November 16, 2012--Chapter 14: A Matter of Factoring

There was my mother’s side--the Malones, an immigrant Jewish family that received its Irish-sounding name at Ellis Island.  Perhaps the official there did not know a better way to transliterate Munya, or maybe he was thinking that he was giving them the gift of viability in this secular rechristening.  They left Poland in the first decade of the Twentieth Century to escape the relentless pogroms and to seek the opportunities that America represented.  

Then there were the Zazlos--my father’s side of the genetic equation.  They were a proud, self-made Austrian family who emigrated to America during the 1880s, from the glittering Vienna, also to seek a better life but, unlike the Malones, trailing no residual feelings of persecution, perhaps because Great-Grandfather Zazlo reportedly read his newspaper, the socialist Arbeiter Zeitung, in the same café as Theodor Hertzl, the visionary founder of Zionism.  This assimilist pedigree inoculated all of his descendents from the stigma of the ghetto; and they thus in all ways felt superior to and curiously resentful of those later-comers whose shtetl-minded Jewishness elicited, even among the Zazlos themselves, a taint of nativist anti-Semitism.  But of course, in the eyes of the goyim, the harder-edge, more obdurate version of anti-Semitism also swept in the Zazlos in spite of their best efforts to ignore their roots.   Even though they had rejected Judaism as well as their Austro-German language as soon as they stepped off Ellis Island and scuttled to neighborhoods of their own as far away as possible from the Lower Eastside where the shtetl had been replicated, and where the Malones settled (in spite of their pseudo-Irish camouflage), like it or not, Jews the Zazlos were and Jews they would remain.

What the Malones thought of the Zazlos and vice versa, before, at, and after my parents’ wedding is best left to the imagination.  There is the brief version: neither side had a very high regard of the other.  And thus as a confused, mongrel product of these two very different families, I sought all means of escape.  

So when Dr. Herman C. Kaufman, Ph.D. said, “Today we will begin the study of factoring.  Actually, ‘Quadric Factoring,’ which is an essential aspect of Algebra,” this seemed like my ticket out . . . and away. 

He was my ninth grade math teacher at Brooklyn Technical High School.  He was also coach of the school’s city champion Math Team.  And although team members did not wear uniforms or have cheerleaders with pompoms doing cartwheels at its matches, at the highly-selective and very competitive Brooklyn Tech, there was more status affixed to being on that team than on the perennially pathetic football team.  Football was reserved for the Technical (read “vocational”) students at Tech, certainly not us College-Prep boys, who were the school’s elite and thus curried Dr. Kaufman’s favor.  

So to be assigned, as I was, to his freshman Algebra class, was a hopeful sign.  Not only would “Doctor-Doctor” (senior wags nicknamed him that because every time he wrote his name on the blackboard or added it to the bottom of a mimeographed homework assignment sheet, he printed it “Dr. Herman C. Kaufman, Ph.D.” with the “Dr.” and “Ph.D.” parts underlined), not only would he be my math teacher but maybe, just maybe he would take notice of my uncanny ability to solve algebraic problems and consider me for the freshman, or junior varsity Math Team. 

That first day of class he continued. “Let me begin by taking you back to Arithmetic,” he smiled, and we newly-minted high school students chuckled complicitously, fanaticizing that he was inviting us into colleagueship.  “In Arithmetic, factors are the two numbers you multiply to get another number.  For instance, the factors of 21 are 3 and 7, because 3 × 7 = 21.   Some numbers have more than one ‘factorization,’ which means ways of being factored.”  All of us, now brought into full fellowship with him, were slouched in our seats with arms folded across our chests, conspicuously not taking notes, and nodded nonchalantly as if to say, “Of course.”  

“For instance,” he continued, “16 can be factored as 1×16, 2 × 8, or 4 × 4.  But then again, a number that can only be factored as 1 times itself is called Prime. The first few Primes are, therefore, 2, 3, 5, 7, 11, and 13.  It’s interesting, isn’t it, to think about a few more Prime Numbers?” 

Plump Milty Leshowitz, who was always looking to distinguish himself, called out, “17, 19, 23.”   He twisted around in his self-assigned front-row seat to look at the rest of us, seeking our acknowledgement.  We knew that though it was still only our first days at Tech, Milty had already declared his intention to go to MIT.  With a full scholarship.  So of course we looked up at the ceiling or out the window in order to, as overtly and blatantly as possible, ignore him.

Dr. Kaufman, equally not impressed, as a form of aside, said to Milty, “If you ever find yourself trying out for my Math Team, Mr. Leshowitz, you’ll probably be given ten seconds to make a list of all the four-digit primes that begin with 4 and 3.” 

Sotto voce, in less than two seconds, Charlie Rosner, still looking out onto Fort Greene Place, could be heard to mutter, “4,327, 4,337, 4,339, and 4,397.” 

“Well done Mr. Rosner,” Dr. Kaufman said, “Please see me after class.  We should talk about the Team’s Junior Varsity.  It’s just for freshman of course.” 

Joey Lombardy, who sat to Charlie’s left gave him a gentle punch in the shoulder, as if to say, “Way to go big guy.”  Milty, on the other hand, initially visibly inflated as if he were filled with helium collapsed into his seat so that only the rings of fat in his neck were visible above the chair back.

“Since you men seem to be taking to this, before we proceed to Quadratic Factoring, let me say another word about Primes.  In a recent lecture, Bernhard Zagier commented, and let me see if I can quote him more or less exactly, ‘There are two facts about the distribution of Prime Numbers of which I hope will be permanently engraved in your hearts.’”  He tapped his chalky hand on his chest where it left a white residue of powder scattered across his tweed vest.  “’The first is that, despite their simple definition and role as the building blocks of the natural numbers . . . ’”  He paused for a moment to clarify, “Natural numbers, you of course know, are those that are most ancient and are used to count things.”  And then he returned to Zagier, still quoting from memory, “’The Prime Numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout.’”  

He looked right at Charlie, “Very nice, no?”  Charlie smiled back at him.  With that we knew that there would be no need for Charlie to try out for the team—a place for him was already assured.  Dr. Kaufman continued, “The second fact about Primes Zagier said, ‘Is even more astonishing, for it states just the opposite: that they exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision.’"   

Dr. Kaufman paused again, this time taking in all of us, “And maybe, just maybe, if you prove yourselves to be worthy, we will together explore those laws of precision.” 

And with that, with the stick of chalk he always kept clutched like a weapon in his left hand, he reflexively twitched, snapping that hand up toward his face where he, without apparent intention, stroked the chalk, in short chopping motions, across his spiky boar’s-bristle moustache.  Three, four, five times. 

That brought things to an embarrassed halt as he quickly gained awareness of what he had involuntarily done.  He just stood there rigid, as if stupefied, staring at us in bewildered silence.

After the first few times we had witnessed this spasm, Gary Phillips, who had a friend in the sophomore class, learned that Dr. Kaufman, who was very old, had been a foot soldier in the First World War, where he had been gassed.  And that the Mustard Gas had affected his brain and left him like this.

This alone would have made him my favorite teacher. 

During our frequent Take-Cover Drills, to protect us from the blast of the H Bomb that the Russians were always threatening to drop on the Brooklyn Navy Yard, less than a mile from our classroom at Tech, when Dr. Kaufman would scream “Take cover!”  though he would run out into the hall in a frenzy, uncontrollably chalking his moustache and face; and then, traumatized by his own memories of war, cringe inside the teachers’ bathroom, the fact that he had commanded us to dive under our desks gave it more credibility, we perceived the drill as more of a real threat because of his own experiences in the trenches of Verdun, more so than when we were commanded to take cover by Miss. Ryan, our ninth grade English teacher, who wasn’t very convincing even when all she tried to do was get us to diagram sentences. 

*    *    * 

“And now,” Dr. Kaufman said, emerging from his seizure, “now we can move on to the factoring of polynomials.”  Joey Lombardy pleadingly glanced over to Charlie Rosner as he did every time Dr. Kaufman brought up a new topic—Joey knew he would need all kinds of help, especially on pop quizzes when he hoped Charlie would slide his answer sheet to the left side of his desk so Joey could catch glimpses of it. 

“Polynomials, of course, are sums of expressions and numbers such as 4x⁴ or 3x. or 7.”  He wrote these on the board with the same chalk he had sharpened on his moustache.   “We can make a polynomial from these expressions.  For example, 4x⁴ + 3x = 7.  But factoring polynomial expressions is not quite the same as factoring numbers.  The concept, though, is similar. In both cases, you search for things that divide out evenly.  But in the case of polynomials, we will be dividing numbers and variables out of expressions, not just numbers out of numbers.”  He smiled at us again, pleased with the clarity and elegance of his explanation.  He always stressed that these two qualities, particularly the latter, were essential to what he called “true math.”

Squirming in his seat, Milty excitedly waved his hand, virtually in Dr. Kaufman’s face.  “Yes?” Dr. Kaufman sighed in a tired voice, looking up at the buzzing fluorescent lights.

“They can be, polynomials, I mean, they can be things like, I mean, expressions like two, open-parenthesis, three-x, plus three, close parenthesis?  I mean.”  Thankfully he remained seated and merely traced the expression in the air.

Dr. Kaufman stared out the window for a moment as if searching there would help him find the answer to when he would have enough money saved in his pension so he could escape from this burden.  Then, without turning back to us, in a tired voice he said to Milty, “Yes Morty (Dr. Kaufman refused to remember his name) that too would be a polynomial.” 

Milty clapped his sausage-fingered hands in celebration, thinking that all for him was still not lost—the Math Team, MIT . . .  

“May I continue?” Dr. Kaufman asked with some sarcasm, of course not pausing for a response.  

On the board he wrote: 2(x + 3) = 2(x) + 2(3) = 2x + 6.

“Previously, recall, we simplified numbers by distribution.  Now we will do so with algebraic expression such as this one, 2(x + 3),” he pointed to the blackboard.  “So much of Algebra is about such simplification.  Here the simplified answer is 2x + 6.   Factoring polynomial expressions is the reverse of distribution. That is, instead of multiplying something within a parenthesis, you will be seeking, through factoring, what you can take back out and put in front of a parenthesis, such as, in two steps:

2x + 6 = 2(x) + 2(3) = 2(x + 3) 

“You see how when we distributed,” he nodded toward the first equation, “we began with 2(x + 3) and wound up with 2x + 6; but when factoring,” he pointed at the new equation, “we began with 2x + 6 and reversed the process, winding up with 2(x + 3).  

“The trick is to see what can be factored out of every term in the expression.  Don't make the mistake of thinking that factoring means ‘dividing off and making something disappear.’  Instead, remember that factoring in fact means ‘dividing expressions out to be placed in front of the parentheses.’   Nothing disappears when you factor.  Things merely get rearranged.” 

Again he turned to face us.  The hand that held the chalk was beginning to twitch.  “You understand?”  He had presented this so clearly that even Joey was grinning. 

*   *   *

Though from the fact that by the second month of my first year I too understood factoring you might be imagining that I had gotten off to a good start, assigned to the fast track Algebra class on the basis of my admission test scores and was making rapid progress in my ability to quickly solve various algebraic problems, you would be only partly correct. 

True, I had done well on the test that determined if I was to be admitted to one of New York City’s elite high schools (I had chosen Tech over Stuyvesant because it was in Brooklyn), but by the middle of just the first week in class, I asked my parents if I could drop out of school, forget about high school altogether, and get a job because, when, during that first week, Dr. Kaufman introduced the concept of “x,” I did not have any understanding of what “x” meant and, as a result of the fear and anxiety that that produced, developed an instant case of literally blinding migraines. 

Dr. Kaufman had begun deliberately enough, as if understanding that for many of us coming to Tech in central Brooklyn, on our own, from the security of our families, neighborhoods, and nearby elementary schools, though we were presumably talented, or we would not have been admitted to Tech in the first place, from his decades of experience he knew that we were still quite tender and innocent and scared and, in truth, unsure of our abilities.  We had been the academic stars at our nurturing elementary schools, but once at Tech found ourselves just one of nearly 5,000 who had done equally well who had gathered in the cool anonymity of that twelve-story brick mountain of a high school on Fort Greene Place.  It was as easy to get lost in Tech’s hundred-yard-long hallways and impersonal classrooms as in the IRT subway. 

But though on the first day of class he ever so gently began by introducing us to the concept of “x,” I was immediately lost.  In my previous schooling, I had been the master of the specific, the concrete.  Anything that could be seen, touched, measured, or hammered had instantly revealed its mysteries to me.  Wood shop thus had been my special preserve.  I fabricated shelves there for each of my eight aunts that were so cleverly fitted and glued together that in every instance they survived moves from New York to Florida and in spite of the heat and humidity continued to support for decades their crystal goblets and porcelain dolls. 

But when Dr. Kaufman led us into the ephemeral, impalpable world of “x,” I began to tremble in my seat and needed to run out to the boy’s room where I promptly threw up in the toilet. 

After I returned to class, when Dr. Kaufman said that not only could “x” stand for any unknown, not just a specific one, we would soon also see how by employing “y” and “z” as well we would be developing the tools to search for multiple unknowns, simultaneously, when I heard about these multiple unknowns, all in concurrent motion, stuck in my literalness, I found myself overcome by the onset of, what was for me, a new kind of headache, one that felt as if it had been caused by a tomahawk having been driven into my skull, splitting it in half, with the left side on fire. 

For that I needed more than the boy’s room, I required the nurse who, after she took just a quick diagnostic look at me, realized what I was experiencing and sent me right home in a taxi, knowing there could be no more school for me that day.  But from the thumping in my head, I realized there could be no more Brooklyn Tech for me.  My problem was not going to be solved by a day of rest in a darkened room. 

Later that afternoon, when I recovered enough to tell my parents about what had happened, I began to make my case to become the family’s first high school drop out, arguing hopelessly that although I was too tall and flatfooted to join the army, in my heart I always wanted to be a carpenter and that this new affliction, which was making it impossible for me to continue my education, was really a “blessing in disguise”—it would help launch me in my career installing drywall well before anyone else who might turn out later to be my competition had moved on to become high school sophomores. 

They of course would have none of this.  If I wasn’t going to be successful at Tech, where they had dreams that I would be transmuted into a nuclear physicist, then I could always switch to Tilden High School, just three blocks from our apartment, which would be a good backup, since many who had gone there went on to become pre-meds in college.  

Not such a bad fate for his son, my father claimed, considering his carpentry skills.  He was as usual explicit, “You have wonderful hands.  Show them to me,” which I promptly did, “So instead of slicing up lumber, you’ll slice up cadavers.”  I cringed at the thought.  “And,” he added, “you’ll make a good living.”  The idea of making a living by dissecting dead bodies sent me reeling toward the bathroom where I again, as at Tech earlier in the day, fell to the floor, having to retch in the bowl.

When I got myself back to the kitchen table to pick up my plea to be allowed to leave school and get a job, my mother was ready with another suggestion.  Before giving up, she said, and my father chimed in in the background, sputtering about the need for me to be a man and show some “intestinal fortitude,” my mother suggested that she see if my Cousin Larry could come by to help me with Algebra since he had been a very successful math student at both Tilden and later at Columbia.  Though dropping out altogether was still my plan, there was no way that I could ignore this one last suggestion.  I knew that Larry had a way with numbers since he had managed to support some of his extravagant youthful habits while in junior high, by serving as the schoolyard bookie—“Three-Batters-Six-Hits” was his betting specialty during baseball season.  

He lived three blocks away; and while waiting for him to arrive, my father made his case that, yes, maybe I was right in seeing my problem with “x,” “y,” and “z” as good fortune.  I should “seize it,” seizing was a favorite theme of his, and become a plastic surgeon.  His argument was strong, he felt, because I had the hands for it and, unlike pediatricians or obstetricians, I wouldn’t have to work nights and weekends.  “Who ever heard of a plastic surgeon having an emergency?” he asked rhetorically. 

In addition, he claimed, there was much opportunity to practice right in our own family, considering the condition of most of the inherited noses, chins, and breasts.  He even volunteered to be my first patient, indicating that in addition to his nose and “turkey throat,” he could use some work on the scar left on his abdomen after his prostate operation.  Thankfully, before he could unbutton his shirt to again show me “the shelf the surgeon built down there,” Cousin Larry came bounding up the stairs, which for him was a great effort since he was overweight and misshapen.  But that evening he took the stairs two at a time because my mother had lured him away from the Yankee game on the radio by telling him when she called that she had just finished making a chocolate icebox cake, his favorite, and after tutoring me he could have some of it.

Always the negotiator, he said if he could have a piece in advance, “just a slice,” to give him energy, he felt certain he could make rapid progress with me.  So with a quarter of the cake and a glass of milk in front of him, we sat together at the dining room table, where Larry inhaled the slice without chewing and drained the milk before lifting his head from the plate and asking me, “So what’s going on?”

In a single breathless and tearful sentence I told him that I-was-already-failing-Algebra (“After just one week?” he asked with more than a little skepticism) –because-I-didn’t-understand-what-Dr.-Kaufman-was-saying-about-“x,”-much-less-“y”-and-“z”-and-beyond-that-I-couldn’t-figure-out-why-he-made-such-a-big-deal-out-of-‘the-power-of-the-equation.’”

“I think I need another piece,” Chuck said, looking up at my hovering mother, “This will take some time.  But don’t worry,” he added as he got up with his empty dish, pushing his way through the swinging door, following my mother into the kitchen, “Don’t worry,” I could still hear him, “I know just how to help you.”  

My mother must have been encouraged by what she heard him say because when he returned he had a full half of the cake and another glass of milk.  She was holding very little in reserve. 

“I suppose your Mr. Kaufman,” Larry began after once again cleaning his plate and licking his fingers, “I’m sure he said very little to you about Algebra itself.  About its history and why it was so important in advancing civilization and why it is still significant today” 

Since Dr. Kaufman hadn’t done that, I felt the need to come to his defense, “But he’s the coach of the Math team.  They’ve been undefeated for three years in a row.  And he is ‘Doctor,’ not ‘Mister.’ So it’s me.  I don’t understand anything,” I whimpered, “Algebra makes my head spilt open, and I just want to be left alone so I can quit school.  I’ll never learn what ‘x’ is or what he is saying about equations.” 

Larry reached over to me and, as a form of assurance, put his lumpy hand on my shoulder.  Perhaps for the first time in our lives he looked directly at me and confessed in, for him, and unusually hushed voice, “I had the same problem when I began Algebra.  I too was good in Arithmetic but struggled at first with equations.   But I had a friend on the block who helped me.  He began, as I should with you, by telling me that Algebra has a history that stretches back more than 2,000 years, to the ancient Greeks; but it didn’t begin to become the Algebra that we have today until the 9^th century when the most important developments were recorded in the book al-Kitab al-muhtasar fi hisab al-jabr wa'l-muqabala , I think I have it right, written by the Arabic mathematician Al-Khwarizmi .”  

I stared open-mouthed at Larrty, who, though his pronunciation of the Arabic words was clearly a Brooklynese version, was nonetheless amazing—how did he know all these things? 

“In English,” he continued without pausing, “the title of his book is Compendium On Calculation By Completion and Balancing.   That I know I’m right about.  In fact, the English word ‘algebra’ is derived from al-jabr, or ‘completion.’  And that is where we will begin—with completion through balancing.” 

He looked around over my shoulder, and wondered out loud, “Do you think I might get that last slice of cake?”  And as if on cue, my mother slid through the swinging door with yet another glass of milk and the remaining slice on a clean plate, shrugging as if to say, “Sorry, darling, but this is the final piece.”  

And almost as quickly as she placed it before him it was gone.  As was his third glass of milk.  He drained the glass, sucked out the last drops, and then emitted a three-toned belch before he continued, “Next, let’s talk about Dr. Kaufman’s famous ‘x’s’ and ‘y’s’ and ‘z’s.’  OK?”  I nodded though I was still distracted by thoughts about how Larry knew so much Arabic and how within fifteen minutes he had managed to gulp down the entire icebox cake and nearly a quart of milk.  

“I know why you are having a problem with this.  You are used to the numbers in Arithmetic that have distinct and specific meanings.  A three is always a three.  You can count on a 17 always being just that—17.  And you can relate those numbers to things you can see and touch in the world—three oranges, 17 points in a football game.  But though you have always thought about those comfortable numbers as representing things that can be found in the world, they are a little more complicated than you might have imagined.  They in some ways are closer to what you are struggling with in Algebra.  For example, that solid 17 can be used to denote any number of things, pardon the pun--a football score, yes, but it also can be applied to the number of students in a class or how many from our family are huddled around the table during Passover.” 

This I was understanding, but not its connection to “x.”  “You may not see any connection to your ‘x-problem.’” Larry said, he also seemed to be able to read my mind, “but it does.  If you think about ‘x’ for a moment as if it were like 17, as having multiple applications, you might be able to understand ‘x’ itself.  All ‘x’ is is a mathematician’s way of representing the answer he is seeking.  What I suspect Dr. Kaufman calls the ‘unknown.’  All that means is that the answer we are seeking, symbolized by ‘x,’ is unknown. 

“But only until we find it,” he smiled with the remnants of the cake clinging to his braces, “Then it’s known.  ‘X,’ the unknown is now known.  The problem is solved.  We have an answer.”  He was so excited, or riddled with sugar from all the cake, that he slammed his hands down on the table with enough force to topple the milk glass.  It was good that he had done such a thorough job of draining it or he would have stained the tablecloth, which had come from my mother’s mother.  She had crocheted it and brought it to America all the way from Poland; and my mother would have been very upset to see it stained, even if it was a consequence of my algebraic awakening. 

“However,” he said, “here’s what’s really special about ‘x’ . . .”  

But before he could continue, as I was beginning to understand, I blurted out, “Like 17, which can refer to many things, ‘x’ can be used again and again since it does not have a fixed meaning.”  

Larry smiled, correcting me by interjecting, “’Value,’ not ‘meaning.’”  

“And,” I raced on without stopping to catch my breath, “it can be used over and over again like 17.  Each time standing for whatever the unknown is that we are searching for.”

This time when Larry pounded the table the glass rolled off and shattered on the floor.  My mother, still listening from the kitchen, said, “Leave it.  Don’t touch it.  You’ll cut yourself.  I’ll sweep up later when you’re done.  And Larry, I found a Danish which I’m saving for you.” 

He did not respond, and though I noticed he took note of the treat that awaited, he pressed on with my lesson, “You now have ‘x’ figured out.  Let’s talk next about ‘the power of the equation.’”  Though he did not know Dr. Kaufman he pronounced that with what he clearly thought was an exceptional German accent, though in fact it was tinctured with classic Brooklynese.

“You must think about equations as just a tool to solve problems or manipulate information.  They are nothing more complicated or mysterious than that.  Really.  A tool, for example, to use to find or solve for ‘x.’  Though mathematicians think about an equation as a form of statement written in numbers and symbols and having two sides connected by an equal sign, for you it will be more useful to think about it as if it were a balance scale.  The equal sign is merely the pivot point or fulcrum in that kind of a scale.  And to use it like a tool, as if you were, say, weighing gold dust, you must always keep the scale in balance.  In Algebra, you always keep what is on the left side of the equation, all the numbers and symbols, the 3’s and 5’s and 12’s and the ‘x’s,’ ‘y’s,’ and ‘z’s,’ equal to, or balanced with what is on the right side.  If you add something to one side, you must add the equivalent to the other.  The same if you subtract.  Being sure always to keep things in balance.”

I was understanding this too and wished my mother had saved a piece of the icebox cake for me.  He went on, “Let me show you how an equation can be used as a tool.  Using a very simple example that can serve as an introduction to how to turn on its full power.”  He reached for the pad that my mother had provided, and on it wrote:  

x – 4  = 10

“It doesn’t get any simpler than that.,” he winked conspiratorially at me, “ And to solve for ‘x,’ as Dr. K would pose it, using the equation as a balance, as a tool, you add 4 to each side which gives you the answer, x = 14.  Voila, QED, solved!  And though, of course, all the equations you will deal with during the rest of your life, even those that fill a whole room of blackboards, they will all work exactly the same way.  Get it?”

As a now potentially reborn nuclear physicist who at last understood these principals of Algebra and could thus contemplate returning to Tech on Monday, I could feel my migraine subsiding and my peripheral vision clearing.  I still, though, needed to ask one last question, even if it led me back into hopeless confusion, “But isn’t E = mc² also an equation?  It seems very different from what we’ve been discussing.” 

For a moment I thought I had him stumped because he sat there tearing at his cuticles, making them bleed—something he did when he was nervous or under pressure.  But he quickly said, “Yes, it is.  Some equations, like that one, express ‘laws of nature.’  We’ll deal with those when you take Physics next year.” 

Twisting in his chair he mused, “In the meantime, I’m wondering about that Danish Aunt Rae mentioned.”

She was halfway into the dining room as Larry uttered these words, and asked, “I have two kinds, cheese and cinnamon.  Which do you prefer?”

He said, “If you have another glass of milk, I’ll take both.”

*    *    *

Back in class at Brooklyn Tech, with my headaches cured and my ‘x-problem’ overcome, I became a demon at solving all kinds of algebraic problems—those involving negative numbers, square roots, adding and subtracting polynomials, and of course factoring.

When Dr. Kaufman asked Aaron Bernard, Joey Lombardy, and me to come up to the blackboard to factor a series of quadratic equations, I did not fear public humiliation.  In fact when he wrote--

x² + 4ax + 3a²

--almost as fast as I could move the chalk across the board I factored it to yield—

(x + 3a) (x + a)

“Well done Dr. Kaufman said even before Aaron much less Joey, unaided by his accomplice Charlie Rosner, were able to write two sets of empty parentheses.

And then when Dr. Kaufman scrawled--

n² – 12n – 35

--as the second quadratic for us to factor, as Aaron picked perplexedly at one of the pimples on his cheek, I grinned toward Dr. Kaufman and said, “It can’t be factored because it’s a prime, and primes can’t be factored.”

“Excellent,” he said.  “Along with Mr. Rosner, please see me after class so we can talk about the Math Team.  The JV as you know.”

Behind me I could hear Milty Leshowitz begin to hyperventilate.  “It’s my asthma,” he gasped and ran out of the classroom toward the nurse’s office, which up to then had been my refuge.

*    *    *

Charlie and I remained uncomfortably at our desks, averting our eyes, as the rest of the class scattered at the first sound of the end-of-period bell.  Though we craved the distinction Dr. Kaufman’s invitation to remain bestowed upon us, we had mixed feelings about being thus set apart from our fellow classmates. 

This was to be a perpetual struggle at the highly competitive Brooklyn Tech, where at the end of each term everyone’s cumulative grade point average, down to the third decimal point, was published, in bold type, for all to see and compare in the Tech student newspaper, The Survey.  We struggled to both compete with classmates, who literally sat to our left and right, while at the same time attempting to remain friends.  Against these “friends” we contested to place higher on the class-standing list where the difference of just one hundredth of a point would determine if you got a free ride upstate at Cornell or were reciprocally exiled to the concrete campus of City College. 

And, of course, the friend on your right was eyeing you in exactly the same way—he saw you crawling up out of the subway in Harlem as assuring that he would frolic on the green hills in Ithaca.  It was all an equation, a balance—to rise on one side, something on the other had to descend.

The one thing our English teacher, Miss. Ryan, was able to get us to pay attention to was her warning not to help anyone with homework.  If we did so, and as a result a classmate got one point more than you on the grammar exam, that measly single point might mean that he, not you, would wind up higher on the GPA list and ultimately in the Ivy League.

“And you know the implications of that,” she would intone portentously.  If we did help a fellow student, she warned, and he as a result surpassed us as “the beneficiary of that seemingly selfless generosity,” there could be lessons to learn since that benevolence and its “ironic” consequences” could serve as “a metaphor for life itself.”  Though as an English teacher she was always talking about things like “irony” and “metaphors,” since these were concepts that very few of us understood, we didn’t think too much about the “irony of benevolence” but rather continued to focus on something we could grasp--how to avoid splitting infinitives, an unforgivable transgression.  Life metaphors could wait until our junior year.

When all our Algebra classmates had departed, Dr. Kaufman summoned Charlie to the front of the room.  I did everything I could to force myself not to listen in on what they were discussing, feeling whatever it might be was private and should thus remain between them.  However, in spite of these intentions, I still found myself inexorably straining forward in my seat so I could take in every single word. 

Even if less-than-conscious, I felt impelled to do so, as if by a force outside myself, thinking that whatever might transpire between Charlie and Dr. Kaufman would be similar to what would be expected of me; and, in my competitive mode, after all wasn’t I being trained for that at Darwinian Tech, I wanted to take advantage of anything that came my way that would give me even the slightest edge.

It was, though, not so easy to hear them since Charlie, tall enough for the basketball team as well as smart enough for the Math Team, Charlie towered over and shrouded the seated Dr. Kaufman, who, behind his battered desk, thus seemed so reduced in size and shrunken in stature.  But it turned out not to matter since Dr. Kaufman did not ask any questions or quiz Charlie in any way; he simply told him to show up for the tryouts next Wednesday, at 4:30, in the Math Team office on the first floor, right next to the nurse’s office.  About this, I would not need directions.

And so, when he signaled to me, I approached him without trepidation, assuming that he would simply tell me as well about the time and place for the tryouts.  But Dr. Kaufman nodded at the old library chair beside his desk, and I sat down.  My heart began to thump.  I felt it throbbing all the way up in my throat.  He looked at me without saying anything for what felt like fifteen minutes.  I was beginning to experience palpitations.

“Lloyd,” he said, using my first name—prior to that he had only once before called any of us by anything but our last names, except when exasperated with Milty, when he called him “Morty”—“Do you remember the other day when I quoted Don Zagier’s thoughts about Prime numbers?  From, I think it was, from his inaugural address at Bonn University, when he became one of the directors of the Max Planck Institute?”

“I remember, what you said Bernhard Zagier wrote about them.  About their distribution.”  I couldn’t believe, without restraining myself, I had appeared to be correcting Dr. Kaufman.

“Ach, good,” he laughed, “I forgot I called him that.  We in the field know him more familiarly as ‘Don.’  He’s an American you know.”  I nodded as if I did.  “Well, Don had more interesting things to say about Primes than almost anyone.  And I am telling you this because you appear to be interested in them and are even beginning to show some signs of promise.  Talent I do not as yet know about.”  I began to fidget.  Which he noticed.  “That is all right.  We will know soon enough.  That is why you are here—for us to find out.”  This sounded both encouraging and ominous.

Without waiting for me to say anything, he continued, “Don, I felt, was always too optimistic, perhaps even a little arrogant when he claimed that ‘though they grow like weeds they exhibit surprising regularity.’  Do you remember that?”  I did.  “Good.  Most mathematicians through the centuries struggled to find that regularity, what Don called their ‘precision.’  From the earliest days in Greece and Arabia.”  Inexplicably, images of Cousin Larry gobbling icebox cake flashed through my mind. 

“Struggled unsuccessfully, I should add.  Indeed, I understand that he is still trying and continues to write elegantly on the subject.”  Dr. Kaufman had again returned to the subject of mathematical elegance.

“For example, though numbers are the simplest of mathematical elements they, perhaps for that reason, do you understand, have inspired the lushest prose.  You find mathematicians referring to them as ‘mysterious,’ ‘stunning,’ ‘diabolical,’ ‘harmonious,’ ‘the Holy Grail,’ ‘divine,’ even, yes, ‘glamorous.’”  His eyes sparkled as he pulled these words as if from the air surrounding us in that barren room.

“This may still be unfamiliar to you, but mathematicians, when considering Primes, they often employ the vocabulary of first love.  To them, they are objects of great beauty.”  He paused to look at me in a way that made me quiver with excitement and nervousness.

“One day, we can hope, soon, we expect, you will participate in the exploration of these mysteries and taste this chaste form of love.”  I was glad to hear him describe it that way, and began to calm down.  “You may have already begun to do so, to have touched some of this, instinctively.  We will discover, together, if you share some of this gift.   When you so quickly, earlier today, perceived that n² – 12n – 35 is indeed a Prime, a very special Prime, exhibiting its own mystery, its own beauty, I thought, if you are patient and work hard, you will find out.” 

I pledged to myself that I would do both.

“Of course, along with Mr. Rosner, you will tryout for the Team next week.”  I indicated I wanted to.  “But before you leave,” I saw under the desk his hand with the chalk begin to twitch, “I need to tell you something that the great Tenenbaum of France said about our numbers.  I cannot recall it exactly, but since it is very important I wrote it down and hopefully have it here in my desk.”  With his right hand a claw clutching the chalk, he rummaged around one-handedly searching for it.

“Ah, here it is.”  He looked up at me with a broken-toothed smile, and read, ‘As archetypes of our representation of the world, numbers form, in the strongest sense, part of ourselves, to such an extent that it can legitimately be asked whether the subject of study of them is not the human mind itself. From this a strange fascination arises: how can it be that these numbers, which lie so deeply within ourselves, also give rise to such formidable enigmas? Among all these mysteries, that of the prime numbers is undoubtedly the most ancient and most resistant.’”

He peered at me again as if he might find within something worth understanding or knowing.  As yet, though I too had recently begun to search, there was nothing worth noting.