What They Say About Math and What We can Learn From It
What They Say About Math and What We can Learn From It
By Dr. Eugene Maier
The following text is the content of Dr. Maier's April 14th, 2000 presentation at the 78th annual meeting of the National Council of Teachers in Mathematics in Chicago, IL.
Quotations about school math experiences, from biographies and autobiographies of the famous and not-so-famous; the picture they paint of societal perceptions of mathematics, motivations for studying math, and classroom practices.
A number of years ago, I read "Dreams, Memories and Reflections," the autobiography of Carl Jung. It contains a long chapter on his school years and within that chapter, a number of pages are devoted to his school mathematics experiences. I found them fascinating. They resonated with much of what I had observed and come to believe about school math. Here are a few passages:
I felt a downright fear of the mathematics class. The teacher pretended that algebra was a perfectly normal affair, to be taken for granted, whereas I didn't even know what numbers really were. They were not flowers, not animals, not fossils; they were nothing that could be imagined, mere quantities that resulted from counting. To my confusion these quantities were now represented by letters, which signified sounds, so that it became possible to hear them, so to speak. Oddly enough, my classmates could handle these things and found them self-evident. No one could tell me what numbers were and I was unable to even formulate the question. To my horror I found that no on understood my difficulty. ... All my life it remained a puzzle to me why it was I never managed to get my bearings in mathematics when there was no doubt whatever that I could calculate properly.
Equations I could comprehend only by inserting specific numerical values in place of the letters and verifying the meaning of the operation by actual calculation. As we went on in mathematics I was able to get along, more or less, by copying out algebraic formulas whose meaning I did not understand, and by memorizing where a particular combination of letters had stood on the blackboard. I could no longer make headway by substituting numbers, for from time to time the teacher would say, "Here we put the expression so-and-so," and then he would scribble a few letters on the blackboard. I had no idea where he got them and why he did it—the only reason I could see was that it enabled him to bring the procedure to what he felt was a satisfactory conclusion. I was so intimidated by my incomprehension that I did not dare to ask any questions. Mathematics classes became sheer terror and torture for me. Other subjects I found easy; and as, thanks to my good visual memory, I contrived for a long time to swindle my way through mathematics. I usually had good marks. 1
Jung's experiences were a clear indication to me that math anxiety is no respecter of intelligence, and I have found recounting Jung's experience to a math-anxious person helps them understand that just because they have been terrorized by a math class doesn't mean they're stupid. Also Jung's experiences are a graphic illustration of how divorced school math can be from one's natural knowledge of number and number operations. Finally, Jung's story gave me a way of describing what I find to be a common phenomenon : swindling one's way through math, that is, getting good marks and not having the slightest notion of what's going on.
Shortly after reading Jung's biography, I was describing his math experiences to an acquaintance who mentioned that they just encountered a description of school math experiences in a biography of Winston Churchill. Churchill too, I discovered, struggled with. mathematics. It took him three tries to pass the Civil Service Commissioners' exam that qualified him for entrance to Sandhurst, the Royal Military Academy. Here's Churchill's description of his nightmarish journey into mathematics as he prepared for the exam:
Of course what I call Mathematics is only what Civil Service Commissioners expected you to know to pass a very rudimentary examination. I suppose to those who enjoy this particular gift, Senior Wranglers [those who obtain first class honors in mathematics at Cambridge] and the like, the waters in which I swam must seem only a duck-puddle compared to the Atlantic Ocean. Nevertheless, when I plunged in, I was soon out of my depth. When I look back upon those care-laden months, their prominent features rise from the abyss of memory. Of course I had progressed far beyond Vulgar Fractions and the Decimal System. We were arrived in an 'Alice-in-Wonderland' world at the portals of which stood 'A Quadratic Equation.' This with a strange grimace pointed the way to the Theory of Indices, which again handed on the intruder to the full rigours of the Binomial Theorem. Further dim chambers lighted by sullen, sulphourus fires were reputed to contain a dragon called the 'Differential Calculus.' But this monster was beyond the bounds appointed by the Civil Service Commissioners who regulated this stage of Pilgrim's heavy journey. We turned aside, not indeed to the uplands of the Delectable Mountains, but into a strange corridor of things like anagrams and acrostics called Sines, Cosines and Tangents. Apparently they were very important, especially when multiplied by each other, or themselves! They had also had this merit—you could learn many of their evolutions off by heart. There was a question in my third and last Examination about these Sines and Cosines in a highly square-rooted condition which must have been decisive upon the whole of my after life. It was a problem. But luckily I had seen its ugly face only a few days before and recognised it at first sight. 2
Churchill provides another example that math anxiety is not a function of intelligence. And also another example of swindling: his fortuitous circumstance of being asked a math question on his Civil Service exam that he could recall from memory. But there was one thing in Churchill's encounter with mathematics that was missing from Jung's; a teacher who opened up new vistas. Churchill credits his achievement in passing the dreaded exam, not only to his own resolution, "but to the very kindly interest in my case taken by a much respected Harrow master, Mr. C.H.P. Mayo. He convinced me that Mathematics was not a hopeless bog of nonsense, and there were meanings and rhythms behind the comical hieroglyphics; and that I was not incapable of catching glimpses of some of these." 3
I found myself relating Churchill's experience to distraught students who had given up all hope of understanding mathematics, along with my belief that they to, along with all other normal human beings, were capable of making sense of mathematics. And it became a personal challenge to help students see "there were meanings and rhythms behind those comical hieroglyphics."
Having found these two biographical excerpts to be particularly revealing and a stimulant for reflecting on the teaching and learning of mathematics, both for me and those to whom I related these stories, I wondered what mathematical tidbits might be lurking in other biographies. Consequently, whenever I came across a biography, I scanned the pages describing educational experiences, picking up a quote here and there, while saying to myself that someday I'm going to make a systematic effort to find out what sort of picture biographies paint of mathematics and mathematics education. Someday actually arrived. Several months ago I began systematically searching biographies to see what I could learn.
So far, I've collected about 150 references to mathematics in biographies and autobiographies. References to mathematics exceed those to any other subject and evoke far more comment. As you might expect, reactions to mathematics run the gamut of human thought and emotion. What's more, these reactions are spontaneous and unguarded, without the inhibitions or biases that occur in the response to and construction of surveys, questionnaires or other externally imposed assessments.
Here is a list of some of the words and phrases used to describe mathematics that I have encountered:
Closed to Ordinary Mortals
Pinnacle of Intellectual Pecking Order
Peculiarly Engaging and Delightful
Earnest and Rigorous
Amusing Brain Stunt
Useful and Substantial
Boring and Obstructive
Rule and Rote
Grab Bag of Diverting Riddles
Sheer Terror and Torture
Bog of Nonsense
Worse than Useless
Curse of My Life
As you can see, the views of mathematics range from the sublime to the ridiculous; the heartwarming to the heartrending. For some, math is an arcane and mysterious subject; for others, it has a magical attraction.
The words and phrases used to describe mathematics may be more indicative of the mind set of the biographer than that of the subject of the biography. For example, the first phrase comes from the opening paragraphs of a John Kennedy Winkler's biography of the financier J. P. Morgan. All I know about the author is that he also wrote a biography of William Randolph Hearst. He appears to be in awe of mathematics, while reinforcing the popular notion that anything beyond the mathematics of the everyday world is beyond the grasp of common people. Here is the beginning of his biography of Morgan:
Perhaps once in a hundred years is born a mind capable of entering a sphere of higher mathematics closed to ordinary mortals. A direct and synthetic mind that cuts across lots and flies straight to conclusions, intuitively and by process unknown to self.
Such a mind we call genius.
Such was the mind of John Pierpont Morgan.
By sheer mental magic, Morgan solved the most complicated problems. He was a mathematical marvel. This quality in itself destined the direction of great affairs. 4
As for J. P. himself, what we know is that he was a very good student of mathematics at Gsëttingen University in Germany; good enough that his professor told him he was making a mistake by going into business and that he should stay at Gsëttingen, "perhaps becoming the professor's assistant, and even possibly—if he worked diligently and fortune favored him—succeed to the professor's own august chair." 5
Biographers' biases frequently show in the choice of adjectives they use when referring to a mathematical subject. Thus Longfellow's biographer reports that Longfellow, while a student at Bowdoin, "mastered the peculiarly difficult principles of geometry." 6 It may be that Longfellow found geometry neither peculiar or difficult. In a similar vein, Huey Long's biographer, when describing the courses Long took in high school, refers to trigonometry and plane and solid geometry as "knotty subjects." 7
I have come across the word "abstruse," i. e., difficult to comprehend, several times in reference to mathematics. One occurrence is in a biography of the economist John Maynard Keynes. Keynes won every possible math prize while a schoolboy at Eton and went on to study math at King's College, finally giving it up for economics. The biographer, pointing out that although Keynes did well in his mathematical studies, "he did not seek out those abstruse regions which are a joy to the heart of the professional mathematician." 8 Keynes ultimately left math for economics, finding math too narrow. He found a fellow student at King's "only a mathematician, a bore and a precise example of what not to be." 9
Keynes' biography also provides an example of a myth that's afloat in the biographical literature, namely, that those with a mathematical bent lack humanistic qualities. One of Keynes' classics teachers at Eton, expressed the hope that "the more accurate sciences will not dry the readiness of his sympathy and insight for the more inspiring and humane subjects: his essay on Antigone was not like the work of one made for mathematics. He has a well furnished and delightful mind." 10 Thomas Paine's biographer, on reporting that he was an excellent student in both mathematics and poetry finds "this combination unusual". 11 The biographer of James Blaine—U.S. Senator and one-time Secretary of State—writing of Blaine's "aptitude in mathematical study" found it to "be wondered at and admired; for the mathematical faculty does not usually co-exist, even in great minds, with the excursive and imaginative faculty which Blaine possessed in so high measure." 12
The converse myth also occurs, that is, that those interested in humanities are, perse, averse to mathematics. The playwright Tennessee Williams' biographer remarks as "one might expect from a budding writer" his marks in English were high while "to no one's surprise" those in algebra were very low, 13 as if that is to be expected of a literary person.
Actually, a number of writers and poets have done well in math. The poet Sydney Lanier "mastered mathematics beyond any man of his class" at Oglethorpe College. 14 The novelist Upton Sinclair, generally a mediocre student at City College of New York, won a prize in differential calculus. 15 The poet Robert Penn Warren did well in math "and inclined ...towards a career in science." 16 The novelist Richard Wright had no trouble with the subject. He said he worked out all his mathematics problems in advance and spent his time in class, when not called on to recite, reading "tattered, secondhand copies of Flynn's Detective Weekly and Argosy All-Story Magazine, or dreamed...." 17
On the other hand, there are those authors who wanted nothing to do with the subject. Lew Walllace "developed a prompt and lasting aversion" to the subject. 18 Gene Stratton-Porter "failed it consistently." 19 F. Scott Fitzgerald found it "boring and obstructive." 20 College mathematics gave George Ade "night terrors." 21 Ellen Glasgow, because of her low standing in arithmetic was put at the foot of the class and, while sitting there, says she "felt a chill crawling up my spine, like a beetle." 22
The greatest accolade to mathematics I have come across occurs in a biography of Charles Proteus Steinmetz. Steinmetz was born in Europe in 1865 and had nearly finished a Ph. D. degree in mathematics at the University of Breslau when he hurriedly left Germany to avoid being arrested for his socialist activities. He came to the states and ultimately ended up at General Electric where he became an expert on the theory and utilization of alternating current. He continued his interests in pure mathematics until his work at G. E. left him little time to pursue his interests in synthetic geometry.
To explain Steimmetz' fascination with mathematics, this particular biographer, Jonathan Norton Leonard, included a lengthy section in his biography entitled "My Lady Mathematics." Again, the sentiments expressed are those of the biographer, Leonard, and not those of his subject. (In addition to the biography of Steinmetz, which he wrote when in his twenties, Leonard also wrote on a diversity of other topics including American cooking, Gainsborough, ancient Japan, Atlantic beaches and the enjoyment of science.)
Here is a portion of Leonard's passage on "My Lady Mathematics:"
There's a certain mystical charm about pure mathematics, a charm which pervades and tinctures the whole soul of the student. It's so totally abstract. You begin with the numbers, 1, 2, 3, etc. You learn that they can be added together, multiplied and manipulated in simple ways to serve the purpose of tradesman and housekeeper. Then you begin to see their more hidden qualities. There are negative numbers, for instance. These are interesting things. You play with them for a while and presently you realize that if you multiply one negative number by another negative number you will get a positive number not only larger than either but of an entirely different order of largeness. It is mysterious. You want to know more.
Finally, when you've juggled with these simple quantities, turned them upside down, turned them inside out, you begin to see short visions of fascinating qualities hitherto undreamed of. Some numbers are imaginary; they don't exist and can't exist. But nevertheless they can be manipulated just like real ones. The answer to a problem done with these unreal ghosts of numbers is just as correct as one done with your own ten fingers. This thrilling revelation is only one of many. Innumerable rules and principles swarm at the gates of the mind and when one of these has become established and naturalized it breeds a host of new ones which in turn present themselves for naturalization. Soon there's a dense population all yelling for attention. Mathematical intoxication is a common disease among students. 23
Whether or not Steinmetz would agree with everything in this passage is difficult to assess. The evidence suggests that Steinmetz found mathematics intoxicating, but I suspect that he didn't have such a mystical view of numbers. Steinmetz used imaginary numbers in his analyses of electrical current and likely they were just as real to him as any other kind of number, as indeed they are, differing from other numbers, such as the counting numbers or the negative numbers, in their mathematical purpose.
Thomas Jefferson is another person who found mathematics intoxicating. Jefferson said that mathematics was the passion of his life when he was young. Later in life, he observed that "mathematics and natural philosophy [i. e., natural science] are so useful in the most familiar occurrences of life, and are so peculiarly engaging and delightful as would induce every person to wish an acquaintance with them." 24
One person it didn't induce was William Lyon Phelps, who is responsible for the two phrases at the bottom of our list. Phelps was a professor of literature at Yale where he taught for 41 years and was voted most inspiring professor a number of times. Mathematics takes a real beating in his autobiography:
Mathematics always helped to keep me back; they were the curse of my life at school and college, and had more to do with my unhappiness than any other thing and I bitterly regret the hours, days, weeks, months, and years that I was forced to spend on this wholly unprofitable study. I shall return to this later with more venom. 25
And 50 pages later is the additional "venom:"
...for those who have no gift and no inclination, mathematics are worse than useless—they are injurious. They cast a blight on my childhood, youth, and adolescence. I was as incompetent to deal with them as a child to lift a safe. I studied mathematics because I was forced to do so, faithfully and conscientiously from the age of three to the age of twenty-one, through my Junior year in college. After 'long division' nearly every hour spent on the subject was worse than wasted. The time would have been more profitably spent in manual labor, athletics, or in sleep. These studies were a brake on my intellectual advances; a continuous discouragement and obstacle, the harder I worked, the less result I obtained. I bitterly regret the hours and days and weeks and months and years which might have been profitably employed on studies that would have stimulated my mind instead of stupefying it! 26
I found it interesting that Phelps had a colleague at Yale who had an entirely different experience in math. Wilbur Cross, was a professor of English at Yale at and later Dean of the Graduate School. Following his retirement from Yale, he served as governor of Connecticut for 8 years. Although he had a "great dislike for intricate problems concerning the time it would take for A to do a piece of work with the aid of B and often with the further aid of C," 27 geometry was another matter:
Euclid ... fascinated me, not because it added anything new to my knowledge of geometry, but by the art portrayed by the old Greek mathematician in proving by a strict deductive method the proof of the propositions which anyone might see true at a glance. It was like traveling over a beautiful road to the foreseen end of one's journey. 28
Cross reminds us that the organization of a subject into a deductive system properly comes after one has a thorough knowledge of the subject matter. A precept that's too often unheeded, especially in introductory geometry courses.
Cross isn't alone in his adulation of Euclidean geometry. Poetess Edna St.Vincent Millay who struggled with math when a student nonetheless wrote a sonnet, the first line of which is "Euclid alone has looked on beauty bare." 29 According to one biographer, "There is a legend that she grew so enamored of her Freshman course in mathematics [at Vassar] that she spent the night before her final examination writing a sonnet about it instead of cramming, and consequently failed to pass." 30 The young Einstein also saw beauty in math, he said that "everything in calculus and geometry is beautifully planned—like a Beethoven Sonata." 31
In the biographies and autobiographies in which I have found references to mathematics, geometry is mentioned more frequently in a positive light than algebra. Dwight Eisenhower "despised" high-school algebra; he said he "could see no profit in substituting complex expressions for routine terms the job of simplifying long, difficult equations bored me." 32 Geometry was another matter.
The introduction of plane geometry was an intellectual adventure, one that entranced me. After a few months, my teachers conducted an unusual experiment. The principal and my mathematics teacher called me to the office and told me they were going to take away my textbook. Thereafter, was to work out the geometric problems without the benefit of a book. In other words, the problems would be, for me, originals. This was a fascinating challenge and particularly delightful because it meant that no advanced study was required. 33
Perhaps if his algebra had set him free in his algebra class he would have found it fascinating also.
The less favorable response to algebra may be that, in contrast to geometry, it is more likely be taught by "rule and rote," a description of mathematics used by the biographer of William Randolph Hearst. "Mathematics he ignored," his biographer writes. "It was ever to be thus, the formal education of rule and rote anathema." 34 While "rule and rote" is helpful in passing math tests, as Churchill pointed out, it's shortcomings are noted. The tutor engaged by the parents of Henry Cabot Lodge to help Henry and his brother overcome their in math found the task harder than anticipated because their training in arithmetic had been exceedingly poor - mainly consisting, he said, of unreasoning memory work. 35 Feynman relates the following anecdote from his student days at MIT:
One day, in mechanical class, some joker picked up a French curve (a piece of plastic for drawing smooth curves—a curly, funny-looking thing) and said, "I wonder if the curves on this thing have some special formula?"
I thought for a moment and said, "Sure they do. The curves are very special curves. Lemme show ya," and I picked up my French curve and began to turn it slowly. "The French curve is made so that at the lowest point of each curve, no matter how you turn it, the tangent is horizontal."
All the guys in the class were holding their French curves up at different angles, holding their pencil up to it at the lowest point and laying it along, and discovering that, sure enough, the tangent is horizontal. They were all excited by this "discovery"—even though they had already "learned" that the derivative (tangent) of the minimum (lowest point) of any curve is zero (horizontal). They didn't put two and two together. They didn't even know what they "knew."
I don't know what's the matter with people: they don't learn by understanding; they learn by some other way—by rote or something. Their knowledge is so fragile! 36
Speaking of calculus, Eisenhower relates an incident where his instructor, but not he, relied on rote:
About midway in our West Point course we began the study of integral calculus. The subject was interesting but the problems could be intricate. One morning after recitations the instructor said that on the following day the problem would be one of the most difficult of all. Because of this he was giving us, on the orders of the head of the Mathematics Department, an explanation of the approach to the problem and the answer.
The explanation was long and involved. It was clear that he was doing his task completely by rote and without any real understanding of what he was talking about. Because I was a lazy student, with considerable faith in my luck, I decided there was little use in trying to understand the solution. After all, with twelve students in the section, only one of us would get this problem to solve, the odds were eleven to one that I would not be tapped.
The following morning I was chosen. Going to the board, on which I was required to produce the solution, and then explain it to the instructor, I had not the foggiest notion of how to begin. I did remember the answer given by the instructor and wrote it in the corner of the board.
I set to work. I had to make at least a good start on the problem, show something or receive a grade of zero which would do nothing for me in a course where my grades were far from high. Moreover, I could be reported to the disciplinary department for neglect of duty in that I had deliberately ignored the long explanation. With this in mind I sought in every possible way to jog my memory. I had forty-five or fifty minutes to solve the problem and I really concentrated.
After trying several solutions that seemed to relate, at least remotely, to the one I dimly remembered from the morning before, I encountered nothing but failure. Finally, with only minutes left, I worked out one approach that seemed fairly reasonable. No one could have been more amazed than I when this line of action agreed exactly with the answer already written on the board. I carefully went over the work, sat down, and awaited my turn to recite. I was the last man in the section to be called upon.
With some trepidation I started in. It took me a short time to explain my simple solution—indeed it had to be simple or I never would have stumbled upon it. At the end, the instructor turned on me angrily and said, "Mr. Eisenhower, it is obvious that you know nothing whatsoever about this problem. You memorized the answer, put down a lot of figures and steps that have no meaning whatsoever, and then wrote out the answer in the hope of fooling the instructor."
I hadn't been well prepared but this was tantamount to calling me a cheat, something that no cadet could be expected to take calmly. I reacted heatedly and started to protest. Just then I heard Major Bell, the Associate Professor of Mathematics (whom we called "Poopy," a name that was always applied to anyone at West Point who was above average in academic attainments) who had entered the room for one of his occasional inspections, interrupting. "Just a minute, Captain."
Of course, I recognized the voice of authority and shut up, although according to my classmates' description that night I was not only red-necked and angry but ready to fight the entire academic department. I would have been kicked out on a charge of insubordination if I had not been stopped.
Major Bell spoke to the instructor, "Captain, please have Mr. Eisenhower go through that solution again."
I did so but in such an emotional state that it is a wonder that I could track it through. The long search for a solution and its eventual simplicity stood me in good stead.
Major Bell heard it out and then said, "Captain, Mr. Eisenhower's solution is more logical and easier than the one we've been using, I'm surprised that none of us, supposedly good mathematicians, has stumbled on it. It will be incorporated in our procedures from now on."
This was a blessing. A moment before, I had an excellent chance of being expelled in disgrace from the academy. Now, at least with one officer, I was sitting on top of the world. 37
One's performance in math meant a lot at West Point. Douglas MacArthur attended West Point a decade before Eisenhower and had an outstanding record in math. According to his biographer, "Math counted most of all. It was at the pinnacle of the intellectual pecking order.... More time was devoted to math than any other academic subject. The surest way of getting on course to be one of the Five [the five students selected as the most outstanding in their class] was to do well in math." 38
West Point began in 1802 as a school for military engineers. In its early years, the beginning West Point student studied two subjects: mathematics and, surprisingly, French. The goal "was to make them, if not fluent, at least to become conversant with French military and engineering treatises." 39 The emphasis on math continued for years, even after the academy broadened its mission to the preparation of army officers in general. Writing in 1928, William E. Woodward, the biographer of Ulysses Grant, questioned the heavy dose of math required of West Point cadets. (Grant was a better math student than French student; in his freshman year—sometime around 1840—he was 16th in math and 49th in French out of 60; the following year he was tenth in math and 44th in French out of 53.)
Woodward was a graduate of the Citadel—at the time, the South Carolina Military Academy—where he lost interest in schooling and graduated third from the bottom in his class. He went into newspaper work, ultimately becoming publicity director of a Wall Street firm and left publicity work to become an executive vice-president and director of 42 banks in which his Wall Street firm had an interest. He grew so bored of banking and finance that he hated the sight of his office. So he quit to become a writer. Despite of his low academic ranking at Citadel, they awarded him an honorary doctor of laws degree. Here's what he wrote about math at West Point:
I have never been able to discover any sensible reason why a military education should be so thoroughly saturated with mathematics. In actual warfare there is nothing in mathematical science beyond arithmetic that is of the least value, except to engineering officers, and these are so few in number that a special education in mathematics might be provided for them without forcing every infantry officer to flounder through Descartes and Newton. It is true, indeed, that mathematics is the foundation of the science of ballistics; but, even so, artillery officers in the field are spared the torture of having to solve differential equations under heavy fire, as printed tables of ranges and distances are thoughtfully provided by the War Department for their use. It is as simple as looking up a number in a telephone directory.
It seems much better, from the standpoint of common sense, to do away with everything in mathematics higher than arithmetic in an officer's education, and devote the time thus saved to such important subjects as the relative nutritive value of different kinds of food, the structure of the human body, and the principles of sanitation and medicine. 40
Surprisingly little attention is given to the utilitarian value of math in the biographies I've consulted. Mention is made of the value of math to engineers and scientists, as in the above comment, but, as above, not by those who actually use it. I'm reminded of those teachers who tell students how useful mathematics is while never using it themselves outside the classroom. Churchill writes he is "assured that [mathematics] are most helpful in engineering, astronomy and things like that." 41 Phelps says "the truth is that for every occupation except one for which higher mathematics are a prerequisite, like civil engineering, Greek and Latin are more useful." 42 George Ade shuddered when he saw engineering students use textbooks that applied math to engineering problems. He said "it was enough to worry through a mathematics textbook without having to think of using such lessons afterwards." 43
Mention is made of the general value of mathematics as intellectual discipline. Often, again, in reference to others. Phelps, who abhorred mathematics, had "no doubt that for those who had a natural aptitude, mathematics are valuable as an intellectual discipline and training." 44 James Blaine's biographer avers that "without doubt, the possession of mathematical ability is of high value to a public man, particularly if he is destined to deal with economic questions." 45 Charles Frances Adams, an economist and descendant of the presidential Adamses, believed he should "have compelled myself to take some of the more elementary mathematical courses simply for the mental discipline they afford." He said he needed 'the regular mental gymnastics—the daily practice of following a line of sustained thought out to exact results..." 46
There are those who, from their own experience, attest to the value of mathematics as an intellectual discipline.. Omar Bradley, a West Point grad who served a four-year assignment as a math instructor at West Point, said he "benefited from a prolonged immersion in math" and that "the study of math, basically logic, stimulates one's thinking greatly improves ones power reasoning. later years, when I was faced with infinitely complex problems, often requiring immediate life-or-death decisions, am certain this mathematics helped me think more clearly logically." 47 Thomas Jefferson maintained that "the faculties of the mind, like the members of the body, are strengthened and improved by exercise" and this is accomplished by "mathematical reasoning and deductions." 48
Several mentions are made of the value of mental arithmetic, a skill that suffers when heavy emphasis is placed on paper-and-pencil algorithms which, in general, are ill-suited for mental calculations. Wilbur Cross said he owed "a lasting debt" to a teacher for the practice he gave him in mental arithmetic, which, he said, was of very great help to him when dealing with budgets while governor of Connecticut. 49 Chief Justice Charles Evans Hughes biographer writes his mother's "exercises in 'mental arithmetic' gave Charles the most useful training he ever had. She would have him toe a mark on the floor and, without changing his position, 'do in his head' the various sums she gave him. He was urged to think quickly and accurately without recourse to paper and pencil—a faculty that would add greatly to his prowess as investigator, advocate, and public speaker." 50 Henry Ford had a teacher who "noted that he was naturally fast at figures and made him do sums in his head instead of on the blackboard. Thanks to him, Mr. Ford in later years seldom had to put pencil on paper when working out a problem." 51
By and large, those who were attracted to mathematics were done so because of its intrinsic appeal and not because of its utilitarian value, something that may be worth remembering the next time one tries to sell math because of its usefulness in other areas. In addition to those we have already mentioned who found math intellectually stimulating and aesthetically pleasing, there are those who enjoyed mathematical puzzles, those who were fascinated by numbers and statistics, and those who simply found it fun. Benjamin Banneker, astronomer and almanac publisher, loved mathematical puzzles and collected them "at every opportunity". 52 Weldon Johnson, an educator and one of the founders of NAACP, found early in his career that "arithmetic is not only an interesting study, it is also a most fascinating pastime." He "tried to discover and prove the principles that underlay 'the rules of arithmetic'" and, for him, "getting at simpler and more understandable methods of solution became an absorbing game." 53 Noah Webster " took delight in figures and statistics. It is said that the collecting of data interested him, even when there was no apparent purpose to which it could be put. He counted houses, examined town lists of voters, and noted weather conditions." 54 Helen Keller said she "could do long complicated quadratic equations in my head, and it is great fun!" 55 She was also "somewhat elated" upon completing a set of geometry problems "although," she added, "I cannot see why it is so very important to know that the lines drawn from the extremities of the base of an isosceles triangle to the middle points of the opposite sides are equal! The knowledge doesn't make life any sweeter or happier, does it?" 56
Once relieved of the demands of school mathematics, many paid little heed to the subject again and seemed to get along just fine. Churchill said he had "never met these [mathematical] creatures since. With my third and successful examination they passed away like the phantasmagoria of a fevered dream." 57 Clarence Darrow thought the aim of all learning was to make life easier, which he found mathematics, beyond simple arithmetic, ill-suited to achieve. 58 Emerson, who viewed math as "pure wretchedness," said "it was not necessary to understand Mathematics and Greek thoroughly to be a good, useful, or even great man." 59 Hjalmar Schacht, president of the national bank of Germany in the Twenties and Thirties, who once "distinguished himself by arriving at a different [incorrect] result from all his fellow candidates" in an arithmetic test to pass out of Sixth Form commented, "In spite of my low marks in arithmetic I have not been entirely unsuccessful in my career as banker and president of the Reichsbank. A bank inspector or manager is not a bookkeeper. His work entails expert knowledge of quite different subjects: for example, psychology, economics, common sense, ability to make decisions, but above all, insight into the intricacies and nature of credit." 60
One wonders what role teachers and others had in determining attitudes towards mathematics. Often an individual is referred to who was helpful, but the nature of the help isn't described. Churchill mentions "the respected Harrow master, Mr. C. H. P. Mayo who convinced him that mathematics was not a hopeless bog of nonsense and that there were meanings and rhythms behind those comical hieroglyphics," and perhaps, most important of all, "that he was not incapable of catching glimpses of some of these." 61 But nothing is said of how Mayo accomplished this.
We do get some glimpses from other biographies of teachers' traits and methods that were valued. Helen Keller, who said that, as a young child, arithmetic was the only subject she did not like, credits Merton Keith, her private math tutor, for changing her outlook. She said it was "much easier and pleasanter to be taught by myself than to receive instruction in a class. There was no hurry, no confusion. My tutor had plenty of time to explain what I did not understand. ...even mathematics Mr. Keith made interesting; he succeeded in whittling problems small enough to get through my brain. He kept my mind alert and eager, and trained it to reason clearly, and to seek conclusions calmly and logically, instead of jumping wildly into space and arriving nowhere. He was always gentle and forbearing, no matter how dull I might be...." 62
Thomas Jefferson credits his successes to Dr. William Small, a professor of mathematics at William and Mary. Jefferson describes Small as having 'a happy talent of communication, correct gentlemanly manners, and an enlarged and liberal mind." 63 The artist John Turnbull mentions a schoolmaster who "had the wisdom to vary my studies, so as to render them rather a pleasure than a task - he mentions how he was given an arithmetic problem that he had difficult solving, but the master would not help him solve it and forbade others from helping him. For three months, he said, the problem was unsolved when "at length the solution seemed to flash in my mind at once, and I went forward without further hindrance."64 Eli Lilly recalls an elementary school teacher, Jane Graydon "who had an unusual element of freshness, and electricity of spirit" and who "inspired me out of my arithmetic slump to make a perfect grade on my last test." 65
Nicholas Tesla mentions a calculus professor at the polytechnic school in Graz, Austria, who ''was the most brilliant lecturer to whom I have ever listened and would frequently remain for an hour or two in the lecture room, giving me problems to solve, in which I delighted." 66 Earlier, we mentioned how delighted Eisenhower was when he was left to work on his own. Richard Feynman, the Nobel Prize winner in physics had a similar experience, albeit in a high school physics class. One day the instructor asked him to stay after class and told him he talked to much, no necessarily about the matter at hand and he believed it was because he was bored. So he gave Feynman a book and told him from now on he was to sit in the back of the room and study that book and when he had finished that he could talk again. It was an advanced calculus book—Feynman had already worked his way through a beginning calculus text. Feynman comments on how much he learned on his own from that book and how useful it turned out to be in later work he did. 67
Of a different stripe, are those teachers who simply accommodated their students. Poet Vachel Lindsay entered Hiram College with the eventual goal of studying medicine. He was supposed to take physics, but was told he needed trig, a subject "with which his mind could scarcely grapple at all even though the instructor, to help him out, worked all the examples himself."68 The historian William Hinkling Prescott was "noted for his horsemanship, his charm, his wit, but never his studiousness" when a student at Harvard. Prescott, as a sophomore, took a required course in geometry from a Professor John Farrar:
For a time he laboriously memorized propositions in geometry and reproduced them in class exactly as they appeared in the textbook. Wearying, however, of the drudgery which stamped him acceptable to his teacher but grossly ignorant of the subject, Prescott confided his secret to Farrar. Convincing the professor of the impossibility of his mastering the subject, he was told that regular attendance, without recitation would suffice. 69
Sometimes it was friends or relatives who saved the day or sparked interest. Jack London's friend Bess Maddern's "skillful coaching eliminated many of the mazes and pitfalls in the mathematics," enabling London to pass the math in the Berkeley entrance exam. 70 Albert Einstein learned mathematics from his Uncle Jake and Max Talmey, a university medical student who came to dinner once a week. Einstein said Max was better at explaining things than anyone at the gymnasium. 71 Later, reflecting on his gymnasium experiences, where he found the teachers severe and the lessons boring, Einstein wrote. "it is, in fact, nothing short of a miracle that the modern methods of instruction have not entirely strangled the holy curiosity of inquiry; for this delicate little plant, aside from stimulation, stands mainly in the need of freedom; without this it goes to rack and ruin without fail. It is a very grave mistake to think that the enjoyment of seeing and searching can be promoted by coercion and a sense of duty." 72
The actress Myrna Loy struggled with grammar-school math. she reports that she went for help to an uncle to prepare for a "big test". When she passed, the teacher accused her of cheating. She walked out of class, reported the teacher to the principal and went home; refusing to return to class until the teacher apologized. 73 Being falsely accused of cheating seems particularly devastating. Eisenhower says the calculus instructor who accused him of cheating "was the only man at West Point for whom I ever developed any lasting resentment." 74
In contrast to those who communicated well, with civility and forbearance, there were those teachers who didn't communicate at all or, if they did, were sarcastic or severe. William Lloyd Phelps had a teacher tell him "In mathematics, you are slow, but not sure." 75 Robert Kennedy, who attended Milton Academy during the second World War at the time the German general Rommel was being defeated in Africa, wrote in a letter home that "on the last day of school the math teacher made a small speech in which he said two great things had happened to him, one that Rommel was surrounded in Egypt and the 2nd that Kennedy had passed a math test." 76
In the 1770's, when Alexander Hamilton was a student at King's College, the forerunner of Columbia University, mathematics was taught by a "testy" professor, Robert Harpur. "More exacting than his colleagues," Hamilton's biographer writes, "the students frequently met his discipline with individual defiance or collective jeers." 77 King's College records reveal that one Edward Thomas, a student, "was ordered before the Governors 'for abusing, along with many others, Mr. Harpur, the Evening before' Thomas proved his innocence, but soon seven more ... were compelled to ask public pardon 'for ill-using Mr. Harpur by Calling Names in the Dark...'" Later a student was suspended "for using Mr. Harpur in the most scandalous manner." 78
A century later, in the 1860's at Harvard, Oliver Wendell Holmes took math from a professor who, in the classroom, "was brief and impatient. Stupid students were terrified of him, the brilliant greeted him with joy." 79 An example of a phenomenon known to most of us, the teacher who is able to teach only those who don't need teaching.
And a century later, in the 1940's, Lee Iacocca tells another familiar story. Iacocco tells how he almost flunked freshman physics at Lehigh University: "We had a professor named Bergmann," he writes, "a Viennese immigrant whose accent was so thick that I hardly could understand him. He was a great scholar, but he lacked the patience to teach freshman." 80
The oft-encountered caricature of the math teacher as a social misfit, living in their own little world, arises. William Woodward, the biographer of Ulysses Grant and author of the comments about math at West Point mentioned earlier, reports that the young Ulysses Grant, who enjoyed math and once inquired if there were any math teaching positions at West point, "built a daydream of himself as a teacher. He," Woodward writes "saw himself standing throughout the years by the stream of life, a half-recluse, sprinkling algebra and calculus generously upon the heads of the passing generation." 81
George Stigler was an economist who spent his professional life in academia. In his autobiography he describes how universities are willing to put up with the idiosyncrasies of experts in their fields. He chose to cast his example as a mathematician: "Universities cater to more highly specialized human beings than most other callings in life. If X is a great mathematician, he will be more or less silently endured even though he dresses like a hobo, has the table manners of a chimpanzee, and also achieves new depths of incomprehensibility in teaching. His great strength is highly prized; his many faults are tolerated." 82
Steinmetz, the GE engineer we encountered earlier, while not as boorish as Stigler's example, is one person who fed the image of the eccentric mathematics professor. Steinmetz, who had all but finished his Ph.D. degree in math before fleeing to the US wished to continue his academic involvement. To satisfy this, a lectureship was arranged for him at Union college in Schenectady where GE was headquartered. His biographer Leonard describes his classroom:
He would write nervously on the blackboard, talking all the time, and then without missing a word whirl round in a tempest of questions. After the first fifteen minutes the minds of the students became rather numb. No one ever followed him in all of his calculations. He'd plunge into a flood of figures like a diver into a whirlpool; he'd struggle furiously with weird symbols which meant nothing to anyone but himself; he'd cover the board with writing too small to be seen beyond the first row, and finally he would emerge with a conclusion which should have been on page 347, two chapters ahead. 83
Despite his students not learning anything, Leonard claims his classes weren't a total failure:
The students got very little mathematical information out of his lectures but they got a good deal of inspiration. And mathematics in its higher forms is inspirational. The sight of the little man on the platform there, bursting with enthusiasm and chalk miracles before their eyes, was enough to put energy into any ambitious young engineer. There aren't many lecturers like Steinmetz. If there were, no one would learn anything definite. But one Steinmetz in the intellectual adolescence of every man would make that man higher minded and less apt to become a mere stodgy technician. 84
According to Leonard, there was one other thing that set Steinmetz apart from ordinary people. Commenting on the fact that Steinmetz never married, Leonard, tells us that "Mathematics occupied completely that central part of his mind which if he had been a normal man would have been dominated by sex." 85
Banneker's biographer also maintains that mathematics also got in the way of romance. He writes that Banneker's "consuming interest in reading and mathematical studies, and his jealous preservation of the little leisure he had, disinclined him to seek a wife." 86
Lest one begins to believe that mathematics is a deterrent to romance, I end with the story of the courtship of Barnes Wallis. Wallis was a pioneer in the British aircraft industry—an aeronautical engineer before the term existed—and a very good mathematics student.
He fell in love with Molly Bloxam, a woman 15 years his younger. Molly was quite taken by Barnes but Molly's father was not in favor of Molly marrying an older man. However, as Barnes discovered, Molly was terrified of taking mathematics exams that were required in her degree program, so he began tutoring her, an activity to which her father did not object. When separated, while Molly continued her education and Barnes pursued his career, he continued his mathematics instruction by letter. Here is the beginning of his correspondence course on calculus:
Now here begins lecture one, from me, Barnes, to you, Molly, on the very delightful subject of the calculus... . The calculus is a very beautiful and simple means of performing calculations which either cannot be done in any other way, or else can only be performed by very clumsy, roundabout and approximate methods. 87
"As some men carry forward their courting with imperfect poetry," the biographer writes, "so Wallis conducted his most comfortably with the perfection of sine and cosine." 88
Molly insisted to her father that passing her exams was only possible if Barnes continued his correspondence course in mathematics. Her "persistence - and the mathematics coaching—began first to circumvent and then eroded her father's obduracy." 89 Molly and Barnes got married and lived happily ever after.
- C. G. Jung, Dreams, Memories and Reflections, pp. 27-29.
- Winston Churchill, My Early life, A Roving Commission, pp. 25-26.
- Ibid., p. 25.
- John K. Winkler, Morgan the Magnificent, p. 3.
- Frederick Lewis Allen, The Great Pierpont Morgan, p. 17.
- Lawrance Thompson, Young Longfellow, p. 38.
- T. Harry Williams, Huey Long, p. 36.
- R. F. Har rod. The Life of John Maynard Keynes, p. 57.
- Ibid., p. 45.
- Ibid., p. 45.
- Samuel Edwards, A Biography of Thomas Paine, p. 7.
- John Clark Ridpath, Life and Work of James G. Blaine, p.47.
- Llyle Leverich, Tom; The Unknown Tennessee Williams, p. 74.
- Edwin Mims, Sidney Lanier, p. 31.
- Leon Harris, Upton Sinclair, American Rebel, p. 11.
- Joseph Blotner, Robert Penn Warren, p. 27.
- Richard Wright, Black Boy, A Record of Childhood and Youth, p. 116.
- Robert E. and Katharine M. Morsberger, Lew Wallace: Militant Romantic, p. 6.
- Judith Reick, Gene Stratton-Porter, p. 83.
- Jeffrey Meyers, Scott Fitzgerald, p. 23.
- Fred C. Kelly, George Ade, p. 52.
- Ellen Glasgo, The Woman Within, p. 47.
- Jonathan Norton Leonard, Loki, the Life of Charles Proteus Steinmetz, pp. 23-24.
- Willard Steine Randall, Thomas Jefferson, A Life.
- William Lyon Phelps, Autobiography, with Letters, pp. 91-92.
- Ibid., p. 148.
- Wilbur L. Cross, Connecticut Yankee, An Autobiography, p. 28.
- Ibid., p. 63.
- Edna St. Vincent Millay, Collected Poems, p. 605.
- Elizabeth Adkins, Edna St. Vincent Millay and Her Times, p. 32.
- Elma Ehrlich Levinger, Albert Einstein, p. 27.
- Dwight D. Eisenhower, At Ease: Stories I Tell to Friends, p. 100.
- Ibid., p. 100.
- John K. Winkler, William Randolph Hearst, p. 27.
- William J. Miller, Henry Cabot Lodge, p. 39.
- Richard Feynman, "Surely Your Joking, Mr. Feyman", pp . 36-37.
- Eisenhower, pp. 18-20.
- Geoffrey Perret, Old Soldiers Never Die, p. 35.
- Stephen W. Sears, George B. McCllelan, the Young Napoleon, p. 6.
- W. E. Woodward, Meet Generall Grant, p. 50.
- Churchill, p. 26.
- Phelps, p. 147.
- Fred C. Kelly, p.53
- Phelps, p. 148.
- John Clark Ridpath, Life and Work of James G. Blaine, p. 47.
- Charles Francis Adams, Charles Francis Adams, an Autobiography, pp 33-34.
- Omar N. Bradley and Clay Blair, A General's Life, p. 51.
- Randall, p.
- Cross, p. 24.
- Merlo J. Pusey, Charles Evans Hughes, p. 7.
- William Adams Simonds, Henry Ford, His Life, His Work, His Genius, p. 27.
- Silvio A. Bedini, The life of Benjamin Banneker, p. 255.
- James Weldon Johnson, Along This Way, p. 127.
- Ervin Shoemaker, Noah Webster, Pioneer of Learning, p. 26.
- Helen Keller, The Story of My Life, p. 197.
- Ibid., p. 193.
- Churchill, p. 26.
- Kevin Tierney, Darrow, A Biography, p. 11.
- John McAleer, Ralph Waldo Emerson, p. 36.
- Hjalmar Schacht, translated by Diana Pyke, Confessions of the "Old Wizard", pp.26-27.
- Churchill, p. 25.
- Helen Keller, pp. 82-83.
- Randall, p. 39.
- John Turnbull, The Autobiography of Colonel John Turnbull, p. 9.
- James H. Madison, Eli Lily: A Life, 1885-1977, p. 17.
- Marc J. Seifer, Wizard, The Life and Times of Nikola Tesla, p. 15.
- Feyman, pp. 86-87.
- Eleanor Ruggles, The West-Going Heart, A Life of Vachel Lindsay, p. 57.
- C. Harvey Gardiner, William Hinkley Prescott, A Biography, p. 18.
- Joan London, Jack London and His Times, p. 130.
- Ehrlich, pp 24-25.
- Herbert Kondo, Albert Einstein and the Theory of Relativity, p. 9.
- James Kotsilibas-Davis and Myrna Loy, Myrna Loy: Being and Becoming, p. 20.
- Eisenhower, p. 20.
- Royce Howes, Edgar A. Guest, A Biography, p. 26.
- Arthur Sclesinger, Jr., Robert Kennedy and His Times, p. 42.
- Broadus Mitcell, Alexander Hamilton, Youth to Maturity, 1755-1788, p. 55.
- Ibid., p. 503.
- Catherine Drinker Bowen, Yankee from Olympus, Justice Holmes and His Family, p. 117.
- Lee Iacocca, Iaococca, An Autobiography, p. 21.
- Woodward, pp. 56-57.
- George J. Stigler, Memoirs of an Unregulated Economist, p. 36.
- Leonard, p. 200.
- Ibid., pp. 200-201.
- Ibid., p. 192.
- Bedini, p. 236.
- J. E. Morpurgo, Barnes Wallis, A Biography, pp. 104-105.
- Ibid., p. 108.
- Ibid., p, 109.