self actualization isn’t appear-ing in the newspapers.


But learning is a subversive activity as well. I am op-timistic that many students, with or without the helpof teachers, will experience mathematics in ways thatencourage their own development.

Larry CopesDirector, Institute for Studies in Educational Math-ematics10429 Barnes Way E.Inver Grove Hights, MN 55075-5016

REFERENCESBrown, S. 1. (200 1) Reconstructing school mathematics: Prob-

lems with problems and the real world. New York, NY: PeterLang.

Copes, L. (1982) “The Perry Development Scheme: A metaphorfor teaching and learning mathematics.” For the Learning ofMathematics, June.

Copes, L. (1993) “Mathematical Orchards and the Perry Devel-opment Scheme.” In White, Alvin M. (ed.) Essays in Human-istic Mathematics, Washington, DC: Mathematical Associa-tion of America.

NCTM. (1989) Standards for School Mathematics. Reston, VA:National Council of Teachers of Mathematics.

NCTM. (2000) Principles and Standards for School Mathematics.Reston, VA: National Council of Teachers of Mathematics.

Perry, W. G. (1970) Intellectual and Ethical Development in theCollege Years: A Scheme. New York, NY: Holt, Rinehart, andWinston.

Postman, N. & Weingartner, C. (1969). Teaching as a SubversiveActivity. New York, NY: Dell Publishing Company, Inc.

It is hard to say what is the BEST thing about AlWhite, but one VERY GOOD thing is that he always has time to hear and consider new ideas.

The Humanistic Mathematics Network and its jour-nal have for many years provided a source and a fo-rum for the new and the different as well as the oldand not-to-be-forgotten.

As a mathematician with strong interests in literatureand the arts, I was delighted to find, through HMN,others with similar interests. Al White has been untir-ing in his support of special sessions at national meet-ings and journal publication so that teachers andmathematicians of diverse interests may exchangeideas.

Now a poet, my memories of HMNJ focus particu-larly on poetry from it and I am ever grateful that Imet on its pages the Czech poet, Miroslav Holub (1923-

98). Both a scientist (an immunologist) and a poet,Holub’s interests paralleled my own and he has be-come a favorite poet for me.

Below I include a brief poem by Miroslav Holub(translated from the Czech by Ewald Osers) that hasbeen included in NUMBERS AND FACES: A Collec-tion of Poems With Mathematical Imagery, an HMNpublication that Al and I worked on together in 2001.

THE PARALLEL SYNDROME

Two parallelsalways meetwhen we draw them by our own hand.The question is onlywhether in front of usor behind us.Whether that train in the distanceis comingor going.

Al’s contributions to mathematics and its teaching arelegion. He has had new ideas when others were dor-mant; he has continued when others were idle. Hehas lived, at least, nine lived. Always a supporter ofthe artists among mathematicians the visual artists,the musicians, the writers. I have appreciated his sup-port of my poetry and close with this brief poem ofmy own.

GOOD FORTUNE

is good numbers,the length of a furrow,the count of years,the depth of a broken heart,the cost of camouflage,the volume of tears.

Alvin White, may you always have good numbers!

Thank you for your years as chief of the HumanisticMathematics Network and as editor of the Humanis-tic Mathematics Network Journal (and, before it, theNewsletter). I am sorry that the years of paper publi-cation have ended for I love the feel of paper in myhands; however, I will find HMNJ on-line, and I hopeeveryone else does too.

JoAnne Growney147 West 4th St.Bloomsburg, PA 17815


Teaching as Though StudentsMattered

A BIOGRAPHY OF ALVIN WHITE AS TOLD TO SANDRA KEITH

I was born 1925 in New York; served in the U.SNavy in 1943-1946 during WWII; was sent to Radio Technician School and then served in the Pa-

cific (the battle of Okinawa). Subsequently, I was sentback to the States for Officer Training School at Co-lumbia University. I graduated from Columbia Col-lege in 1949 with a major in math but I took manyother subjects, of course. Among my most memorableprofessors and courses were Moses Hadas teaching"Greek and Roman Mythology" and Bernard Stem in"The Family Past and Present." I feel I learned the most(and got the most enjoyment) from one course thatrequired me to write a paper independently in Au-tumn and Spring term; in that course, I chose to learnand write about the history of public housing in NewYork City. Special apartment houses had been built inNew York City for the waves of immigrants comingto the USA in the early part of the 20th century. Theseearly houses were instant slums. Eventually, a law waspassed that required every room to have a window;even that minimal ruling was barely satisfied. But Ispent many happy hours in the Columbia Architec-tural Library reading Jane Jacobs and other architectsand historians on the history and progress (albeit slow)toward improving public housing.

I received an MA in math from UCLA and went on toPalo Alto where I enrolled at Stanford. I worked withStefan Bergman on partial differential equations re-

lated to fluid dynamics and received my Ph.D. 1961.1 was hired by Harvey Mudd College, but spent theyear 1961- 1962 at the Math Research Center at theUniversity of Wisconsin, Madison.

While teaching traditional courses at HMC, I hadfound and read "Freedom to Learn," by Carl Rogers.As an attempt to apply my understanding of Rogers'book, I offered a seminar on Calculus of Variations inmy living room. I received a small grant to purchaseportable blackboards (green actually) and bean bagsfor sitting. (I visited the California Institute of Tech-nology some time later and found that they had a roomsimilarly furnished that was used every hour by manydepartments to teach various courses!) I wanted thecourse to be "student centered", as I understood thatconcept. At the first meeting I introduced ten or twelvebooks on the subject and explained the problems andtopics from the early history of the subject. Each stu-dent was to report on a topic. I met with those sixstudents once a week for three hours. As it workedout the students' reports paralleled the usual course.The reports were also distributed and used. But at theend of the course some students remarked, " I enjoyedthe seminar and learned a lot. But I think that I wouldhave learned more with lectures and a textbook." Iwas not aware of anyone teaching mathematics in thisway at that time.

I tried to publish my experiences in that course with"The American Mathematical Monthly" but was re-jected with the remark, "No numerical data." How-ever, Carl Rogers, while guest editor of the journalEducation, published my article under the title "Hu-manistic Math: An Experiment." A physicist fromPurdue was inspired by another book by Carl Rogersand also taught a laboratory course in physics in thesame student-centered manner. His students had asimilar response:

"We enjoyed the course but we would havelearned more with a text and lectures." Hiscomment, in an article, was: "I doubt that theywould have learned more ... although theymight have suffered more."

In the early 70's, I began to organize informal, inter-disciplinary discussions among faculty of theClaremont Colleges. The discussions drew not onlymathematicians but physicists, philosophers, histori-


ans, and psychologists. At this time, writing unsuc-cessful grant proposals was to become a repeated oc-cupation of mine. In 1976-77, however, I was one often Danforth Faculty Fellows awarded year-longgrants to study at various universities. I studied atStanford, took a short stay at Northwestern, and spentthe spring semester at Massachussetts Institute ofTechnology (MIT)-Harvard, where I taught a seminarat the Division for Study and Research in Education(DSRE). The seminar was described by a set of ques-tions:

"How does one acquire knowledge? What arethe limits of certainty? What is the relation-ship between scientific knowledge and gen-eral knowledge? What is the role of beauty,simplicity or intuition in creative discovery?Our present knowledge in the arts, humani-ties and sciences is the legacy of creative imagi-nation. How can this legacy influence educa-tion at all levels?"

There were twelve students in the course, represent-ing Math, Art, Linguistics, Electrical Engineering, Bi-ology, Artificial Intelligence, and Computer Science.We sat around a table discussing our readings andthoughts. We met twice, for an hour and a half, thefirst week. After that, everyone dropped their othercourses and we ended up meeting the whole morn-ing until we were eventually displaced by lunch hour.We became a community that cared for one anotherand learned from each other. Students invited otherprofessors to participate. Visitors to MIT-DSRE wouldask permission to observe quietly, although they usu-ally joined in the discussion. One student remarkedthat the popularity of our seminar among visitors wasprobably due to the openness, honest listening andcaring that were evident. We accepted all contribu-tions in a non-judgmental way. No one was forced tospeak, and everyone had a chance to speak. In thatcourse, we examined writings of Dewey, Kant,Polanyi, Russell and others. The last week of the termbecame a time to discuss and evaluate the seminar.Why was it so successful? What had happened to us?How had we been transformed from a mere collec-tion of strangers to a group of friends and colleagues?It was as if we had chanced upon a semester-long cel-ebration, and, like Alain-Foumier's "Wanderer," wehad been caught up in the "spirit of the place." A stu-dent observed that this was the first course where her

presence in the room had "made a difference." Wewondered: what had we learned and what should wedo if we wanted to find that spirit of celebration again?An unexpected answer emerged, one that simulta-neously addressed some of the questions of the semi-nar as well as questions about the seminar. The an-swer from one of the students was : "we learned withlove and trust." This was more food for thought: whatdid that response mean? In my opinion, the meaningis well discussed in E.A. Burtt' s book: " In Search ofPhilosophic Understanding."

After I returned to HMC from MIT and the exciting,warm relationships that I had experienced there withstudents, I was disappointed by the relationships thatstudents assumed or insisted upon between them andme as their teacher at HMC. To them the classroomcould not be a community of mutual learning andteaching, and for some there was an invisible line be-tween teacher and students. In an attempt to over-come some of these barriers, I started teaching mycourse in Calculus upon returning to HMC by read-ing with the class, J. Bronowski, "Science and HumanValues." I wanted to investigate with my classBronowski's statement that "the society of scientistsis more important than their discoveries. What sci-ence has to teach us here is not its techniques but itsspirit: the irresistible urge to explore." I have to ad-mire the students' tolerance. Their fear was that theywould not understand calculus and that we wouldnot complete the syllabus . As it turned out, we com-pleted the syllabus with time to spare, and studentsand I agreed that they understood the calculus moredeeply than they would have in a traditional courseof straight lectures. In addition to homework prob-lems, students were encouraged to modify problems,for example, by increasing the dimension of the spaceor by changing a constant to a variable. They inventedoriginal problems and challenged classmates to solveproblems. We also consulted several textbooks al-though we followed, more or less, the traditional syl-labus of the "official" department text. Instead of onetext, however, the students consulted at least twobooks for each topic in the course outline. The hopewas that the difference in approach and notationwould motivate the students to a deeper understand-ing of the material. Most students commented thatthe multiple text approach resulted in a deeper, morecreative understanding of the material although it tookmore effort.


Although some calculus texts have over 500 pages,these books may represent a narrow path of theknown. The problems students invented, for whichsolutions were known or not, were given to the classas challenges which many students accepted. Theystruggled, guessed, reasoned by analogy and testedtheir tentative solutions. Many students were highlymotivated by these challenges. We were a society par-ticipating in doing mathematics. It often happenedthat a problem that was impossible to solve was thetopic of a future chapter, which led us into the newmaterial naturally. The most controversial (at the time)innovation was the cooperative exam where the wholeclass could discuss the problems and solutions. Stu-dents could not catch on to this idea on the first exam,but the second exam was more successful with stu-dents learning more by cooperating. My hope was touse cooperative exams to bring intellectual excitementinto the mathematics classroom. One student, in a laterevaluation, remarked that the cooperative exam was"one of the best of the innovations but is not suitablefor performance evaluation."

In those experimental classes at HMC , in addition tocooperative testing, term papers were required. Stu-dents were asked to write a term paper on any topic.There were essays on creativity, responsibility of sci-entists, fractional integration, computer generatedpoetry and other topics. Class time was spent answer-ing questions, comparing different approaches, anddiscussing invented problems and non traditionalproblems. The non traditional material and approachgave us insights about mathematics and learning thatwere wholly unexpected. Many students respondedpositively (one a year later!) and some were indiffer-ent. A few were hostile. But class attendance was con-sistently high. Section A of the class then sent a mes-sage to section B, inviting them to meet in a large lec-ture hall on the weekend to discuss some solutions toproblems. One student presented a solution that wasrejected. Another student presented a different solu-tion. There was arguing and some shouting. Finally asolution was presented that survived criticism. A con-sensus was achieved. The students were exhilarated.Most students remarked how much they had learnedand how good they felt about themselves and theirrelationship to the material. In " A New Paradigm forthe Mathematics Classroom" (Int. J. Math Educ. SciTechnol 1976 vol7, no2) I raised a number of issues..

Mathematics is exciting. Can it compete with a litera-ture course where students are caught up in the intel-lectual clash of ideas? Can students recognize math-ematics as a creative activity? After the article waspublished, I learned of others who had tried coopera-tive testing, some of whom were inspired by my ac-count.

Before I embarked on my Danforth Fellowship, mycolleagues and I submitted a proposal to FIPSE in 1976:"New, Interdisciplinary, Holistic Approaches to Teach-ing and Learning." The beginnings were informal, in-terdisciplinary discussions that I had organinzedamong the faculites of the Claremont Colleges. WhileI was at MIT, I learned that proposal was denied. Onereviewer said the idea was trivial; another said it wasimpossible! I traveled to Washington, DC, to consultwith the FIPSE staff, who encouraged me to reapplythe next year. The second application to FIPSE wassuccessful and resulted in a three-year grant. Facultyfrom six colleges (all of the Claremont Colleges, anearby state college and the neighboring communitycolleges) participated. The goal of the project was tomake every participant an interdisciplinary scholar/teacher—to introduce scientists to the humanists'viewpoint and knowledge-and visa versa. Each fac-ulty member was encouraged to understand and ap-preciate the viewpoints and problems of the wholerange of knowledge. Prominent scholars from all overthe US came to speak to us. We had as many as twospeakers with discussions per day, every academic dayfor three years. A project which had begun as a per-sonal vision of a few became a major part of facultyculture. Approaches that were first viewed with skep-ticism became accepted and even expected. Ideaswhich were on the radical fringes of academe aboutthe benefit of integrating the sciences and humanitiesmoved into the main stream of desired educationaloutcomes. Some faculty members who were timidabout trying new modes of teaching or introducingnew content, such as values and ethics, were sup-ported and encouraged by the newly created setting.For example a chemist introduced humanistic themesinto his classes and began holding discussions in hisclasses. We ended on a high note, with a three-dayconference addressed by some of the nations' leadingeducators. Among the presenters were Nevitt Sanfordof Berkeley's Wright Institute, Benson Snyder of DSREand Harold Taylor of Sarah Lawrence. A participantcommented that she had never expected to see any of


these educational leaders in person, but here she wasseeing so many altogether!

After the FIPSE project I was elected president ofSIGMA Xi, AAUP, and co-president of the faculty sen-ate (not concurrently!) The FIPSE project, 1977-81 wasan exciting time for me as well as for many facultycolleagues.

My interest in interdisciplinarity le d me to solicitauthors and articles for one of the Jossey-Bass series," New Directions in Teaching and Learning (0)" titled"Interdisciplinary Teaching," which I edited. The otherauthors --Geoffrey Vickers, Ralph Ross, KennethBoulding, David Layzer, C.West Churchman, RichardM. Jones, Arthur Loeb, Owen Gingerich, BarbaraMowat, CArl Hertel, Miroslav Holub --were well-known people from many disciplines: Business, Law,Education, English Literature, General SystemsTheory, Astronomy art and Environmental Design.They included a professor of design science at Harvardwhose degree in chemical physics led to a collabora-tion with R. Buckminster Fuller, a professor of art andenvironmental design, and a Czech poet who becamechief research immunologist for Clinical and Experi-mental Medicine in Prague. My article in this seriesdiscussed my seminar at MIT, and gave the title, forthe Jossey Bass Series (#21): "Teaching as Though Stu-dents Mattered."

All my experiences contributed to my seeing math-ematics as one of the humanities. I wrote to the ExxonFoundation for a grant (of a million dollars) to de-velop the concept of Humanistic Mathematics; to cre-ate an approach to teaching and learning mathemat-ics that would be nonthreatening, but inviting to stu-dents, who would participate in a cooperative spiritwith each other and with teachers. My dean was verydiscouraging about the whole idea and the possibil-ity of receiving the grant. Within a week of sendingthe letter to Exxon, however, I received a phone callfrom the program director, who wanted to discuss theproposal and negotiate the particulars. After threeweeks of conversations on the phone he suggestedthat he could give me $10,000 within a month to havea "Planning conference" for my colleagues and I totalk about the ideas and program of humanistic math-ematics. There was no time for complicated arrange-ments and advertising. I asked some friends if theywould come and recommend other friends for a three-

day conference. For me this would be a singualarmoment in time. As I recall, the attendants wereStephen Brown of SUNY Buffalo, Marion Walters ofOregon, Phil Davis of Brown, Reuben Hersh of NewMexico, Jeremy Kilpatrick of Georgia, Bob Borrelli ofHMC, Paul Yale of Pomona College, Tom Tymozko ofSmith College (who said at the 1992 Quebec meetingof ICME, "Alvin White started all this"), Sherman Steinof UC Davis, Ed Dubinsky of Purdue, a psychologistfrom Boston, two mathematicians from Cal Poly SanLuis Obispo, several others, and myself -- thirty inall.

After three exciting days, we decided to start a news-letter; this was the birth of the Humanistic Mathemat-ics Network Journal. Exxon Education Foundationgraciously agreed to support this newsletter-journal.The first issue was sent out to thirty people, one yearlater in 1987. The second issue was sent to sixty people,and then distribution grew rapidly. The first three is-sues were unedited pages which I xeroxed and stapledtogether.. The fourth issue was edited and retyped bySusie Hakansson's staff at the Graduate School ofEducation, UCLA. I arranged for it to be printed witha blue cover with a painting by Blake and a poem frag-ment by Milton. Subsequent issues have been typedby a student from Harvey Mudd College and printedby a local printer. It now became possible to edit thearticles before printing. The Foundation continued topay for hardware and software, typing, printing, andmailing the issues. The typing (production manager)has been a student. Proof reading and refereeing etc.are done by worldwide colleagues and myself. Wehave a minimum of promotion, but people hear ofthe journal by word of mouth or by mention in anarticle in another publication; they eventually sub-scribed by contacting me, usually through e-mail.Most articles have been sent to me unsolicited, al-though I have invited some articles and made requeststo reprint some.

The launching of the journal was followed by a num-ber of conferences and workshops on HumanisticMathematics across the country-a session in Bostonand another in New York. At the 1987 winter meetingof the math societies in San Antonio, I organized anafternoon panel on " Mathematics as a HumanisticDiscipline," that lasted 4 or 5 hours. We held some-what shorter sessions as well as poetry readings atsubsequent national meetings as well as conferences


with the same title at a small Catholic college and acommunity college in New York.

A strong encounter in my life, although I'm not abso-lutely sure where it fits in, chronologically, was withthe book "Personal Knowledge," by Michael Polanyi.I discovered it from a footnote in a paper by AbeMaslow that a psychologist at the campus counselingcenter had given me..

Every time I read Polynani's book, I underlined cer-tain lines in colored pencil. When I reread it, I felt com-pelled again to underline many of the same passagesin different colors. That cherished book is now quitepersonalized, in a colorful way. I cannot summarizethe more than 400 pages of that book but I would liketo close with a few tidbits and phrases From thepreface"I start by rejecting the ideal of scientific de-tachment. In the exact sciences this false ideal is per-haps harmless, for it is in fact disregarded there byscientists. But ... it exercises a destructive influence inbiology, psychology and sociology and falsifies ourwhole outlook beyond the domain of science.... "Tacitknowing is more fundamental than explicit knowing:we can know more than we can tell and we can tellnothing without relying on our awareness of thingswe may not be able to tell ... ... Thomas Kuhn in "Struc-ture of Scientific Revolutions" expresses similar ideas.J.Hadamard in "The Psychology of Invention in theMathematical Field" discusses the role of the uncon-scious in discovery and problem solving. Accordingto Polanyi---" Such is the personal participation of theknower in all acts of understanding. But this does notmake our understanding subjective. Comprehensionis neither an arbitrary act nor a passive experience,but a responsible act claiming universal validity. Suchknowing is objective in the sense of establishing con-tact with a hidden reality; a contact that is defined asthe condition for anticipating an indeterminate rangeof yet unknown implications." Polyani mentions tacitskills of which we are not aware - such as swimmingor riding a bike. "The complete objectivity as usuallyattributed to the exact sciences is a delusion and infact a false ideal. "To learn by example is to submit toauthority. You follow your master because you trusthis manner of doing things even when you cannotanalyze and account in detail for their effectiveness.By watching the master and emulating his efforts inthe presence of his example, the apprentice uncon-sciously picks up the rules of the art, including those

which are not explicitly known to the master himself.These hidden rules can be assimilated only by a per-son who surrenders himself to that extent uncriticallyto the imitation of another."

When I first started to teach, my ambition was topresent a complete and masterful lecture (and I havewitnessed several such lectures.) After I read Polanyiand others, my ambition became not to present a per-fect complete lecture, but to inspire my students (orrather, conspire with them) to become interested inthe subject and learn the subject for its own sake. Per-haps I have made some progress toward that goal andhave had some partial success, but I am continuing tolearn in my teaching, now, at age 77.


The first online issue of HMNJ will feature a major article by Stephen Brown. To whet your appetite, wehave included his introduction here.

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ing about the concept of humanistic mathematics. Ido so by setting my first publication on the topic inbas relief against my writing that emerged some thirtyyears later. I will not in this brief space (shades ofFermat and his marginal notes!) have a chance to paintthe variety of self-portraits that emerged over this timespan. I will, however, point to a number of contribut-ing factors that influenced the change. I propose thisact of introspection as a case study of one person’sstruggle with new ideas.

My first article, published in 1973, that explicitly high-lighted the word “humanistic” was playfully entitled“Mathematics and Humanistic Themes: Sum Consid-erations” (MHT ). The evolved book, published in 2001is less playfully entitled, Reconstructing School Math-ematics: Problems with Problems and the Real World(RSM). As I look back at MHT, it becomes clear to methat this article planted the seeds for much of my sub-sequent writing. Perhaps the most dominant theme—maybe an obsession—has been a focus on problemsand their educational uses. As will be obvious when Idiscuss some of the humanistic categories, part of thatfocus is ameliorative with regard to problems. Thatis, I point out “near relatives” of such concepts asproblem solving, and indicate the educational short-sightedness of excluding them from the educationalscene.

Humanistic Mathematics: Personal Evolution and ExcavationsStephen I. Brown

Professionals from many different disciplines and per-spectives who frequently do little more than greet eachother politely, have come to appreciate, acknowledge,and even communicate with each other. Those inter-ested in exploring a diversity of fields in relation tomathematics have set up tents around a bon fire thatwas lit by Alvin White’s newsletter, Humanistic Math-ematics Network of 1986—a newsletter that officiallybecame a journal in 1993. Fields as diverse as cogni-tive psychology, education, history, literature, linguis-tics, history, philosophy, and poetry are representedin the journal. This journal has also inspired the hu-manistic mathematics movement, now represented bya well attended topic group at the annual meeting ofthe American Mathematical Society and Mathemati-cal Association of America. In addition, it has beenacknowledged as an emerging force in a recent inter-national handbook of mathematics education pub-lished in The Netherlands (Brown, 1996a). From apersonal point of view, the journal has meant a greatdeal, not only because of the direct impact of its ar-ticles, but more importantly because it was a contrib-uting factor in giving me the courage to write aboutand integrate a variety of fields—a feat that stretchedconsiderably the bounds of my perceived expertise.

With much appreciation for such encouragement, Ireflect in this essay on the evolution of my own writ-

History of Math

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