Wise Words of Mr Spooner

Part 2

The Importance of Math

"...which is kinda' funny, because computers came from math. And speaking of math, computers helped win the Second World War. The helped, as you should know, break the secret codes of our enemies. We broke the Japanese code. We broke the German code. We therefore won the battle of Midway. Now, I'm not sure of my facts here, but I know we won the battle. There was a lot of violent destruction. There were a lot of people dying. But we won anyway. That stopped the Japanese from taking Australia -- they were set to invade; could've taken the whole northern half of it. And they were shelling islands all over the Pacific. So the Second World War wasn't as cut and dried as you might think, and you should pay attention to calculus."

The Factor Theorem

"What goes up must come down. It's our old friend the parabola. How about this: it goes up and down and up again! It's Canada's Wonderland! And the roller coaster's goin' and people are screaming, so learn the factor theorem."

"Now you think about this: In Mexico, workers make $4.00 an hour assembling cars. In Windsor, they make $40.00 an hour! Now is that fair? Imagine if workers in Kingston only made $4.00 an hour. You'd have to be insane not to move to Oshawa. And in seconds, I can get information to or from any place on the planet with a modem or fax machine. So there is a real problem. It's in process and the economy. Now you think about that. It all comes back to numbers. Now, back to the factor theorem..."

Number of Variables

"Now let's talk about the stock market. Usually the curve is steady, but lately it's been going up, up, up, and those greedy little beggars at Queen's who're gonna take their pensions are rubbing their grubby little fingers together. Congratulate any of your parent who have the Queen's pension plan: it's a really good one. But really suddenly, last week I think, the stock market plunged down from its nice even curve like this: [draws a gradual curve upward, then slashes it down] So we can't do equations for really jiggly curves like the stock market, 'cause there are just too many variables. Actually, the stock market never looks nice like this. It's more like [begins slashing a very disjointed squiggle all over one chalkboard]. Get it? Too many variables!"


"We have a highway here, and a curve. It's January 15th. Mr. Spooner, in his car, has just hit a spot of black ice on a curve on highway 15. Now Mr. Spooner goes out of control and flies off the road. In physics, we'd be interested in finding his velocity-speed and angle -- as he zooms off the road into a tree... this reminds me: When I was a boy, I had a paper route which had this really good looking girl on it, and I always wanted to get to know her, but I never did, I just saw her occasionally while doing my paper route. Now [laughs, while shaking and twitching his head] I found out later that she thought I was a good-looking paper boy! Now, [still laughing] it's kind of tragic, but we both grew up and she married a guy at RMC. And they were really happy together. One day [giggles a bit] they were driving down this stretch of highway 15 during the winter, and they flew off the road into a tree and DIED. [Dead silence from the class for about 5 seconds while they absorb this. Then the silence is interrupted by a few nervous chuckles as everybody checks their watch.] Eventually they rebuilt that section of highway 15, making the curve a little less sharp, and moving it away from that nasty tree, but the important thing is: whenever a car, let's go back to the example of Mr. Spooner in January, flies off the road like that, it leaves the road on a tangent to the curve."

Cost Lines

"Now, which cost line would you rather have? The lower one! Now how do you cut costs in a stagnant economy? You cut people. So what happens to the Comp. Sci. graduate who ends up driving a taxi for 6 years? Would you hire him? And that's why this graph is so important."


"The United Nations are running into trouble in Somalia. Why? Because it's more profitable to elements in Somalia to resist the U.N. The U.S. is reluctant to send in more troops and I don't blame them. The U.N. should never have gone into Bosnia in the first place. The Germans couldn't do it with five divisions led by Marshal Zukov. Some of the problems with the Serbs in Bosnia go back to the Second World War. So these problems are a lot more complex than they look. What is the fundamental problem? What is the domain? The U.S. had a solution. They sent in helicopter gunships to shoot up the Somalis. They fired rockets into crowds and blew up buildings and sent bodies flying, and all sorts of people were dying. The gunships were killing women and children along with the bad guys, buy I guess they think that's okay. They haven't put a limit on their actions. So you see, some limits can be different depending on how you look at it, and, as your mother always said, you can't divide by zero."

Cancelling the Denominator

"This problem is like teaching a class with one bad kid. If you pick the kid up and get rid of him, everything becomes a lot easier. But you have to be sneaky about it or you go to jail. I mean, we want to get rid of that h=0 in the denominator, but we can't just erase it from the equation and continue on; that would be like using one of those helicopter gunships I'm so fond of on that bad kid. It would be bad."

The Importance of Stating Limits

"Occasionally, you know, I get a little philosophical. And people say: `Look, Mr. Spooner went off the deep end.' But if you forget to put the lim x-> 2 in on this step, you're telling a lie! And we don't want to do that. Remember: in math you never really know what you're doing. You see, the lim x-> 2 shows that the two expressions aren't equal. Mathematicians sometimes come to blows over this sort of thing. Sometimes even I want to say: `What the blank blank blank is the difference?' But most evidently there IS a difference. Look! [picks up two pencils and starts fencing with himself and bashing them together] Here's Mr. Spooner and some other math guy fighting over limits! Do you want this to happen to you? Then write in the limits. Remember: if you're gonna get elected, your heart is pure, but you're telling a lie!"

Rate of Growth

"Those of you who've grown up in Kingston may have noticed some changes around here. I'm not terribly pleased with the changes. I'd like to see Kingston stay under one hundred thousand people, I don't want to see it turn into another Toronto. I mean, who wants to live in Toronto?"
Student: "I do."
"Well, you can have it. Kingston will be a better place without you, Jeremy."
[much laughter]
"I take that back. Now, back to the secant of this curve, take a look at Mexico City. Let's find the average rate of growth."

Shortcuts in Taking Derivatives

"If you find a shortcut now, you're slitting your own throat... Hey! Did you see that thing on TV last night with the girl who was psycho? She slit a mother's throat and then delivered her baby by Cesarian section and stole it. Anyways..."

Implicit Differentiation

"Now this is important. How many people in here have taken chemistry? Come on, don't be shy." < six or seven students reluctantly raise their hands> "Then you should know this." < writes the formula P1V1/T1=P2V2/T2 on the board> "Now this relationship is important. What happens to a hot-air balloon if it goes way up to where the outside air pressure is much smaller? It expands. And this has important applications in diving. Divers can get what's called the bends if they change depth too quickly. This reminds me -- you know, when I was young like you people, I hadn't really considered being a mathematician. I wanted to be a chemist, and my father thought I'd make a good one. But when I realized how much chemistry depended on mathematics, I switched fields. So diving can be very dangerous if something goes wrong. I mean, especially in terms of pressure. When you're way down under the water, say at 150 feet under, there's a tremendous amount of pressure on your lungs. I mean, imagine if you had, oh, a big tower of water about this big around," < holds hands about six inches apart> "no, let's say it was about one metre by one metre, and 150 feet high. Now imagine that huge amount of water resting on your chest if you were lying down. Could you breathe? I don't think so. And divers, out there in Lake Ontario, even, are constantly facing this dilemma. How could an ordinary human survive at this depth? He couldn't. He'd be crushed. So how do they solve this problem? Well, they do it by pumping air into people's lungs, so that the pressure inside their bodies is the same as outside, and they can swim about underwater all they want. Just like there was that woman who was diving out near Amherst Island a while ago. Now she was swimming around checking out a shipwreck or something. And she had some problems, I'm not exactly sure what. And she died. If you're 150 feet below the surface, and your lungs are all pumped up, and you suddenly rise to say, ten feet below, do you think you'll be okay? No! You'll EXPLODE! It's like, have you ever been fishing, and reeled in a deep fish too fast, then decided to throw it back because it was too small? Well, what do you think happens to the poor fish? It hasn't got a chance. The air bladder that kept it afloat has burst, so that fish is dead. Anyway, let's get back to implicit differentiation. But you know don't just dismiss everything I just told you. I mean, if you got a question about all the stuff I was talking about on an IQ test, you'd be able to answer it. And then your IQ would be higher, and one day you'd come back and say: `Thank you, Mr. Spooner.'"

Rene Descartes

"Reenee Day-cart -- an interesting man. He was a philosopher, scientist, mathematician, but what most people don't know is that he was deeply religious. And he wrote this long book on religion that was absolute crap. I mean, except for twenty pages on math that he added on the end, the whole book was just crap. And he's well known today just because of those twenty pages. But other than that, he wrote a lot of crap."

Slant Asymptotes

Student: "Are we going to be covering slanted asymptotes?"
Spooner: "Yes, I think we'll be going over that tomorrow."
Student: "Geez, it looks pretty hard."
Spooner: "Oh, no, it's not hard. It's like... say you were polling people to see who wanted to vote for Kim Campbell and who wanted to vote for Jean Chretien. It wouldn't be very clear at first, but once lots of results started coming in, it would mean something. You know how a line might look like a curve at first, but then turns into an almost straight line? Sorta' like with drugs. You know how they say that some drugs have an 98.7% safety level? Aspirin, for example. There's a lot of things we don't know about Aspirin." < notices that student has been laughing wildly for some time now> "Anyhow, it's a curve that appears to be a curve, then appears to be a straight line."

The Number Zero

"Unfortunately, the good Lord didn't give us brains that could deal with zero very well. Now, the human brain is still a very interesting piece of work. It's got billions of brain cells, all working together. Actually, the brain is just a very primitive brain with a whole bunch of really special things added on. But, on its basic level, the brain is identical to that of a worm, turtle, salamander, and duck. Just bigger. And, unlike these other animals, humans destroy their brains: drown them in alcohol. If you dring a whole twenty-six ounce bottle of Seagram's VO, you'll die. Now this is why you bring it up. It's a defence mechanism. And that's why the sidewalks are always so messy over in the Queen's ghetto. Because people are always bringing up the alcohol. Now the really macho guys, they keep it down. Then they collapse and have to go to the hospital to get their stomach pumped, or they die. So the human brain is actually a great fragile gift that we should be careful with. And it can't handle zero over zero."

Simplifying the Chain Rule

"I'm trying to think what I could relate this to... It's like Cher. She had a little work done on her body, and figure's she's a little easier to look at now. It adds a few years to her career. If it's worth a million, go for it!"

Big Business

"If I can sell my orange juice in carboard cans for ten cents less than your orange juice in tin cans, I won't care if sometimes it leaks through and destroys my cheap can. I'll be driving a Cadillac, and you'll be driving a Dodge Acclaim."

Inflection Points

"It's as easy as bread and jam. Or we could put ketchup on it. I dunno, is ketchup a jam? Hee, hee, probably not. But you know, ketchup is best on fries anyway, with salt. Of course, the fries consist mostly of fat. You know, when you eat fast food, you're getting a lot of sugar too, which, when combined with all the fat, can cause heart trouble. Yeah, that burger and fries is almost entirely fat and sugar."
Student: "And salt."
"And salt! Yeah, I mean, you might as well just eat styrofoam with butter, sugar, and salt on it, maybe drink some oil. I mean, this stuff is terrible for you! But mosst importantly, it tastes great! Yessiree, I love a burger and fries for lunch. Anyway, back to the fish finder and concavity problem..."


"We should try not to use negative numbers with this. For example, where were you before year zero? Before you were born? Was there a you? Did you exist? Do you exist? Think about that. It's a good question."

"I don't know if any of you have ridden elevators around here, but they're very well manufactured. You get on one, say at the KGH over here, and, except right as it starts and slows down, you don't notice the velocity! If you're engaged in conversation, it might stop and you'd still be talking, thinking it was still going. It's also like, if you're in your friend's '93 Covette, and he pulls out of the light really quickly, you can really feel it! Some of those new cars can really move! Of course, it can be pretty dangerous, Heh-Heh!"

"Boxing can be a very brutal sport, but it's also a very honest sport. If you're playing chess or tennis, you have to put on this thin show of politeness even though you might want to shove that ball down that guy's throat! Boxing is very honest. Velocity can be like lots and lots of sports, from football to windsurfing..."


"This is where computers come in. The just LOVE gobbling up numbers, like x=1.1, x=1.2, x=1.3! They just gobble them up. And they LOVE it!"

The Calculus Textbook

"This is a good book. My only objection is that it weighs too much. My left arm is stronger than my right arm!"

Accuracy of Decimal Answer

"Hee hee... what a great answer! Isn't that a joyful answer? [begins to sing in a happy voice] Joyful, Joyful... Joyful, Joyful! Heh, Christmas time is coming up, and here's the proof! Boy, this'd make a great present, eh? All Praise the mathematical Gods! You know, if a heart surgeon saw an answer like that, he wouldn't let it faze him, if he made a little mistake he'd just grab that artery, strech it a bit, and sew it on!"

Norman Windows

"There are some really beautiful cathedrals in France, worth visiting if you're there. But you know, about half of them fell down, killed a lot of people, because they didn't use Norman windows like in this problem! Of course, they didn't have calculus to maximize their calculations, so a lot of them fell down anyway."

Curve Sketching Problems

"Number 7. An excellent problem. As we say on TV, `two thumbs up!' So, who wants to be the reviewer and do the question on the board?"


"There's a beautiful sacred mountain in Japan. Does anybody know what it's called? No? Fujiyama. It's a volcano and they do a pilgrimage to the top. Well, if you've been unsuccessful in love, or something, you can throw yourself into the lava that's bubbling and burbling down below and smuck! Well, that sounds like quite a horrible way to go, but some physicist or mathematician figured that what with all the excitement and so on, you'll probably unconcious, therefore it's actually quite a pointless way to commit suicide."

Mr. Spooner's Advice

"Substituting in the number 2395 is like beating yourself on the head with a big stick."

"The tenth commandment, of course, is `Thou shalt not wash with mixed colours.'"

"I have an interesting book. It's sketches by Leonardo da Vinci of people's hands. A young person's hand, a baby's hand, an old person's hand. You wonder why he was so good? Behind the genius was hard work and practice."

"Somepeople think that math is like the Ten Commandments handed down to Moses. Here it is! Follow it! But that's not the way it is. You have to find your own path."

"Think about how you used to count. Using Roman numerals. Think about how on Earth you'd do [writes XIIIIV¸IX on chalkboard]. See? It doesn't mean anything. It's crazy. So we're going to do things the easy way."

"Infinity over infinity. We cannot handle that. Even writing it down is blasphemy."

"Yes sir... some people didn't do so well on this question. It was a pretty easy question, if, instead of the quotient rule, you used a common denominator. That's why I wrote `No! No! No!' on some people's papers. I mean, people wonder why I got up, went to the fridge, and downed an entire beer in one long pull... marking this question was that painful."

"Now, if you want to solve this problem, you're going to have to have a connection. Just like if you want to be a drug dealer, you're gonna need a connection... Pays better than math teaching."

Say huh?

"Say you have a boy and a girl, and they're going' out, and he's got so many carbon atoms, so you count them, and she's got so many carbon atoms -- you can calculate the chemical force of attraction. But it doesn't mean anything."

"This is a very good Calculus textbook... written by a guy at McMaster... a very good college, I know they have an excellent physics course..."

"So why are we going through the pain and agony of these mathematics? Well, I know most of you all just want your credit, but let's forget about that. Why are we torturing ourselves? To find the slope! [grins, laughs, and claps his hands together] The slope! It's all for the slope!"

"Must have some pretty vicious doctors -- they knocked off an extra 517 boys!"

"If you've got a friend who's gone off their rocker, just say `Joe, you're crazy! You're irrational!' So these square roots, like root three or root one point five, are irrational."

"Really what it is, is if you're playing a chord on piano or guitar, there are many ways you can do things. D7 can be played in many different ways, and if you're a really brilliant guitar plater, you can play all of them, while if you're an ordinary guitar player, you can only play a few of them. Having small fingers makes things even harder. So that is what we're doing in math."

"Is that a smooth, continuous curve? The Gillette shave type?"

"An interesting thing happened with the Nazis. They'd take prisoners of war to concentration camps and try dropping them in ice cold water with sensors on their body to monitor how long it would take them to die, so that if something happened to a destroyer, they'd know how long a person could last before they needed to be picked out of the water. Of course, in the Second World War, they never picked anyone up, so if someone fell off a convoy, the convoy would keep on going and leave the person in the water to die... I don't know how I keep on coming up with these crazy examples."

"For the two hundred and thirty-three lives I've lived on this life, will I be able to meet the minimum and maximum on this one?"

"A duck flying in a straight line overhead might actually be flying on a curve. Or a hawk."

The Problem With Teaching Math

"I'd ask people who are taking Calculus to look at the newspapers, look at anything, and see how it relates to Calculus. It helps make the mathematics more real. It's much harder if you isolate the mathematics and do it in a vacuum. That, I think, is the big problem with math teaching in North America."

Brett Allen (allen@cs.washington.edu)