Support Materials

2. Learning As We Grow - Development and Learning


Linda Darling-Hammond: Watching children develop and learn is like watching a miracle in action. One day they're talking and walking; the next, they're reading and writing; and the next, they're hypothesizing and inventing.
How can teachers support each student's developing abilities? How can we ensure that our teaching supports the whole child?
Hello, I'm Linda Darling-Hammond and that's our challenge for this session of The Learning Classroom.
Teaching in a developmentally appropriate manner means working with, rather than against the child's natural learning process.
Learning takes place along several critically important pathways.

James P. Comer, M.D., Yale University: There is the physical, the social interactive, the psycho-emotional, ethical, linguistic, intellectual cognitive. And it is development along all of those lines that’s really important. Up until recently, the school focused on the linguistic and the intellectual cognitive. But it is growth along all those developmental pathways that is important.

Linda Darling-Hammond: Good teachers start where their students are and build upon what they are able to do. But how do we know what our students are ready for and when? The concept of the zone of proximal development helps us here.

Roland Tharp, Ph.D, University of California, Santa Cruz: The zone is an important concept because to teachers it’s absolutely vital, because it helps the teacher understand what is the basic act of teaching. And that is this – to locate that point in the zone of proximal development in which this learner needs the assistance and then to provide it. Good teaching means constantly stretching, moving, rising in the developmental process, and that means always providing more assistance.

Linda Darling-Hammond: Psychologist Jerome Bruner described a spiral curriculum that returns to important concepts at different stages when children can understand them more deeply.

In this half hour we will see three teachers guide their students through an increasingly sophisticated understanding of momentum. They provide wonderful examples of developmentally appropriate teaching.
Fe MacLean, a 1st grade teacher at Paddock Elementary School, uses a variety of concrete tools and draws on her students’ life experiences to address the concepts of mass, speed and momentum.

(classroom scene)
Fe MacLean: What do you have there?
Girl: Surprise.
Fe: Surprise? Okay. Today we have a very exciting thing to talk about. You’ll have to help me think, this is a thinking kind of activity. I think before we came to school it snowed a lot, remember? Did anybody go sledding?

Fe MacLean: This morning is just an introduction of several cycles which will help them understand concepts of motion, for example, uh, the relation of weight, or… mass, with speed, the relation of incline, of a ramp to speed and momentum, or the relation of weight with momentum.

(classroom scene)
Boy: We went in my sled together down the hill.
Fe: Oh, you used one sled?
Boy: Yeah!
Fe: Oh, okay.
Fe: Is there anyone else sledding with an adult in a separate sled?…… So who got there first to the bottom?
Girl: Both of us.
Fe: Really? Oh, okay, well this is interesting, class. 'Cause I have a book here. What does it say Colby?
Colby: Sledding on a Hill. Rolling down a Ramp.
Fe: Rolling down a Ramp. So our story starts with a child and a grown up. And they’re gonna go down the hill. How do you think they’re going to get down the hill?
Girl: I think the littler kid will go down first, because it probably has more, much more energy.
Boy: I think the grown up will get down first because more weight makes the sled go faster.
Fe: So we have all these different ideas. Look what happens in the story. They go down and go Swoosh!
Students: Whoa.
Fe: They get to the bottom of the hill together. That’s what the story says.
(scene changes)
Fe: You are going to help me do this.
Student: Yeah!
Fe: Let me see, okay, how about if half of you go on this side and half of you go on that side? We’ll call this the ramp, okay? Now, how are we going to know or how are we going to remember how long it takes for a ball to go down? Remember? We can’t just keep it in our heads 'cause everybody forgets especially after recess. We want to make sure we remember.
Girl: Write it down?
Fe: Write it down.

Fe MacLean: For these age of children it is necessary that the material is chosen so that they see not just the, the abstract time but they see it with their own eyes how the ball rolls down the ramp. So I have a six-foot ramp instead of a small ramp because I want to make sure the numbers are big enough to see a differences with the children .

(classroom scene)
Fe: It’s the same ball. We didn’t change the ball. I wonder what’s happening here? Okay, let’s try it again.

Fe MacLean: I want to make sure that in their participation they are very clear of what we would call controlling of variables. We would call it fair, so that it starts from the same place, and we time it the same time until the end.

(classroom scene)
Students: …STOP!
Fe: We call this our data. What do we call this?
Class: Data.

Fe MacLean: When we made the graphic organizer that I used while we were taking down the data that the children are writing in there, that is a very abstract way of representing what we were doing, so that’s no longer concrete. When I plan my activities or units of study, I make the activities or the context rich enough so that it will benefit the children who are quite competent and those, the children who are not quite so competent in certain areas.

(classroom scene)
Fe: If you think you know what you’re going to do, come and get a big paper and start.

Fe MacLean : I want to make sure that children who have, are close to mastery will be able to have tasks that will…them to be to that mastery and in their interaction with the children who are just entering the zone they will solidify or stabilize their competence, and the children who are entering will advance and become more stable in that zone.

(classroom scene)
Fe: Look at Mrs.MacLean, I forgot one thing. Would you write a sentence at the bottom of each illustration and say what you think in words this time?

Fe MacLean (interview): Hopefully in the next investigation they will be at an even higher level and in the zone it’s advanced.

The next thing we’re going to do is, is to look at all of their drawings and what they wrote and their conclusions of how many seconds it took for the balls to go down the ramp. So they will discuss that, which is part of oral language and literacy.

(classroom scene)
The lower the ramp, the lower the ball, oh the slower the ball goes.

Fe MacLean: When they have to write it down they really have to think about it, and that’s what we want the children to do, not just in science, but to understand informational text, which is what they really wrote this information about what they did.

(classroom scene)
Fe: You’re going down there. It’s going to hit it. What’s going to happen?

Fe MacLean: When we use the long ramp, that’s a physical symbol of reality, which is the hill. So I think of that as a concrete operational tool, and it is a symbolic tool, it’s a physical tool representing something. When we made the representation of that ramp, and the children drew illustrations of that, that then became a higher level of symbolic tools inter…a graphic illustration or a graphic representation. Which they did on their paper and on the chalkboard.

(classroom scene)
Fe: And the question that we are asking is that, is it true that when the ramp is high you would move an object farther than when the ramp is low?
Boy: Ready?
Fe: Wait, wait. Look at this. Make sure they are all in the same line.

Fe MacLean: From my experience children are not going to be able look at the data using numbers of measurement to really understand the concept in this context of the level of the ramp relative to the momentum, and that is how far the can will move…

(classroom scene)
Chris: Come to Papa, ball! Come to Papa!

Fe MacLean: The tracing is more pictorial and is more appropriate to their age.

(classroom scene)
Fe: Now, can you look at it and think about the speed of the balls as they went down the ramp? And look at our picture. Actually, people would call that a graph, a line graph. I’d like you to make a picture of it, though.

Fe MacLean: I tried to relate it to a to a form of a story, a narrative that hopefully can relate to their own lives, for me to understand or to assess their understanding the concepts. In one instance, for example, two children working together…

(classroom scene)
Fe: Oh I see your pictures go together!

Fe MacLean:…They drew this picture where one of them won the race with a snowboard, and the other one got a bronze metal according to him, because he started from a lower hill so he couldn’t go as fast, and the other child who started on a higher hill went faster. So to me they are understanding that the height of a ramp, or the steepness is related to momentum or speed in this case.

Linda Darling-Hammond: Fe MacLean finds many different ways to assist individual students within their zones of proximal development.
Now let's take a look at how George Mixon presents a similar lesson to his eighth graders at Birmingham Covington School. Notice how he extends his students thinking by asking them to form hypotheses and examine multiple variables.

(classroom scene)
George Mixon : If I have these items, okay, talking about one of them dropping or both of them dropping. Are they going to fall at the same rate or are they not? What do you think Alex.
Alex: They’re going to hit at the same time.
George: Hit at the same time. How many of you agree with Alex, how many of you disagree with Alex? You all agree with Alex. So, okay, if I drop them they should all about, they should all hit this table at the same time, correct?
George: What was the rate at which they fell? About 9.8 meters per second squared or what?
Boy: 33feet.
George: Or about 33 feet per second squared, Okay, that is the acceleration in which they fall down. You guys are going to do that same kind of experiment but you are going to do it with an inclined plane.

George Mixon: I just wanted them to look at a piece in calculating the acceleration of a free falling object, getting the speed of that object, graphing that information and collecting data and putting it in an organized manner into a data table.

(classroom scene)
George: I got two cars, I got two tracks. You’re going to go in two groups. Make sure you get a graph of the speed of your car. And also, we’re going to see if we can calculate the acceleration of your car. So by the end you should have two graphs. Calculators, stop watches, you guys have got the sheet, read it, figure it out, get going.

George Mixon: With this age group you have to start these kids off with something that’s a little more concrete and more solid for them to understand and then you can kind of branch them off into the abstract and get them to formulate ideas and almost, what I call taking intellectual risks.

(classroom scene)
George: We got our incline okay. Check to make sure you're tight.
Boy: Don’t we have to measure the angle, though?
George: That’s why you got to read it.
Boy: How, wait, how are we going to find out the acceleration? Are you going to tell us the formula?
George: Most definitely, but you got to find the speed first. Good question, dude!

George Mixon: I think I started them with the ramp primarily because it’s kind of like, most of these kids sled, they all snowboard.

(classroom scene)
George: You really got to be on the trigger when it crosses your mark to get your time, okay. So this is why you have to make sure you do a couple trials before you just start collecting data.

George Mixon: I toss a lot of variables at the kids because I think one of the goals as a scientist is that they’re going to be bombarded with variables that will hinder experiments or procedural steps, and they have to learn how to control those and identify what is an independent and what’s dependent variable.

(classroom scene)
Boy 1: We need find figure out where it.s coming off.
Boy 2: I got 303.
Boy 3: Allright, well, that’s why you do more tests.

George Mixon: I think when they, when they realize that, they can say I need to control this, control this, control this, to test for just the one thing that I need to test for.

(classroom scene)
George: Hey, guys, Doddy and Steph have a pretty good data table that we can use. If you guys in the other group, check it out. Once you have your distances of one meter, two meters, three meters, four meters, divide that time into your displacement and what you will get is your speed or your velocity.

George Mixon: If I gave them the table, they don’t think. They need to be able to figure out ways in which to formulate and organize their information. That shows me how well they’re thinking.

(classroom scene)
Student: We’re done.
George: Once you get done, you guys need to start getting this information and start trying to graph it. So we’re trying to find out where that car is picking up speed. And what is changing its speed.
Boy: What was the average?
George: You’ve got to make it clear. So let’s just make it very simple.

George Mixon: I think that some of the kids were kind of confused on how to collect and organize the data. But that’s an eighth grade piece, and I think that one of the things that they will start to, to develop as they get older and become better scientists is start to organize that information…in a format that, that would be presentable for somebody else to duplicate and be able to read and understand.

(classroom scene)
George: If you guys look at Christina’s graph we get .82 meters per second, and then our last one we got .85 meters per second. Which means our car did what as it went down the ramp?
Class: It accelerated.
George: It accelerated. What we’ll look at next is we’ll look at adding the mass in there and then from the mass we’ll look at the momentum and see how well these cars will actually, what will happen to these cars when they have more mass.
George: Now how would we find the momentum of these cars?
Boy: It’s velocity times mass.
George: Velocity times mass. What I need you guys to do for me is, I need you to find the speed of these cars, okay? Once you find the speed of them, okay, we then got to figure out how we’re gonna get the momentum.

George Mixon: I think if you can get kids active, and motivated, and involved, and get their hands in stuff, they’re focused. I think that’s what kind of pulls them in and kind of gets them motivated, plus just knowing who they are and having a relationship with them.

(classroom scene)
George: Who needs me?
Boy 1: Do we need fractions on this thing?
Boy 2: I don’t know, let me ask. Can we use fractions?
George: Decimals.
Boy 2: Decimals dude, decimals.
George: Deci-ma-mals.

George Mixon: I sometime have to go outside my realm, and you know, the kids have to understand too, that there some, they can be flexible in their thought process, and formulating data tables, because not every kid is going to think alike.

(classroom scene)
George: See now, he’s going totally different than what I just did. You figured it out, you figured it out one way, and there’s nothing wrong with that. I figured it out totally different. Listen, I’m a simple minded guy, so I figured it out this way. You’re a genius, dude. So if you figured it out that way, God bless you kid, okay. And if it worked, it worked. That’s great.

George Mixon: Kids have unique ways in which to organize information and collect data and control certain variables. It’s just a matter of how well they’re able to collaborate with a group to come up with that ultimate goal.

(classroom scene)
George: Darin, if I threw this ball at you, can you catch? Are you sure you can catch? You positive? Okay. I throw this ball.
Darin: That’s a bad throw.
George: Would you try to catch this chair? If it was coming at 30 miles an hour?
Steve: Too much momentum.
George: Why is it going to have too much momentum Steve?
Steve: Because it has huge amount of mass and it’s going at the same amount of speed as all the other things, but it has more mass.
George: Beautiful. It’s gonna have much more mass, okay. And that’s one of the things that I wanted you guys to really try to get and understand when you are looking at these different cars. So on the back of that sheet of paper, what I want you to try to do is, I want you to try to answer those questions. And then I wanna show you one demonstration, 'kay?
George: Let’s just say that I got this as a concrete wall. I got these little sand barrels, okay. If you’re driving in a car. ‘kay, we got that little bounce back there. If I do this, kay, this kind of, it bounces back a little bit, but it also absorbed, this is gonna absorb more momentum. Would you rather run into a nice concrete wall or some sand filled barrels, if you’re in a car?
Class: Sand filled barrels.
George: Why Darin?
Darin: Well the concrete wall is going to be denser, it’s gonna have more mass.
George: Ok, it’s going to have more much more mass.
Darin: And it will exert a lot more force back on you.
George: Good, it’s going to exert way more force back on you.

George Mixon: I just kind of wanted to see if they could make that transition and see that connection, and I think some of them did. But I still, there is something that something that you have to revisit to make sure that they understand it.

Linda Darling-Hammond: George Mixon pushes his students' thinking by asking questions that get them to analyze data and test their hunches with one another.

Roland Tharp: Vygotsky pointed out that that kind of assistance that will help development in the zone can come from more capable peers. It doesn’t really matter where the assistance comes from. And the most competent teachers, I think, provide the assistance themselves when they need to, make sure that a good, rich diet of assistance is available from other class members, and outside resources, and the web, and wherever assistance can be provided to make sure that’s available to the student. That’s the orchestration of excellent teaching.

Linda Darling-Hammond: At the Detroit High School for the Performing Arts, Ken Gillam’s physics students study the same concepts, drawing on even higher levels of abstract reasoning. Through experimentation they move into evaluating evidence, drawing inferences, and predicting outcomes.

Ken Gillam: I started what I knew as prior knowledge for them. We had done situations, we worked in situations where they had the opportunity to evaluate velocity, acceleration of a ball rolling on a ramp.

(classroom scene)
Ken: We’re going to use little bitty cars. Since we're going to do momentum studies and we're going to look at the momentum of the car, we have to have a mass to go with the velocity we determine. We’re going to run it off the end of our ramp, let it go for a meter and determine its velocity. We’re going to take five readings we're going to take five stopwatch indications, so we know how fast the automobile is going when it exits the ramp. And I am going to show you a collision.

Ken Gillam: Their initial thought was, it’s going to be the same lab again, well, it really wasn’t going to be the same lab again, because the minute I put a barrier there and crashed it, they said this is not going to be the same. So the hook was I think it’s going to be, no it’s not. And so I hooked them by getting, giving them something they knew, but then giving them something new to look at.

(classroom scene)
Ken: I want you to gather data on it when that car bounces back a little bit it moved a certain distance, can we determine or measure that distance? I’ve chosen for you to work with 5 objects and you're going to get to predict if you where going to be in an automobile crash, and you were going to hit a barrier of some sort, what would you want the barrier made out of? Think about what occurs in a crash. What occurs in a crash, someone tell me. Daniel, tell me what occurs in a crash?
Daniel: Pain, 'cause of ah-
Girl: Force of momentum.
Ken: What else?
Girl 2: Pieces can fall apart off the cars, somebody could get hurt or somebody might die, depending on the type of crash it is.
Ken: So death can occur in these.
Ken: One person from each of your groups come, take the car with the crash material that are there, stop watches are here, meter sticks are here, your towers to build your things are there. What we're going to do now is once you get that, the first thing you got to determine is what? The velocity of your vehicle.

Ken Gillam: They are ready to go into college… But in a lot of ways they are still just kids and they like to see things that happen.

(classroom scene)
Ken: When’s the last time you got to play with toy cars in class and they called it work?

Ken Gillam: So if you give them something on a concrete basis, this is concrete, you can take this car, you can roll it down this ramp, and you can make informational observations, you can collect data, you can use that data to develop information that is solid, meaningful in a problem solving sense.

(classroom scene)
Boy: So when it hits right here, we start.
Ken: Yes!
Boy: Mr. Gillam?
Girl: This is the acceleration from here to the end of the ramp.

Ken Gillam (interview):
What you’re trying to do with these kids is you’re trying to keep them involved in what they’re doing, and you're trying to, every time, deepen the level of thought.

Girl: It moved back a little.
Ken: Can you determine a little?
Ken: You’ve got a distance, can you time it from impact?
Student: Ohhh!
Ken: Over a certain…Okay.

Ken Gillam: Once you see them beginning to fall into this pattern that says we’re all beginning to get this, then it’s time to challenge them again. Move them up a level.

(classroom scene)
Boy: We concluded yesterday that since the energy and momentum in a car could not be destroyed, that it was transferred from, the momentum of the cars energy was transferred from the car into the foam block that we had.

Ken Gillam : So you take the solid concrete, then you take them into the problem solving area and into the analytical, analyze what you’ve seen. Then, once you begin to analyze it, how are you going to use this information in a real world? How do you build this meta skills of thinking? How do you think in a broader context?

(classroom scene)
Ken: Now you’re going to design something to break down the momentum of an object as it is…being hurtled down the road at 70 miles an hour, with a mass of 2,000 pounds. Highway engineers have spent lots and lots of time trying to figure out what’s the best way to slow the vehicle down that is about to have a devastating crash. How many of you have been driving down the road and have seen the yellow barrels? Now, do you know that barrels are filled? With what?
Boy: Water.
Ken: Yeah they’re filled with… why?
Boy: 'Cause the water will absorb most of the impact, like, like in our experiment or whatever, you know, like the bounce back from the wall. The car has so much energy going at a certain speed, that once it hit it, you know if it hit it dead on, it would stop.
Girl: If you hit something I would prefer my car to go into it, rather than to just hit it and bounce back, because that would hurt. But if you, if your car goes into it, it will stretch. I mean if the object stretches, like some form of a putty or something, once it hits the car will go in, slowing it down instead of just hitting it head on.
Ken: Ok, now we need to look at our equation one more time. We’ve got to remember that if I’m going this way, and this way is positive. And then I turn and I go back in this direction, and this way is negative. What is my change in velocity?
Class: Negative?
Ken: V final minus V initial… My final velocity is positive, minus a minus velocity makes the whole thing?
Class: Positive!
Ken: Oh!

Ken Gillam: And you begin to put together a structure, a pattern into not only abstract, but into, not only being able to bring it all together and synthesize something that may be totally unique in their analysis.

James Comer: And so understanding that you are really an instrument of learning, and that you can help the child grow all the developmental pathways, and that growth along all the developmental pathways is what makes academic learning most possible. If you can think that, then you will find all kinds of opportunities to help children grow, and develop, and learn, both what it takes to be successful in school and as an adult, and to get the academic material they need to be successful as adults.

Linda Darling-Hammond: Fe MacLean, George Mixon and Ken Gillam taught similar concepts using similar materials, but adapted their lessons to the developmental needs of their students.They created intellectual challenges to support increasingly complex thinking.
As a result, their students grew in both their competence and their confidence as learners.
This has been The Learning Classroom. Thanks for watching.

Return to the Support Materials for Session 2

Contributors to this Session

Linda Darling-Hammond
Charles E. Ducommon Professor of Education, Stanford University

Dr. James P. Comer
Maurice Falk Professor of Child Psychiatry, Yale University

Roland Tharp
Director, Center for Research on Education, Diversity and Excellence, University of California, Santa Cruz

Fe MacLean
first grade teacher, Paddock Elementary School, Milan, Michigan

George Mixon
eighth grader teacher, Birmingham Covington School, Michigan

Ken Gillam
Physics teacher, Detroit High School for the Performing Arts, Michigan