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Speaker 1: The following is a videotape module from the learning system Seymour Papert, on Logo. The tape series has two parts. The first, New Mindstorms, focuses on the process and the principals of learning. The second, Logo Hurdles, focuses on specific technical aspects of Logo. The New Mindstorms module teaching looks at Logo as the instrument teachers need to appropriate the computer to their own personal styles of work.
Seymour Papert: I’d like to read you a favorite passage from Mark Twain. Huck Finn is talking. “Sometimes, we’d have that whole river to ourselves for the longest time. It’s lovely to live on a raft. We had the sky up there, all speckled with stars, and we used to lay on our backs and look up at them, and discuss about whether they was made or just happened. Jim, he allowed they was made, but I allowed they happened. I judged it would’ve took too long to make so many. Jim said the moon could’ve laid them. Well, that looks kind of reasonable, so I didn’t say nothing against it, because I’ve seen a frog lay most as many, and so it could’ve happened. We used to watch the stars that fell, too, and see them streak down. Jim allowed they got spoiled, and was hove out of the nest.” Huck’s use of frogs to think about stars is a marvelous example of syntonic thinking.
Thinking about frogs making their eggs gives us a term in between the making of simple objects and just happening. In enriches the discussion, and then besides the conceptual side, it resonates with strong images of spawning that form a central part of Huck’s personal culture. I wonder what Huck would’ve done if he had had a computer. Perhaps something like what I did. I wrote a Logo program, in which a procedure spawns more stars than I would’ve had like to have made one by one, by hand. Making a star field like this doesn’t fit the stereotype that’s been established in tens of thousands of classrooms all over the world, of what children do with Logo in the elementary school. The star field isn’t more difficult than drawing squares and triangles. It’s just a different style of programming. Not better, just different. I’ll call it Huck Finn style, because I’m sure that hat profound young philosopher would’ve preferred to do it this way.
As a matter of fact, when I first wrote this program, it was rather more complex. I learned to simplify it when I used it as a theme for a teachers workshop. I wasn’t surprised when teachers showed me a simpler way, and I was truly ecstatic when I discovered that this simpler way was rooted in techniques that many children in second grade and even younger have rediscovered, in the course of their first encounter with Logo. Let me show you how this works by recapitulating how somebody might meet the turtle for the first time. An object on the screen is introduced as the turtle. We’re going to learn a language for telling the turtle to do our bidding. Write is a command in turtle talk. It orders the turtle to turn, that is to stay in the same place but change its direction. 90 says how much it should turn, that much. Left has exactly the opposite effect. Left 90 will exactly undo what right 90 did. Notice the turtle stays in the same place. It changes the way it’s facing.
Forward makes it move to a different place. It’ll move in the direction that it’s facing, up there. Pin down doesn’t produce an immediately visible effect. Turtle’s still there, but now the instruction back 50, back is the opposite of forward, will move the turtle and leave a trace, a line, so you begin to see how the turtle can be used as a drawing instrument, to draw almost anything you can imagine. The word clean wipes the screen clean. Right 90. This time I’m doing it. Right 90, left 30. The turtle can turn and so can I. In fact, I can carry out the exact same instructions I would give the turtle in turtle talk. Usually when you learn about angle, you use instruments like set squares and protractors, wonderful instruments that ought to be part of the culture of everyone. The turtle gives us an extra root, an extra set of connections for an idea like angle. It gives us what you can think of as a pipeline from body knowledge, knowledge of how we move our own bodies, into the study of geometry.
This idea is so important that it deserves a name. In psychoanalysis, there’s a word egosyntonic, which I’ve adapted, and I call body syntonic learning, the kind of learning that is rooted in this way in our own knowledge of our own bodies. A small personal touch in teaching the ways of the turtle make the turtle bump my thumb. LT is an abbreviation for left. A child will need some exploring before hitting on left 30, but that does it, and then forward something. Here we practice little problem solving. Try anything, forward something 20. Not enough? Try again. Forward 20, but it’s almost there, so let’s use a smaller number this time. Forward 10. Got it. To bring it back home, we need back how much? Back 50. BK is an abbreviation for back, as FD is for forward. Right 150. See? I’m learning to do it in one go. Children gradually learn through experience, which for me means thinking about and talking about what you do as much as doing in the hands-on sense, to associate numbers with particular amounts of term, and numbers with particular distances.
The amount of turtle talk you’ve seen is actually enough to open the door to the wonderful world of turtle graphics. Many children delight in going on from here to exploring these associations of numbers and angles and distances to make squares and triangles, and put them together in increasingly complex figures of their own design. In this tape, I’ll be talking about two roots into Logo. One, that’s become dominant in the turtle graphics literature, I call the hard edged style. One way into that is by practicing instructions like repeating forward 50 right 90. You get a square, and then you get a triangle by repeating that, and you get a star by repeating that, and we know how to make a triangle and a square. You learn how to put them together to make a house. Then you put together houses to make a village, and so on and so forth.
In a while, we will rejoin this root, the standard one into Logo. In the meantime, I’d like to take a peep at the other root, the Huck Finn style. Let’s see what Huck might’ve done. This is the beginning of an imaginary scenario. The turtle has been set up with a left turn with pin down. It’s about to do forward 50. That could be the beginning of almost anything, a rooftop, a mountain. When we see another forward 50, we begin to wonder, and in fact, this is not the beginning of a drawing. It’s the beginning of a more mischievous enterprise. The learner is wondering what happens when the turtle bumps the end of the screen, and another forward 50 is going to tell him. There it is. It went off the screen and came back on the other side. That’s a phenomenon known as wrapping, something you need to understand by playing at the computer. For children who do so, it opens the door to exploring instructions like this. Forward 2,783, a number chosen only because it’s big, then turn a little bit.
Let’s set the turtle up with a right 37, and do this. Even if you understand the idea of wrapping, the actual pattern is a little bit surprising. So is the place where the turtle reappeared. Let’s do the same instruction again, and I’ll do it in a different color so you can see the pattern more clearly. Note where the turtle is, and once more. In their early encounters with Logo, many children are fascinated by these rap patterns. Spend a lot of time with them, and some teachers wonder, “What could a child be learning through all that?” I think a child could be learning many things, and it’s very hard to know which. What is most relevant to my star story is the amazing fact that repeating one command can produce said surprising results. Besides surprise in the pattern, there’s surprise in where the turtle dams. Guess where it will appear next. Did you guess right? Guess again. There it is.
Two factors make this the kind of situation where, as a teacher, one wonders whether and how to intervene. One fact is the evocative nature of the situation. It has a fascination, and it could hold a child longer than you think appropriate. The other is the objective interest of what’s going on there. There is a lot you can say. The question is whether you should let the child explore, whether you should interpret what’s happening, why the patterns come up as they do, how the situation can be used to produce other effects. It’s a kind of very personal judgment for a teacher to make. I can give you rules, but it is good to have a bag of tricks, things you know to say when you think the time opportune. For example, in this case, it’s interesting to clear the screen and run through the same set of instructions again, this time with pen up, so you can separate interest in the pattern from interest in the way in which the turtle comes up in surprising places. We do pen up so the pattern won’t be drawn.
We set up the turtle with right 37 as before, and then we’ll do the same instruction, forward 2,783, right 10. Turtle pops up there, and there, and there, and there, and there, just as before. This play on the edge between control and randomness is useful as well as evocative. It shows a way to our stars. For example, if we know how to do repeat 12 times, do this forward and right, it’ll pop up somewhere, and then run a procedure draw, which is going to make a star but could do almost anything. Let’s do it. That star could’ve been a little procedure the child wrote, could’ve been a procedure you provided. That’s enough to show you that the stars program is within reach. Of course you don’t understand all the details. You won’t until you’ve tried it at the computer. Do so. Meantime, I’d like to tie together some threads of this discussion, threads about syntonicity and making learning personal.
I’d talked about personalizing angle by relating it to your person, your body. That’s body syntonicity. I talked about personalizing learning by allowing people to do it in their own style The Huck Finn style and the hard edge style of doing turtle graphics are just two examples. That’s closer to the sense of ego syntonicity of psychoanalytic theory. When you do it in your style, it sits more comfortably. You feel yourself. At the opening, I referred to cultural syntonicity. For Huck Finn, arguments from frogs and spawning are culturally syntonic. They belong to his culture. They’re well-rooted there. When I think of learning Logo, I think of two sides of cultural syntonicity. I’d like Logo to be supported by the learners culture. This is the only way it can be solidly learned, but I’d also like it to enrich, to enlarge the learners culture. This is the only way in which it can be meaningfully learned.
The main purpose of Logo is not what they call computer literacy. Of course it serves that better than anything else I can think of, but the real purpose is not to have better understanding of computers, but through computers, to have better understanding of everything else, including, I’d like to say, yourself. One way to encourage syntonic learning is to be imaginative about exploring the world around the school. Forms, or city, or perhaps a great river. A rich kind of exploration is the investigation of the people who make it all work. For example, who drives a riverboat like this?
Speaker 3: Basically, it’s easier to navigate a small boat because it’s … The fact that it is small, it turns quicker, you could stop quicker if you had something to do. With a boat like this, you have to be thinking about two miles ahead, because if you see something right now, it’s too late. You can’t make it, so you have to be looking ahead the whole time. You have to figure out. I look way down the river and I see a boat coming up, I have to make plans on what I’m going to do, how I’m going to meet, so it’s like almost anything. The farther ahead that you plan, the better off you are. You can’t wait until the last minute with a big boat.
Speaker 4: What fascinates you? Why do you this job?
Speaker 3: I’m from the central part of Missouri. I don’t know if you all know where it is. Mexico, Missouri. It’s flat as a pool table there and there wasn’t even a creek that water was deep enough to get to my knee, so I didn’t swim. I’d never been in a boat of any kind, not even in a row boat, and they drafted me, and put me in the Navy. I fell in love with boats, and so when I got out of the Navy and I came back to the middle of Missouri, there wasn’t any ocean to play around in, so I came to work on the river.
Seymour Papert: If the river is a theme of activity for your class, it makes sense for Logo to be integrated into their work with rivers, and rivers to be integrated into their work with Logo. The picture you see on the computer screen is a Logo picture. It’s written in Logo. It might be a lot of work for an individual student at a young age, but it could be done by a group of students, even by a whole class. It’s also a game that can be used for beginners at Logo to exercise their first attempts at controlling the turtle. Let’s see how this works. This is a turtle game, in which the turtle starts at a starting pier, has to find a path to a landing pier where it seeks the treasure represented by the diamond. The user or child types commands in turtle talk, and the turtle does whatever is typed. Forward 50 took it where you see, then right 90 turns it. Forward 50 again. The child working at it is practicing the commands, practicing the idea of angle, practicing what it’s like to use turtle commands.
Also, the child is learning that you can correct your first guess. You can back off again. You don’t have to be right at the first shot. Teachers often try to give their students this kind of exercise by overlaying a transparent maze on the computer screen. It’s a lot easier, but I think it’s worth the trouble to write a Logo to give a background for this kind of exercise. There’s several reasons. The first is that we can give the game several levels of difficulty just like real computer games. This level of the game is really very different. First, there are no forward commands. The turtle moves inexorably, so you don’t have time to stop and think. You have to think, but you must think on the fly. The turning commands are written in this unusual form, L 45, but it’s not the turtle’s LT. It does the turtle’s heading by 45 degrees, but only gradually, as you see from the near misses.
We’re taking off here on the captain’s remark that he has to make his decisions at least two miles ahead in order to steer his heavy boat. I find that the kind of real time thinking we need here brings you into a different kind of relationship with the work. You really have to make the turtle an extension of you, and you have to make the ideas underlying the turtle’s movements part of your intuition of the world and of yourself. The fact that this game is a Logo program means you have an open-ended project of adding levels, levels of variety or levels of difficulty. For example, you will soon know enough to add, and without any trouble at all, the effects of wind. Adding the effects of river currents is very instructive although more challenging. However, what I think is truly powerful about basing it on a Logo program rather than a passive overlay or something like that is that the learner can take charge.
I’d like to talk about the steps that could lead a beginner, an absolute beginner with no knowledge of Logo from playing the game in order to learn the fundamental actions of the turtle to making it. Let’s look at the scene. It’s made up of simple parts. That’s one part, that line, top bank, and that’s another part, and then these river objects are parts. A first step for a beginner could be to add a new part, and if when you program the game you had the foresight to program it in such a way that it stopped between the drawing of the scene and the navigational action of the turtle, you’d give the beginner learner a chance to jump in in order to add something new. For example, suppose that the learner wanted to add a raft. I’ll set things up so that Logo commands appear in the center of the screen for visibility. Show turtle makes the turtle appear. Now we’ll position the turtle to drive it over there, where we’re going to place our raft. Left 45, forward 100, FD is an abbreviation for forward. Right 45 straightens it. Pen downside will drawn and here’s the instruction that draws a square.
Repeat four time, forward 20, right 90, and we’ve got a raft. The command swim will place the turtle in the starting position for the navigation game, and we’re ready to go. The Logo technique used in adding this raft is called driving the turtle because the turtle is driven one instruction at a time, so much forward, so much right, and so on. Turtle driving can produce quite fancy effects. For example, as soon as you master the use of color in Logo, you can fill in the raft. I’ve put the turtle back at the raft. I’m going to give this instruction to drive it into the raft. We’re now going to set a color that we’d like to fill it with, put the pen down, and fill the square, so we’ve got a colored raft. Hide the turtle, HT, and there is a pretty raft. The next step into Logo beyond turtle driving brings us to the most important intellectual contribution of the computer to contemporary thought, the idea of program.
In Logo, we introduce the idea of program through a metaphor, the metaphor of teaching the turtle, or of teaching Logo a new command. We just saw how to make a raft by a whole sequence of Logo commands. Suppose we’d like one command, raft, to do that whole job. This is what we tell Logo. To make the word raft into a command, we have to do several things. First, we type to raft. Then we type some instructions. These are the very same instructions that drew the raft, and when we give the command raft, these are going to be carried out and end means this is the end of the lesson on the teaching analogy, or the end of the procedure that defines the command raft. Think of this by analogy with a recipe. You might say, to bake a cake, sift a pound of flour, separate six eggs, and so on. In order for this to become a command, we have to do one more thing. We have to do a certain, I suppose Huck Finn would’ve called it an incantation.
You do it by typing certain keys that differ from version to version of Logo, so I’ll leave you to find out about it by looking in the study guides, and I will get on to using this command. First, I’d like to tidy it up. Open a hole there and put in pen up home. Home is the position where we found the turtle, and it’s good practice when you write this kind of procedure to leave the turtle in the state in which you found it. Let’s see how to use such a command. Game is the command that caused the whole show to happen of the navigation game, and it has two parts. A sub-procedure river. This causes the river to be drawn. Swim, as you saw already, causes the navigation action to begin, and the true power of having this procedure comes when we see how to insert raft in another procedure. Let’s make a hole, move swim down, put raft in there. Now when we give the command game, river will be drawn, raft will be drawn, swim will make the navigation action begin.
What’s an ankle got to do with a turtle? I’m sure you’ll admit that angle has got a lot to do with a turtle, but I bet that a lot of you didn’t know that the word ankle and the word angle are etymologically the same word. Both derived from the ancient Greek for bend. In preparing for this tape, I explored a number of such lighter connections between turtles, angle, Logo, and elements of the culture, as I suggest you should when you think about teaching Logo in your classes. The one that astounded me most was the relationship between angle and English, or England. I knew that people called angles had something to do with the name England. What I didn’t know is that the angles are supposed to owe their name to the shape of their country. The scavenger hunt later on from the etymological dictionary to an atlas of old Europe. The country of the angles is shown in red. Some members of your class might be skeptical about the dictionary maker’s story. Others might want to defend it. Great. Nothing’s better for learning than a good fight.
You might be asking, “What is angle?” It’s exploring the idea of making connections, an idea that you might pick up by declaring an Angle Day, on which your students will report the results of their own research on finding angles in every subject. English is rich in them. Besides etymology, they’re idioms. For example, what’s his angle, or the boast. Hey, I’ve got a new angle on that problem, or the complaint. What you’re saying now is 180 degrees from what you were saying before. You’re contradicting yourself. In the same spirit, there’s collecting jokes, for example. Two wrongs don’t make a wrong, but three rights do make a left. A new angle on geography is measuring the angles between rivers and looking at the distribution of these angles. Why is it so different from the distribution between the angles at intersections in a city? Now I’d like to demonstrate some activities related to maps and children’s games.
Speaker 5: There’s the tree house.
Speaker 6: Okay, let’s go.
Speaker 5: [crosstalk 00:29:09] there.
Seymour Papert: The children read their scavenger hunt crew are saying, find the tree house, and take a bearing 30 degrees east of a certain tree.
Speaker 5: Right 30. We’re not turtles, but I’m still [crosstalk 00:29:25].
Seymour Papert: A turtle has built-in procedures to handle 30 degrees. [crosstalk 00:29:28] Children draw on their experience. One uses a fact about her body.
Speaker 5: Starting there, we could go 30 degrees because my fist is 10 degrees wide. One, two, three, and there we would have 30 degrees.
Speaker 6: I have a better way. Let me find some sticks and show you. Can you move the map?
Seymour Papert: Most people can recognize 90 degrees. The proposed better way is based on recognizing 30 as a third of 90, [crosstalk 00:30:01] and drawing on experience in dividing a pie slice into three equal parts.
Speaker 6: Then we take another stick and put it about a 90 degree angles.
Seymour Papert: It’s not an accident that play often provides [crosstalk 00:30:15] the best connections between concepts and experience.
Speaker 6: … and put it out at a 30 degree angle, and there you have it.
Speaker 5: Why do we make it so difficult? Why don’t we just use a compass?
Seymour Papert: The last connection takes us back to where we started this tape. Huck Finn was looking up at the heavens. I’d like to go to the computer and show you a program that brings a little piece of astronomy down to earth. I’ve made a micro-world in which the turtle is an astronomer. He’s watching the sun go around him. It’s a turtle-centric world. He’s interested in the size of the sun. He’s also interested in how long it’s going to take to set, so time it from there, when it touches the horizon, to there when it’s completely set. You probably missed it, but maybe you can get it on the rise. It’s now night. He’s waiting for the sunrise. When the sun rises, he’s going to freeze the sun in place and use an instrument he’s invented for measuring the angle of an astronomical body. It’s rising. How long is it taking? It’s there.
The turtle faces the sun. He turns on his instrument. There’s its counter. The instrument is nothing but a simple Logo procedure. It does forward 100, back 100, and then right one so that it scans the angle of the sun. Each time, it steps the counter. That’s an angle of nine, of 10, of 11, 13, 14, 17, 18, 19, and it looks like the size of the sun is pretty close to 20 degrees. How can we measure the angular size of our real sun? If you had a sextant, a real world version of the turtle’s instrument, you could do it just like the turtle did, but if you don’t, you can use a second method, using a watch, timing it. Have you ever wondered how long it takes the sun to rise and set? If you knew, you would calculate how many suns would fit around the whole sky, like this necklace of suns drawn by another simple Logo procedure. The turtle used the necklace to confirm his measurements. If each sun is 20 degrees wide, 18 should fit into the 360 degrees of a whole circle.
You can see that the turtle’s sun would take 18 times as long to go all the way around as it takes to rise or set. What are these numbers for the real sun? I am a mathematician. For me, and I think for human history, mathematics has played the role of intellectual glue, making connections between diverse chapters of human knowledge. I don’t think we succeed very well in using it like this in our classrooms. In this tape, I’ve tried to show how a teacher can use Logo to play the role of intellectual glue, the role that mathematics has played for me. I’d like to end by some reflections, unpacking the intuition that every teacher has, that it’s good to make connections. Why? There’s a cognitive side. Connections help you understand. You understand the new by referring it to the old. They help you remember, but there’s a deeper side, one that has to do with how you feel about knowledge and how you feel about yourself.
Connecting new knowledge to things you know and love, and things you think you can do, makes you feel good about it, makes you take it in in a form that is your own. By taking in knowledge in a form that feels to you as you, you change your feelings about yourself as well. You no longer think of yourself as somebody who can do math but doesn’t really understand poetry, or can draw but doesn’t have a head for numbers. Instead, you appropriate all knowledge