I was a graduate student in math at Dartmouth College. I wound up teaching an introduction to linear algebra course that was also the first course where students were asked to do proofs. The class was somewhere in the range of 15-20 students. If I remember correctly, this was in the fall of 1996.
我是达特茅斯院数学系的一名研究生，有幸教线性代数入门课，这是第一个要求学生作证明的课程，班上有15至20名学生，如果我没记错，那时正值1996年秋季。
In preparation for the class I set myself goals around how well the students would learn the material taught. After some thought I settled on four ideas that I would use:
在备课时，我会根据学生们对我所讲内容的吸收程度来制定目标。经过深思熟虑，我决定使用以下四种方法：
1. Homework not present at the start of class would not be accepted. However students were only graded on the best 20 out of 27 possible homework sets.
2. All homework sets were c.lative. Generally 1/3 was the current day’s material, 1/3 from the last week, and 1/3 from anywhere in the course. Those thirds were in increasing order of difficulty.
3. Every class would start with a question and answer session to last no less than 10 minutes.
4. Every student could expect to be asked at least one question every other class.
1. 上课后还没交的作业就不再收了。另外，假设一学期总共有27次作业，只把其中最优秀的20次计入最终成绩。
2. 布置的所有作业均遵循积累原则。一般来说，作业分为三分之一的当天内容，三分之一的上周内容，剩下三分之一复习原先学过的任意内容。这三部分难度递增。
3. 每节课均以提问和答疑的形式开始，并且持续时间不得少于10分钟。
4. 每两节课每位学生均有至少被提问一次的机会。
These ideas may seem odd, but there was a method to my madness. Here is each idea explained.
这四种方法看上去可能挺奇怪，不过倒很适合我。下面将详细介绍每种方法。
1. Homework not present at the start of class would not be accepted. However students were only graded on the best 20 out of 27 possible homework sets.
1. 上课后还没交的作业就不再收了。另外，假设一学期总共有27次作业，只把其中最优秀的20次计入最终成绩。
The point was to make sure that class started on time, with everyone ready to pay attention for question and answer time. I also didn’t want to deal with people doing homework during lecture, evaluating sick excuses, etc. The leniency of not having to turn in 7 homework sets compensated for the rigidness of the policy. And c.lative homework sets meant that I didn’t have to worry about students not practicing any given day’s material.
这一点的关键在于确保准时上课，以保证每个人都能集中精力为提问和答疑做好准备。我不想处理上课做作业的问题，也不愿意评估病假理由，为此，我给予每个人7次不交作业的机会。由于布置的所有作业均遵循积累原则，这意味着我无需担心学生会因为不交作业而错过某个知识点的练习。
This worked even better than I hoped. The downside was that I had an argument on the second day when someone came in 2 minutes late and was not allowed to turn in his homework. But the first complaint was the last, and the students liked the freedom to decide when something else took precedence over doing homework.
它的效果比我预想的还要好。不好的是某天有人迟到2分钟，我不准他交作业，为此而发生争执。还好那是第一次也是唯一的一次争执，因为当其他事情和完成作业发生冲突时，学生们喜欢拥有决定哪个更为优先的自由。
2. All homework sets were c.lative. Generally 1/3 was the current day’s material, 1/3 from the last week, and 1/3 from anywhere in the course. Those thirds were in increasing order of difficulty.
2. 布置的所有作业均遵循积累原则。一般来说，作业分为三分之一的当天内容，三分之一的上周内容，剩下三分之一复习原先学过的任意内容。这三部分难度递增。
This was the most important idea I wanted to try. I had long been aware that research on memory had demonstrated that when you’re reminded of something as you’re forgetting it, it goes into much longer term memory. As a result periodic review at lengthening intervals is very effective in increasing long term recall. A typical effective study schedule being to review after half an hour, the next day, the next week, then the next month.
这是我愿尝试的最重要的方法。我很早就知道对大脑记忆的研究表明当你快要忘记某样东西时，经他人提醒后，你就能记得更久。因此，定期复习对长期记忆非常有效。典型的有效学习计划是半小时后、第二天、下一个星期、下一个月。
Now of course you can tell students this until you’re blue in the face – but they won’t do it. However when the study schedule is disguised as homework, they don’t have a choice.
当然，你可以直接跟学生这样讲，直到你口干舌燥，他们仍然不会这么做。不过，当你把学习计划暗含于作业中，他们就毫无选择了。
This really seemed to work. What I noticed on tests is that students were noticeably shaky on material they had learned in the previous week, occasionally didn’t remember stuff for a half-month before that, but absolutely nailed every concept that they’d first learned at least 3 weeks earlier. I credit the forced review schedule from c.lative homework sets for much of that.
这个方法确实很有效。我注意到，考试的时候学生们对上周的内容记得马马虎虎，偶尔会忘记半个月前的内容，但是绝对记得至少3周前学过的每一个概念。我觉得这主要归功于遵循积累原则的作业设计。
3. Every class would start with a question and answer session to last no less than 10 minutes.
3. 每节课均以提问和答疑的形式开始，并且持续时间不得少于10分钟。
For me this was the most important part of the class. The questions that came up in this session were my opportunity to refresh people on what they were forgetting, and were how I kept track of what topics should come in for more review on future homework sessions. Given my knowledge of how critical review is to learning, I honestly felt that time spent answering questions was more valuable than lecture. As long as there were questions, there was no maximum on how much time I was willing to spend on this.
对我而言，这是每节课中最重要的环节。这一环节的提问和答疑是我帮助学生回忆他们正要忘记的知识的机会，同时我还可以借此总结出哪些知识点将会以作业的形式出现在今后的复习当中。我了解复习对于学习来说有多么关键，我确实认为花在答疑上的时间比讲课更宝贵。只要学生有疑问，我可以将整堂课用来答疑。
Of course the challenge is getting students to ask questions. My strategy was simple: I told them that someone will ask questions and someone will answer them, but they don’t want me to be the one asking questions. On the second day nobody asked me any questions and I had to demonstrate. I picked a random person and asked her to explain a key point from the first day’s lecture. She couldn’t. I asked another student the same question. Again difficulty. I asked if everyone was sure that they had no questions. Someone asked me the question that I had been asking everyone else. I answered the question, answered the follow-up, and the point was made. I never again had to ask a question during question and answer period.
当然，挑战在于引导学生们提问。我的策略很简单：我告诉他们，有人提问就有人回答，但别指望我提问。转天，没人提问，我必须亲自示范。我随便点了名学生，请她解释第一堂课中的某个要点。她答不上来。我向另外一个学生提了同样的问题，同样答不上来。我问是否所有人都确定自己已无任何疑问。有人问了我我刚才问他们的那个问题。我作出回答，随后又回答了几次提问，这算是引导成功了。今后的问答环节，我再也不必提问了。呵呵。
4. Every student could expect to be asked at least one question every other class.
4. 每两节课每位学生均有至少被提问一次的机会。
My goal here was to be sure that every student was awake and following the lecture. It was never my goal to embarrass anyone or put them on the spot. To that end I developed a rhythm. Every few minutes I’d stop, say, “Let’s make that a question,” ask the question, pause so everyone could think through the answer, then ask a random person the question. I made sure to rotate people around so that everyone got their turn fairly.
这样做的目的是确保每位学生都能保持清醒，并且跟得上课程进展。让任何人难堪或丢脸，绝不是我的本意。为此，我制定了一个节奏。每过几分钟，我会停下来，说“让我们就此提问”。提问，暂停，这样每个人都可以思索答案，然后随便点一名学生回答问题。我确保轮流点名，保证每位学生都得到公平的机会。
The questions I’d ask were always straightforward. They were things like, “What is the result of this calculation?” Or, “Why is this step OK?”
我问的问题都很直截了当，比如“这个计算结果是什么？”或“为什么这个步骤没问题？”
I treated failure to get the answer as my failures, not theirs. If they couldn’t get the answers then they weren’t following the lecture, and I needed to slow it down, figure out the rough spots, etc. It might seem that the constant interruptions were slow. But I found that having everyone pay attention more than made up for it. The class as a whole moved as fast as any other class – but with far greater comprehension. And the interactivity made the class become very open about asking questions.
假如学生回答不上来，我会视其为自己的失误，与他们无关。如果他们答不上问题，那么他们就跟不上课程，我需要放慢节奏，弄明白难点在哪，等等。这样不停的中断看上去可能很低效，但是我发现，让每个人都能集中精力补偿了这一不足。班级整体的课程进度和其他班级一样快，但比较而言理解得更为深刻。互动使得班级的提问氛围活跃起来。
As a bonus I managed to convince the entire class that taking notes was not worthwhile. I learned this lesson about math in first year undergrad. What you do is read ahead in the textbook. If you really want a set of notes, you can make them from the textbook before class. Then show up at class having read the day’s material and ready to pay attention. Then if anything that the professor says doesn’t make sense to you when you’re paying attention and have already read the day’s lesson, then ask the question then and there. If you don’t understand it, then probably nobody else does either. Add to that periodic reviews, and you’ll have a huge edge in any math courses.
此外，我总算说服全班同学上课时不要记笔记，这个经验是我在大一数学课上获得的，相反你应该做的是课前预习。假如你真的需要一套笔记，可以在预习时做。这样，上课时你已经对我要讲的内容有所了解，便可以全神贯注地听讲。如果上课时你专心听了，也提前预习了，而老师所讲的内容你还是不明白，那么就要立即提问。如果你不懂，说明其他人也可能不懂。如此这般，再加上定期复习，你的数学成绩（包括高数和线性代数）都将会非常出色。
Nobody ever believes that that works. But this class had no choice because there is simply no way to take notes and pay attention at the same time. Which meant that the note takers couldn’t answer questions. But within a few days they learned to not take notes, and I believe did much better for it.
没人相信我这法子能行得通。可是我们没有其他选择，因为根本不可能又记笔记又专心听课，记笔记时无法回答问题。还好，过了几天，他们学会不记笔记，我自信这样更好。
So how well did this package work? As far as my goals were concerned, much better than I had dreamed possible. What really brought this home was the final exam. Based on class performance I drew up a test that I though was a fair test of what I thought they understood. I showed it to some fellow graduate students. They thought I was crazy. They thought the class would bomb, and were willing to bet me on whether anyone would get the bonus question.
这几种方法效果如何？就我制定的目标而言，比我预期的要好很多。真正说明问题的是期末考试。根据课堂表现，我出了份难度适中的试卷。我把它拿给我的几个研究生朋友看，他们觉得我疯了。他们觉得班上同学会傻掉，而且还愿意跟我打赌看看到底有没有人能做出加分题。
The class aced the test. That bonus question? 70% of the class got it. I don’t remember what the bonus question was, but I do remember another one that I thought was cute. It went like this. Let V be the vector space of all polynomials of degree at most 2. a) Prove that d/dx is a linear operator on V. b) You can put a coordinate system on V by mapping p(x) to (p(0), p(1), p(2)). (Please imagine that flipped 90 degrees so it is a column.) Find the matrix that represents d/dx in this coordinate system. My fellow grad students got me worried that this might be too advanced for an introductory linear algebra courses. But I needn’t have worried – the only significant errors were minor arithmetic mistakes in the calculation. And I think I dinged someone for not having enough detail in the proof.
结果，全班同学均考试通过。加分题？班上70%的人都做出来了。我忘记加分题是什么了，不过我记得有道题挺有趣，题设大致如下：假设V是所有多项式的向量空间，最大2°。 1)证明d/dx是V上的线性算子；2)你可以将p(x)映射到(p(0), p(1), p(2))，从而给V加上坐标系（想象将其翻转90°，形成柱体），找出在此坐标系中代表d/dx的矩阵。我的研究生同学把我也弄得担心起来，害怕这对线性代数入门课来说过于难了。其实，我无需担心——唯一的重大错误不过是计算过程中小小的算术错误。我记得我还因为某人证明不够详细而扣了他的分。
Furthermore I was lucky enough to talk to some of my students about the experience a few months later. The general consensus was that the material really stuck. Furthermore nobody studied for the final. No joke. As one girl said, “I tried studying because I thought I should, but I gave up after a half-hour because I already knew it all.” That is how I think it should be – if you study properly through the course, then you won’t need to study for the final. Because you’ve already learned it. And you’ll have a leg up on the next course because you still remember the material that everyone else has forgotten.
几个月后，我有幸和部分学生谈论了这段经历。大家一致认为课程内容都记得很牢，毫不夸张的讲，期末考试没人再学习了。有位女生说，“我试过再最后复习一遍，因为觉得理应如此，可半小时之后我就放弃了，因为所有知识点已经了然于心。”我觉得就应该这样——如果上课时你努力学习了，考试前就无需再学，因为你已经掌握了。除此之外，你在下一阶段的学习中也会有优势，因为其他人把学过的都忘了，你却记忆犹新。
So were there any downsides? Unfortunately there were some big ones. I had set goals around learning. I failed to set any around happiness. Having to pay attention during class was hard on the class. Also it motivated them to work hard. Since everyone worked hard and they thought that I was going to grade them on a curve, there was a lot frustration that they wouldn’t properly be recognized for their work. (In fact I gave half of them A’s in the end.) This frustration showed up the teacher evaluations at the end of the course.
我的方法有没有不足之处？有，且很致命。我总是围绕学习效果制定目标，从未考虑过学生是否快乐。要求课上时刻集中注意力，这对学生来说不易做到。这也激励他们努力学习，每个人确实学得很认真，他们认为我会按照分布曲线来评定成绩，担心自己的最终成绩与实际付出得不到正确匹配，让他们很有挫败感（其实，最后有一半学生我都给了优秀）。这种挫败感在学期末的教师评价中有所表现。唉~
Therefore if I had to do it over I’d ask somewhat fewer questions, hand out a lot more compliments, make it clear that I would not grade on a curve, and if they performed anything like that first class, I’d be even more liberal with good grades. Of course the point is moot since I’ve found myself profitably displaced from math to software development. But if anyone decides to replicate my experience, I’d recommend paying more attention than I did to those issues.
因此，如果我能重新来过，我会略微减少提问，给予更多鼓励，事先讲清楚我不会按照分布曲线来评定成绩，如果有人像第一节课那样表现，我对“优秀”的评价标准将会更为宽松。当然，这也是事后诸葛亮了，因为我有幸从数学专业调到软件开发专业。不过，若有人想复制我的经验，我建议对我遇到的问题和疏漏多加以关注。
