Seymour Papert:
I’d like to take a different tack. Uri has talked about some good things that happened through work that I’ve been associated with, and I think it’s wonderful, and it makes me extremely happy. But I’d like to talk today more about the roller coaster and the problems and what has not been done.
And I thought I’d start by talking about Latin. And so, I put that little quotation up there, which I want to use in two different ways. One is content, and one is language. Now, as for the content, we’ll come to that. That’s most what I want to talk about. As for the language, well, it’s become– it’s fashionable to laugh at Latin and to laugh at the idea that people should learn Latin.
But let’s remember that not so long ago, for example, when I went to school, some very intelligent, caring, sensible educators were quite convinced that any child who did not learn Latin was a child left behind is. Now, we tend to laugh at that, and I’d like to explore some ways in which perhaps we’re laughing too soon.
Now, laughing too soon, not because I want to go back to learning Latin, but laughing too soon because those reasoning, that reasoning for those excuses, those justifications given for Latin are probably much better than the justifications that are being given today for teaching other things. And I want to try to distill out a view of some of those justifications, and not in order to necessarily support Latin, but to maybe we can invent a new Latin or maybe we can identify something that could more effectively play the role that those educators wanted to serve through Latin.
Now, also, for the content of that, I’m going to assume that everybody knows about enough to recognize that it’s Latin, that some of you went through all that time spent and can actually read it. But it’s interesting, I think, that almost anybody who speaks English can look at that and spend a little bit of time and puzzle out kind of what it means, and just good enough kind of what it means.
Obviously, it has to do with in mathematical circles, you know, some sort of asking questions being compared with answering them. And that’s all I really need for this for this for what I want to say today. Because what I would like to do is to emulate, although hopefully modestly and not trying to put… I mean, not– This is a demigod for somebody like me, David Hilbert, who at a congress of mathematicians early in the century made perhaps the biggest contribution to shaping 20th century mathematics by posing 12 problems. 12 problems unsolved and it wasn’t that he thought that knowing the answers to these problems was going to be so momentously important.
What he thought his deep intuition was that thinking about those problems, working on those problems, would give rise to great intellectual richness, and he was right. Among other things, these computers came out of it. Not that he anticipated or invented anything about the computer, but he recognized ideas that were going to mature, or seeds that were gonna grow into the ideas of Gödel, Turing, von Neumann, and eventually produce this machine.
And that image of not inventing it, but finding the seeds that will grow and finding how to nurture them is one of the, the themes that I’d like to sort of bring out in the course of, little less than an hour. So, it’s the asking of the questions and not the, the giving of the answers. I’m afraid that what I’m gonna do is gonna be a little bit dense. So, I’d like to refer to to place some places where you might find a some more details if you’ve got patience. One of them that doesn’t need much patience is my, that website, which was created by Dieter Burrell.
Those are titles of two books on which I’m working simultaneously, and I thought I’d mention the titles, the traditional titles because they also illustrate a kind of theme of what I want to say. The one called Fiddling While Rome Burns: Computers and the Future of Children- I believe that this is very accurate.
Rome is our education system. Maybe it’s learning, maybe it’s the children, maybe it’s the whole world. I think that we are in the midst of a learning revolution, and I think that the education establishment is fiddling. Is watching it and continuing to play their little games, and I mean fiddling in kind of two senses.
One reminiscent of Nero, and the other fiddling with, well, details in a system that is radically obsolete. It’s going to collapse. And I dare say in the many thousands of people giving papers here, if I saw them, which I’m not going to Because I don’t want to be unkind, I’ll bet that most of what is being done in educational research is gonna be obsolete before it can get into practice.
And so I’d like to give some indications of where some other direction, where else could we go. Now, the other book I’m working on … Now that, so that’s a sort of manifesto book. The other book is meant to be serious, and it’s about where are the deep ideas we can draw on, and slightly playfully title, provisional title is Piaget, Minsky, Chomsky, and Me: Three Seminal Thinkers and How I Translate Their Ideas into a Vision of Learning.
And I’d like particularly to emphasize Piaget, as Uri spelled out, I’ve worked with for a long time. I think Piaget has been given a rough deal because of the widespread idea that somehow, he’s been disproved or demolished by new discoveries. And I’m gonna say something about those, those new discoveries.
But not much. I think the important point is that he was being … He had a raw deal long before, and he had a wrong deal with he was translated by the psychology community into something that he wasn’t and that something is certainly refutable and probably refuted. But there’s a different side of Piaget that has been much less influential in, in psychology circles, in American psychology circles in particular.
And I think that is very much alive in its force and should be in important aspects of it should be reasserted rather than put aside into historical courses on the past history of our subject. Chomsky is the one of those that I haven’t actually worked with very much, but he’s a constant presence of, I think, another point of view that is often seen to be opposed.
Certainly, he sees himself as opposed to the position of Piaget. I’m gonna suggest ways in which because that, that apparent opposition between these two comes about because of a very limited, narrow view on education and learning that was imposed on us by the very primitive knowledge technologies that we had before the coming about of digital technologies and computers.
And that I think that we’re now in a position to say they were both right, and we can resolve those those conflicts, and I want to give some hints at how to how to do that. Well, I’ve been talking for a while, and well, so that’s a lot to want to say in a short time, and but since I’m gonna content myself today by only really asking the questions.
Now, I’m gonna make a lot of assertions, but I’d like you to take them in the spirit that what I say, I believe, of course, but my real intent, what I think is even more important about what I say, is not whether it’s true or false or its specific content, but the question that lies behind it. Behind the question, behind the assertion is the question.
I want you to ask those questions. I want you to encourage other people to ask those questions. We’ll now I’m going to look at a big span of how we could think about a different world of learning. And I’m going to have to look at this from three different points of view. From point of view of the individual learner and of the knowledge, and the point of view of the school, because good or bad, schools are there as an institution, and unless we take account of their dynamics and how they work and how they don’t work, we are not going to be able to really effect change.
And I’m going to get a little critical of the way in which people thinking about the dynamics of school, sort of as sociologists and people thinking about the dynamics of learning in individuals as psychologists are very often so disjoint from one another that each is bound to, to have– each is bound to fail in their purposes.
And I’m going to really advance a, an idea about why it is true that why education reform has proven to be impossible. So I will keep that dangling. So I’ll look at the individual learner, the school, and the society. They’re these three, three different themes. And so I will discuss them separately and then try… And they’ll probably come together after that. But before even discussing anything in detail, I’d like to indulge in my habit of in using parables and metaphors, and I’d like to tell my story briefly in the form of three parables. And that’s– One of them is an old one used often, but it’s grown, and one of them is sort of new.
A new one is about a country in which, for reasons lost in history, people’s diet consisted almost entirely of suet. Now, this, as you imagine, led to a lot of health problems. It was very difficult for people to survive. But somehow they managed, thanks to very brilliant doctors and medical specialists who really had figured out how to cure the diseases that came from overindulgence in suet, and add little bits here and there to make it work.
Well, one day somebody said, “Oh, you know, those conditions that made it impossible for us to get a greater variety of foods have gone away. We can now have all sorts of things because there’s now all sorts of technology and everything. Why don’t we make a new diet?” Well, this is a very radical suggestion, and all sorts of objections were raised by, “Sewer is good enough, my grandfather’s good enough.”
But the real reason, the real deep reason why it was very hard to escape the hole is that these very brilliant medical researchers and doctors, what they were expert about was sewer and how to deal with that. They would know nothing about how to make another diet, how to do something else. And I guess I don’t have to spell out the translation into relevant terms. You know, I think we are feeding our children sewer, intellectual sewer, and we don’t really know how to make another diet.
And this is our problem. How do we go about making another diet? And this can’t be a simple thing to do. It’s too big a thing to get your PhD on, to get tenure on, to write a paper on and be accepted by the conference. It’s just everything is stacked against the modern intellectual sewer doctors tackling what is really the, the real problem. So, this is the, what I’d like to sort of try and plunge us into. How to go about at least making some inroads into identifying what are the things you have to do to, to make this other diet.
The second parable is one that I’ve used politically, and Uri mentioned how I persuaded the governor of Maine to embark on a program that’s been picked up all over, create creating waves all over the world, I’m glad to say, of giving every kid a computer. And I think what really got him was this little parable of a country which I’ll call Foobar, which where a huge, a very high level of civilization had developed.
Great poets, great philosophers, great mathematicians, great sci– all sorts of people. But nobody had ever thought of writing. You know, taking an instrument and… They didn’t write; they didn’t draw. They sculpted. They didn’t paint. One day, somebody invented writing, and very quickly, the businesspeople grabbed onto it because they could communicate at a distance, and the scientists, and this became known as ICT.
And the idea eventually dawned on somebody, “Why don’t we put ICT in schools?” So, then this started. But these were cautious people, so the debate was, “Well, how would we do this? How do we start? Should we put a pencil in every classroom?” “Or should we– each school have a room where there’s a mountain of pencils so-” “-everybody gets a fair chance?”
So they tried all sorts of things, and there’s no doubt at all that, you know, ingenious teachers and enthusiastic, bubbling, creative kids found very interesting things to do with these pencils. But the important point is that what they could do with those pencils had nothing in common with the role of writing in our culture.
And what we can do with computers by putting a computer in every classroom, or even six or even ten, still has nothing to do with whether it’s possible for the computer to become a new medium other than– could displace the pencil as the medium of expression So, that hit Angus King, then governor of Maine, and we the next two years, I had no idea that was gonna happen. I, next two years was mostly up to here in local politics and fighting with legislators and writing columns in newspapers. And it was a very interesting experience, and it taught me one more lesson about if you’re to change this thing called education, you can’t do it by confining yourself to proving that this sort of way of teaching biology is better than that way of teaching biology or this is a big social problem that fits many dimensions. And I’ve tried, I mean, in my life’s been a bit scattered, but I’ve tried to work with mathematicians and philosophers, with toy companies, with Lego and Nintendo and Disney, and with politicians, and with at least three governors, now ex-governors and so on. Well, that’s the simple form, short form of the of that parable. But I’d like to mention something about the deeper form that, that I didn’t even suggest to, to Angus King. And I’m not even sure that I’d thought of it by that time, of extending it in this way. Probably not. But we laugh too quickly at these people in Foobar who are get– to cancel out.
But if you really think about it, there were very good reasons why they should oppose this idea of making such a huge transformation to the medium through which that they developed their education system. Some objections, let’s try and imagine, some were some that really had happened in our own tradition from Greek times of people saying that we work very hard on kids developing memory power. Let them write and they won’t develop memory power.
And look, we’ve got these wonderful curricula, and I want to show you one other thing here Okay. Was they had– of course, they did mathematics, but they didn’t do mathematics only by– They didn’t do it by making marks on paper. They made it in lots of other ways. First of all, their mathematics was mathematics in action. These people learnt mathematics by doing things that involved mathematical ideas, and that was a, a much better way of doing mathematics.
And then I was gonna show you on this slide, I don’t know why it isn’t coming up. It doesn’t matter. This is chisanbop. This is a- Sorry … Korean way of, It’s, You can find it on the net. It’s really fascinating actually. A very elaborated way of doing complicated arithmetic with your fingers.
And if you learn to do that, you can do the kind of algorithms that we teach. Well, they’re not the same algorithms at all, but they are– we think they’re as effective So this new thing wasn’t gonna help these algorithms. It wasn’t gonna help the kids pass their tests in finger math, if that’s what they were doing.
On the contrary, that could confuse them about that. Well, and so on. You can see, if you try and use some imagination, why there were many obstacles, and the many obstacles really are real disadvantages. But we know that writing had such a powerful, overwhelming advantage in other respects that it overwhelms all these, these difficulties.
However, they don’t know that, and it was gonna take time for the advantages to develop. And the real difficulty there in getting the transformation to happen was that developmental process. How could it develop? You know, we could say pencils would be better, writing would be better, but what about the transition from where they were Monday to where they could be one day?
And that developmental process is something that is as important as the content of what’s taught.
We could have another example, a more real one of that developmental pro- of that same sort of situation that I mentioned because I’d like another example to emphasize the need for a word or words for talking about learning You know, I’ve used this slogan for a long time. It’s why instead of trying to make the kids love the math they hate, why not make a math they will love? Well, why not indeed? That’s the simple answer. Two kinds of answers. One– or three kinds of answers. One, we might not know how to do it, and it’s taken a long time for, for me to ma- get little toeholds on some ways of doing it.
I’ll be talking about that. And some people here, Uri, you know, of Northeast Uri, has also worked very hard to make toeholds of what it would be like to have another math So it’s hard work. It’s gonna take time to, to take root and grow. It’s gonna have all sorts of obstacles, like we saw the фувариумs We’ll come to some other obstacles. But it’s not something that can happen simply. Well, one of the reasons is we don’t have a language for talking about it now. Because when I say I’d like to do another math, I mean, “Oh, you mean make another curriculum?” No, but I don’t mean make another curriculum.
Doing mathematics like we do it on pencil and paper is not another curriculum for what the Foobarians do with their finger math. It’s a different thing. Another example of a different thing that’s a real one in this case If you think, imagine that we all used Roman numerals and we hadn’t invented Arabic numerals.
Under those conditions, you can do with Roman numerals, you can do everything you need to do. You can do calculation, keep your records, and, you know, but it’s much harder work to do it and much harder to learn. Now, it so happened that whoever invented Arabic numerals did not do it, I think, did not do it as an educational enterprise, but it could have been, couldn’t it?
Let’s imagine that there’s this world that uses no Roman numerals, and it’s an educator who gets this wonderful idea, “Here’s Arabic, the kind of numerals that we use.” So this is a different what? Different curriculum? No, it’s not. And it certainly wouldn’t help those kids pass the, the standardized tests of their time, which were all about X’s and V’s and R’s.
But it serves the real purpose that the arithmetic serves. So we need a name for something, creating some different thing, and I know what the name is, but I’m gonna take off from the fact that it doesn’t seem so bad to say make them do math. And then I always like to make a huge emphasis, difference between math, which is what you learn in school, and mathematics, which is this great jewel of creation of the human mind.
And these are not to be confused. Math, this thing we do at school, it’s a particular field. Take that tiny little sliver of mathematical knowledge, selected. Well, why? Who selected it?
Nobody knows, but it was selected in times long ago And but it’s become Kind of sugarless And we think all kids should know that. And when I said we laugh at the people who said we should learn Latin, but we laugh too soon, I think one could laugh just as much and imagine people in the future laughing just as much about what we teach in kids at school. Because, well, let’s introduce a word for distinguishing there.
I’d like to distinguish between Latinesque justifications for putting something in and let’s say driver risk. Driver risk because here’s an example if you need. Talk about driving a car, learning to drive a car, and passing a test for your license. It’s fairly clear that there’s knowledge that you need that you’re actually going to use, and you need that knowledge Even then, it’s not necessarily.
Maybe somebody could invent a car that could be driven. But basically, I think that thinking of the spectrum it’s driven by what goes into the curriculum of driver education is what you need in order to drive. As opposed to Latinist justifications where I’ve not talked about in previous centuries, where Latin was the actual language that they spoke in universities.
But when I was a kid, nobody was learning Latin in order to speak Latin. We were all learning Latin because it would do certain other things. And I’m gonna discuss whether it does those other things. I’m interested in the form of the question. Now, if we think about what we learn in mathematics at school, it’s all Latinist. You– Almost all. Ninety-nine percent of it.
You know, who ever uses that formula for solving a quadratic equation? Or who ever factors polynomials? Or who ever– You can go run along and go look at it all. It’s not stuff that people are learning ’cause they actually want to use it. And there’s a semantic thing there that it’s interestingly hard to break through, that people say, “But you need to know math because the modern world, you can’t survive there.”
But that’s talk as if math, mathematics, was some unitary thing, which all or none, you have it or you don’t have it. And then, yes, I’m not gonna question, mathematics is a useful thing. But the question is what mathematics, and which mathematics, and which parts of it? And when you start breaking it down like that, it becomes less obvious that math is something that you have to know for this for this to exist in this world.
So I don’t know what to… Alternative maths, other maths. Let’s call them mathoids. A mathoid is something like– The use– Familiar example for mathoid is math in school. And what I mean by a mathoid, it’s it’s going to represent a certain slice of knowledge, the way it’s represented, but also a culture of how you think about it, how you talk about it, your attitude towards it How it reflects on your attitude towards yourself and people.
Like it’s part of the math culture, I would say, that we believe that some people are good at math and some people are not good at math, or that there’s such a thing as mathematical intelligence. This is part of a cul– That all goes as part of math. And there’s a moral aspect of it too, because there’s a hidden part of it that I think is terribly important, and that is that because it really is useless stuff in the real world, and because teachers are not brought up to believe that the Latinesque justification is a good justification, and they don’t know how to present the Latinesque justification to kids, when kids ask them, “Why should we learn this?”
The answer is always as if it was driveresque knowledge. That you’re gonna use it some way. But everybody knows you’re not gonna use it . So the net result of that is something extremely bad, and that is kids say, consciously or unconsciously, “There they go again, lying to us. It’s teacher double talk.” It’s undermining a sense of moral integrity in the education system, and nothing could be worse than that.
So all this gets together, that’s what math is. It’s that whole cluster of stuff. So we want to talk about what would we have to do to make another different mathoid?
Well, I am gonna say, let me tell my third, third parable, which is actually in Children’s Machine. That parable is it’s a, not quite a parable, but it’s a paraboloid interpretation of history. Leonardo da Vinci made some very ingenious sketches of flying machines. Now, Leonardo was a practical person.
He made many practical things. He took them and realized them, but he didn’t make a flying machine. How come? Now, if you look at his sketches, you can say they wouldn’t have worked, but that’s not why he didn’t make a flying machine. He didn’t make a flying machine because he didn’t have the technological infrastructure to implement his own ideas about making a flying machine, or else he would have found out they were not so good, as everybody does in first ideas on working, and he might have improved them.
But the reason why he didn’t do what the Wright brothers were going to do was not that his ideas were inferior to theirs. Well, they were, but I don’t know what the Wright brothers’ ideas were either when they first started. That’s kind of simply inferior to the Wright brothers’ ideas of implementing. But he couldn’t because there weren’t machine tools, there weren’t fuels, there weren’t engines, there weren’t materials.
There– nobody could invent all this stuff. It had to grow and develop before you could make a flying machine. And I think that there are many people who try to make methods or educoids or other forms of education. I mean, I think Dewey well, well, everyone, all the great thinkers about it.
But I think they were lucky enough. I think that before we had this digital technology, it was not feasible. There wasn’t the technological infrastructure to implement a method And let me say, as I should have said before, I’m using mathematics only as an extreme case, but I said that mathematics applies to everything.
So we now have this technology. How can we use this technology in this structure? Well, now, those in my parables, in a moment we’ll come back to sort of use them as reference points. But now getting into some more technical, some specific- One thing that I got from Piaget is his use of the word epistemology.
Because Piaget refused to call himself a psychologist. He didn’t call his, use the word psychology in his department. He was a fundamentalist man of genetic epistemology at that time, because he thought it was about knowledge And that’s how I came to be in this, because mostly you would think somebody who wanted to investigate how three, four, five-year-old children think about number, who would you find as your assistant for that?
Well, you’d find a psychologist, of course. Piaget wasn’t a psychologist. He was a mathematician. Because he was convinced that not that the mathematics in the child’s mind would emulate the development of mathematics in real mathematics being taught by mathematicians, but that the sort of same sort of nodal points, the same sort of obstacles come up for both of them.
And so he dared to think in the same breath about the foundations of mathematics and the mathematical thinking of children. And that baby has been thrown out with his bath, bathtub of throwing Piaget out because they found out that children have rudimentary forms of number perception and a few other things, and that Piaget was a little bit mistaken about the extent to which he thought babies came into the world with with no prior or very little prior knowledge.
Well, so we want to talk about, we’re going to be talking about ideas, about knowledge structures. And we find very little of this in, well- In, in, in education. I mean some examples, but there are a few other people I think who do deserve my list. I’m recently much impressed by rereading Karl Popper.
Popper, and I think that K- you know, Karl the writer… it’s not Karl Popper, is it? It is Karl Popper I think the writer is maybe the educator who has taken this most most to heart that when Popper talks about three worlds, there’s the world of physical things, then there’s the psychological world that these things are represented, and that’s world two.
And then his world three is this world in which ideas or not, in a way, intellectual constructs have a reality, and that’s the subject that we’re talking about. So I’m going to talk a little about the sort of developmental and, you know, the dynamic of intellectual constructs. Just one little facet of that’s easy to talk about that I’ve written about quite quite extensively, is a process that I call disempowerment of powerful ideas.
Incidentally, in parentheses, I don’t think Uri mentioned my book, Mindstorms, and it was very pleasing that many people read that book. I don’t think there were millions, but there certainly were hundreds of thousands. And it was most pleasing of all that I had no idea that teachers would like it, and that the most pleasing thing is large numbers of teachers who felt that somehow it articulated what they really thought and allowed them to think what they didn’t dare to think because they didn’t know the words to put it in, or they didn’t have a, some sort of external voice that was saying similar things.
But the sad thing about that book was that the book was subtitled Computers, Children, and Powerful Ideas. And all the reviews of the book, mostly, you know, good, some were bad, but they were all as if it was about children and computers. And in some ways, very painful to find a review about, “This is a wonderful book about children.”
I didn’t think I was writing a book about children and computers. I thought it was a book about ideas. And I thought the point about the computer is that it changes the relationship between children and ideas, the ideas that a child can get at different times. And if you throw that away, the point isn’t there anymore, or a point, whether this one, which for me is the central point.
So, pursuing this theme of powerful ideas let’s take an example. I’ll take an example that Uri has has pursued in depth as his PhD thesis and in a lot of the related work, and that’s the idea of probabilistic thinking or stochastic thinking Now probabilistic thinking, certainly in the history of science, is one of the all-time most powerful ideas.
It’s transformed physics, transformed biology, trans- made possible social sciences. It’s certainly, and however you define powerful, it is powerful. So it’s not surprising that when people think about introducing new stuff into the math curriculum, they say, “Well, let’s put in some probability.”
But putting in some probability NCTM style- … is giving some little problems like I got this one out of one of their publications. Well, how many girls, count how many girls and boys and girls, how many like chocolate ice cream, how many like vanilla ice cream? What’s the probability that Mary prefers chocolate?
That sort of stuff. Well, okay. But that, you would have no idea from that, that this was a powerful idea. It’s one more little ritualistic calculation, and maybe it’s fun because it’s about chocolate and vanilla or, you know, I don’t know. But it’s not, it doesn’t give that vibrant sense of here’s something that changed history Now how could we re-emp– so but, so what is this process, and it’s a systematic process that, that ideas as they come into the school world are disempowered.
Once you recognize this as a process and you can see the stages through it, and it’s a subject could be studied as a kind of opposite of a topic that has been studied by some philosophers of science and some psychologists of the development of, you know, conceptual development, how an idea starts simple and grows more complex.
Here, an idea starts complex and grows trivial. So if you recognize that, we might say, well, couldn’t we re-empower it? Under what conditions could it be re-empowered? Well, here are some conditions under which it can be re-empowered. For example, in the work we’ve done, it wouldn’t have been possible in Dewey’s time, okay?
We have children build robots out of Lego. It doesn’t matter that it’s not Lego. They build robots with computers in them, sensors, and they can program the robot to do something like go to a light.
Or anything of that sort.
Now so if I’m imagining this properly, how does it go to the light?
The next powerful idea I’m gonna come to is anticipated by this, that you might think the right way to do it is somehow it locates where the light is in XY space, XY coordinates, and it plots a path from where it is to where the light is, and that’s perfectly possible mathematically However, it’s normally really happens.
I’m sorry. It’s not what children do. But there’s another way of doing it, which is often snuck in as if it were a little thing that just gets us to work, but is really one of the other all-time powerful ideas called calculus, which Newton invented. Which is you don’t have to know that whole path. You just have to think of a law of motion in which at each instant the– you know what it’s going to do.
So you make a little algorithm which says, “I don’t, I mean, I don’t know where that light is. If I just know if it’s more to the left or more to the right, and turn a little bit in that direction, take a little step forward, and keep on doing this, eventually I’m gonna get there.” And that’s an incredibly important idea in engineering called feedback.
That’s a more important idea. It’s the core of what calculus is really about. It’s the core of what Newton did when he found out how to really understand the orbit of the moon and the Earth. One thing different from Kepler was Kepler looked at the whole orbit and its properties, and Newton said, “We don’t have to think the whole orbit and its properties.
We can think differentially what happens every little moment,” and that does it all. So with those– that idea, you can program this little robot here. Now, probability. What if there’s an obstacle? A sort of slippery pair. Let’s suppose there are physical obstacles. The robot can still see the light, but it goes up against it.
Again, you could imagine algorithm that detects the fact we’re running up an obstacle, that we can jump back off it, go around it, and, Well, you could, but there’s a terribly simple algorithm that doesn’t need to do that. And this terribly simple algorithm is just put a little bit of randomness in So the algorithm for the, for this robot is see whether the light’s more to the left or to the right.
If it’s more to the left, turn left. If it’s left, turn a little bit right. But every now and then, make a big turn at random.
It will still get there, provided you get your probabilities right. The probability of making the, the turn at random mustn’t be too big, and the turn mustn’t be too big. But there’s a right balance. And working with the quantitative probability, you can get at– You can make an algorithm that will… Well, if there’s no obstacle, it’ll make a little funniness in the turn, and it will get there a little less efficiently, but it will still get there.
And if there is an obstacle, it’s got a much better chance, hugely better chance of getting there if there’s an obstacle. So there’s a little bit of probabilistic thinking in a tiny little situation, but it’s powerful probabilistic thinking. And because even more powerful if you now recognize that’s the way nature did it too.
You know, all simple creatures operate like that. That’s why the butterfly goes flutters by. And so this kid will now understand it. So this kid, who could be a five-year-old, can be using probability in a powerful way, probabilistic thinking in a powerful way, in a quantitative way, in a way that gives a sense of I’m solving the problems in the way God has made to do it.
And I can see connections. And that’s just a little glimpse we could, with, by putting it into this kind of context So we get two ideas, two great ideas of mathematics. The differential idea and the probability idea, which I think are the two great ideas that made modern mathematics happen, which we can capture and translate into a form that’s accessible. So that I call re-empowerment, and that’s showing a kind of dir– an compass to use for looking for the right things to do with making this new math world.
And so looking for these powerful ideas I’d like… Can I– I’ve forgotten what the time- We could probably, you know, take some questions then Yes. Maybe talk for about four minutes and then take questions? Okay.
So, I’d like to one more– So that’s one kind of ingredient. Let’s look at ideals, and let’s look at… I mean, this is just a sample. The sample is empowerment, re-empowerment, disempowerment as a kind of dynamic category of thinking about what happens to ideals in a dynamic form. And as we can do that, we are groping in the dark The second kind of ingredient which is closely related to this is that the concept that we ear- introduced early into Logo called the turtle Should know. Who knows about the other two? Well, anyway. So, so, so just quickly for someone who don’t, that the– this idea was that the You have this, this entity, and it could be a physical thing, like this robot could be one of them, or it could be an abstract thing on the computer screen. But the important thing about this turtle is that it’s a mathematical entity that has a position and a heading, and it can be made to move in its heading or change, change its heading or move in this direction.
And you can program– once you– if you can program a computer even in simple ways, you can program this thing to do all sorts of geometric constructs So for example, if you wanted to, if you wanted to make a square, you program the turtle to go forward somewhat and get a right ninety If you wanted to make a triangle instead of a square, what you do instead of repeat four times that, then you repeat three times, and you have to change the direction. And most people guess, which is probably some of your heads, turn 60 degrees. And you would turn 60 degrees. It’s not enough.
Actually, you have to turn more than 90 degrees. But the difference between that and and what happens in the math, in the real, in the math, traditional math class is if you ru- you’re right or you’re wrong. But here you can try it and see what happens. You can think about it, and you can explore and get to think right answer.
But more important Is the fact that there’s– we’re looking for the entities that can put you inside the, the mathematics. This turtle is something you can identify with, and you can internalize it. By internalizing this, this turtle, once you internalize this turtle, you’ve got access to a huge range of mathematical knowledge that you can get by running in your head, if not on the screen, experiments with the turtle.
And that raises an interesting epistemology or question about kinds of knowledge. It’s become fashionable, I mean, under the influence of computers, to say, “Well, we should distinguish between knowing how,” that’s skills and procedures, “and knowing that,” that’s propositions and facts. But there’s another kind of knowing that’s vastly more important and maybe more deeply rooted biologically and neurologically in our brains, and that’s knowing somebody.
You know, I can get to know a person, and I internalize that person. And this is a kind of knowing that’s not knowing that or knowing how. It’s an important kind of knowing, and children do it, babies do it, and animals do it. And so this is another kind of dimension on epistemology. Let’s have an extra kind– another look at more kinds of knowing.
Knowing a thing, knowing this turtle and internalize. And we found looking where do you find discussion of that sort of thing? Well, in sci-fi and movie literature and in odd places that aren’t usually connected with math education. There’s a lot of knowledge, there’s a lot of thinking about internalizing objects.
And then it connects with some of our modern ideas about object-oriented programming and making the object the important thing. Again, you see that it’s in this context of the computer and the digital that we can open up dimensions of of epistemology, thinking about knowledge and ideas that weren’t there before.
So these are directions in making math books. And I’m going to say one other thing about about math. I was going to read this piece.
But I would like to read this part of it because- … because it’s quoting somebody, and this is where Chomsky comes into it. Let me quote. No, I can’t, excuse me. I’m gonna quote it. Yeah, I was gonna quote this, you know, after five minutes. I’m gonna quote this inaccurately The question, why is mathematics considered hard? There are some reasons, I think. In the Chomsky world, there’s a good reason.
It’s s- explicitly formulated by Pinker in his book, How the Mind Works, and that’s becoming extremely widespread as part of a kind of biologization of thinking. According to Pinker, language is easy to learn, and children learn it naturally because in the course of millennia, the genes were stamped by language, languages.
So we’ve got these innate mechanisms that put them in. School mathematics, he says, has not had the time to stamp the genes. Now, I’d like to just make the point about the mathoid as a, as an object to create a little– give it another dimension by suggesting a different kind of hypothesis. That it’s not that the genes didn’t have time to get changed by the mathematics, but the mathematics was not allowed to adapt itself to the genes.
The point is, we don’t have genes that make us talk English. We have genes that, that give rise to– that do not create the conditions where language can develop. So the development of language is a process of something forming that’s coherent with– rooted in the– that makes use of the human. Now, why didn’t this happen with mathematics?
Well, now first of all, it happened with some mathematics, and there’s all this stuff with Piaget and L5. But why didn’t it happen with algebra, which is what Pinker– Well, it didn’t happen with algebra because we haven’t allowed it. Because algebra was the creation of grown-up intellectuals who made this thing in the form that they thought was the right way, and you just have to do it the way that they thought it.
So there’s not been allowed the process of evolution of a mathoid that would be consistent with the genetic structures. So, that’s the dimensions of mathoid. And now, just quickly on two other themes. Quickly. First is, I was complaining about schools And of course it’s the school where the rubber hits the road, so to speak, when you make changes. If the school wanted to change, you’re gonna have a terrible time getting it in. It strikes me as very amazing that an example of a disconnect of ideas that many people who say, “We’re developmental, developmentalist, constructivists, and we would never tell the children the right answer.
What we must do is create the conditions where the children will find it and develop it,” and so on. And that’s obviously– I mean, we can use a few minutes to– I don’t need this oversimplification. But once they’ve formulated this, they go and tell the teachers and the school system “This is the thing that, that you ought to be do…” So they’re not applying their ideas of constructivist learning to the people they really want to change, which is the school. So this is yet another thought, that we have to find ways of finding for the school to be able to do. Now, ideas for that, I just mentioned one, and that is, again, from Piaget, the idea of assimilation and accommodation.
For Piaget, assimilation always comes first. Now, in educational reforms, if you read books like Tyack and Cuban and others, they make much of the fact that there are these great reformers who come up with an idea, and the reform wants to change school, but change– school changes the reform. Great phrase of Tyack or Cuban or whoever did it.
It’s in the Tyack and Cuban book what’s it called? Tinkering toward Utopia.
Tinkering towards Utopia. But now, I think that they got the wrong end of the stick. We shouldn’t complain about assimilation. Assimilation has to happen. What we’ve got to do is to design our interventions in such a way that they can be assimilated, and where the design criterion is once assimilated there in a form that we know they’re gonna be assimilated, and they’re going to start a disequilibrium inside and give rise to a process.
And so this is really what I’m trying hard to do, to create kinds of elements of mathematical and other kinds of learning that would go into into the school system with the idea it’s going to be assimilated, but good, not bad. And, but we calculated the kind– So we’re trying to look at the kind of assimilation so that will, in the long run, turn into something different