Outliers, The Story of Success

5.

A few years ago, Alan Schoenfeld, a math professor at Berkeley, made a videotape of a woman named Renee as she was trying to solve a math problem. Renee was in her mid-twenties, with long black hair and round silver glasses. In the tape, she's playing with a software program designed to teach algebra. On the screen are a y and an x axis. The program asks the user to punch in a set of coordinates and then draws the line from those coordinates on the screen. For example, when she typed in 5on the y axis and 5on the x axis, the computer did this:

At this point, I'm sure, some vague memory of your middle-school algebra is coming back to you. But rest assured, you don't need to remember any of it to understand the significance of Renee's example. In fact, as you listen to Renee talking in the next few paragraphs, focus not on what she's saying but rather on how she's talking and why she's talking the way she is.

The point of the computer program, which Schoenfeld created, was to teach students about how to calculate the slope of a line. Slope, as I'm sure you remember (or, more accurately, as I'll bet you don't remember; I certainly didn't), is rise over run. The slope of the line in our example is i, since the rise is 5 and the run is 5.

So there is Renee. She's sitting at the keyboard, and she's trying to figure out what numbers to enter in order to get the computer to draw a line that is absolutely verti cal, that is directly superimposed over the y axis. Now, those of you who remember your high school math will know that this is, in fact, impossible. A vertical line has an undefined slope. Its rise is infinite: any number on the y axis starting at zero and going on forever. It's run on the x axis, meanwhile, is zero. Infinity divided by zero is not a number.

But Renee doesn't realize that what she's trying to do can't be done. She is, rather, in the grip of what Schoenfeld calls a “glorious misconception,” and the reason Schoenfeld likes to show this particular tape is that it is a perfect demonstration of how this misconception came to be resolved.

Renee was a nurse. She wasn't someone who had been particularly interested in mathematics in the past. But she had somehow gotten hold of the software and was hooked.

“Now, what I want to do is make a straight line with this formula, parallel to the y axis,” she begins. Schoenfeld is sitting next to her. She looks over at him anxiously. “It's been five years since I did any of this.”

She starts to fiddle with the program, typing in different numbers.

“Now if I change the slope that way...minus i... now what I mean to do is make the line go straight.”

As she types in numbers, the line on the screen changes.

“Oops. That's not going to do it.” She looks puzzled. “What are you trying to do?” Schoenfeld asks. "What I'm trying to do is make a straight line par?

allel to the y axis. What do I need to do hereI think what I need to do is change this a little bit.“ She points at the place where the number for the y axis is. ”That was something I discovered. That when you go from i to 2,

there was a rather big change. But now if you get way up there you have to keep changing."

This is Renee's glorious misconception. She's noticed the higher she makes the y axis coordinate, the steeper the line gets. So she thinks the key to making a vertical line is just making the y axis coordinate large enough.

“I guess 12 or even 13 could do it. Maybe even as much as 15.”

She frowns. She and Schoenfeld go back and forth. She asks him questions. He prods her gently in the right direction. She keeps trying and trying, one approach after another.

At one point, she types in 20. The line gets a little bit steeper.

RICE PADDIES AND MATH TESTS She types in 40. The line gets steeper still. y “I see that there is a relationship there. But as to why, it doesn't seem to make sense to me What if I do 80?If 40 gets me halfway, then 80 should get me all the way to the y axis. So let's just see what happens.”

She types in 80. The line is steeper. But it's still not totally vertical.

“Ohhh. It's infinity, isn't itIt's never going to get there.” Renee is close. But then she reverts to her original misconception.

“So what do I need100Every time you double the number, you get halfway to the y axis. But it never gets there...”

She types in 100.

“It's closer. But not quite there yet.”

She starts to think out loud. It's obvious she's on the verge of figuring something out. “Well, I knew this, though... but... I knew that. For each one up, it goes that many over. I'm still somewhat confused as to why...”

She pauses, squinting at the screen.

“I'm getting confused. It's a tenth of the way to the one. But I don't want it to be...”

And then she sees it.

“Oh! It's any number up, and zero over. It's any number divided by zero!” Her face lights up. “A vertical line is anything divided by zero and that's an undefined number. Ohhh. Okay. Now I see. The slope of a vertical line is undefined. Ahhhh. That means something now. I won't forget that!”