September 9, 2006--Saturday Story: "A Matter of Factoring"--Part Two

In Part One, our 9th grade hero finds himself proudly placed in Dr. Herman C. Kaufman’s Algebra class. Proudly, because this suggests that he might eventually be invited by “Doctor-Doctor”’ to join the city’s winningest Math Team. (Forget football—the Brooklyn Tech team was a perennial loser.) The subject at hand was Factoring—the ancient art of seeking all possible combinations of numbers or, in the segment that follows, polynomials, which when multiplied produce the same result.

Let us then, without pause, proceed directly to Part Two, where Dr. Kaufman is about to introduce the subject of . . . .

“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,” and he then wrote 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 asked 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, that too would be a polynomial.”

Milty clapped his tiny sausage-fingered hands together in celebration, most likely 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, remember, 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 migraine.

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 public schools, but once at Tech found ourselves just one of nearly 5,000 others who had done equally well who had been gathered in the cool anonymity of that ten-storey 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 abstract 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 on 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 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 day, when I recovered enough to tell my parents about what had happened, I began to make my case to become the family’s first 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 get a job, my mother was ready with another suggestion. Before giving up, she said, and my father chimed in in the background, muttering about the need for me to be a man and show some “intestinal fortitude,” my mother suggested that she see if my Cousin Chuck could come by to help me with the 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 he had a way with numbers since he had supported some of his extravagant youthful habits while in junior high school, by serving as the schoolyard bookie—“Three-Batters-Six-Hits” was his 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, obstetricians, and other kinds of surgeons 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 quite a bit of 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,” Chuck came bounding up the stairs, which for him was a great effort since he was always 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 TV by telling him when she called that she had just finished making a chocolate Icebox Cake, his favorite, and maybe 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 Chuck 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, “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 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,” Chuck 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.” I said, “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.”

Chuck 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 9th 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 Chuck, 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, 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, Charlsy, 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 Chuck 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.’” Chuck 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. 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 was from my mother’s mother. She had crocheted it and brought it to America from Poland; and my mother would have very upset to see it stained, even if it was a consequence of my algebraic awakening.

“However,” he said, “here’s what’s 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!”

Chuck 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 Chuck 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 the pieces when you’re done. And Charlsy, I found a Danish which I’m saving for you.”

Chuck did not respond, though I noticed that he made note of the treat that awaited, but 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. But again, tinctured with 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 ‘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.”

I was getting 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. And to solve for ‘x,’ as Dr. K would pose it, using the equation as a balance, as a tool, you subtract 4 from each side which gives you the answer, x = 6. 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 now 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 always 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 your mother mentioned.”

She was halfway into the dining room as Chuck uttered these words, and asked, “Charlsy, 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’ solved, I became a demon at solving all kinds of algebraic problems—negative numbers, square roots, adding and subtracting polynomials, and then of course factoring.

To be continued . . . .