PLATO wrote over his door, "Let only geometers enter." Times have changed. Most of those who seek to enter Plato's intellectual world neither know mathematics nor sense the least contradiction in their disregard for his injunction. Our culture's schizophrenic split between "humanities" and "science" supports their sense of security. Plato was a philosopher, and philosophy belongs to the humanities as surely as mathematics belongs to the sciences . . . .

The status of mathematics in contemporary culture is one of the most acute symptoms of this dissociation. The emergence of a "humanistic" mathematics, one that is not perceived as separated from the study of man and "the humanities," might well be the sign that a change is in sight. So in this [study] I try to show how the computer presence can bring children into a more humanistic as well as a moe humane relationship with mathematics . . . .

I have asked many teachers and parents what they thought mathematics to be and why it was important to learn it. Few held a view of mathematics that was sufficiently coherent to justify devoting several thousand hours of a child's life to learning it, and children sense this. When a teacher tells a student that the reason for those many hours of arithmetic is to be able to check the change at the supermarket, the teacher is simply not believed. Children see such "reasons" as one more example of adult double talk. The same effect is produced when children are told school math is "fun" when they are pretty sure that teachers who say so spend their leisure hours on anything except this allegedly fun-filled activity. Nor does it help to tell them that they need math to become scientists -- most children don't have such a plan. The children can see perfectly well that the teacher does not like math any more than they do and that the reason for doing it is simply that it has been inscribed into the curriculum. All of this erodes children's confidence in the adult world and the process of education. And I think it introduces a deep element of dishonesty into the educational relationship . . . .

I see "school math" as a social construction . . . . Like the QWERTY arrangement of typewriter keys, school math did make some sense in a certain historical context. But, like QWERTY, it has dug itself in so well that people take it for granted and invent rationalizations for it long after the demise of the historical conditions that made sense of it. Indeed, for most people in our culture it is inconceivable that school math could be very much different: This is the only mathematics they know. In order to break this vicious circle, I shall lead the reader into a new area of mathematics, Turtle geometry, that my colleagues and I have created as a better, more meaningful first area of formal mathematics for children . . . .

Turtle geometry is a different style of doing geometry, just as Euclid's axiomatic style and Descartes' analytic style are different from one another. Euclid's is a logical style. Descartes' is an algebraic style, Turtle geometry is a computational style of geometry.

Euclid built his geometry from a set of fundamental concepts, one of which is the point. A point can be defined as an entty that has a position but no other properties -- it has no color, no size, no shape. People who have not yet been initiated into formal mathematics, who have not yet been "mathematized," often find this notion difficult to grasp, and even bizarre. It is hard for them to relate it to anything else they know. Turtle geometry, too, has a fundamental entity similar to Euclid's point. But this entity, which I call a "Turtle," can be related to things people know because unlike Euclid's point, it is not stripped so totally of all properties, and instead of being static it is dynamic. Besides position the Turtle has one other important property: It has "heading." A Euclidean point is at some place -- it has a position, and that is all you can say about it. A Turtle is at some place -- it, too, has a position -- but it also faces some direction -- its heading. In this, the Turtle is like a person -- I am here and I am facing north -- or an animal or a boat. And from these similariteis come the turtle's special ability to serve as a first representativa of formal mathematics for a child. Children can identify with the Turtle and are thus able to bring their knowledge about their bodies and how they move into the work of learning formal geometry.

To see how this happens we need to know one more thing about Turtles: They are able to accept commands expressed in a language called TURTLE TALK. The command FORWARD causes the Turtle to move in a straight line in the direction it is facing . . . . To tell it how far to go, FORWARD must be followed by a number: FORWARD 1 will cause a very small movement, FORWARD 100 a larger one. Many children have been started on the road to Turtle geometry by introducing them to a mechanical turtle, a cybernetic robot, that will carry out these commands when they are typed on a typewriter keyboard. This "floor Turtle" has wheels, a dome shape, and a pen so that it can draw a line as it moves. But its essential properties -- position, heading, and ability to obey TURTLE TALK commands -- are the ones that matter for doing geometry. The child may later meet these same three properties in another embodiment of the Turtle: a "Light Turtle." This is a triangular-shaped object on a television screen. It too has a position and a heading. And it too moves in response to the same TURTLE TALK commands. Each kind of Turtle has its strong points: The floor Turtle can be used as a bulldozer as well as a drawing instrument; the Light Turtle draws bright-colored lines faster than the eye can follow. Neither is better, but the fact that there are two carries a powerful idea: Two physically different entities can be mathematically the same (or "isomorphic").

The commands FORWARD and BACK cause a Turtle to move in a straight line in the direction of its heading: Its position changes, but its heading remains the same. Two other commands change the heading without affecting the position: RIGHT and LEFT cause a Turtle to "pivot," to change heading while remaining in the same place. Like FORWARD, a turning command also needs to be given a number -- an input message -- to say how much the Turtle should turn. An adult will quickly recognize these numbers as the measure of the turning angle in degrees. For most children these numbers have to be explored, and doing so is an exciting and playful process . . . .

Since learning to control the Turtle is like learning to speak a language it mobilizes the child's expertise and pleasure in speaking. Since it is like being in command, it mobilizes the child's expertise and pleasure in commanding. To make the Turtle trace a square you walk in a square yourself and describe what you are doing in TURTLE TALK. And so, working with the Turtle mobilizes the child's expertise and pleasure in motion. It draws on the child's well-established knowledge of "body-geometry" as a starting point for the development of bridges into formal geometry.

The goal of children's first experiences . . . is not to learn formal rules but to develop insights into the way they move about in space. These insights are described in TURTLE TALK and thereby become "programs" or "procedures" or "differential equations" for the Turtle . . . .

The same strategy of moving from the familiar to the unknown brings the learner into touch with some powerful general ideas: for example, the idea of hierarchical organization . . . , the idea of planning in carrying through a project, and the idea of debugging . . . .

Typically in math class, a child's reaction to a wrong answer is to try to forget is as fast as possible. But [here] the child is not criticized for an error in drawing. The process of debugging is a normal part of the process of understanding a program. The programmer is encouraged to study the bug rather than forget the error. And in the Turtle context there is a good reason to study the bug. It will pay off . . .

These examples show how the continuity and the power principles make Turtle geometry learnable. But we wanted it to do something else as well, to open intellectual doors, preferably to be a carrier of important, powerful ideas. Even in drawing these simple squares and stars the Turtle carried some important ideas: angle, controlled repetition, state-change operator. To give ourselves a more systematic overview of what children learn from working with the Turtle we begin by distinguishing between two kinds of knowledge. One kind is mathematical: The Turtles are only a small corner of a large mathematical subject, Turtle geometry, a kind of geometry that is easily learnable and an effective carrier of very general mathematical ideas. The other kind of knowledge is mathetic: knowledge about learning . . . Of course, the two overlap.

We introduced Turtle geometry by relating it to a fundamental mathetic principle: Make sense of what you want to learn . . . . Turtle geometry was specifically designed to be something children could make sense of, to be something that would resonate with their sense of what is important. And it was designed to help children develop the mathetic strategy: In order to learn something, first make sense of it.