She types in 80. The line steeper still, but it's not totally vertical.

她输入80，线更斜了，但是还是没有变成垂线。

"Ohhh, it's infinity, isn't it? It's never going to get there."

“数值是无穷大是吗？我永远都划不出一条垂线是吧？”

Renee is close. But then she reverts to her original misconception.

雷妮的发现已接近事实的真相，但是她又绕回了她最初的“思维误区”。

“So what do I need? 100? Everytime you double the number, you get half way to the y axis. But it never gets there..."

“那我该怎么办呢？100吗？每一次我输入加倍的数字，它就会更接近，但是却总是达不到我想要的结果。”

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.

她输入100。“接近了，但依旧不大理想。”她开始更大胆的想象，很明显她就要得出事实的真相了。

"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..."

“好吧，我知道了，尽管……但是……我知道了，每加大一个数字，线就更接近Y轴一点。至于为什么会这样，我仍然感到困惑不解……”

She pauses, squinting at the screen. "I'm getting confused. It's a tenth way to the one. But I don't want it to be..."

之后她停顿了一下，紧紧盯着屏幕，“我有点糊涂了，这里只是十分之一，但我没有意识到 ……”

And then she sees it. "Oh! It's any number up and zero over. It's any number pided by zero!" Her face lights up.

然后她突然明白了，她的脸闪耀着光芒：

"A vertical line is anything pided by zero. And that's an undefined number. Ohhh, Okay, now I see.

“噢，任何一条垂线都被零整除，因此就会无穷大。嗯，好，现在我明白了，

The slope of a vertical line is undefined. Ahhhh.That means something now. I won't forget that!"

垂直的斜率是无穷大的，嗯，现在总算是明白了，我永远也不会忘记这堂课。”

Over the course of his career, Schoenfeld has videotaped countless students as they worked on math problems.

在肖恩菲尔德任教时，他曾用录影机记录了许多学生的数学课。

But the Renee tape is one of his favorites, because of how beautifully it illustrates what he considers to be the secret to learning mathematics.

但是雷妮的录影带是他最喜欢的一个，因为这个录影带贴切地诠释了学习数学的秘密。

Twenty-two minutes pass from the moment Renee begins playing with the computer to the moment she says, "Ahhh. That means something now."

从雷妮开始操作计算程序，直到第22分钟，她才说了声：“嗯，现在有些眉目了。”

That's a long time. "This is eighth grade mathematics," Schoenfeld said.

那是一段很长的时间，肖恩菲尔德说道：

If I put the average eighth grader in the same position as Renee, I'm guessing that after the first few attempts, they would have said,'I don't get it, I need you to explain it.'"

“这是道八年级的数学题，如果我把给雷妮布置的数学题目，同样布置给与雷妮程度相同的八年级学生，我想，在他们试试之后就会说，‘我做不出来，我需要你来讲解一下。’”