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Following is the text of a talk I gave as part of the 2004 Project NExT Panel on Using Class Time Well.

I'm supposed to explain what I do in the classroom, and why. So that I don't forget what I want to say and how I want to say it, I'm going to read to you, as though I've written you a letter. And, the quickest way to say what I do is that I don't lecture (in complete contrast to what I'm doing now). The first question I'm usually asked when I say this is, Why don't you lecture? I think that's a negative approach though, and frames my teaching as though there's something wrong with this deviation from the norm. So here's what I'm going to tell you: First, I'll answer that question briefly. Then, I'll take a more positive approach, and explain the questioning and philosophy that led me to use my classroom time as I do. Aparna, in her email, flattered me by naming my approach as "student-centered learning." I'd love for that to be true. But it, like any other teaching goal I have, is probably unattainable in reality.

Negative approach: Why don't you lecture?
Well, what good does lecturing do? If all we want is to convey information to our students, we might as well just read a text to them, and in that case, they might as well just read the text themselves. If we don't like any of the texts, we can just write one and then we're done. Right? Lecturing presupposes that students can pay attention for an hour, and that we are the primary source of material (so what's the point of having a text then?), and that they understand, absorb or at least write down everything we say. Unfortunately, only one of those things is true. Students do view us as the primary source of material, but that's really because they're not accustomed to viewing texts as the primary source of material. If our goal is to help students learn, and to help students learn how to learn, then it seems to me that lecturing doesn't help much. It's rare to be able to pay attention to anything for an hour, and by lecturing we have no idea of how much material they are absorbing, understanding, or even writing down.

Positive approach: How did you develop your classroom format?
My method (if you can call it that) was born out of thinking deeply about very basic questions: are my students really learning? what are they learning? how can I know whether or not they actually understand? I began by wondering first about assessment, actually -- what exactly is it that a timed in-class exam measures?, for example, and how do I design an assessment tool which really shows what students have learned? -- but then this led me to ask the much more basic questions of how I could help my students learn material, and how I could help them to learn how to learn.

All of us in this room are very skilled learners. We can attend an hour lecture and come away with new knowledge and/or understanding. We can pick up a mathematics research journal and read any paper. You think I'm kidding, don't you? If you're a functional analyst, you can't just read an algebraic geometry paper directly! But you do know how to figure out which words are specific to the paper and which are presupposed knowledge of algebraic geometry, and you know how to follow the references backwards and how to locate appropriate-level graduate texts and, at worst, who to ask if you get really stuck. It might take a year or two, but really, you can read any mathematics research paper.

Our students don't come to us this way, or at least most of them don't. Most of us picked up all sorts of skills subconsciously, through observation of others, imitation, experimentation, trial and error, and lots of practice. We trained our minds to not be afraid of the difficulty of learning. Our best students don't need our help. In fact, we didn't need the help of many of our professors! Sure, our best students can benefit from our help, and we benefitted from the help of our professors -- don't get me wrong, gifted education is extremely important and it's what I spend my summers doing -- but that's not the majority of our students. And even the most gifted students have their flaws.

What I'm leading up to is that on the whole, the content we "teach" our students is not as important as "teaching" them about learning. They can pick up the content later, and need to figure out how to learn now. But we also organize most of our curriculum around particular topics, and we have some courses which are prerequisites for others, so we do actually need to pay attention to what material our students learn in our individual courses.

Note! Pay attention! I did not say that we need to pay attention to what we cover! In my completely unhumble opinion, we should never think about covering material. We should be thinking about students learning material. See, just because you say or write a mathematical statement doesn't mean that your students understand it. You could be flawless and they might not be listening. They might be tired, or sick, or distracted by Spring. And really, class is for them, not for us.

Also, notice that I put quotation marks around the word "teach." Just because we teach, that doesn't mean students learn, yet we often equate the two. We say, "well, we did that in class," or "I told them three times," but that's not what's important. That's why I use the quotation marks, to remind myself and perhaps make clear to you that our actions do not directly cause results in students. They do their own things, in their own ways, and we can try to guide them, but that's about it.

This is all a huge preamble to explaining what it is that I do in my classroom. The reason I went through all that is to help you in designing your own classroom environment: what I do will certainly not work for you in an identical fashion. But by thinking about the things I've said, and forming your own answers to questions like What are my students learning? How are they learning it? How can I tell what they've learned? you can develop methodologies that work for you.

So what is it that I spend most of my class time doing, then? I listen to my students. I watch my students. I respond to my students. If I'm at the board, I'm writing what they tell me to write, or setting up a question to ask, or drawing something to answer one of their questions. If students are at the board, I'm listening to what they say and watching what they write, checking for accuracy and completeness, intervening if they go too far astray or correcting their grammar. If students are working together, I'm circulating among them, looking at what they write, listening to what they say, and asking them questions. What don't I spend class time on? Lecturing.

There are two big goals for my classes: (1) get students unstuck on whatever they're working on and (2) elicit from students expressions of their understanding (or lack thereof). Let me give some examples of how this works in practice, by giving sample days for different classes.

First: Calculus. I begin by asking for questions on the reading, or the daily homework (which is mostly short skill questions). If someone is having trouble with problem 6, I don't do it for them. For one thing, that doesn't tell me where their difficulty is. For another, it doesn't allow them to express their understanding or to practice speaking mathematically! I try to make the student start the problem, and show the class where s/he got stuck. Then the rest of the class can help. A document camera saves a lot of time here that's otherwise used in writing things on the board. Sometimes I instead tell them to put up all the daily homework on the board (and cajole individuals into doing that), and we check it over as a class. After the daily homework has been dealt with in some fashion, I ask if they're ready to attempt harder problems. If yes, I break them into groups and assign 2 or 3 difficult problems chosen to bring up the major concepts in the reading for the day. Circulating and listening lets me answer questions that folks were too shy to ask publicly, and to make sure the quieter students are at least interacting with their peers via writing. Then we re-cap the solutions to the problems by my standing at the board and asking them to tell me what they found out and why it's important etc., and it's time to assign the next day's reading.

Second: Real Analysis. I sit down and ask for questions over the reading in the section assigned for that day. Sometimes students ask about definitions or examples or request to go directly to the next step, which is talking about the problems assigned. (I use a text in which there are about 5 problems per section, and I ask the students to do all of them each day.) Students choose what order to discuss the problems in, and either describe their proofs or write on the board whatever they know and ask for help from the rest of the class. I correct, rephrase, and guide. This class is generally small enough that we can check each individual's understanding at every juncture. Then the students choose a problem which they'll write up formally (and I have to consent). If there's time, I give a 1-minute preview of where the difficult parts are in the next section's reading.

Third/Fourth: Math for Liberal Arts. Or, a major elective, like Graph Theory. I begin by asking what's on the students' minds. They know that I'm really asking them what they want to talk about that day---if they're silent, I ask majors to pull out their running list of open questions or give gen-ed students a menu of topics to choose from. Then, we discuss. That often means that I ask them about recent material and get them to make conjectures of various kinds. Then I try to get them to prove those conjectures. This is pretty easy in major classes. They conjecture wildly, and automatically generate examples and counterexamples and use prior knowledge to prove things and run collaboratively up to the board, or decide that they need to split up example generation amongst them or whatever. They end up deciding the direction of the course by their questions and investigations, and I sometimes have to spend a lot of time outside of class making sure I know about the exact questions they've brought up. Gen-ed students need more prodding. They do everything the majors do, but slowly and sometimes hesitatingly. I have to remind them about what I expect them to do pretty frequently, in order to get them to develop the habits of mind our majors already have about inquiry.

So now, I'll answer the questions Aparna posed to me:

How do you choose what gets read before class and what questions to assign?
I have a syllabus that's pretty close to what people who lecture use. I tell them to read the part of the book that other people would lecture on, and choose very basic/definitional problems for them to do at home. In class, I choose hard problems so that they'll need to grapple with the essential concepts in order to make headway.

What do students get from this approach?
They retain the material pretty well because they're so active with it. They learn to speak mathematically and to write mathematically and develop some of the habits of mind that mathematicians commonly use. They also learn how to read texts and how to be persistent in figuring things out.

How do students respond to this method of learning?
They never fall asleep in class. They think of class as a social activity and they usually like being involved with the material rather than lectured to. However, first-year students find it really difficult. They don't think it's an efficient use of time---which sometimes is a reaction to it being not what they're used to, and sometimes is because they don't know how to use the time well themeselves.

How do you get students to pay attention to each other, to make critical comments, etc.?
I tell them to. And if they don't, and want me to give an answer, I redirect them to the student who originally gave the answer. Also, I sometimes lie to them and say false things to make sure they're reacting when they need to.

With students running things, how do you make sure the content is accurate, and replete with the context that an experienced teacher/mathematician could provide?
Accuracy is no big deal, as I correct students when they're inaccurate. But context is a difficult balancing act for me. I know that I have to make sure that I provide that context, so I try to comment on what students say and repeat what they say in different words, and give a few sentences here and there on what's going on in the big picture. But it's hard for me to do that while not interjecting too much of me into the class. In keeping in mind that the class is all about them, I sometimes lose sight of the fact that for them, I am part of the equation.

How much of your time does this method take?
At first, a lot. Now, with practice, hardly any preparation at all. It's mostly energy during class rather than before.

How much time does a student spend?
For easier classes, .5 - 1 hour reading and the same again on homework for each class. Of course, they do less than I intend. There's additional homework and stuff, but that's for an assessment commentary, not a class time commentary.

Is there a class size for which this method would not be feasible?
That depends on how good you are at it. At first, I couldn't do more than 30 students. But now, I can do more like 60 if the class level is low enough. Having student assistants helps with the class size. 30 students per assistant will do it. There are people who have success with this sort of thing in huge classes, but they do it really differently than I do.

How can you ensure that the not-so-vocal students are getting a lot out of this method?
I check in with them frequently during each class, and make sure they don’t just say “uh-huh, I get it.”

What are some of the problems that might arise?
Low student evaluations. Students often think they learned a lot, but that I didn't teach them. And in the traditional sense of the word "teach," that's true. But I set up and manage the environment in which they learn, and that's teaching too. They just don't know it yet. And that's all I have to say for now!

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