Video @ https://www.youtube.com/watch?v=4iIqLc0sjjs
George Markowsky: All right. Why don’t we get started then maybe people are drifting in. It’s still a new facility, but it gives me great pleasure today to have the team of Seymour Papert and Robin Jettinghoff, I need to practice that to get the pronunciation, they’re going to tell us about some of the new approaches to teaching mathematics that they’ve been working on. Seymour, as you know has been a long-time advocate of reform in teaching and teaching mathematics. He’s world-famous, and I think it’s a privilege to have him here speaking about some of his greatest work.
Robin has been involved in the Maine schools for a long time and has been working with Seymour for a long time, and has been facilitating many things. It’s great to have her join us as well. Some of this work has been funded by a grant that we received from the Bill … from the Gates Foundation by way of the Maine Department of Education, so they deserve some credit for supporting this work [project 00:01:03]. Without further ado, let’s welcome Seymour and Robin.
Seymour Papert: Okay, hi. Maybe I should give a table of contents first especially since some people might be … This is being streamed?
George Markowsky: Yes.
Seymour Papert: So, some people might look at it and this is like they say, “Don’t go away.” I’m going to talk for about 10 minutes on very general things about trying to set up a kind of perspective, and then I’m going to describe some of the concrete things that we’re doing to develop some new mathematical materials. Then, I hope we can have some time for discussion.
So, first of all, the generalities. Two weeks ago, I was asked to give a lecture at the American Education Research Association, AERA and I decided to be even [punchier 00:02:06] punchier than usual and I invented the following parable. It was just a theme for the next … The parable is about a country where for reasons that have been lost in history, they live on a diet consisting almost entirely of suet. Now, as you can imagine, this is not a very healthy diet nor a very pleasant diet. However, in this country, they had developed an extremely competent and profound tradition of medical research and doctors who had found ways to remedy the results of this unbalanced diet and cure the diseases that came from it, and added little additives, so that in the end, people more or less survived. More or less, most of them, not all of them.
Then one day, somebody said, “Hey, we can import all sorts of foods from all over the world these days. We don’t have to stay with the suet. Let’s invent a new diet.” But they couldn’t get the new diet adopted. They were a lot of reasons, a lot of obstacles they’re still battling with the new diet. Some are fairly obvious obstacles like people said, “What is good enough for my grandfather is good enough for me,” and all that sort of stuff and how can we change. There’s all sorts of vested interests of people. Businesses had invested millions in developing suet manufacturing and all sorts of stuff.
But the deepest reason was ignorance, and ignorance of their researchers who were not ignorant; they were extremely well informed and very smart people, but what they were informed about was suet, and how to deal with suet. None of them had the slightest idea about how to go about inventing a different kind of diet. You need some other kind of knowledge, and they didn’t have it and there was never time to develop it because everybody needed to publish or perish, or get his PhD thesis written, and so in getting whole other different area took too long, so, they went on dealing with suet.
Well, I’m told by people who were observing this quite large audience at the AERA that he could see on most people’s faces, people are saying, “Yes, he’s right. The education researchers are suet doctors.” Everybody except me. I mean, everybody would say it, all the others … But I think it’s profoundly true. There were 8,000 people at that AERA meeting, [pod 00:04:58] card-carrying education researchers. They’re all great experts in studying, improving, changing the diet, the intellectual diet that we give our kids, but if we consider that to be intellectual suet, there just are no people around in the world whose job is really make a very different kind of diet.
What I’ll spend the rest of my time talking about is the things you’d have to do, the kind of knowledge you must need to have in order to make this diet. I can’t do all that now. I’m going to give a few indications but I’m going to have to be more concrete on some specific things. But, we need to deal with one objection that people might say, “Oh, but we aren’t feeding the kids suet. We’re teaching them what they need to know.” To deal with that, I think I want to make a … introduce some terminology. And let me say, I’m concentrating on mathematics, I wasn’t so much concentrating there on mathematics but this work we’re going to be talking about here is mathematical work.
The same sorts of ideas apply to other subjects. I’m taking mathematics just because it’s a good, simple example and I guess I have a bit of credentials as a mathematician for talking about that. Well, do we teach our kids the math? Yeah, everybody agrees that kids need to know mathematics. You’ve got to learn mathematics especially in this new world, you’ve got to know mathematics. But, I’d like to plant some seeds of doubt not about the truth of that, but what on earth does it mean? What does it mean to “know mathematics?” Nobody knows mathematics. There’s so much mathematical knowledge that nobody knows more than a tiny fraction of it. What we teach in school is a tiny fraction of a tiny fraction of a tiny fraction.
It’s hard to measure. If I were to estimate, it’s somewhere between a millionth and a billionth of the total amount of mathematical … There’s a little sliver—why do we teach that sliver. That’s the question, not whether they need to know mathematics. Now, people might say they need to know the mathematics they need to know and at this AERA talk, I used another analogy and that’s with an idea that used to be very prevalent when I went to school that any child who did not learn Latin was a child left behind. So, we aql.@lAaa@@@@aqⁿl,ll had to spend many hours learning Latin and Greek, and this wasn’t like many centuries ago when they spoke these languages in the universities.
We were told we had to learn them because it encourages analytic thinking. It makes you more logical. It makes you more rigorous. You get a background to a cultural knowledge, all sorts of indirect reasons. So, I’m going to use the word Latinesque for the justification of teaching something on the grounds, not that we are going to justify itself as [inaudible 00:08:23] valuable, but it’s good for something else. To have a contrasting word, let’s say Driveresque. In driver education, there are for sure, there’s a lot of stuff where its justified to itself. If you don’t know that that hexagonal thing means stop or red means stop, and green, you’re in trouble. So, we don’t have to say that learning that is going to justify learning …
George Markowsky: Maybe it justifies learning Latin.
Seymour Papert: Maybe it justified learning Latin. So, driveresque justifications are direct justifications. Now, let’s look at our mathematics curriculum and look at it piece by piece and say, is this is justified in Latinesque or Driveresque ways? My argument I think is quite clearly, almost all Latinesque. Just consider how often you as adults, even those of you who might be in mathematical kind of professions ever do the kind of things that you were trying to do with high speed and accuracy at school, and did you ever factor a polynomial? Did you ever add fractions by finding the common denominator or divide them by turning one upside down, or did you in fact ever solved a quadratic equation using that famous formula?
I don’t think that there’s hardly anything in that curriculum that anybody ever does.
George Markowsky: Well, I do.
Seymour Papert: You do what?
George Markowsky: I do all those things.
Seymour Papert: No, you don’t. I don’t believe you do all those things … I don’t think you do.
George Markowsky: No, I even use square roots when I balance my checkbook.
Seymour Papert: But you didn’t use that formula, A squared, B squared plus and minus the square.
George Markowsky: Yeah, I do that. I do the squared.
Seymour Papert: Well, he’s special. I don’t think that the pamount of time [inaudible 00:10:19] the number of times that even you use it to justify the cost of our mathematics education in our schools. The only justification is Latinesque, that well, learning that is going to teach you something else and prepare you for something else, and develop this mathematical way of thinking, rigorous thinking, and so on, and all these other nice things.
Well, I don’t want to argue with that on this occasion whether it does that or not. But, I do want to say that if you support that there are two … There is an onus on you to prove two things. One, that it achieves that effect and as I said, I will pass that over. Two, that it is the only way or the best way of achieving that effect. Now, nobody even addresses that question because in the literature on mathematics education, as far as I’ve been able to find out, nobody raises questions like, could we have some other mathematical content that would serve those purposes better would even give, even George some better skills that he might use more often in his later mathematical career?
So, this is my framework for what we’re talking about there or one of the frameworks that the … We can justify this. We can’t even understand what it’s about unless we got some sort of comparative study, some other something to compare it with. Well, in other… what sort of things so let me imagine one more situation by getting kind of little bit closer to the real content. Let’s imagine, since we’re talking about Latin that we were still using their system of numerals with these Xs, and Vs, and Is, and all that.
Now, it’s quite possible, it didn’t happen that way but it could’ve happened. I think it’s conceivable that it would be educators who decided, “Look, we’ve got a much better way or doing arithmetic with these Arabic numbers, and it’s much easier to learn, incredibly much easier to learn, why don’t we teach that instead of these Xs, Vs, and Is in school?” Again, you’ll get lots of oppositions to it because all the tests are formulated in terms of the… Romans learning results are formulated in terms of whether you know the difference between XV and VX.
But, that’s really not very important to know. What’s important to know is what you do with it and so, we need a name for something else. What is this other theme? What’s the different between ours … It’s not a different curriculum. It’s something deeper. Different curriculum seems to imply a different way of teaching the same thing, but this is not just the same thing. It’s teaching something else which serves the same purposes, and that’s a different kind of enterprise which is the kind of enterprise that we should be talking about. But, again, it’s outside the framework of the usual discussion.
I’m just going to throw a word out for that, that I hate. I don’t like it, but somebody will suggest a better word. I’m going to say that it’s making a different … Well, I’d like to say a different math, but that throws people off…[throws people off 00:13:40]. I’ll say make a different Mathoid. So, this stuff, this thing that they did in Rome with all the Xs, and Vs, and Is together with the whole culture that went with it; the way they taught it, the kind of books they used, the kind of metaphors that they used is one Mathoid. What we do in Arabic now may take another Mathoid. Can we invent another … other Mathoids, and this is what we will be talking about.
In the course of history, they have been some attempts made to invent other Mathoid. I would say the famous new method in the ‘50s and ‘60s could count reasonably as another Mathoid, and was a total flop, I think everybody would agree. I think unfortunately, the reaction to it being a flop was people saying, they’ll stick with what they’ve always had because changing it has led to bad consequences. A different way of reacting might have been, “Well, maybe we changed it in the wrong direction, and we could change it exactly in the opposite direction.” What I’m talking about is really almost deliberately in every respect, changing this in the opposite direction.
The new math those days was … There was a Mathoid based on making mathematics more abstract and more analytic. I think that we should go in the opposite direction. We should make mathematics more rooted in its origins, in solving other problems outside of mathematics and I don’t just mean conning children to thinking that it’s good in the supermarket. It’s think about physics, think about science, think about economics and social sciences. We should see mathematics as coming out of these methods, these areas where we use deep mathematical knowledge.
Well, why didn’t people do it before? Because there were very few opportunities of using mathematics in a deep way in the era before computers. Historically, it grew out of the history of the pyramids. People are building pyramids. They were sailing the oceans. They were predicting the stars. They were doing all sorts of very complex things which we can’t afford or it’s not practical to have kids do those things. So, it’s not possible to have kids go through anything like the process, that was the real process of growth of mathematics where it grew as mathematical thinking for solving real problems and gradually became more and more abstracted until this beautiful jewel of the human mind called pure mathematics came about.
Now, because we can’t give these things to kids, we’ve taken a reversal of that order and teach them mathematics in an incredibly formal, abstract way. You first learn to do these operations as if they’re on black marks and paper, and then one day, you’re going to learn to do some applications, but not much. The applications like to the ones we have in the supermarket is just plain nonsense and only encourages children to believe that teacher talk is double talk, and they don’t believe them, going out and measuring the school yard.
Well, these things, they aren’t any deep things. There’s an extremely limited number of deep things you can do with mathematical, good mathematical ideas if you’re in an elementary school. Computer changes that because we can now do an infinite number of exciting kinds of projects of all sorts. So, that’s what is different now and that’s what the computer should be about. However, our education researchers unfortunately are suet doctors. They’re very smart people. I’m not putting them down. They’re doing great work but they’re doing work in suet medicine, and very, very few people are devoting themselves to creating an alternative Mathoid.
So, one of the roots for creating an alternative to Mathoidis in looking at computational mathematics, looking at how some mathematical ideas come up in computation. So, I’d like to start off with some examples of … Starting with something close to what happens in schools. Some of the things I’ll tell you are sort of compressed on pictures. Some of the things I will be talking about, I was doing 30 years ago, and some of them are new developments that we’re doing now and trying to concretize it in a way that could be adopted in schools.
So, I’ll start with a story. In England until some years ago used to have a big, sort of No Child Should Be Left Behind kind of test at age 11 that really shunted people into two streams. One day, they gave in this 11 test the following problem. Somebody has to send 100 letters to different destinations. Some of them require, I’m just translate into American: Suppose you got to send out a hundred letters and some of them will need the 5 cents stamps, and some of them will need the 10 cents stamps. You know that you spent $140 dollars … I’m sorry.
You know that you spent $14 in total, and how many letters had the 10 cents stamps and how many had the 5 cents stamps? Well, this test item produced a a storm of protest from the math teachers, in Britain namely this was an algebra question and these kids are not supposed to know algebra before age 11. So, it was unfair question. Well, of course these math teachers had been well disciplined. The way you solve that kind of problem is you say, let X be, and you set up this equation, you solve the equation.
It didn’t faze the kids though. A very large number of kids developed a solution that was on the following sort of lines. Imagine that you’ve got the hundred letters put in front of you, and then put a 5 cents stamp on each one, and now you spent $5. Then, put a 10 cents stamp on until you’ve run after your total of, whatever I said it was.
George Markowsky: $14.
Seymour Papert: $14. Well, I should’ve actually said $7, whatever, so that it would work out, and then you got it. You don’t need any equations. I would call that concretizing the problem. I’m going to just flash up here an example how I might do it in a computer context that we’re trying to develop. Not that the computer is needed for that problem. Those kids did much better. But what I’d like to do with it is to develop and more generalize the methodology for that so that you can use it in much more general situations, and so we should include problems that could be solved quite easily by everyone and put them in a more general way, and so you can remember this is how you conceive the new, the more general ideas that’s rooted in which you already know how to do.
I’m going to introduce to you our turtle. Who knows our turtle? Who doesn’t know our turtle? Everybody knows our turtle. Oh, good. So, I just have …
George Markowsky: [Crosstalk 00:21:56] why he was famous, because the people who are watching this might …
Seymour Papert: People watching this might not. Well, for the people watching, everybody here knows the turtle but … I’ll just say a few words about it because I’m also talking about how we’d introduce this thing to children. Also, I’m going to say some things about what this turtle means which is not exactly the way it is usually understood when it’s presented in elementary schools where it’s quite widely used.
So, the turtle is a little programmable object. It’s probably the first example of object or a tiny, little, sliver of object programming, and in fact played a role, Alan Kay says, in the development of thinking that led to object oriented programming. By the way, some might note, Alan Kay has just been awarded the ACM Turing Award. This is a little programming project that’s extremely easy to do in LOGO which … It makes this model that if we’ve got this situation, these shadowy turtles there represent the letters and this turtle represents the five cent stamps, and then you’re going to see a purple turtle that it represents. If we’re going to write a project tree, it plays out this process.
Now, here we’ve got another turtle who is called the counter turtle. The counter turtle keeps track of how much of what the total is. The counter turtle of all turtles knows the x-coordinate of its position. You might in fact write down here, show x-cor.
Female: The recorder?
Seymour Papert: For what’s its name? A recorder it’s called. Come, I’ll show x-cor, and it should say, “Oh really?”
Female: Oh, we can fix it.
Seymour Papert: Say back 100 and then … You see the turtle can … I’ll now x-cor again and it’s going to show zero. The turtle are base commands like forward and back, and tell us where you are, is one of the commands it knows. This is a way we’re going to do addition with these turtles. We’re going to reflect this sort of thing way back right into doing this kind of problem in the 1st grade or earlier. We don’t have to have variables that can … The Xs and Ys to name the positions, the quantities. This turtle is going to move five steps forward for each 5 cent stamp, 10 steps forward for each 10 steps, and the total amount that it’s moved forward is going to be the total amount that we’ve spent.
That’s the way these turtles are going to go. So, let’s…Now, what we’re going to do is simply try it with all the different numbers, the 20 stamps, we’re going to try it with putting one which comes first, one 5 cent stamp and 19 10 cents stamps, and two 5 cents stamps. Now this is going to be laborious if you had to do it by hand but since the computer can do it in a flash, why don’t we do that? Do it slowly first with …
Female: [Inaudible 00:25:46]
Seymour Papert: Yeah, okay. Do it for two. You see this turtle is going 10 steps forward now and when it gets to the end, with the program that we wrote, it’s going to say, how many … ?
Female: The first turtle [inaudible 00:26:11] turtle so ignore that in your count.
Seymour Papert: Yeah. So, now it’s doing two and it’s going to … There’s a problem … Don’t worry about problems there. We’re talking about the problems at the moment that one of the great things about doing programming is that you run into problems that are solvable, that you can think about, and this idea of debugging as a key, really deep mathematical process can be introduced much more concretely than in any other way that I know.
We get all that, and if we knew that it was 140 … We can read it off there and … Now, I’m not saying that if the total amount was $14 … We did something wrong with our units. But you see … So, that’s great for us. Then, we got to correct the units and it’s sort of right, and this is one of the things that’s so wrong with mathematics as it’s taught in schools. That you can’t be sort of right. If you don’t have it right, you’re wrong; most of the time. One of the things about doing it with computers is that something happens, and then you look at what happens and you say, “Well, that’s not quite what I wanted but I’ll try to understand what happened there,” and then, that will lead me to understand more deeply what’s going on and so what I did.
That’s an important theme of how having the computer context makes a … not a superficial but a very deep difference to how we’re teaching mathematics and learning it. But of course, it only does it if you’re doing that. Unfortunately, I noticed that 90% or more of teachers who are using this in schools don’t emphasize that because they don’t want to emphasize the getting things wrong as they think, but our point is to emphasize that it’s much more valuable to get it wrong than to get it right. If you got it right, you might get a good grade but you haven’t learned anything.
If you got it wrong and could correct it, that’s when you really learn. This has to be going out into the education system. For example, a reasonable way of testing would be not to see if you got the right answer but to see how you fix a wrong answer which you got … Well, anyway, that’s getting off too…. So, this is an example of simple turtle programming and I’m looking at the programming here. This programming can be done at different levels. I only have time to go into two. But we’re trying to think of it at different levels as programming suitable for even kids who can’t really write yet with iconic things and then for advance programming but for …
Now, if you think about middle schools, level … This is what the program might look like. You see, you’re telling a … You create these two objects. You give them names. One is called Stampo, one’s called recorder. It says, state your position such and such, repeat so many times, go forward so much, stamp, stamp is the command that makes it stamp its color.
George Markowsky: [Crosstalk 00:29:36]
Female: Too big?
George Markowsky: No. It’s just you can widen the window
Seymour Papert: You could almost … I’m sure you could if you wanted to spend five minutes on it which we won’t. Now, you would be able to read that program and understand what it’s doing and be able to do similar programs. We know from a lot of experience that middle school children don’t have any trouble with those. So, at the same time, we’re trying to develop a … advance the art of programming that’s accessible and introduce new mathematical ideas of which I’ve only touched on an old one, debugging, that we should bring into mathematics and some more mathematical ideas we’ll come into a moment, but these two things go together.
In terms of advancing the idea of programming, I’d like to mention that there are computer scientists here, I think that … of that using this recorded turtle to keep track of the number by moving along is a very powerful technique for introductory programming. It’s a new data to you. You’re using the screen as a data type. Instead of having a variable that stands for whatever … This is a tricky concept, the value that you talk… … You don’t have to use terms, or value, or variable. You can represent the thing by something concrete or visible like a position, and it’s perfectly rigorous and it enables, opens whole big doors of doing some very complex programming in a much more simple way.
Not that I don’t want them to have these other ideas but I wanted to get at the more sophisticated ideas through, I shouldn’t have said, sophisticated. I think these are just as sophisticated but the ideas that you usually teach in computer sciences courses via these ideas. And in some ways, George has always done that using LOGO in not quite this respect but in other respects as a way into programming in more “serious programming languages.”
George Markowsky: You can avoid a lot of detail if you [crosstalk 00:32:10].
Seymour Papert: You can avoid a lot of unnecessary detail that’s conceptual.
George Markowsky: [Crosstalk 00:32:13]
Seymour Papert: So, let’s leave this one and go to … Just to learn a little more about the turtle, since you know the turtle … Oh, we’re onto page one. Just very quickly, just how one might introduce this. I’ll just run super quickly over how would we introduce the turtle through making some programs that can do some amusing things that are good for elementary school or beginning of middle school child. Here, I’ve written a program which is called make an arc “to-arc” and it’s going to … That’s a radius and that’s how much of the thing.
It says, do a certain number of times, go forward a certain amount and turn a certain amount, and just run it. Let’s do arc 260, so that did that. Now, the next thing I wanted you to do was, once we’ve taught it that new word, and here’s another thing that’s glossed over often in presenting, but the idea of defining your own terms is something that’s vitally important to real mathematics and real science but real mathematicians are doing this all the time, define a such and such kind of function as …
But in school, they have to digest definitions given to them by other people and never have the occasion to define their own thing. So, this isn’t only that they are being consumers instead of producers but they’re poor consumer first as they don’t know what it’s like to make one of these things. So, we’re making a mathematical operation because now when we want to draw a petal, we use this as something that the computer knows how to do, and so we will know how they do it. I’m going to make a petal by having the turtle go there, and then do another one, but in order to do it the other way, I just have to turn around, and how much of a turn? It turns 60 in the course of that.
Now, how do you think about doing that other turn? I think this is where we introduce another I think, big idea and that’s we introduce a theorem that’s really very pretty. It’s a very classical mathematical theorem, but we give it a different name and formulate it differently. I call it the Total Turtle Trip Theorem, and it says that if a turtle makes a trip and in the course of the trip has turned completely around and it’s facing the same way at the end as it was in the beginning, this total turn must have been 360 degrees. More accurately, that’s only if it’s a closed, simple curve. In general, it might have been a multiple of 360 degrees traversing.
George Markowsky: Do you call that a theorem or is that … I mean, to me …
Seymour Papert: It’s a theorem.
George Markowsky: … it’s called motion of degree is somewhat arbitrary, right?
Seymour Papert: Well, it doesn’t depend on the measure. I mean, if you wanted to say it’s 2 pi radians, it’s that …
George Markowsky: No, but what I mean is when you start out with a circle, what’s the definition of the degree? It’s one 360 of a circle, right?
Seymour Papert: Well, so, if you do any trip that’s not a circular one, of whatever shape it is, in the end, you would’ve turned the same exact amount as one circle. You would have turned one circle worth. Maybe you’re right. Maybe that’s a better way to introduce it, and this is what degrees are. The circle is taken as the measure.
George Markowsky: Yeah, but the whole thing about 360 into one of these historical, strange oddities as to why you would divide it into 360 degrees as opposed to something decimal or something[crosstalk 00:36:07].
Seymour Papert: Absolutely. I got that, yeah. That’s a very good …
George Markowsky: [Inaudible 00:36:12]
Seymour Papert: It’s not only that though. It’s also in children’s culture. But since as these boards and skateboards and so on, everybody knows what doing 360 means. That’s another kind of link that’s valuable that one can take that thing and show talk about. They don’t have much … But now, how much of the turn … If the petals were all the way around, it must do 360. If it does 60 on this one and 60 on that one, 120. 240 is left. So, it must be 240 at the 2 pointy bits. It must be 120 at each. That’s how we’ve come to this one. Do an arc, and do a right 120, and then we made the petal.
Then, we can put the petal ones … We now have this term petal in the system. It’s building up a language here. We can define a flower by this procedure, and there’s a flower which … We’re doing a lot of prettier things and all sorts of interesting … graphically interesting things too but we’ve dealt with a lot of that in the past, so I’m going to skip over just emphasizing though that you can make some pretty things using a powerful mathematical idea namely the idea of a theorem, how do you formulate it.
The idea of theorem is a piece of mathematical knowledge that has great generality, can be used in many, many different situations and has practical use.
George Markowsky: It seems though in fact, that if you can even have kids operate with things like these, you’ve already achieved a lot of the stated goals of what you want them to do in education which is to think in terms of building blocks, and getting precise names, the things, and defining things. There’s a lot of benefits [crosstalk 00:38:19].
Seymour Papert: A lot of the benefit is …
George Markowsky: From being able to write, you know, a three-line program
Seymour Papert: Yes. A lot of the stuff that’s hidden behind the three-line program. We’re giving Latinesque justification for the programming. I’d also give Driveresque justification so it gets it from both sides. Driveresque is that you can use it. I’ll just jump to another example. Oh, let’s have this … Go to page two. I just … You told me mathematical ideas, and we didn’t even put the add definition. We forgot about it.
Female: Add definition [crosstalk 00:39:02].
Seymour Papert: Yes. See, mathematical ideas that we are trying to incorporate in concrete form there, idea of a theorem. Now, the idea of a heuristic, and that’s a good distinction there. You saw me walking around here and it really is a good thing to do that they are a simple problem. What’s the internal angle of a triangle? I saw some of these triangle where; walk one, play turtle, put yourself inside it. Then, the goal of being able to put yourself inside the mathematical problem is an important, really important goal that is not addressed. Again, for many people, they never get into mathematics because they’re always at arm’s length and they’re outside. They never put themselves in it. They don’t feel it viscerally, and that’s an important here; algorithm, variable, proof, rigor, debugging.
Okay, I want to talk about the idea of calculus. I’m talking about the idea of calculus in relation to a process that I think is really important as a way of thinking. That certain key ideas in the history of mathematics and science have gone through a process that I called disempowering as they come into school. Our problem is to identify those and re-empower them. I’m going to talk about two examples. The second one is I’ll mention first because it’s easier to see this general idea.
The second one is the idea of probability. The idea of probability, it’s one of the half dozen ideas that really changed mathematical thinking, and probably of probabilistic thinking in the last say, two or three centuries. The other one is the best idea of calculus that came from Newton and Leibniz, and those guys. Now probability, it’s what made possible statistical mechanics and indeed, I mean, Einstein’s own work came out of probability of the Brownian movement and biology is full of it, and evolution wouldn’t be conceivable without ideas of randomness and thinking, and so on. It’s everywhere.
George Markowsky: In the business world and insurance.
Seymour Papert: In the business world and insurance. It’s not surprising that when people say, “What new things could we put into the mathematics curriculum?” They say, “Well, how about probability?” So, they do, and if you look in the National Council of Teachers of Mathematics who have appointed themselves, and often considered to be the main component or defenders of standards in mathematics, they say, “Yes, you should introduce probability.” The typical kind of example they give is this. This is quoted from one of their things that, well, you could by counting or doing a survey in the school, you could find out whether how many boys like vanilla ice cream and chocolate ice cream similarly girls.
We could find out the probability of boys and girls liking the two kinds of ice cream, and so we can say, is it more likely that Mary like chocolate or vanilla? Well, it’s ridiculous for all sorts of reasons. If you really want to know, then ask Mary. But it’s a trivial app, from doing that kind of stuff you’d never imagine that this was a powerful notion that changed the whole course of civilization. That, I’ll call disempowerment. It’s drifted into school in a form that sounds like the great, powerful idea but it doesn’t have the sense of power associated with it.
We must find situations where it can be re-empowered, and again I come back to the computer because it enables us to do a lot of things. It gives us many more opportunities, not only to make it powerful in the sense that you can use it in lots of ways but also and that you can connect to a lot of other things. I’m going to give some examples of that in probability. But first, I’d like to touch on another thing which is the idea of calculus.
To introduce that, I’m going to take a concrete thing and Robin and I have been working on, and I see Max Crain back there has been contributing, too to this a lot. I’ll tell you specifically an example of a calculus-like thing that is included in the standard, kind of middle school curriculum, and this included in the context of making… of the area of irregular figures. One of the better of the standard curriculum books called CMP, Connected Mathematics Project, they approach finding areas in the same simple way.
They say, “We’ll draw grid lines across the area and count how many are inside and how many are outside, and this is an estimate of the area.” That’s fine except, well, how many grid lines might you draw? Robin has put up on the screen there something that falls under that, but nobody in this school would ever dream of doing that if you had to do it by hand. You couldn’t possible count how many or to just be beyond the even the level of tedium that’s often acceptable in schools.
What they do in schools is some big blocks. But what we can do in the computer in less time that it takes to draw the grid lines on this on a piece of paper, you could’ve written a program using very elementary LOGO and turtles to draw grids, and then program it to find out, to count how many of those squares are in or out. What we’re going to show you here is extension of this idea that you do it using the computer, and then instead of just taking one grid, what we can do take successively finer grid lines and see what happens, and so we’re get into the idea of limit and a lot of things are going to arrive. We’re also getting to this idea that sometimes you can cheat yourself, you can think you’ve seen some things precise when you haven’t.
Like, count how many grid lines, grids are and the squares that are in. What do mean by that? How do you decide whether it’s in or out? If the computer’s going to do it, you’re going to have to be more precise and that’s an important lesson that … You could be more precise by saying, “Well, the whole of it must be inside,” or you could say, “Three of the four corners have to be inside,” or “At least one corner.” You give a lot of different definition. Then, that raises a question. Does it matter which one you give? What difference does this make in the end?
Now, we can do experiments and these experiments are … Oops, what do we have there? Can we pause it on the graph?
George Markowsky: I think it gives the turtle a chance to count.
Robin: A chance to count.
Seymour Papert: It’s doing all sorts of counting, but now what we’re doing is we’re going to use a number of different criteria and draw the graph on the [inaudible 00:47:05] are finer and finer as the number of grid lines gets bigger, and what’s the estimate of the area to date. These are going to be these different lines, and you can see that they are very different from one another, but when you run it … Did you have that one, on the … ?
Okay, well. There we are. As you run it on successive, as they get closer and closer, you can these shapes which go up and down, and all over the place, because there’s a random, certain element of kind of randomness in it, but they are all converging and they’re all going to be in the limit. They’re going to come to the same result. We’re able to get a deeper connection between this way of getting at area as the limit of these … of counting squares.
Two big advantages of … When two big things are just on there in the usual school curriculum that they’ll come up automatically here. One is that you not only got a tool for a school exercise but you’ve written this program, you can use it to measure areas of all sorts of things. For example, you can scan it a map and use it to measure the area of some country, or some area, or how much of Maine is above some altitude.
You got a usable device and you’ve got at least one deeper connection with important mathematical ideas, and one other is that, you know, this is something that hadn’t struck me before but we’re looking for things where you can get the idea of proof. It’d be interesting to see there are a lot of questions. Suppose I double the size of my figure. What do you mean by double the size? Well, expand and blow it up by two. I suppose you can do that somehow. What does this do to the area?
It’s not obvious to most people, probably is to the kid but it’s fairly obvious is you got a square and you blow up the square, double each side it will … The area will be multiplied by four. So, since the area of our complicated figure is just a collection of squares, if each of those squares is multiplied by four in its area, so will the whole thing. We’ve got an approach into proving things that couldn’t be attacked and proven in the usual context. I see we’re taking more time that I thought, so. Well, you get the idea. Do you have the thing for the Monte Carlo?
I’ll skip from this and another way of doing something very similar and getting into better uses of probability is, well, one way to estimate the area of something is to pin it up on the board and throw a lot of darts with your eyes closed, and see how many get in it and out of it. The bigger the area, the more darts will get into it. An interesting question, whether it’s only the size that matters or the shape, but this is something we can do.
Now, what’s happening here is we haven’t done grids, we’re just making the turtle jump around at random and counting how many times it’s inside and outside, and getting an estimate. Do we have the graph? Here’s the graph of a couple of attempts that as it goes towards the asymptote, a good notion to introduce there. There are a lot of interesting things that can be discussed in that with an alerted teacher. Now, isn’t this interesting that … Look at that, how it got down there and it stayed down for a long time. Do you expect that to happen?
You can go into some … Get really thinking about unlikely probabilistic events and how the unlikelyhood works there. It needs an unlikely event to get far off, but unlikely events will happen from time to time. Once you fall offit would need another unlikely event to get back quickly. The chance of two unlikely events is pretty low.
George Markowsky: It’s even less likely.
Seymour Papert: It’s even less likely. What going to happen usually is, if it does get far off by an unlikely event, it’s going to take a long time to drift slowly back to being close to the average number and that’s interestingly connected with the psychology of what makes people think they have gambling systems and encourages them to lose a lot of money because you can get easily into that kind of situation where you get a so-called winning run which isn’t a winning run at all but it’s just this phenomenon.
George Markowsky: What’s ironic is that the proceeds of the lottery go to fund educations. Wouldn’t it be ironic if you could educate people that that’s a stupid use of money
Seymour Papert: Yeah. Okay, so that’s a kind of connection and what we’re trying to do specifically with this connection, with this piece of the CMP curriculum is make substitute units [inaudible 00:52:40]. We might make a substitute unit on areas which is designed so that a teacher can adopt that while staying with the whole rest of the curriculum but here’s another piece that you can plug in there which is guaranteed to cover or covers all the essential ideas that were in the original, but to do much more as well.
Also, to give the teacher the opportunity of learning, of acquiring comfort, skill, and understanding of doing programming because this one involves extremely simple programming but with the experience doing the simple programming, that teacher on another round might be able to now pick up another unit that involves somewhat more sophisticated ideas, and so that instead of saying we’ll make a new curriculum that can be plonked down all or none, the idea is to have these pieces of curriculum that can be adopted bit by bit.
It’s like living in the house as you’re remodeling your house as you live in it. It’s living in your old curriculum but changing it and changing it until it’s turned into something radically different, and that’s a fairly big project to do that whole thing. What we’re trying to do is to make samples of that, this is one of them and the probabilities and another one of them that we can use to look for the resources to get more people involved so that we can move it along faster.
George Markowsky: Do you have these prepared booklets or units?]?
Seymour Papert: Yes. So far, we’ve got booklets on the areas that have been distributed to a certain number of teachers who have been trying them. Our goal is that by the next school year to have half a dozen or so. Each one comes a little like a 16-page booklet that could be used to pick out a piece of this curriculum and …
George Markowsky: Some people are interested in seeing this booklets [inaudible 00:54:48].
Seymour Papert: It’s going to be in the mailing list, yes.
George Markowsky: You’re mailing it?
Seymour Papert: Our first one which is sort of being tested by a small number of teachers. We don’t want to circulate too widely until it’s got through. If you send us your name on a mailing list, we will send you the revised version of it which might be early in the summer or before the end certainly because we want to have them out in the world by the beginning of the next school year.
Male: Could I just ask from the audience who we are and who you’re talking about sending it to?
Seymour Papert: I’m sorry, sending the book. I’m sorry. I’m sorry, we have … George mentioned earlier that there was a small grant that came from the Gates Foundation through the Department of Education through the Computer Science Department here which is supporting this development, and in fact most of the concrete work is being done by Robin, and me, and between other things, and a few volunteers in which Max Crain sitting at the back is one, a couple of other volunteers and we’ve just … Some teachers we knew, we put out the word and they volunteered to look at …
Male: [Crosstalk 00:56:16]But in this case…
Seymour Papert: But anybody who wants to join this … Anybody who wants to join in this is more than welcome. If we’re talking about people who’d like to become actively involved, send us an e-mail … Send an e-mail to [papert@media@midname.com 00:56:32] and we will or robjet@midname.
George Markowsky: Eventually, you know, maybe when the booklets would become available, I could post something on my website[crosstalk 00:56:51].
Seymour Papert: Yes, we will post it on the website.
George Markowsky: [Inaudible 00:56:54].
Seymour Papert: Just a last example of … See, already I think we’re taking the calculus idea. We’re bringing it together with this probability idea and present them in a more powerful sort of way. I’d like to just end with another kind of example of getting at the essence of the … the big idea of calculus. This is my favorite example, and this is something that I’ve been doing for a long time, and some of the people here in the Computer Science Department, Larry Latour and people working with him are also doing a lot on, which is kids building robots that can be programmed.
There’s a connection … They sponsor to this competition every fall open to increasing number of schools compete in a challenge to build a model, a robot that would do some particular challenge. My simplest example, I always use the same one but it’s just too good not to, and that is that if you … Supposed you got this job, you’d like to make a robot that say, will walk around this table, circumnavigate an object. Suppose it’s got a touch sensor and you can program it to know if it’s touching or not touching. Supposing you start it off, so it’s touching and you wanted it to go around the table.
Now, there’s one way you can do it. You could measure the table, and then make it go forward that amount of distance, turn right angles, go that amount of distance, and hope it will go around the table. Now, two things are wrong with that. One is it won’t work because you wouldn’t get it exactly right and it will probably turn too close and run into the table and jam itself up. In any case, all that trouble would’ve been spent on just one table.
Instead, we can have another idea which exemplifies some really fundamental mathematical ideas. One of which you might call generality. Because it applies not only to this table but to all tables and to a lot of other things well, and that is just this. It’s using another powerful idea called feedback but it’s the idea of thinking locally instead of … Global thinking is I think of the shape of the whole table. Local thinking is like this. If I’m touching the table, there’s a risk that I’ll jam myself against it. So, I will turn away from it a tiny little bit, and I’ll take a tiny step forward. If I’m still touching it, I must’ve been jammed pretty tight before, I’ll turn away a little more and as soon as I’m not touching it, I’ll turn this way and take a little step. I’ll make a slightly, wiggly path along the path. When I come to the edge, boom, nothing there so I turn, still nothing, turn, still nothing and eventually, I’m going to get it and I will go around the table again.
Now, it doesn’t matter if the table is square or round or whatever. It will work for all tables of all sizes, of all shapes. This is basic, fundamental idea that Newton really discovered and introduced this scientific thinking. Maybe Copernicus and Kepler were responsible for breaking away from the earth-centeredness, but they still taught in global terms that here’s this ellipse and that the properties of the ellipse are matched against observable property. Newtons said, Hey, no. Newton didn’t start thinking about the properties of a whole thing. He started thinking, well, here’s the moon, say, and what does the gravity to it? It makes … In a tiny incident of time, it will make a tiny little effect on it, and this will move to a new position.
Then, to make another tiny, little effect on it which might be different if it’s getting further away or getting closer. The accumulation of all these tiny little bits will make the thing move around. There’s a whole, big, new way of thinking that opened up, of thinking locally, differentially a little bit at a time. The smooth reality is seen as the limit. The same idea that we had there. That idea is present in this littl robot, and you can use it in a lot of ways, to measure a linear curve, a line’s length. How do we do that?
Again, like we broke up the square … the shape into squares. We’re going to break up the line into tiny, little segments. We might program a turtle on the screen or this turtle to move along that line and measure as it goes, how far it moved and if it can move along … Like, this thing could be adding up how many steps it took, and so it could get the circumference of this and actually Max Crain wrote a really nice program for getting the circumference of things like ellipses using this idea on the screen that this turtle could move around the circumference by using this idea of the little bits …
Now, that’s the essential calculus but try finding out from students who’ve taken advanced program in calculus in high school what it’s about, and you’ll get very little sense of anything like that. Maybe just flash Kiri on the screen, I don’t have time to talk about it but following through these ideas, we take … You can do all sorts of problems like, Robin has a dog called Kiri who likes … She like to take him to on the beach, and she noticed that he also likes to retrieve things.
Now, if she throws the ball into the water, what does the dog to? Is it going to dive in and go straight to the ball or is it going to run along the beach until it’s opposite the ball and then dive in? This is a calculus test. It’s a classical calculus problem that, where would you go? There’s the ball. The dog starts where? Here? Running a little, you see. He starts there. There is the place where he jumped in. He ran then, this … Once he gets there, this is going to jump down there. This graph is the total time that he took.
We’re simulating the dog and with various assumptions about his feet in the water or his feet out of the water, we can study the … to find out in this case. You see it gets the shape of the curve. But if he obviously, he runs past there that’s going to take a longer time, but somewhere around here is going to be the minimum, and this minimum would be in different places. The point is that we can take this kind of classical problem and not only simulate it but get an analytic grasp on what lies behind it and how to think about the shape of these curves, and how to deal these problems in maxima and minima, and linear length and areas, and so on.
Kiri is going to be one of the units I think we’ll have about for this math thing. Not only about this one but about the huge range of problems, all sorts of trade off problems when you could decide, I can run fast on the beach and I swim slower in the water. How do I divide that? If I ran all the way on the beach, because I think that will maximize my time of, my speed on the beach, that doesn’t give me the shortest time to the object across. If I’d jumped into the water a little earlier, I would’ve had less time on the beach because I … well, obviously, I mean. This whole class of trade off problems is what becomes central to an area of study for kids.
So, I’ll stop. I don’t know how we’re thinking about time but I think that’s give you a taste anyway, of the kind of things that we do.
George Markowsky: Why don’t we have a few questions, Seymour? We’re going to have some refreshments upstairs on the second floor. I have a few questions here and then we can go up for refreshments, and we can have some detailed questions. Any questions from the audience. Here we go. One right here for Seymour …
Seymour Papert: Oh, better than questions. Ideas.
Male: How much of a … Well, George’s 250 class requires that they pick calculus before they take it further.
George Markowsky: For mathematical maturity.
Male: For mathematical maturity.
George Markowsky: For Latinesque reasons.
Male: For Latinesque reasons. So that answers my question, never mind.
Male: I was just wondering how difficult teaching them programming is when they’re having mathematics in order to use programming to teach mathematics.
Seymour Papert: It’s got to go like sort of each one supporting the other one. That there’s as I said, some very simple programming that you can do with very little sort of formal mathematics. But once you got that simple programming, we can now introduce, for example, once you do some programming, we can introduce the idea of a variable and …
Male: In your simplest program, your flower, you had division. They have to understand division sort of to …
Female: Division is just here.
Seymour Papert: Yeah, actually. They had to understand division in a better sense that, well, if 360 were the five petals, we got to divide 360 into five parts. This division thing is doing that for us. Basically, I think the kind of mathematical knowledge that you need to presuppose is what everybody who worked as cooking in the kitchen and picks up automatically …
Male: [Crosstalk 01:08:04] syntactical details like whether there’s a colon or a square bracket or a comma, or all these other kinds of notational short hands that you used to allow the compiler to parse the statement.
Seymour Papert: Well, we do. We work hard at … And I don’t think we really solved that completely, but we work hard at trying to partly minimize how much syntax you’ve got to know but especially, make syntax meaningful so that the words required is what you would guess. Thirdly, make it so that if you get a drawing, you’re not going to get just error, something’s going to happen. You’re going to get a chance to jump in and fix it.
George Markowsky: [Crosstalk 01:08:47] most of the languages? I mean, unless you want to first element you say car, and the rest is cutter because these are assembly language instructions. LOGO has really advanced concepts like if you want the first thing, you say first and if you want the last thing, you say, last you know, other stuff like that, and you get error messages that are actually in English and not like memory dumpe tests
Female: Error 404.
George Markowsky: Yeah, error 404.
Male: I’m Pete Nicholson and I realized that this is a Windows program that I’m looking at your Windows.
Seymour Papert: It doesn’t matter. It runs on anything.
Male: Pardon? The question I had was the proverbial name laptops. I have a 7th grader who has one. Do they include this program as part of the software package?
Seymour Papert: I hope they will make at some point though they don’t now, and partly because of the process of requisition to get the contract that …
George Markowsky: And they got LOGO.
Seymour Papert: LOGO is available for those machines. No, they don’t. All LOGOs are available for free download. The more complex LOGOs, can be purchased, they would run on the machines. I hope we’re struggling hard to get sooner or later our programming language is going to be compulsorily included.
Male: What percentage of the kids you deal with know what a turtle is?
Seymour Papert: In Maine? Actually …
Male: Okay, in Maine.
Seymour Papert: Actually, a fairly high percentage because the CMP program uses some very trivial, some extremely simple forms of LOGO programming but it is included in the CMP content. Quite a lot of them have met the concept of … or quite a lot of teachers. It’s not totally off. Do you have any idea what proportion might have actually met it in some … ?
Male: [Inaudible 01:10:59]
Seymour Papert: I’m sorry I couldn’t …
Male: [Inaudible 01:11:15]
Seymour Papert: Oh, that’s true. I think it’s absolutely true that one reason why we’re choosing CMP … this isn’t the major. The major reason is that it is the curriculum that’s most widely used by the schools that I think are more open to innovation. There are lots of exceptions. The secondary point is that although it doesn’t give enough guidance to make anything profound out of it, it treats it very superficially. But it’s there as a little toe hold.
Male: You misunderstood my question.
Seymour Papert: Oh, sorry.
Male: Your paradigm is that you know in turtles in terms of LOGO. My question was, how many 7th grade kids know what a turtle is.
George Markowsky: A real live turtle.
Seymour Papert: Oh, a live turtle. I’m sorry. I’m sure quite a lot. That’s something they can learn.
George Markowsky: How many have heard the turtles … ?
Male: [Inaudible 01:12:19].
George Markowsky: The ninja film?
Seymour Papert: They probably know the word but they might know what …
Male: My kids are pretty intelligent. They’ve never had the opportunity to see or touch a turtle. I’ll take one to class or … And we are in a fairly rural area, I grew up in Connecticut and I used to see them there but it was a warmer climate.
George Markowsky: I don’t think they really know have to know if a turtle is [crosstalk 01:12:39].
Male: But the turtle is, excuse the term, a logical choice for your LOGO program. You know, it moves slowly and can be [crosstalk 01:12:51].
George Markowsky: Oh, the turtle can move fast. They’re having a super charged turtle.
Male: I’m well aware of that. [Inaudible 01:12:58]. It’s just that you use the turtle to embody the principle that LOGO is extremely simple and stepwise. Forget about the speed, a simple stepwise. That’s one the things that evokes a turtle for you. But conversely, in terms of simplifying the language, if you’re saying turtle to an audience, among then eight people know what a turtle is, and 24 do not; I’m not sure whether that’s a really good metaphor. It is for us because we’re older. The 7th grader today, in today’s world has probably seen a lot more Britney Spears than turtles.
Seymour Papert: Certainly, I’m afraid. But I think she doesn’t quite serve as a good metaphor On the whole we find that kids like the idea the turtle and get the point?
Male: What educational levels are you trying to target with this app?
Seymour Papert: One day, I think we’re going to see the whole concept of the mathematics curriculum content changed, and we’re trying to influence that. In the short term like we’re going to do on Monday, we’re very focused in Maine on the middle school because there’s a growing demand there. There’s a growing demand of teachers to want to do something better and deeper than they know how to do with their laptops. I think it’s essential because this laptop program is not going to survive unless deeper things are foundan initial kind of honeymoon period of great excitement about just having them but, I doubt if it will survive long term if we don’t infiltrate into the system. This Apple, this is just one. I’m not saying it’s the panacea. We’re concentrating on middle school because Maine gives a great opportunity for doing that.
Male: I agree. It’s good to target the younger kids because if you wait until you’re older, it may be too late to …
George Markowsky: Oh, it’s definitely too late
Male: You’re imagining that if it would go as well, then down the road, you might be doing the similar things in college curriculum?
Seymour Papert: For younger kids?
Male: For older kids.
Seymour Papert: Oh, older kids, yes.
Male: I do similar types of simulations where I have thousands of agents running around, and I use that as a way to teach like concepts about differential equations. Kind of like the starLOGO but more mathematical and from a computaional point of view [inaudible 01:15:48] think you’re pushing that towards that direction as well?
Seymour Papert: No. I think it’s very important here.
Male: Even earlier than that. If there’s anybody here whose ever had a child and I cannot speak from personal experience but I’ve seen it in my children, before they came out of the womb, they are doing what you’re turtle did going around the table. They are already learning. If I push here and I contact that or if I listen here and I could do that where it’s warmer or colder, that’s directing their neurons and their entire learning center, if you will.
Male: Age is really quite independent on it. It’s part of being a learning organism.
George Markowsky: Maybe we can stop here.