評論
打印
She types in 80. The line steeper still, but it's not totally vertical.
她輸入80，線更斜了，但是還是沒有變成垂線。
"Ohhh, it's infinity, isn't it? It's never going to get there."
“數值是無窮大是嗎？我永遠都劃不出一條垂線是吧？”
Renee is close. But then she reverts to her original misconception.
雷妮的發現已接近事實的真相，但是她又繞回了她最初的“思維誤區”。
“So what do I need? 100? Everytime you double the number, you get half way to the y axis. But it never gets there..."
“那我該怎么辦呢？100嗎？每一次我輸入加倍的數字，它就會更接近，但是卻總是達不到我想要的結果。”
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.
她輸入100。“接近了，但依舊不大理想。”她開始更大膽的想象，很明顯她就要得出事實的真相了。
"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，盡管……但是……我知道了，每加大一個數字，線就更接近Y軸一點。至于為什么會這樣，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..."
之后她停頓了一下，緊緊盯著屏幕，“我有點糊涂了，這里只是十分之一，但我沒有意識到 ……”
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.
然后她突然明白了，她的臉閃耀著光芒：
"A vertical line is anything divided by zero. And that's an undefined number. Ohhh, Okay, now I see.
“噢，任何一條垂線都被零整除，因此就會無窮大。嗯，好，現在我明白了，
The slope of a vertical line is undefined. Ahhhh.That means something now. I won't forget that!"
垂直的斜率是無窮大的，嗯，現在總算是明白了，我永遠也不會忘記這堂課。”
Over the course of his career, Schoenfeld has videotaped countless students as they worked on math problems.
在肖恩菲爾德任教時，他曾用錄影機記錄了許多學生的數學課。
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.
但是雷妮的錄影帶是他最喜歡的一個，因為這個錄影帶貼切地詮釋了學習數學的秘密。
Twenty-two minutes pass from the moment Renee begins playing with the computer to the moment she says, "Ahhh. That means something now."
從雷妮開始操作計算程序，直到第22分鐘，她才說了聲：“嗯，現在有些眉目了。”
That's a long time. "This is eighth grade mathematics," Schoenfeld said.
那是一段很長的時間，肖恩菲爾德說道：
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.'"
“這是道八年級的數學題，如果我把給雷妮布置的數學題目，同樣布置給與雷妮程度相同的八年級學生，我想，在他們試試之后就會說，‘我做不出來，我需要你來講解一下。’”
