A r c h i v e d  I n f o r m a t i o n

MS. NANETTE SEAGO: Thank you. Before I get into the presentation and into our work and looking at teaching today, I?d like to bring us back a little bit into our own experiences and think about where it is that we might have images or pictures of teaching. We think back to movies. There are lots of famous movies that have images of teaching. Think of a few. "To Sir With Love," "Up the Down Staircase," "Dead Poets Society," "Educating Rita," "Blackboard Jungle." Can you think of others that come to mind when we think about images, media images that actually display teaching?

MEMBER: "Mr. Holland?s Opus" and "Stand and Deliver."

MS. SEAGO: Okay. There?s a multitude. Remember "Welcome Back, Mr. Cotter?" There?s quite a bit of images.

The point is that, out there, we have both these images of teaching and we also have our own experiences as students for many years, in which we were participants in a classroom, and in which we also have images of teaching. I?d like to have us take a second to think about those images, both in our media and in our experience, that actually can seduce us into believing that teaching is simple and straightforward, and that we know a lot more about teaching than we actually do.

So today what I?m going to do is actually challenge us to dig underneath the surface of what we might think we know about teaching and on what might be considered much like an archaeological dig, in which we?re going to dust off and uncover some of the things around teaching -- the relationships involved in, the students and teacher interaction, around learning a concept.

[See Slide 1] So today in my presentation I?m going to be using a small video clip of a mathematics classroom for us to look at and have some discussion. I know that your charge is to think about not just mathematics teaching but also science teaching. And in doing that we could think a little bit, prior to digging into the mathematics piece, what might looking below the surface entail seeing? If we actually did try to dig a little bit and look at teaching and the process of teaching, what might that look like?

Well, it might look more complex than we might think it would be. It would be more complex and in complex practices there are lots of different decision points. Some things are unpredictable. We can prepare for some things, and there are other things that you can?t prepare for. There are a lot of different points at which teachers make decisions around students and the subject matter, and actually try to bring those together for learning in the classroom.

We also might find that a variety of teaching approaches can work. Now today we?ll have time to look at one small segment within a lesson so that we?ll see one approach, but by seeing one approach, by no means does that mean that only one approach works. There are many approaches that work. I believe that you?ve talked about that already in this Commission. So the point is that there are lots of different ways in which teaching can work.

But in saying that, it also doesn't mean that anything goes. It does depend on whether the teaching approach is helping kids learn mathematics and science. That is the point. To teach in ways that help kids learn the content in which you want them to learn. So studying the various approaches and learning which approaches at which time, in which situation, might be helpful for students? learning is something that we can study and learn more about.

Prior to going in, we have a problem that I?m going to pose, but I also would like to talk a little bit about the big points, the big points that we have made around teaching is complex. We?re going to have a chance to look inside, study a little video clip of a particular lesson. We?ll have a chance to have a discussion and bring to the forefront some of the dimensions around teaching. But I also would like to say that while we have these big broad points about the complexity of teaching, the variety of methods, those could be thought of in terms of any teaching, if you think about that in terms of mathematics or science teaching.

What we then begin to think about is for teachers, if in fact -- let?s say emphasizing reasoning skills is important in both math and science, then figuring out the reasoning inside the discipline of either science or math takes on a different meaning once inside the discipline. So you can say that those are important, emphasizing reasoning and skills in both subject matters, but when teachers have to do the work, the work is actually inside the discipline in which they?re trying to teach. And so what we?re going to do is recognize that these are principles in which one could look at both math and science teaching, but in order to really dig inside them, dig underneath, we?re going to have to look at specifically the content inside which is being taught by the teacher.

Before viewing the video segment it?s always helpful to get a little bit of a sense of what is the problem that is posed. In this particular case we?re going to see a clip of a third-grade teacher, and she?s teaching a lesson. We?ll see seven minutes of it, so we?ll see a small piece of video. But it?s helpful to get a sense of what the problem is that?s been posed. What is the problem the kids are working on, that the teacher is working with the students on, what?s going on in terms of this particular segment.

[See Slide 2] So here?s the problem that?s posed by this third-grade teacher. Joshua ate 16 peas Monday and 32 peas on Tuesday. How many more peas did he eat on Tuesday than on Monday?

So I have two different ways that I could go with this, and I do need for you to let me know which are the ways you want to go. I can either show you a couple of ways, or talk to you about a couple of ways that kids might solve this, thinking of third graders in particular, or I can have you, amongst yourselves, spend a few minutes figuring out what it is you think kids might do with this problem.

So do I have a preference? It might have been quite a while since you were in third grade, and so I was wondering if we might for a second put our hats on for what it might be like to be a third-grade teacher, thinking about giving this problem and thinking about what kids might do with it. What is one way you think kids might solve this problem?

MEMBER: Count on their fingers.

MS. SEAGO: Count on their fingers how?

MEMBER: What their reference is if they?ve been taught tens or borrowing or if they just count, ten to twenty appendages they could count on.

[Off-mike discussion]

MS. SEAGO: Okay. When you say using manipulatives, what do you mean they might do with that? They might have some peas or beans you mean to represent the 32 peas and the 16 peas? Both? Somebody say "both"? How might they represent both? How might they solve that if they actually had peas or beans to represent peas?

MEMBER: Two piles. One that is 16 and one of 32 and then they pair them up and see whatever is left over.

MS. SEAGO: Does everybody, do people understand what he just said? If they have two piles, one of 32, one of 16, and actually pair them up or match them up and then what?s left over would be the amount that was eaten, or the amount more that had been eaten on Tuesday. That?s one way to do it. Can people think of another way that students might do that? What if they decided not to use peas or beans to represent it? What might they do? Hash marks? Does everybody know what?s meant by a hash mark?

MEMBER: That's what Senator Glenn said he would do.

MEMBER: Make marks and count them.

MS. SEAGO: Make marks and count them? A graph of what? Okay, so Monday there would be 16 peas; Tuesday, there would be 32?

MEMBER: ?and then they have something they can visualize and compare them.

MS. SEAGO: Okay. So a comparing across those.

MEMBER: Compare and contrast.

MS. SEAGO: Okay.

And so a possibility is that another way to solve this is they could subtract 16 from 32. As someone said, they could count on. They could start at 16 and count up to 32. They could count back. They could start at 32 and count back to 16. They could match, as some said. So there are quite a few possibilities in which you?ve been able to pull out of ways that kids might solve this particular problem. And we?re going to get a chance to look at what some third graders, in this particular class, did with this in a few moments.

[See Slide 3] Okay. Before I show you the video, it helps to situate this particular video segment inside some of the context of this particular lesson. It helps to get a sense of the seven minutes since we're going to see. And in your handouts, it was my understanding that you got what has some of the context. The title of it is "Third grade classroom video segment." You have some context, and then you?ll have the transcript that follows this particular video. So this is the part at which I am going to start talking a bit about it.

I also have some visuals that might help both in the context of the video segment, as well as when we see the video segment that might help you get a sense of what it is that the students are doing.

So this is a third-grade classroom. It?s in November, the beginning of the year. This particular time that we're going to drop in on this lesson is immediately after lunch. For those of you that remember coming in from lunch in the third grade, or have been there, there is something that is unique to a group of 20 to 30 students coming into a room right after lunch. Sometimes it?s helpful to know that.

This class is culturally, racially, linguistically, heterogeneously diverse. Half of the students speak English as a second language. Some have limited English, and there is high mobility in and out of this school. Although we see only a few children speak, participation is generally wide and shared across this and other class periods. A caution is to make a lot of statements that are more general in nature. This is just a few minutes of a classroom that we?re going to have a chance to visit. And so it?s helpful to know that participation in this class is generally wide and shared.

In this particular lesson, students are working on subtraction with regrouping -- i.e., borrowing. Borrowing is a term that may, in fact, be one that you?re more familiar with. These students? basic facts skills are good, but still they sometimes can make what might be called "bookkeeping" errors. So their handle on that is pretty good, though, the basic skills, as well as they had been taught the conventional way to solve a subtraction problem -- i.e., borrowing, and I?ll actually use a student?s work to show you what that means. They were able to carry this procedure out. [See Slide 4] I'll use Bernadette's work to show this. Bernadette will be a student that you?ll get to meet when you watch this video. And here is her work prior to this actual lesson, in which she had taken these particular numbers and displayed that she had facility in using the conventional method. So in terms of borrowing, if you look at 92 and subtracting 65 from 92, what is it that -- if we talked about borrowing, what is it that is the conventional method's asking for students here? What is the borrowing?

[Off-mike discussion]

So somebody said regroup. What is it that you?re regrouping? The tens and the ones. So you?re actually taking a 10?.

MEMBER: Taking a 10 and adding it to the 1.

MEMBER: So you have a larger number.

MS. SEAGO: Okay. So the point -- so, actually, that?s a really good point, and I want to highlight that, that we often forget. You?re taking it and regrouping it. This term "borrowing" is an interesting term and has some English language parts, and that can cause a little bit of confusion as well, with what does it mean to borrow somebody?s clothes, let?s say, but now we?re also talking about mathematics. And so what does it mean to borrow? So you?re taking these and regrouping them because in fact in the ones column -- in this case five from two, you need to borrow or regroup the tens so that you have 12 in the ones so that you can subtract.

I want to highlight this. For third graders, this seems to be probably pretty simple for us. This is a pretty straightforward procedure that ought to be pretty quickly and efficiently taught and learned. This is not easy for third graders. I want to just highlight that. It's very difficult for kids to get a sense of -- especially around -- what does it mean? I can't take -- I'm trying to get a sense of what it means to take a bigger number from a smaller number in these positive integers.

So if you look over at the common student error, one of the very common student errors that teachers see happen over and over again with students who have been taught and have some facility with procedures. They will do what you see in the common student error. What has this student done in this common student error? What has the student done?

MEMBER?S DESIGNEE: Factored the smaller number from the larger one.

MS. SEAGO: So they subtracted the smaller number from the larger one because they're still struggling, it could be said, with this notion of taking a larger number from a smaller number. So they simply just reverse them and say, I can?t take five from two, but I can take two from five. So, that ought to work. This is a very common error, and this is something that teachers in third grade, as they?re trying to help students learn, are confronted with in ways in how they might come to help kids know more than that.

[See Slide 5] Okay. The students in this class have used bean sticks, which I now understand you have in front of you. Some of you have already taken them out of the package to look at them. These are bean sticks that have been used to understand grouping by 10s and the basic trading process that underlies the standard procedure. So you have singles and you have sticks that have 10 beans that represent the tens. So trading in, regrouping, is something that the students had used in this class. What you have in front of you is somewhat of an artifact of this classroom so that you have a sense that this is part of the tools which students had access to.

Could someone just very quickly -- how might you model 16 with these bean sticks? What might that look like, if you could model that? One stick, six beans? Is that what people have? Okay. And then to model 32, you would obviously have what? To model 32, what might that look like with these bean sticks? Three tens and two beans. Okay. So three sticks, three bean sticks and two beans.

Okay. The teacher?s goal in using this particular lesson, I think it?s helpful for you to get a sense of it, is to let the students recognize how subtraction arises in a story context and to provide practice in using subtraction with regrouping in solving a problem, and to make the mathematical connections among representations, conventional procedures, bean stick models, some number line models, or other models and explanations. These are the goals that the teacher had as we enter into looking at this particular lesson and what unfolds in the actual online moments of teaching.

[See Slide 6] Okay. So I told you that this lesson is approximately -- it's about an hour-long lesson and we?re going to see seven minutes of it, because I think we?d spend quite a bit of time, and it's a lot to talk about, as you?ll see. We drop in after the teacher's asked the students to read the problem, and she has one student read it aloud and then asks who can say what the problem's asking. So the point at which she asks that question is when we will enter into this classroom and observe it.

I have a bit of a task. I?m going to divide you into fourths so that we are going to look at four dimensions inside the seven-minute segment. I?m going to ask that this group is divided into fourths and I?m going to give each group a task, a lens, a frame in which to view the segment. So I?ve looked around and thought that if we take about the first six people, I think it's about that. We may end up with six or seven. You?re officially group one.

[See Slide 7] Group one, you?re going to be the content focus. And you have some people who'll have some time to have a discussion together. We're going to try to have you zoom in, in this particular segment, on what kinds of mathematical ideas and processes do you see, what might the students be learning here. That'll be your focus.

Group two will be this corner. So what are the students doing here. You?re going to take the student focus. What are the students doing here, what can we say about their understanding of subtraction.

And then the next corner of about six people will be an environment focus. How would you describe the environment for learning in this particular segment? What could you say about the environment?

And then the last group, which is the group over there, last but not least, you?re going to take the teaching focus. What do you think this teacher is doing to facilitate the students? work, what might the teacher have had to teach students so they could do what they are doing.

Okay. Any questions on these focus? We'll view the video. I'll put these questions up again before you have some time amongst your group to discuss what you saw around these questions, and then we?ll have a whole-group discussion around these different dimensions.

And so now we'll watch the video.

SHEA: It was asking, like, um, he ate?he ate?sixteen?sixty-one?sixteen peas on Tuesday?on Monday. So, it?s like I?m telling you. It?s just telling you that, um, he?s?how much?many more did he eat than on the other day to get thirty-two.

TEACHER: Mmm-hmm. And what do you think about that? What do you think? How many more did he eat?

SHEA: He ate 16, because I count?I added them up. I went sixteen plus sixteen is thirty-two.

TEACHER: So, you?re saying he ate that he ate sixteen more peas on Tuesday than he did Monday?

SHEA: I used the?

TEACHER: Excuse me just a second. Let?s let Shea finish his thought.

SHEA: I used the number line. Can I go up there?

TEACHER: Yeah. Could you show it? Could people look to see how he?s explaining how he got sixteen, because other people got different answers to this.

Shea goes up the front of the room , and using the pointer, points at the numbers on the number line above the chalkboard.

SHEA: I went sixteen?one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen and I ended up on thirty-two.

TEACHER: Okay, comments either about his answer or about his method?

SHEA: Can I call on someone?

TEACHER: Yes, you can.

SHEA: Liz?

LIZ: I agree with you. I got that same answer and I did it the same way.

SHEA: Rania?

RANIA: I have another?I know?I know what I would say. I did it the same way, but I would say something else. I want to prove that his answer is right, because I got it.

TEACHER: You would like to prove that his answer is right?

RANIA: Yeah, because a half of?a half of thirty-two would be sixteen.

TEACHER: Uh huh. And how does that prove that his answer is right?

RANIA: I?because ?it?s ?it?s a half of thirty-two. Sixteen is a half of thirty-two. That proves his answer.

TEACHER: So, did that help you know that the difference between Monday and Tuesday had to be another sixteen?

RANIA: Yeah.

TEACHER: I see. That?s interesting?

SHEA: Bernadette?

BERNADETTE: I disagree with you.

SHEA: How come?

BERNADETTE: Because, what I did is I used the sticks in order to do it.

TEACHER: And you came up with a different answer?

BERNADETTE: Yeah. I got fifteen.

TEACHER: Okay. Would you like to see it?her show how she got fifteen?

SHEA: Okay.

Bernadette comes up to the overhead projector and uses bean sticks on the overhead to present her solution.

BERNADETTE: See, what I did is?

TEACHER: Ogechi, can you see if Shea stands there? Would like him to sit down?

OGECHI: I would like him to sit down please.

TEACHER: Shea, could you sit down so that Ogechi can see. Thanks.

BERNADETTE: Okay, so there?s sixteen, and here?s thirty-two. And what I did is I went like this and I matched them together. And see?then I couldn't match anything else together but these two. So, I took one of these and I traded it in for ten beans.

TEACHER: How many did you trade it in for?

BERNADETTE: Ten beans.

SHEA: What?

BERNADETTE: Because, see, I matched?see, I had?

TEACHER: I think people aren't sure why you?re matching. Could you back up and start over and explain why you are matching things? Could you put the two back down for the thirty-two also?

BERNADETTE: Okay, see, it say, "How many more did he eat on Thursday than on Monday," so if you match them together, you?ll get how many is left over and how much he?

SHEA: But it looks like more?

TEACHER: Okay, let?s stop before we go farther and see if people think matching up is a sensible way to compare the two days. She?s matching how many?the two amounts together to see what?s left over for Tuesday to see how much more it was. What do you think about that as a method? Lin?

LIN: I disagree.

TEACHER: With the method?

LIN: Yes, ?cause if you do that you?ll?it only?and I think that?if you want to do thirty-two take away sixteen or something like that, you?ll need to take away only sixteen and?when you?you have?and you shouldn't be putting on thirty-two and sixteen up there.

BERNADETTE: Well see, this is what he ate on Mon?no, this is what he ate on Thursday and this is what he ate?

STUDENTS: No, Monday! He ate sixteen on Monday.

BERNADETTE: Yeah, this is Monday?s and this is Thursday?s.

TEACHER: Tuesday?Tuesday?

BERNADETTE: Tuesday?s. So, what I?m doing is I?m seeing how much more he ate by putting them together. And when you put them together, you?re matching it up just like?just about the same way Shea would. But, see instead of adding them together, I?m putting them together like this. And then, since it has a match, I?m putting I down here. So that means you don?t count these ones because those are the one that have match. So, I keep?I did this and then see I can?t take four away from ten. So, what I do is take this in for ten beans and then I match these together. Then, I counted how many I had and I had fifteen. One, two, three, four, five, six, seven? This is seven? (pauses) I did something wrong.

LIN: You?ve got eleven beans right there.

BERNADETTE: (pauses) I think I did something wrong.

TEACHER: Why don?t you do t one more time, because some people are thinking about the matching anyway. Go through it one more time and see what you think. Can I ask a question about it?

BERNADETTE: Yes.

TEACHER: So, are saying you are matching them up so that you see how much was the same both days and then the stuff that is left over on Tuesday is how much more he ate on Tuesday?

BERNADETTE: Yeah. Because, if you match them together, it means that he didn't eat that much more. See, like when I take these two away, it?s just like saying, well?this?no, I need two here?these two are ones that have a match so that you can take them away because they have?because they are ones that?you?re trying to take this away, so you count one of these and you?ll see how many you take away, and how many you are left over with is how much he ate more.

Bernadette goes on to prove to herself that 16 is the answer using her matching method with the bean sticks.

[Video ends]

MS. SEAGO: [See Slide 9] I?ll ask you to spend a couple of minutes probably thinking individually. And then in your groups, if you could just sort of move so you can have a bit of conversation here, I?ll keep these up so you can be reminded of your focus. To think about this particular segment, you have the transcript for this video with you, and so if it helps, to reference that when we talk about your particular focus.

I?ll let you now work together for a few minutes, and then we?ll come together as a whole group discussion around this segment.

[Group discussions]

MS. SEAGO: So I think what we're going to do is go ahead and start with each of the sort of focus questions in the focus group and have some discussion. That group can raise some of the things that they began with. And some groups are still involved in their discussion. I know there's no quite a lot of time, but it's helpful to have other people hear from each other.

Okay, so, thanks. So why don?t we start a bit with group one, the content focus. And I?ll ask if group one can say a bit about what they were talking about and others can also here anyone else that would like to add to the content focus, the question on what kinds of mathematical ideas and processes did you see and what might the students be learning here.

Yes.

MEMBER: Well, I?ll go through a few things, and maybe others in the group will want to add then. But what we talked about, what we saw in the video were concepts of doubling. It wasn't exactly multiplication or fractions, but it was at least that concept of doubling. And then matching, comparing two sets, and looking at the remainder. There was simple counting and number line concept, and addition, using representations or manipulatives, I guess.

And also then also there was the concept of proof, which seemed to be, if I agree with you, then I?ve proved it. But at least there was a stab at the concept of proof.

Others want to add to that?

MS. SEAGO: Others that were maybe not part of this group that would like to add anything that they noticed around the content?

MEMBER?S DESIGNEE: I?d like to make a comment there that I thought the content was extremely complex. Distraction, to coin a phrase -- subtraction is really a difficult process, and you could see the kids really attacking this from a number of different content perspectives. So it is really instructive as to how difficult the teacher?s job really is when it comes to really simple issues.

MS. SEAGO: Can you give -- I mean can you give a little bit -- can you talk a little bit more about what you saw in terms of complexity, especially around the content?

MEMBER?S DESIGNEE: Well, from the standpoint of -- if I was to critique -- and I think it was Rita -- was it Rita? -- who mentioned in our group that it was hard to gauge on some levels what happened later on. We didn't see what happened later on, so it would have been nice to see that. It might have helped. But when we were comparing some of the students and their answers as to what it really meant in respect to the content there, because I think that?s the issue we?re addressing here, is when the last girl there, her process in how she was addressing what you actually do in this was really complex. And it really showed that she understood it, even though she got the wrong answer. Which was really interesting.

MS. SEAGO: How did it show that she had the understanding?

Actually, you know, I forgot to give this little piece of information that might be helpful, that Bernadette goes on to prove to herself that 16 is the answer by her matching method. She continues on with the matching method in the class and proves to herself that 16 is it.

MEMBER?S DESIGNEE: I think, as Representative Holt mentioned, it was a matching issue. She showed that she matched things. And when they had done it equally the same, she took away those two sticks, which was showing, in a strange sort of way, what some processes in the content knowledge of subtraction really is. You can actually take these out of the picture now. And it was really strange because it was part of a larger picture, and still part of the problem remained, being the other 16.

Then the other student, when he reacted to the question -- and this is something that one of the members here mentioned also was that you didn't really know where he went with that afterwards because you didn't really know how much he really knew of the content. Because he gave the right answer, but the way he framed the answer was in a way that you weren't quite sure whether it was a rote response, just 26, or 32 or whatever.

MS. SEAGO: Do you remember which student you?re talking about?

MEMBER?S DESIGNEE: That was the first student.

MEMBER?S DESIGNEE: That?s what I would do a lot of times when I was in math. I knew you were supposed to subtract one from the other, and I had a 50-50 chance of getting it right. So I?d go?32 minus 16. Of course that was one of the examples in borrowing; you just reverse the actual order and you get the wrong answer.

MS. SEAGO: Okay. Any others on the content before we move into the student focus?

MEMBER: I thought what was quite powerful with Bernadette was when she can?t take the beans off of the bean stick. She goes in there and takes them out and switches. The whole notion of regrouping now had a physical reality, and that was really powerful to see her do that and as students saw what she did, it allowed her then to go on and do her further matching. But the simple act of breaking apart that group was very powerful as a metaphor for what regrouping is all about, or "borrowing."

MS. SEAGO: Yes.

MEMBER: As far as content, one thing that struck me was that this was more than a math class. It was a class on group dynamics and it was a class on communication because just as -- I think it was Bernadette -- she made the statement, "I can?t take 4 away from 10." What she meant was, "I can?t take the stuck ones off the stick." In other words, by the way that they were interacting, where they didn't just say, you?re wrong, they said "I disagreed," they had to articulate what was in their mind through their mouth. So simply getting the right answer wasn't enough. They had to say why, and in order to do that they had to learn how to work together, had to learn how to communicate. And that math became only the primary element of content, as least as far as the class was concerned. But I?d say the kids learned a lot more about something beyond math. I guess is the way to put it. That math is taught in the context of other human experiences, and that was one of the key things about how this class was run. That may have more to do with environment, but in terms of content, I think there was a lot done in terms of communication.

MS. SEAGO: Communicating the content?

MEMBER: Communicating why they learned something.

MS. SEAGO: So somewhat of a reasoning, you saw a lot of focus around some of the reasoning?

MEMBER: Much of the learning process is visual. That?s why some of them used the number line; it?s easier to use than fingers and toes. But I think that?s part of the point of this, that that was not the classroom that I had when I went to third grade, where everything went through the teacher.

MS. SEAGO: So this is an alternative image of a third grade subtraction.

MEMBER: I think the key thing there is that there is much more to content than just mathematics.

MS. SEAGO: Okay.

MEMBER?S DESIGNEE: But one of the elements of the content, I think, was that communicating mathematics correctly in that although Bernadette didn't back up when she said, I can?t take four from 10, she did try to clarify her statement that she was putting things together. But I don?t mean adding things together; I mean putting them together in this matching way so that you get a sense that they really have gone through some classes and some lessons where it has been necessary to clarify their thinking so other children really understand what it is they're saying, and so that the language becomes more precise.

And when we think that these are third graders, these are beginning steps, and they?re very rough. But they?re beginning to take those steps.

MS. SEAGO: So the student focus, the particular student focus, the group that focused in on what are the students doing here. What can you say about their understanding of subtraction. You can see that these dimensions are not discrete, that we?re blending as we even talk about content, talking about Bernadette and Shay. And so I imagine there will be some sort of overlap.

But in terms of the student focus group, what were some of the kinds of things that you talked about?

MEMBER: I think that one of the things we talked about was the fact that this was a process they were going through, rather than just simply taking an algorithm or a formula and saying, okay, here you are. You do this and you do this and you get that and that?s all there is to it. The children were learning, they were using multiple solutions to problem-solving. They were learning to listen and to share information, to build on each other?s information. They were learning to justify their answers, to explain them, to take time to analyze problems.

And those ultimately are the kinds of things I think that are going to make the biggest difference. I think we talked about maybe some of the applications, what kind of students are we trying to turn out. Are we really turning out the kinds of students that we need, and if not, maybe some of these processes will help.

Now in terms of their understanding of subtraction, there are a lot of different ways to look at it and means you can have a different view. We didn't really reach consensus on that because our understanding of subtraction is that you take 10 -- you do the borrowing thing, when I was there, or the regrouping thing now, and that?s what we understand subtraction to be. Maybe that?s not all there is to it. I?ll let some of my other colleagues speak.

MS. SEAGO: So others around this student? Anybody else around sort of the students? One of the things it sounds like you?re still thinking about in that group is understanding and focusing in on the students also means that you?re also sort of looking back in your own understanding of subtraction. I?d like to sort of highlight that that?s what teachers get faced with all the time, which is thinking something is somewhat simple and then actually finding out that it?s not so simple when they?re using it with real students. A really nice way to sort of highlight the work of teaching in this group.

Any other things that people that were outside this group would like to make note around the students? Yes.

MEMBER: In terms of what they were learning about subtraction, I?d say that the student that used the number line was not thinking in terms of the decimal system. He was thinking more in terms of a linear way of counting. And the girl who was pairing things up was thinking, I guess, in binary, maybe is a way to put it. But the most prevalent way of teaching subtraction is decimal. We learned to count in tens. Even abacus was based on tens. And at some point when they go to work for Intel, they have to learn to count in binary, octal, hexadecimal and others. That becomes a more challenging concept. But what they?re being introduced to here is a decimal system of subtraction. And you could tell with the one who used the number line, he was thinking a lot more in terms of just a linear system without borrowing.

MEMBER: I was actually disappointed that none of them came up with the difference between two to the fifth and two to the fourth. (Laughter)

MS. SEAGO: Well, this was just after lunch, remember.

MEMBER: It seemed to me that there were three absolutely different ways and different levels of sophistication in solving that problem. There was the number stick was just, he ate 16 today and how many more is 32 from 16 and count. I actually thought the young lady doing the bean sticks was doing subtraction decimal-wise, powers of 10 and such. Then there was one little girl who very quickly said, everybody knows that half of 32 is 16. So there were really three levels of different sophistication. Subtraction in the physical sense, the number stick, and then somebody who knew their numbers reasonably well and kind of did it without even thinking about it.

I guess maybe that represents different levels of sophistication that teachers have to deal with on an ongoing basis. Some understand quickly and others don?t.

MS. SEAGO: And then also, when you think about these various ways, the work is also reconciling these differences. How are they similar, how are they different. When would one draw on these different kinds of ways to look at things. So it's also the work in reconciliation.

Yes, Diane?

MEMBER: Yeah, going to that point, I just wanted to mention something about Lynn, the one student we don?t seem to have mentioned. She?s the one who had questions with the matching method, kind of quoting that you want to do take away. She was seeing subtraction as take away. It seemed like this situation was an interesting one, really allowing students to confront this notion that?s the same arithmetic procedure, but it fits different models. And that student was one who had not yet put things together, matching and take away, using the same procedure for both of those different kinds of situations.

MS. SEAGO: Now Lynn was the student that appeared to be -- I mean she disagreed with why she had both 32 and 16 when you were just taking away.

MEMBER: Right. Right.

MS. SEAGO: Yes. Okay, the environment. If we could talk a bit about the environment in terms of this. How would you describe the environment for learning?

MEMBER: This is what we came up with. First of all, we noticed that there was a large room and on the walls there was a lot of student work displayed, which is nice to see. We also noted that there was a sofa in there, which is not typical in third grade classrooms. But basically what that means is that it?s probably a home environment, nice warm environment where, in fact with an adult sitting there. So this was probably a classroom with a teacher?s assistant.

Also I noted that as the camera went from one side to the other, it was a small classroom, probably a 20, 25-1, as opposed to 30, 35, 36 that we?re used to across our nation. The teachers use multiple methods of teaching, or have it available for the students. That meant that the first boy who went up there was probably understood the number line, and therefore was a spatial learner and used a number line quite well. Yet the other children used a visual and kinesthetic. That means that she went up and used the manipulatives, the beans. So different children learn in different ways, and that was displayed by the children there.

The seats were arranged to work in groups. If you noticed, there were I think either groups of 3 or 4. That means that cooperative learning was going on, even though at this time the children were talking to the teacher and not to each other. So children talking to children may have happened at a different time.

When the students demonstrated their answers, they were able to do it on different mediums. Number one, the number line on the wall. Number two, the blackboard was available. And number three, the overhead projector with the manipulatives.

The children displayed a non-threatening environment when they said "I disagree," not "You're wrong," which is nice to see. It was respectful. The teacher obviously prepared her lesson because all the groundwork was done. The number line was there. The beans were there, so the children were able to display their answers in different ways.

The teacher was not in front of the classroom. She was probably somewhere on the side. We never got to see her, which again is non-threatening. And there were no rigid time constraints on finishing the thought process. That means that each child had the opportunity to actually demonstrate what his or her answer was without the teacher pushing. The child just kept going and going.

And somebody noted that a child, a female, Bernadette, actually felt comfortable answering her question, and we?ve read so much research as to why girls don?t excel in mathematics. And this is one case where the girl felt comfortable doing that.

MEMBER: "Girls," plural. It's the atmosphere, right.

MS. SEAGO: Okay. All right, on to teaching. We're running a little bit into time. So let?s make sure we have some discussion around the teaching. And I hope that you represented all your group?s comments. Teaching. What do you think this teacher is doing to facilitate the students? work?

MEMBER?S DESIGNEE: We agreed that the teacher had created the kind of environment that allowed for open discussion and the ability to disagree. And that in fact that had to have taken place. They had to have been taught that in order to get to this point. She was doing questioning and giving them time to give their full answers in order to facilitate their learning. And she had available many of the things that they said needed to be there that the other group mentioned.

We thought that she had to have taught them the number line, the matching, how to disagree with one another and how to bring forward their answers in order for this discussion to have taken place.

MS. SEAGO: Any other comments on teaching? Yes?

MEMBER: The thing that I find very comforting that you mentioned is that Bernadette goes on to prove to herself that 16 is the answer and the validity of all of the strategies that she used?. And my interpretation of that is that the teacher knows her students. We may take this for granted, but we can?t. Knowing the abilities of students and the way to motivate them and all the subtleties that have to be made in this case, the teacher did the right thing and it paid off.

MEMBER: I think it would have been interesting to see what happened at the end of the class because thinking of the first young man who just counted up the number line, did he take anything away from this lesson that would have helped him understand that Bernadette was really onto something here? If that didn't occur then I kind of worry about sort of the end result of all of this. If there was some way in which the teacher summarized at the end and students got the opportunity to learn from, sort out from all these different ways the most constructive ways to proceed.

MS. SEAGO: It?s probably a good place to sort of also wrap this back up. [See Slide 10] But this notion, this reconciliation of these multiple ways is something that in fact teachers work on. This particular teacher talked about that as being part of her goals, connections amongst the different ways, how are they similar, how are they different. My sense is that that?s not just the work of one lesson but multiple lessons in which teachers will work and rework that with students, asking these questions around, how are they similar and different, and how one might draw on those.

We saw seven minutes of the beginning of a lesson. We can already see that it generates quite a bit of discussion and analysis amongst at least these four dimensions. And that can highlight what it was that we were able to I think just begin to see, which is that teaching is complex, it?s multi-dimensional. The tools, the strategies, the knowledge that teachers need in actually bringing students and content together so that students learn something is complex and involves a lot of different decisions, some of them unpredictable. Some of them you can have some ways in which you can prepare a bit for them. But teachers will tell you there?s always something that happens that I didn't plan on. And that?s sort of what part of teaching is.

So the more you have a sense of your subject area, knowing your students, which was raised in here, as well as a sense of what it is you?re aiming for, then you can try to shoot for that aim with those students in mind. In teaching you have your students and you have this aim, which you could think of it as two points. In teaching, one straight line between these two points isn't the most efficient way in which one can teach something to students so in fact you have you aim, have your students and figure out what it is they?re telling you, and reworking a lot of those aims in which you can get to, to learn something significant and worthwhile. Both in mathematics and science, and in this case we dug into mathematics.

Again, this is one example you are able to see one example of one teacher and one approach. There are multiple approaches, and it?s important to acknowledge that, I think, in teaching, that there are other ways in which one can view this work of teaching. But it also isn't true that anything goes. There are some teaching methods that don?t produce children?s learning, and we ought to study those and understand that when we draw on what it is we?re using to teach students, that we spend time finding out which methods help us get to where we want to be. And that is the ultimate aim, is having children in this country learn mathematics and science.

MEMBER: Without throwing this into a complete time warp here, but I was thinking back to Dr. Stigler's presentation from the last, our first meeting. And I was thinking in particular about the Japanese model and how they would have gone at this same problem in that the teacher would probably, if I remember correctly here, and with all the collective expertise around here, correct me if I?m wrong, but the teacher probably would have given the problem, given it to each kid for several minutes, and said, now combine with three or four of your buddies here and try to see what direction we?re going.

Finally we would have gotten around to the end product of putting this all together and having the kids make their presentations as a group. I don?t know that that?s better or worse. But what do you think? Is this as good a method, or is this a better method, or are they just different methods with equal likelihood of success? The Japanese kids in the testing programs apparently do very, very well compared with our kids. Not necessarily at third grade level. Our kids test, as I understand it, pretty well up to about the fourth grade level, then we start dropping off, and by the time we get out of high school, we?re not doing well. That?s one of the reasons we?re all here.

So is this the best method, or would the Japanese be better, or are they equal? Would we have any comment on that from anybody?

MEMBER: Senator, one thing I think that?.

SENATOR GLENN: Could everybody pull the mikes up. I think maybe some people in the gallery back here would have trouble hearing, and I have trouble hearing once in a while here too. So if everybody could start using the mikes a little better.

MEMBER: I think we have to pay close attention as we look at your objectives for tomorrow, to really recognize that this was a third grade class, and the complexity of the responsibility of the teacher, even in a third grade classroom, and the methods that she was using in this particular instance were somewhat closely aligned to what Jim showed us on the Japanese classroom, which was allowing the students to be engaged, to question themselves. There was polite disagreement.

And I?m not sure that we?ll resolve whether there?s a single method that was best, but certainly with one of your charges, talking about the core set of practices, I don?t think we can afford to lose sight of the fact that teaching young children is a difficult responsibility as well, and that this wasn't simply a third grade teacher getting in front of a class and posing a problem and making the students go through one procedure.

MEMBER: I?d like to add that in this particular process, maybe two or three of us have the opportunity to demonstrate the thought process completely, as opposed to if you were to do the Japanese style so that everybody gets involved and everybody is thinking. Then at the end there are a few spokespeople who actually get to present.

This model here, two or three actually had the opportunity to go up to the board while everybody else listened, and they maybe didn't have the opportunity to think. So I would -- although I enjoyed the lesson, I would hope there was at the end cooperative learning before the students would work in groups so everybody had the opportunity to think about the problem and not just tackle the problem and only two or three students get an opportunity to answer.

MS. SEAGO: I?m going to jump in here a second. I?m very familiar with the study and the videos which Dr. Stigler brought last time. I think that one of the things we have to keep in mind is, what he was able to show us was that you could think about a study of teaching at a much more macro level. In other words, they looked at patterns, themes, scripts that emerged culturally in this country, in Japan and in Germany, and they actually cross-compared those. They looked at big, sort of underlying themes.

In this particular instance, what I?m bringing to you is a macro level of also looking at teaching, which is the details inside some specific moments. We only saw seven minutes of a lesson, so we can?t make a comparison to say it?s exactly like or not like sort of the underlying structure or scripts that come out.

I can make a slight point, though. I think that in this particular instance we were able to see multiple methods being used, presented to the class, and work on reconciling those methods, which I think is similar to the way in which there is talk around the ways in which the Japanese lessons are taught. That is not typical of the way in which we have come to learn mathematics in this country. So I think there are some things we can say and other things we can?t.

What we are trying to do today is go pretty deep inside of one moment. In Jim?s work, he?s looking across a large data set and cross-comparing some patterns. We?re going inside of another level. Both are important to the study of teaching. What we are doing here is a little bit different today.

MEMBER: I?d like to make one point. As already many people pointed out, teaching is very complex. To measure the teaching effectiveness is perhaps even more complex. The Senator mentioned about teaching scores, or tests. That should be only one, you know, gauge or one indicator, not necessarily entirely reflecting the learning because I think in mathematics and science a lot depends on the skills. We can teach skills, but not necessarily really in terms of creativity or deep comprehension.

I learned mathematics a certain way in China. I was very good and learned a lot of skills, but not necessarily in terms of different methods. That?s what impressed me today. You know, you come from very different methods of dealing with things, give you a lot more kind of creative thinking or comparative modes and so on. I think that?s very important. Sometimes we carry too far in skills. Skills are very important, but still that?s not all. We have to consider other elements?.

This page last modified March 17, 2000 (gkp)