When I was asked to write this Education Forum, I was in the process of correcting the proofs of my book, Mindstorms: Children, Computers and Powerful Ideas. There I struggled to present in two hundred pages a vision of a few ways in which computers might affect how children learn; it is challenging now to find the right 3000 words to convey something of the same vision. What images, what metaphors best capture for me the essence of the computer as it might enter the lives of children?

I start with an image, more general than the computer, that has helped me to think about how the world takes up any new technology. The first movies were made by setting the newly invented motion-picture camera in front of a stage where a play was performed just as plays always had been. Only after some time did cinema become more than theatre plus camera. When it did, what emerged was something original and unique, a whole new culture with new modes of thinking and new breeds of people-stars, directors, scriptwriters, cameramen, critics, and audiences whose sensitivities, expectations, and ways of seeing were quite different from those of the theatre-goers of the past.

So too with the computer. The first instinct of educators is to couple the new technology to their old methods of instruction. My vision is of something much grander. So I dream of using this powerful new technology not to "improve" the schools we have always known (and, to be honest, hated) but to replace them with something better. I do not believe that this something will look anything like what is now known as "computer-aided instruction" (CAI). I think it will be more like the growth of a new culture, a "computer culture" in which the presence of computers will have been so integrated into new ways to think about ourselves and about the subject matters we learn that the nature of learning itself will be transformed.

In thinking about the nature of such potential transformation, the LOGO group of the Massachusetts Institute of Technology (MIT) Artificial Intelligence Laboratory has been guided by the idea of creating computer-based environments in which mathematics and other areas of "formal" learning can be learned in a natural fashion, much as a child learns to speak; and applying concepts from artificial intelligence to children's learning, to help children become articulate about, aI;ld thus gain control over, the learning process. Before developing these ideas, I would like readers to clear their minds of a misleading but common image. People generally think about computers in schools as a scarce resource to which students have occasional access. It is time we learned to think in terms of a computer for every child, and we should think about children having access to computers from infancy. If we think in these terms, we begin to recognize that there is a clear discontinuity between the current ideas about using computers in schools and the situation of the future. I really believe that almost everything being done today is only relevant to the future in that it sets a bad example so that people become accustomed to primitive models.

A natural place to begin a search for "something new in education" is to look for examples of highly successful learning. For me the most dramatic image of successful learning is the way children learn to talk. This learning contrasts with school learning in many ways, of which I think two are most important. First, it is highly successful: all children learn to speak the colloquial dialect in which they grow up. Second, it has none of the technical paraphernalia ofschooling-no curriculum, no set lesson times, no quizzes, no grades, no professional teachers. It is part of living. I call it learning-without-teaching or "Piagetian learning (after the Swiss philosopher-scientist Jean Piaget who has done more than anyone else to show us how very much children learn ift this way).

Much of the work done to date in the whole area of computers and education -- eg: CAI -- has promoted a style of learning that gives the impression of a child being "programmed" by the computer. Our approach has been diametrically opposed to that. By striving to make the computers processes as transparent as possible and creating activities in which children "teach" (ie: program) computers in a well-structured, procedural language like LOGO, we have aimed toward putting children in control of their own learning. Obviously, I cannot hope to explore these ideas in much depth in a short space. What I shall try to do is to describe a couple of learning environments we have created which I believe challenge the fundamental assumptions our society makes about children and learning.

### Mathland

The belief that only a few people are mathematically minded is a truism in our culture and a cornerstone of our educational system. It is therefore sobering to reflect on the f1imsiness of our reasons for believing it. In fact, the only evidence is crass empiricism: look around and you will see that most people are very poor at mathematics. But look around and see how poor most Americans are at speaking French. Does anyone draw the conclusion that most Americans are "not French-minded?", that they are not capable of learning French. Of course not. We all know that these same people would have learned to ;peak French perfectly well had they grown up in France. If there is any question of lack of aptitude, the aptitude they lack is not for French as such but for learning French in schools.

Could the same be true of mathematics? Could there be a place, a "mathland," which is to mathematics as France is to French, where children would learn to speak mathematics as easily and as successfully as they learn to speak their native dialect? I believe that the answer is Yes. In Mindstorms I suggest that the world we live in contains pockets of mathland, which explains why all children learn some mathematics spontaneously (eg: one-to-one correspondences, conservation of number, reversibility of logical operations) and some children become very good at it. Here I have space only to talk about some ways in which the world could become much more of a mathland for everyone.

Computers are the Proteus of machines: they take on many different forms. One of their manifestations is as mathematics-speaking beings. If children grew up surrounded by such beings, the learning of mathematics might very well be much like the learning of spoken language. Developing and testing this image has become a central research question for us at MIT: under what conditions will children talk in mathematical languages to mathematics-speaking computers? The results have already convinced us that the idea of mathland is fundamentally sound and that, indeed, what the mathematics schools fail to teach can be learned successfully on the model of picking up living languages.

But computers do not automatically create that result. For example, instructing computers in FORTRAN to manage inventories is of no interest to the average child. Babies brought up in IBM computer centers will be no better at mathematics than any others. They may even be worse {and their other lapses of culture might be more disturbing). In order for computers to play the role of mathland for a child, two conditions are necessary: the computer must understand a language a child can learn (and love to learn), and the computer must be able to do something for the child.

### Euclidean Geometry-Cartesian Geometry-Computational Geometry

Turtle graphics is this kind of mathland. It was first developed in our laboratory as part of the programming language LOGO and then taken over by several other languages including Smalltalk and UCSD-Apple Pascal.

A lot of experience has taught us that computer graphics can be a great turn-on. People of all ages enjoy putting images on the screen, and when these images can be made to move and change color, they acquire a dimension completely lacking in conventional pencil-and-paper drawing. At the heart of the work on turtle graphics is the idea of developing a new kind of geometry--"turtle geometry"--which provides powerful and yet easily accessible means to manipulate shapes and motions. To put this in perspective, recall that you probably encountered at school at least two styles of doing geometry: Euclid's style (primarily logical in structure) and Descartes' style (primarily algebraic). Turtle geometry is a new style matched to the computer: it is a computational style of thinking about geometry. The difference in spirit is illustrated by how one thinks about a familiar geometric object in Cartesian and in turtle geometry. Descartes taught us to think of the circle as an equation such as:

x<sup>2</sup> + y<sup>2</sup> = R<sup>2</sup>

In turtle geometry it is possible to use such equations, but the natural way to think about a circle is as a process. To do this, turtle geometry adopts as its fundamental concept an entity called a "turtle" whose properties include its position (as does the point in Euclidean and Cartesian geometry) and also its heading. At any particular time, it is at a position and is facing in a particular heading. The position and the heading are changed by commands that are built into a programming language. Among these are **FORWARD <some number>** which causes the turtle to move in the direction of its heading without changing the heading, and **RIGHT <some number>** which causes the turtle to change the heading while keeping the position fixed; ie: to pivot in place. Given these commands, a program in LOGO to draw a square of a certain fixed size takes the simple form:

TO SQUARE FORWARD 100 RIGHT 90 FORWARD 100...etc

A slightly more sophisticated program to draw squares of varying size takes the form:

TO SQUARE SIZE REPEAT 4 [FORWARD :SIZE RIGHT 90]

Now we can think of a circle as generated by:

TO CIRCLE REPEAT 360 [FORWARD 1 RIGHT 1

More sophisticated programming leads to circles of variable diameter and even to letting the number of steps go to the limit, but the simple example will illustrate the main point I want to make here. Children can solve the problem of drawing a circle by using a very powerful heuristic principle: play turtle, walk out yourself what you want the turtle to do and describe what you did in turtle language. The children are practicing a lot of powerful ideas. They are exposed to the idea of using heuristic knowledge, they are learning to think of formal mathematics as rooted in (not opposed to) intuitive body-mathematics, and they are using mathematics as a language; moreover, they are learning to think about mathematics not as a ritual to be learned by rote but as an instrument to be used for personal ends.