September 2, 2006--Saturday Story:"A Matter of Factoring"--Part One
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.”
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. That team 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 version of the Math Team.
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 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 Solly 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 second month at Tech, Solly 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 Solly, “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 the first five four-digit primes.”
Sotto voce, Charlie Rosner, still looking out onto Fort Greene Place, could be heard to say, in less than two seconds, “1,001, 1,003, 1,007, 1,011, 1,013.”
“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, you know.”
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!” Solly, on the other hand, first visibly deflated as if he were filled with helium and then 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. We knew at once 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 always brought things to an embarrassed halt as he quickly gained awareness of what he had again involuntarily done. He just stood there, as if stupefied, staring at us in perplexed 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, of course, 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 cringe inside the teachers’ toilet, undoubtedly traumatized by his own memories of war, the fact that he had commanded us to dive under our desks gave it more credibility, we perceived it as a real threat because of his own experiences in the trenches of Verdun, than, say, when we were directed to take cover by Miss. Ryan, our ninth grade English teacher, who wasn’t even very effective when she tried to get us to diagram sentences!
To be continued . . . .
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. That team 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 version of the Math Team.
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 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 Solly 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 second month at Tech, Solly 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 Solly, “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 the first five four-digit primes.”
Sotto voce, Charlie Rosner, still looking out onto Fort Greene Place, could be heard to say, in less than two seconds, “1,001, 1,003, 1,007, 1,011, 1,013.”
“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, you know.”
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!” Solly, on the other hand, first visibly deflated as if he were filled with helium and then 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. We knew at once 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 always brought things to an embarrassed halt as he quickly gained awareness of what he had again involuntarily done. He just stood there, as if stupefied, staring at us in perplexed 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, of course, 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 cringe inside the teachers’ toilet, undoubtedly traumatized by his own memories of war, the fact that he had commanded us to dive under our desks gave it more credibility, we perceived it as a real threat because of his own experiences in the trenches of Verdun, than, say, when we were directed to take cover by Miss. Ryan, our ninth grade English teacher, who wasn’t even very effective when she tried to get us to diagram sentences!
To be continued . . . .
posted by Steven Zwerling at
12:50 PM
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