Sean and the Professor: 1. How They Met

Sean and the Professor

Chapter 1: How they met

Dusty from his winning slide, carrying his bat and glove, twelve-year-old Sean Smith made a selection from the cold drinks, and sat it down on the checkout counter.
"That's ninety-five cents, please."
Sean pulled a dollar coin from his pocket, received his nickel change, nodded to the clerk, and popped open his soda. He went over to check the sports magazines - the new issue wasn't out yet. Then he heard the clerk tell the next customer, "That's two-oh-five."
The customer sighed, then in a deep, rumbly voice, said, "Oh, I have no change with me..."
With the speed that had just got him the winning homer, Sean was at the check-out. "Here, mister," he said, handing the clerk his nickel.
"A-hem! Why, thanks, young man," came the deep voice. "Very kind."
Sean looked up at the man - tall, rather fat, unkempt gray hair, worn brown suit over a white shirt and dark purple tie - nodding at him with a beaming smile and twinkling eyes.

"Your team win?" he asked as he followed Sean out the door.
"Yeah; 4-3. I got the winning homer."
"Great; good work. Then why so glum?"
Sean took a swig of his soda. "Homework. Our math teacher gave us a bunch of addition problems."
"Hmmph, addition!" The man chuckled. "You're probably in sixth grade, right?"
Sean nodded as the two walked along.
"What kind of addition - fractions, decimals...?"
"All kinds."
The man nodded. "Really? All kinds? Well, of course not all... A-hem! Say, if you're going up the hill, you can stop in my office, and I'll give you your nickel. It's just ahead, in Alsace Hall."
"You a professor at Howell?"
"Itinerant, son, itinerant. For a little while, anyway. I go where I'm needed."

Sean followed the professor up a few steps. At a large wooden door, almost hidden between holly bushes, the old man pulled out a big old-fashioned key, opened the door, and led the way into the basement of Alsace Hall.

"They put me down here this year," the professor explained as Sean walked with him down the hall, passing offices with students working quietly at desks. "Some don't like the basement, but I don't mind at all. Lots of students to talk to, learn from... and it's much easier to get to by this back door - and as long as I have a desk and room for my books, I'm quite happy." With a different key, he opened the door to his office, and Sean followed him inside.

"Addition, eh? Have a seat," the professor gestured. "I'm going to make some tea; would you like some?"
"No, thanks; I've plenty of soda left," Sean said, staring around the office. Most of the walls were lined with rows and rows of shelves stacked thick with books; there were four-drawer filing cabinets with papers sticking out; a big wooden desk with more books and papers, spotlighted by a lamp on a complex little derrick clamped to its side; a computer on its own separate desk; and in a corner under a cupboard, a little table with jars, bottles, canisters and cups.
The professor soon had a steaming cup of tea, and put a plate of cookies on the desk by Sean.
"Have a cookie. Cheap at $2.05 a package. Oh, and here's your nickel."
"Thanks," Sean put it in his pocket and took a cookie. "What do you teach here?"
The professor took a sip of tea. "A little of this, a little of that; right now I'm, er, doing some research - on how addition works."
Sean wrinkled his face. "What?"
"Yes, " the professor smiled, "that's why I thought it might be interesting for you to come by. Of course, A-hem! if you have to get home..."
Sean checked his watch, fascinated by the office. There was a tall glass mirror-backed cabinet full of rocks, bottles and other strange things, and he wanted to ask what they were. "No, supper isn't till six today, and Mom knows I like to relax after a game. Sometimes I sit in the garden, up by the President's house..."
"Yes, very nice; I like that too."
"And besides I live just up the hill, in the Agnesi Apartments. Won't take me two minutes to walk home from here."
"Well, as long as you won't be late - because I have a tendency to talk on and on."
Sean took another cookie. "I'll keep an eye on the time."

The room was silent as the professor drank his tea. Sean kept glancing around at the books and glass cabinet. Almost without thinking he blurted out, "So many books... I have so many questions..."
"Yes," he nodded, "they have that effect on me, too. A-hem! But let us not get distracted right at the beginning of our journey." He finished his tea and put the cup on the corner table.

"Archimedes," the professor intoned.
"Yes master," came a voice. Sean's mouth dropped open.
"Where is my wand?"
"Your wand is in your desk drawer."
"Thanks."

Seeing the question on Sean's face, the professor smiled, shook his head, and murmured "I'll explain later." He opened his desk drawer and pulled out a clear red stick, just about a foot long. He waved it and the ceiling lights dimmed. "Ah, good," he nodded. "Let's start with the basics - integers."
"You mean whole numbers?"
"Even more basic - just the digits. Let's say," he flicked the wand, "Two. Plus. One. Equals." The numbers appeared in white, as if there was a ghostly blackboard hanging in front of the bookshelves.

2 + 1 =

"Three," answered Sean, bored. "First grade stuff."

2 + 1 = 3

"Are you sure?" asked the professor.
"Huh?"
"Why isn't it - twelve?" Another flick.

2 + 1 = 21

"That's twenty-one."
"That's because you're speaking English, and we read from left to right..."

1 + 2 = 12

"But that's not addition!"
"Sure it is. I started with a one, and added a two to it. One plus two is twelve."
"No, one followed by two is 12." Sean thought the professor was nuts. "If I have an apple and buy two more, I have three apples. Not twelve."

"Oh, I see," the professor said. "But I can't draw apples. I will just write little lines, though they will look like ones."

1 + 1 1 = 1 1 1

"That's right," Sean said, "but now it looks like one plus eleven is one hundred eleven."
"Yes, it does, doesn't it?" He waved the wand again and the lights went back to normal. "I told you I am studying addition. And all those things are just the beginning."
"So what do you mean by addition?" Sean asked.
"There are different meanings, and different - er - strengths - of addition. You've already seen two or three. And I will show you more." He waved the wand and the lights dimmed again.
"When you learn arithmetic, you first learn how to write the ten digits," and these appeared in a row,

+

0

1

2

3

4

5

6

7

8

9

"and then you learn numbers. Addition follows along. The first kind of addition in arithmetic is addition of the digits. That's so important you will have to have it as a basic skill, just like you learn how to stand - or you couldn't stand in right field." Sean nodded, all the while the professor had been waving the wand as if writing. Soon the following table was visible:

+	0	1	2	3	4	5	6	7	8	9
0	0	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9	10
2	2	3	4	5	6	7	8	9	10	11
3	3	4	5	6	7	8	9	10	11	12
4	4	5	6	7	8	9	10	11	12	13
5	5	6	7	8	9	10	11	12	13	14
6	6	7	8	9	10	11	12	13	14	15
7	7	8	9	10	11	12	13	14	15	16
8	8	9	10	11	12	13	14	15	16	17
9	9	10	11	12	13	14	15	16	17	18

"So you learn this little table, but not as a table, but by memory - so when I say five plus eight, you can answer thirteen, quick as quick." As the professor said the numbers, the row with five at the left lighted in green, and the column with eight at the top lighted in red, making the 13 where they crossed light up in yellow.

+	0	1	2	3	4	5	6	7	8	9
0	0	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9	10
2	2	3	4	5	6	7	8	9	10	11
3	3	4	5	6	7	8	9	10	11	12
4	4	5	6	7	8	9	10	11	12	13
5	5	6	7	8	9	10	11	12	13	14
6	6	7	8	9	10	11	12	13	14	15
7	7	8	9	10	11	12	13	14	15	16
8	8	9	10	11	12	13	14	15	16	17
9	9	10	11	12	13	14	15	16	17	18

"Wow, that's neat!" Sean said.
"Yes, isn't it? A friend of mine helped me with that."
"So - you need to know that, just like you learn your street address or telephone number - cold, by heart. No fingers!" he said, chuckling.
"OK, yeah, so I learned that long ago," Sean laughed. "How can there be other ways of adding?"
"I'll show you, but we're not yet finished with the first way, are we?"
"What do you mean?"
"That table is nice if all you wish to do is add digits." The table shifted over to the left of the room, leaving a large dark area. "But what if we have, say, 11 plus 11?"
11
+ 11
------
"Easy; twenty-two!"
"Yes... but how about 81 plus 91?" The numbers appeared, one over the other:
81
+ 91
------
"Uh... a hundred seventy two?"
"Right, but not as easy, right? How about this?"
478393
+ 48
--------
"Not as easy, no?"
"But professor, there's something wrong - we don't write it that way."
"You're right. I broke a rule - do you know what it is?"
"Uh - no. But the numbers have to be against the right side."
"Correct. After you learn to add digits, you learn the rules on adding numbers containing more than one digit. The first rule is to add like to like - units to units, tens to tens, and so on. Which means that the numbers must be aligned - each decimal point, even when it is not written, must be all in the same vertical column - so our problems must look like this:"

478393

+ 48

" And then?" the professor paused.
"We use the table..."
"Nine plus four..." As before, a row and a column lit up within the table. "Thirteen. Then three plus eight... eleven. So the answer would be..."

478393

+ 48

--------

1311

"One thousand three hundred eleven."
"No!" said Sean. "All wrong. The answer can't be between the numbers you're adding!"
"Very good. You have pointed out another rule. But I broke one as well, didn't I?"
"Yes. You have to start at the right."
"Correct, just as you always run counterclockwise, going to first base, then second..."
Sean smiled at the analogy. "But you also have to do the carry..."
"Also correct... that means we may have to add three digits at once! So that means..."

478393

+ 48

--------

141

"No, that's still not right. You have to bring the other numbers down," Sean stated.
"Why?" asked the professor. "Oh, you said the answer can't be smaller than either of the numbers being added. But I think there's a better explanation. We left out some zeros, because we don't really need them - but if I put them in, we'll be able to see the rule..."

478393

+ 000048

--------

"Oh! Sure," Sean said excitedly. "Now you just work from the right, column by column, each time using that table..."
"Ah, very good. You fielded that one nicely," smiled the professor. "So that gives us..."

478393

+ 000048

--------

478441

Sean nodded. "I know I could have done all that, but I didn't realize all the separate steps."
"Of course not," replied the professor. "You've just been doing it mechanically, because you were taught that way. Now you see, from just this one problem, you actually know two different kinds of addition already! The simple kind, where you add digits by knowing the table," and here he pointed to the glowing array of numbers hovering on the left side of the room. "You also know the series of steps - we professors call it an algorithm - to perform addition of two numbers of multiple digits."
"Wow. Sure, just like first I learned to catch and throw; later I learned field strategy."
"Yes, A-hem! So." He looked at his watch. "Well, it's getting close to six, so I'll just jot down the two ways we've looked at today:"

I. Addition of digits
a. can add any pair of numbers between 0 and 9
b. must use the "addition table":
c. pick row of first digit, column of second digit, and sum is at the intersection
d. have to learn the table "cold" (keep in memory)
e. Notice that the biggest result is 18!

II. Addition of two multiple digit integers
a. make sure numbers are aligned (units under units, tens under tens, etc) - push both to right side.
b. assume zeros on left if one number is shorter
c. working from the right, going column by column, add each pair of digits for that column. If that sum is more than 9, set down the units in that column, and write the carry above the next column to the left.
d. Notice that the sum can get bigger by no more than one digit, which has to be a one!
e. Also note that we can check that the answer is correct by seeing that it must be bigger than both of the numbers being added.

"Wow, that's a lot to know - but I learned all that in first and second grade," Sean said as he stood up.
"Yes, and that's just the beginning. There are ways of adding faster, and ways of adding different things... many other interesting things to talk about," the professor nodded, rising. As he put the wand down on his desk, the room's light returned to normal. "Maybe you'd like a copy?" He turned to the computer desk, picked up a piece of paper from the machine, and handed it to Sean.
"Thanks for a very interesting time, young man," the professor said with a smile. "Perhaps we can talk again another day."
"Sure, professor, thanks," Sean replied. "And it looks like some of my homework will be fun for once!"
Sean turned and went out of the office, down the hall, out the big wooden door and past the holly bushes out into the sun. He ran up the hill and reached his home just as the chimes of Old Main began to ring six o'clock.

(to be continued)