Sean filed out of class with the rest of the herd. The crowd spread out like distributaries of some great river as students headed to their respective destinations.
"Hey dude, wait up," the low deep voice didn't belong to his tiny circle of acquaintances.
"Oh... oh," Sean glanced over his shoulder and picked up his pace when he saw it was Brandon Cox towering a head above other students. Strategic withdrawl was Sean's ingrained response to being pursued by one of Jason's buddies. Oddly enough he was never inclined to flee whenever he ran into Jason himself. The sight of Jason with Tiffany and Carmen brought out the worst in Sean, a primeval rage that drove out any instinct of self-preservation.
"Why're you running?" Brandon kept pace effortlessly on longer legs.
"Don't want... to get punched," Sean slowed to catch his breath. Even dodging between students hadn't slowed down the halfback.
"Not gonna punch you," Brandon sounded aggreived and held up a backpack, "Just want to talk... about my studies."
"What?" Sean stopped.
"Can we go somewhere quiet?" Brandon looked up and down the corridor uneasily.
"Um... whatever," Sean started walking, "Stay atleast ten feet from me. And no texting your buddies."
Brandon followed after a moment. Sean led him on a tour of the ground floor and finally chose an empty classroom with two doors.
"Sit down," Sean pointed a few rows down. There was enough furtniture in between that Brandon couldn't rush him. Sean could flee through either door if needed, "Talk."
"I can't do it anymore, dude," Brandon emptied his backpack on a table and stared morosely at the heap of textbooks that tumbled out.
"Start making sense," Sean said testily. His limited patience was reserved for Jason's pretty girl friends.
"It... it's just too much," Brandon gestured at his books, "Coach says if I don't meet minimum GPA, I can't stay on the team. He isn't going to intercede with the principal anymore, like he did in my freshman and sophomore years. Says he's used up too much goodwill on my behalf."
"Get a tutor," Sean suggested, "Look up online tutorials. I can recommend some good ones."
"There's no way I can catch up on my coursework," Brandon seemed close to tears, "I've been neglecting my studies for too long. My folks hired a math tutor, but there's too many rules to remember. Nothing makes sense."
"So you came to me?" Sean frowned.
"Kyle told me what you did for him," Brandon nodded, "Maybe you can help me figure this out?"
Why should I? Sean stared resentfully at Brandon's broad muscled shoulders, his dark curly hair that women loved, even his excess flab that they described as 'snuggly'. Pretty girls came up to Brandon all the time to have long chats, to laugh with him and give him a hug. Why should Sean keep helping guys like Brandon and Kyle who had hit the genetic lottery? Sean suddenly felt ashamed at the turn of his thoughts. His own academic position would be precarious if not for the whim of an alien God trapped in dead cosmos. How often had Sean felt guilt at the unfair advantage that his power gave him over other students. Besides Brandon was part of Jason’s circle that Sean was driving a wedge into, one person at a time. Tiffany would have nudged Brandon towards Sean sooner or later.
"Tell me the parts that you do understand,” Sean nodded, “and we’ll go from there.”
Sean paced before the whiteboard as Brandon talked, his dismay growing as he realized how far behind the other boy was. Sean continued to pace in silence after Brandon stopped talking.
"Since we are short on time,” Sean picked up a dry-erase marker, "I'm going try something drastic. Your problem is that you look at math as a bunch of rules to be memorized instead of seeing the big picture. Like missing the forest for the trees."
Stolen novel; please report.
"What big picture?" Brandon looked skeptical. Sean drew a square on the whiteboard, marking it with sides one unit long. Then he connected two corners with a diagonal.
"How long is the diagonal of a unit square?" Sean tapped the board.
"Square root of 2, of course," Brandon frowned, "I'm not stupid, you know."
"I know you aren't," Sean nodded, "Now why is it root 2?"
"The Pythagorean Theorem states that..." Brandon began.
"I'm not asking you to quote the theorem," Sean held up a hand, "I'm asking you why it is so."
"You expect me to prove the fucking Pythagorean Theorem?" Brandon demanded.
"I am asking you to see it," Sean sketched three more unit squares touching the first square in a 2x2 grid, then drew diagonals for each such that every diagonal was a mirror of the diagonal of the adjacent squares. The four diagonals together formed another square standing on its tip like a diamond, "See these diagonals? They are connected together like a diamond, right? If these diagonals are d-units long what's the area of the diamond?"
SKETCH: https://www.cut-the-knot.org/do_you_know/SqRtOf2.shtml
"d-squared," Brandon was looking bored.
"Right," Sean nodded briskly, tracing the biggest square formed by the 2x2 grid, "what's the total area of all four unit squares?"
"Four," Brandon shrugged.
"Each diagonal splits its own square into two halves, right?" Sean tapped the diamond that the diagonals formed, "So this diamond is really made up of four triangles that are each half of their unit squares. So the diamond must be half the total area of all the unit squares. Half of four is two. Do you see now?"
"So... the diamond is two square units in area," Brandon nodded slowly, "which also equals d-squared. That's why the diagonal length is root-2 ! Fuck... never knew it was that simple... that's really cool, dude."
"It is that simple," Sean grinned as he erased the sketch and drew a slightly more complex mosaic involving right trangles, "and we can extend the same reasoning..."
Understanding dawned on Brandon's face as he watched, "Shit... that's why the Pythagorean Theorem is true."
"Yes," Sean nodded earnestly, "Math isn't about numbers or formulas. It's the language of thought itself... a mosaic of reason that you can see for yourself. A theorem is true because it can't be any other way, anymore than a diagonal of a unit square can be anything other than root-2. Once I'm done with you, you'll no longer see coursework as boring crap to slog through, but as mathematical structures you can hold and play with in your mind."
"You... you can do that?" Brandon blinked, "For all of them?"
Sean nodded, "If you are willing to put your mind to it."
"Bullshit," Brandon snorted, fliping through a book, "drawing squares and triangles is one thing, but you're saying everything can be boiled down to pretty pictures? What about this... this calculus stuff? This chapter goes on and on about integral of something dx and dy/dx and crap like that. What does any of it mean?"
"What about it?" Sean laughed, "dx is just another way of saying a tiny piece of 'x'. Integral is just a fancy way of saying "add it all up". What do you get if add up all the tiny pieces of 'x'."
"x, probably," Brandon raised a brow.
"Correct," Sean smiled, "and if you split the area under a curve into tiny slivers and add it all up again?"
"The total area under the curve?" Brandon ventured hesitantly.
"Correct again," Sean nodded, "If a tiny change in 'x' causes another tiny change in a function, lets call the function 'y'. Then what is the ratio of their changes?"
"dy/dx, I think," Brandon said after a moment's thought.
"Bingo," Sean raised his arms, "dy/dx is simply the ratio of tiny changes on the ouput and input of a function. It's basically algebra dressed up to look fancier. The language of thought."
"It can't be that simple" Brandon stared, "You are shitting me."
"Brandon," Sean sighed, "None of this is supposed to be hard. Isacc Newton invented calculus just to figure out some other problem he was trying to solve. Granted, most of us aren't Newton, but we don't need to be. The reason all this seems intimidating and boring is because the school curriculum hasn't been updated since the dark ages. In the days before computers the only way to do calculus was by complicated analytical methods on toy problems. With computers none of that shit is necessary, but the school still expects us to go through the motions. It sucks, but once you start thinking in terms of higher dimensional structures, it'll get easier, I promise. "
"Higher dimensional?" Brandon gaped.
"Some problems require N-dimensions to describe properly.," Sean nodded, "but there are tricks to visualize them as 3-D. And once you do, no problem in your textbook will be beyond your reach. Here, I'll show an example..."
As Sean sketched, Brandon acquired a far away look, "A thousand-fold thought."
"Huh?" Sean raised a brow.
"They say a picture is worth a thousand words," Brandon barked a laugh, "but, dude, your sketches are worth a thousand thoughts all connected to each another. It just clicks."
"Many-fold thought," Sean smiled, "I like it."
END OF CHAPTER