Discovering Assumptions 

Adapted  from 101 Words I Don't Use by Paul Niquette
Copyright 1996 Sophisticated:The Magazine. All rights reserved.

The instructor strode into the classroom empty handed. He nodded to the class and checked his watch. Seven P.M. Every chair was occupied, fifty in all. Many students wore business suits, ties pulled loose. A dozen stood against the back wall. Murmuring stopped.

Donning his spectacles, the instructor groped in his pocket for a scrap of paper. Without saying a word, he clicked his pen and wrote briefly. Students exchanged glances. The instructor folded the paper and returned it to his pocket.

Parenthesis: Customarily, at the beginning of the first lecture, a professor will take up chalk, write the name of the course on the blackboard, then turn to face the class. 

This is the time instructors use to deal with obligatory administrative matters: the title of the textbook, the date of the midterm, the policy on homework. All quite collegiate. 

Not this time.

The instructor, an oversized, stern-looking character in his thirties, grasped the lectern with both hands and gazed intently at his new charges.

"I have just written a number on a piece of paper," he said. "What is the number?"

The sound of scuffing feet.

"I'll give you a hint," the instructor spoke solemnly. "The number is between one and a thousand." A few in the back are heard to chuckle aloud. Then only the muted sounds of university life beyond the classroom windows.

"Do you understand the question?"

No answer.

"I have written a number between one and a thousand on a piece of paper," said the instructor. "What is the number?"

A bearded chap, probably a graduate student, hesitated. "One ninety-eight?" A companion laughed.

The instructor cupped his hand behind his ear. "Did you say 'one hundred ninety-eight'?" The laughing stopped.

Bearded Chap shrugged. "It's the number of the course."

"One moment please." The instructor pulled the slip of paper from his pocket and unfolded it. He shook his head. "Wrong."

"One?" asked a student wearing a gray suit after a long pause.

"Was that a guess, too?"

Gray Suit grinned. "No, that was a guess: One." A collective groan rolled over the class.

"How about 999?" someone asked.

The instructor squinted. "Who said that?"

A young man in checkered shirt raised his hand timidly.

"Why didn't you say 'two'?"

"Thought I'd save some time," chuckled Checkered Shirt.

"Sorry to disappoint you but the number is not 999. Perhaps you would like to explain how your guess saves time."

Checkered Shirt cleared his throat. Before he could speak, however, a student wearing tattered jeans blurted: "Is it less than 500?"

Someone on the other side of the room cried out, "Aha!" All eyes focussed eagerly on the instructor.


"Is it less than 750?" asked Tattered Jeans triumphantly.


A cheer rang out. Other students joined in: "Less than 600?" No. "Less than 700?" Yes. A handful of questions later and the answer was found: 666. Tattered Jeans stood and took a bow. The applause subsided.

The scene described here took place in the middle sixties. The course was called "Logic Design." By the tenth time I taught it, I hit upon that number exercise as a means to introduce 'heuristics.' That's the fancy word for 'discovery.' {Definition}

About that time, a subversive movement in education emerged, called 'discovery method.' Not new, really, it's what infants do on their own. What teachers do to children in ordinary classrooms is called 'prescriptive teaching.'

As a dabbler in self-reference, I found especially pleasing the idea of using discovery methods for teaching -- well, teaching discovery methods.

The instructor removed his jacket. "Before you leave this evening and every time I see you, I will assign a problem. During each classroom session, we will work together to analyze your solutions. You may find information in the textbook useful. Often, however, you will have to develop your own methods. Discovery is like that."

At about this point, an uneasiness arises among the doctoral candidates scattered about the room. They viewed this course as one of the Stations of the Cross. To graduate students, grades mean everything.

"However," continued the instructor, "you might just as well know, we will be using 'discovery methods' here for all aspects of the course. Shall I explain?"

Eager nodding identified the would-be PhD's.

"Getting the right answer assures you a grade of C," said the instructor. "Now, I am not unmindful that in graduate school, a C is tantamount to failure. So, for a grade of B, you must discover a method for solving all problems of the same type as that assigned." The instructor paused for effect. "And to get an A? -- well, that requires the discovery of a new problem."

Dozens of working engineers car-pooled to the campus from as far away as Vandenberg Air Force Base, 80 miles to the North, from San Diego near the Mexican Border, and from Palmdale in the High Desert. They came to learn discovery methods for their immediate work needs -- logic design of controllers, automatic machines, computers.

Mixed in were grumbling graduate students getting their tickets punched for advanced degrees, invariably right-answer habituated.

Every semester, a standing-room-only crowd showed up for the first lecture. Most did not stay the distance, though. By the third session, there were two chairs for each student. And I never graded more than twenty finals.

My favorite moment was about to begin.

The instructor pulled the slip of paper from his pocket and held it up. "Earlier you succeeded in solving our little number problem. What was the first thing that came to your mind?" The Class Jester revealed himself with a rendition of the opening theme from "The Twilight Zone."

Every class has its Humorless Expressive, the best straight-man one could hope for. "You didn't give us enough data," he will complain.

"Not enough data? That was your first thought? Prepare for a surprise. Many of the problems I will assign during the coming weeks do not have enough data. Better get used to that. Oh yes, and some will have too much data -- more than you need." The instructor managed an amiable smile. "Which is worse, not enough or too much?"

"Not enough," someone called out.

"What about contradictory data?" the instructor asked.

The class inhaled.

"That's another cruel fact: Some of your problems in this course will have contradictory data -- for which I make no apology."

"How will we know we have the right answer?" asked Humorless Expressive.

"You won't. But it's only worth a C anyway. Indeed, there may not be a right answer. Try to remember: In this course, the answer -- right or wrong -- is not the end of anything. It's the beginning of something else. Usually the hard part. Back to our little analysis: So, you did not have enough data -- yet you solved the problem anyway. How did you manage that?"

"Guessing," said one of the working engineers.

The instructor turned and wrote "Guessing" on the blackboard. "Contrary to what you have been taught all your life, guessing is not bad. How many times have you been reproached for guessing? -- at home, at school, in examinations -- 'subtract the number wrong from the number right' -- that's how to punish guessing.' Some of you, I'm afraid, are afraid to guess. Well, in this class, guessing is rewarded. But watch out. For the number problem, you had only one chance in a thousand, didn't you. Unless you are exceptionally lucky, guessing can waste a lot of time. In this case, how much?"

Checkered Shirt raised his hand. "On the average: 500 tries."

Bearded Chap spoke up brightly. "Mine was an educated guess: The course number."

"It sure was. On the other hand, you might have guessed that I had heard that one before. In previous classes, people have used the number of students in the room as their educated guess. Also, '77 Sunset Strip,' 'Route 66,' and '101 Dalmatians.' If those are educated guesses, welcome indeed to the 'The Twilight Zone'."

A full-lung laugh lingered in the room, the first such of the evening. The instructor nodded toward Checkered Shirt. "Your 500 guesses would take a long time."

"Wouldn't my method have been better?" asked Gray Suit. He grinned and pressed his thumb to his chest. "I'm the one who started with 'one'."

Checkered Shirt waved his hand. "No different than guessing."

Eyes turned toward the instructor. "There is a fancy name for his method, though. Anybody know what it is?" The instructor turned and printed the word "Determinism" on the blackboard. {Definition} "Some might call it 'brute force.' For many problems, however, it may be the only practical way to struggle through, from known to unknown. You don't look like a brute to me."

"My method is best," said Tattered Jeans.

Class Jester clapped his hands sarcastically. "Give that man his B."

The instructor became serious. "As an exercise, now, try to recall that part of the experiment. Re-run it in slow motion. Jot down the sequence of events exactly as you witnessed them." The instructor waited for the scribbling to stop. "What was the first thing you wrote down?" he asked, pointing at Gray Suit.

"That guy over there asked you, 'Is it less than 500?'"

"Sure enough. What happened next?"

"You said, 'No.' Then he asked if it was less than 750 -- "

"Wrong!" exclaimed the instructor. The class froze. Here and there, a nervous smile. "Excuse me, but that is not what happened next. Anyone!"


The instructor banged the lectern. "Aha!" he shouted. "Someone said that. Didn't any of you hear!" More silence. "It was you, was it not?" he gestured toward a blushing student. "Didn't you cry out 'Aha!'" Blushing Student nodded.

"That, my new friends, is the sound of discovery. It comes from deep inside somewhere. St. Thomas Aquinas nearly 700 years ago described the feeling as 'a sudden release in the tensions of inquiry.' How many of you felt it?" The instructor waited for an answer, which was slow to come. A few hands are raised, including Tattered Jeans'.

"You better raise your hand. You were the one who made the discovery. What is your name?"

"Aquinas. Call me Tom."

"Is everybody here a comedian or what! All right, Aquinas, what was your discovery?"

"Process of elimination," replied Thomas Aquinas, trying to erase a smirk.

"Successive approximation," Bearded Chap corrected.

"Ever hear of one of these?" the instructor asked while printing "Algorithm" on the blackboard. "You discovered an algorithm, Mr. Aquinas. As Casey Stengel would say, 'You could look it up.'" {Definition}

"Just more guessing," protested Humorless Expressive.

"The discovery was the algorithm -- the procedure," explained the instructor. "Hot dog! Something new was discovered. Getting the answer, 666, after that was -- why, it was downhill all the way."

It is environments which do the selecting of attributes, not vice versa, according to Charles Robert Darwin. The holes pick the pegs. Richard Hamming of Bell Labs once told me that education fits the Darwinian model: Schools select more than teach.

"If," asserted Hamming with a smile, "if you take the students who pass the entrance exam to, say, MIT and simply lock them up in a gymnasium for four years, occasionally tossing books in through the windows, they would graduate with the same qualifications as if you subjected them to the ministrations of the classroom."

Hardly encouraging to the conscientious educator. Moreover, conventional wisdom proclaims the unlikelihood of ever teaching some things: aesthetics, creativity, judgment, sophistication, wisdom -- hah! -- and discovery. To be sure, from the courses I taught, only a scant few students emerged with discovery skills, doubtless assured by congenital traits -- or, at least, attributes possessed long before showing up in my classroom. That first, bizarre lecture probably afforded the most selective value.

The instructor passed out a sign-up sheet and collected registration cards. He wrote the name of the textbook on the blackboard, then turned suddenly to face the class. "What has two wheels, seven letters, and starts with the letter B?" he asked.

Too obvious. There had to be a catch.

"Buffalo," said the instructor solemnly. "I lied about the wheels."

In the middle sixties, that joke repaid its telling with a reluctant laugh. Most students joined in -- except, of course, Humorless Expressive.

"Do you ever wonder about what makes things funny?" the instructor inquired. "Our course, by the way, is entitled 'Logic Design' not 'Fundamentals of Funny Things.' Just the same, can you think of one word that most explains what makes people laugh?"

Class Jester frowned in concentration. The best kidders have a gift for synthesis -- more than analysis -- of the absurd. The first suggestion came from Checkered Shirt. "Surprise."

The instructor wrote the word on the blackboard. "What does it take to be surprised?" he asked.

"A crooked joke," grumped Humorless Expressive.

"On the part of the listener, I mean," prompted the instructor. No answer. "How about..." He printed in large letters the word "ASSUMPTIONS" and tossed the chalk into the tray. "What did you, the listeners, assume?"

"That you were honest," someone said.

"Sorry to betray your trust about the wheels. Fact of the matter, every person in this room makes assumptions all the time. It's part of being human. We are automatic assumption machines. How far would you drive if you couldn't assume that on-coming cars are going to stay on the other side of the road? Strolling across an intersection on a green light requires an assumption, too, doesn't it. Same for when you buy something. Or marry somebody. Only we don't always think about our assumptions. Ever hear of 'hidden assumptions'? Our thought-lives are bathed in assumptions. Authors, playwrights, artists -- and yes, comedians -- exploit our assumptions, set us up and then surprise us. Sometimes we laugh, but not always."

The instructor pulled the folded paper from his pocket. "Every one of you assumed I was honest when I posed the number problem. Crazy, maybe, but honest." He handed the paper to Humorless Expressive. "Was I?"

The student nodded and passed the paper along the front row.

"Discovery is nourished by assumptions. If you don't have enough data, you what? -- assume something. When there is too much of the stuff or it's riddled with contradictions, what else can you do? Assume something."

"But!" bellowed the instructor while commencing an exaggerated scrawl across the top of the blackboard: "Make your assumptions explicit!"

Invariably the class would be too stunned to write. Damn, what fun! I relished the moment then and have savored the memory for decades.

The instructor lowered his voice. "Make your assumptions explicit. Now, as an exercise, please write down all the assumptions you made."

"In finding the number, you mean," confirmed Bearded Chap.

"You may assume that, yes," replied the instructor, winking at no one in particular. He strolled around the room patiently.

"Who wants to lead off?" The instructor handed the chalk to St. Thomas Aquinas in tattered jeans.

The student rose reluctantly, head bowed over his notebook. "That you wanted us to ask questions," he wrote on the board.

Checkered Shirt raised his hand and took the chalk. He inserted a carat and the words "Yes or No" in the statement.

"Like the game 'Twenty Questions'?" the instructor mused.

Checkered Shirt nodded.

"But you assumed I would answer them, too, did you not? Another hidden assumption. Did anyone write that down?" No answer. "I thought not." The instructor wrote "Instructor will answer" on the board.

"Incidentally, how many questions would it take, using the Aquinas Algorithm, to solve the problem? Anyone!"

Gray Suit raised his hand. "No more than ten. There are 1,024 combinations of 10 yes/no answers."

"So you didn't need 20 questions, you're telling us."

"With 20 questions," said Gray Suit, "we could have isolated any number between one and over a million!" He buffed his nails on his vest

"That so?" asked the instructor. "Tell the class what you mean by the word 'number'."

"Uh-oh." Gray Suit's expression darkened. He stepped to the blackboard and wrote "Integer!" An audible gasp came from the front row.

"What's your problem?" the instructor taunted. "How many of you assumed 'integer'? Whole number?" All hands rise slowly. "Obviously Saint Thomas Smarty Pants over there did. But did anybody write it down?"

Humorless Expressive was not amused. "The problem would have been impossible if you didn't mean whole numbers." There were good-natured "Yeah's!" from all over the room.

The instructor's eyebrows rose toward his hairline, his eyelids drooped. "Impossible?"

In all the years I performed the experiment perhaps only a half dozen students made explicit that assumption about whole numbers without prompting.

But there is a more critical assumption, a preconceived notion so deeply hidden in the basement of foregone conclusions, that nobody ever found it. Nobody.

The instructor chalked up a list of the course highlights and delivered an introductory lecture that ended with a problem assignment. He checked his watch. Nine P.M. "See you next week." He slapped the chalk dust from his hands. "Some of you." The instructor donned his jacket and strode toward the door.

"If you didn't mean 'whole number,' -- "

It was St. Thomas Aquinas. He was just about to ask the question which was most earnestly hoped for by the instructor.

"-- how would tonight's problem ever have been solved?"

"Assumptions! The key was right there in the assumptions." Not one student moved to leave the room. "You assumed I would only answer 'yes or no' questions. If, in fact, you had asked me for the first digit of the number, I would have answered '6'."

Long silence. "Same for the second digit?" inquired Checkered Shirt.

"Yes. And the third -- and any number of digits after the decimal point."

"Aha!" cried Blushing Student, grinning.

The instructor stood in the doorway, face iridescent. "For that matter, if anyone had asked me for the number itself, I would have told him."

The only sound was that of palms slapping foreheads.

"Do me a favor," said the instructor over his shoulder. "Don't tell anybody."

heuristic adjective. Helping to discover or learn; guiding or furthering investigation.

[From Greek heuriskein, to discover, find. Related to "eureka," which is used to express triumph upon finding or discovering something. From Greek heurika, "I have found (it)" (ostensibly exclaimed by Archimedes when, while bathing, he discovered the means of measurement of volume of an irregular solid by the displacement of water, and thus was able to determine the purity of a gold crown belonging to the tyrant of Syracuse).]

    As used here, heuristics (with an s) is applied as a term of distinction for a teaching method in which the student is allowed or encouraged to learn independently through his or her own investigation -- as opposed to prescriptive teaching methods. {Return}

determinism noun. The philosophical doctrine that every event, act, and decision is the inevitable consequence of antecedents, such as physical, psychological, or environmental conditions, that are independent of human will.

    As used here, determinism denotes a problem solving method that exploits an exhaustive procedure making the solution inevitable. {Return}

algorithm noun. A mechanical or recursive computational procedure. {Return}

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Background: "Discovering Assumptions" is a composite 
of vivid recollections from an engineering course on 
logic design taught by Paul Niquette at UCLA,
             1960-1970. {Return