Uses of Technology to Enhance Education

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Artificial Intelligence Laboratory

Artificial Intelligence Memo No. 298  |  LOGO Memo No. 8
by Seymour Papert
June 1973

Source: Papert, S. (1973). Uses of Technology to Enhance Education (MIT Artificial Intelligence Laboratory Artificial Intelligence Memo, Issue. 

This paper is the substance of a proposal to the N.S.F. for support of research on children’s thinking and elementary education.

This work was supported by the National Science Foundation under grant number GJ-1049 and conducted at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology.

NSF PROPOSAL — SEYMOUR PAPERT

Organization of the Four Sections:

Section 1: Schematic outline of project and what we want. Hardly any intellectual content.

Section 2: Statement of our goals in general terms. This statement is intended to have serious intellectual content but lacks meaty examples. Readers who find it too abstract for comfort might like to read at least part of §3 first.

Section 3: A series of extended examples intended to give more concrete substance to the generalities in §2.

Section 4: This is the real “proposal”. It sets out specifically a list of concrete “goals” on which we want to work in the immediate future.

Appendix: Papers by Jeanne Bamberger, Marvin Minsky, Seymour Papert, and Cynthia Solomon.

SECTION 1: WHAT WE WANT

1.1 We Want Time

Over the past five years a certain style of research on Elementary Education has developed within the Artificial Intelligence Laboratory at M.I.T. The funding of the research has been on a relatively short term and projectoriented basis. We feel that the work has matured to a point that requires somewhat longer term planning and that the research style has proven itself sufficiently to justify this confidence. The need for longer term stability comes from three sources:

1.1.1 People Need Time To Develop

A key feature of the research style is a much deeper than usual melding of competence and creativity from different areas, such as mathematical sciences/cognitive science/computer science/educational practice. In order for this to take place satisfactorily, individuals need to be immersed in the project for a sufficient period of time.

We feel that less than a three year period is inadequate.

1.1.2 Ideas Need Time to Mature

Our most successful concepts, such as LOGO itself and Turtle Geometry, have typically taken about three years to progress from inception to maturity.

1.1.3 Children Need Time to Develop

We need badly to study the effects of exposing children to our learning environments over larger periods of time. We have often noticed in working with children over periods of a half or whole year that their development into the kind of thinking we try to foster goes at an increasing rate over the period.

1.2 We Want More People

We have demonstrated the richness for Education of a thorough integration of imaginative competence in several fields. Our team is still below critical mass for this. We believe that three new research associates (postdoctoral) and more research assistants (graduate students) would make a vastly more than proportional pay-off in productivity and in writing about what we produce. The possibility of attracting high calibre people in these categories has been enhanced by the recent formation at M.I.T. of an “Education Division” which will make it much easier for graduate students at M.I.T. to choose Research in Education as their primary academic focus. This research project is very closely associated with the new Division through overlap of people and of intellectual focus.

1.3 We Want a Workshop

We have built some effective educational devices using general facilities of the Artificial Intelligence Laboratory. Several factors make this arrangement increasingly unsatisfactory and we feel that this aspect of the project has proven itself sufficiently to deserve its own facility.

1.4 We Want Our Own Experimental School

We have been able to develop ideas and materials through working with children “lent to us” for a few hours a week by their normal public school. We are ready to move to the next stage of testing how the intellectual development of children will respond to a totally redesigned learning environment.

SECTION 2: GENERAL GOALS

We want to make progress on the general goals described below. The sub-sections numbered 2.x will convey the form of our research style and the structure of general subgoals on which work is either actively in progress or in an advanced stage of planning. Each section describes in general terms a kind of research goal, illustrated, where possible, by examples drawn from our previous work.

More specific information will be found in subsequent sections (3 & 4). Although we claim the right and the need to work in an open-ended spirit, we do, of course, have a clear and very specific map of what we shall be doing in the immediate future. But to communicate one’s reasons for thinking there is more gold in those hills is a much more complicated and uncertain business than to show the nuggets one has already found. However we must try. Section 4 will sketch specific research projects in progress or advanced states of planning. Sections 2 and 3 will develop enough of our philosophy of education to provide a perspective for these projects. The research plans mentioned in section 4 contain some projects we would not be able to carry out entirely with funds requested in this proposal. We include them because they contribute to the conceptual coherence of our plans and because they do overlap projects for which we are here seeking support from the N.S.F.

The general goals will be described in this section under the following sub-headings:

• 2.1 Compelling Examples of the Uses of Technology to Enhance Education
• 2.2 New Conceptualizations of Knowledge
• 2.3 A Cognitive Theory About and FOR Children
• 2.4 Studying Heuristics
• 2.5 Relationship of Our Work to Schools

◦ 2.5.1 Material Suitable for Use Within the Traditional School

◦ 2.5.2 New Concepts of “School”

◦ 2.5.3 Intellectual Centers Parallel to Schools

◦ 2.5.4 Opening Avenues for the Severely Physically Handicapped

◦ 2.5.5 Remedial Mathematics for Adults

◦ 2.5.6 Paradigm for Experiments on Developmental Psychology

• 2.6 A New Kind of Professional for Research on Education
• 2.7 Mundane Aspects of Computers

2.1 Compelling Examples of the Uses of Technology to Enhance Education

One of our explicit goals is to provide compelling examples to show how technology can be used in education more profoundly and more imaginatively than has previously been done. It is frequently asserted that informational technology ought in principle to be a great boon to education but that current ideas on how to use it are insufficiently developed if not frankly superficial. (See for example the Carnegie Commission Report: “The Fourth Revolution”; James Koerner’s recent article in Saturday Review and many others.) We offer our work as an example to show that while this stricture might apply to current practice in C.A.I., the bottleneck to progress is not a lack of ideas.

The main thrust of our examples has been to show that the experience of program controlled devices can be used to give children, to a quite unprecedented degree, a sense of the power of ideas in general, of science in particular, and especially of mathematical science. In a suggestive aphorism we might say: we have been able to give children a mathematical experience more like an engineer’s than like a bookkeeper’s.

To do this it is, of course, not sufficient merely to have a computer. It is necessary to develop contexts in which the computer can be used by a child to serve real, personal purposes. Such a context needs be both material and conceptual. The material facet of our work consists of making computer-controlled devices a child can use for projects with a high potential for personal involvement, intellectual adventure and cognitive enhancement. Devices of this nature already constructed are: a music generator which enables a child to embark on experiments in composition, musical games, etc.; a graphics system with the capability necessary for simple animated cartoons; cybernetic “animals” — turtles, spiders and worms, motors, relays, etc., etc.

The conceptual facet of our work consists of making intellectual tools designed to give children the power to use the devices. The obvious item in our tool kit is a programming language; but this is far from enough; to make a computer generate music, pictures or mechanical processes one must also have the mental tools to think about temporal, tonal, geometrical and physical matters and much more beside these.

A big component of what one needs besides specific technical knowledge (of music, geometry, etc.) is general heuristic knowledge related to the skill of carrying out a complex project. Concepts related to this include: planning, debugging, modular structure, hierarchical structure, model. All this is what is common to the scientists’ task of making a theory that works, the engineers’ task of making a machine that works and the administrators’ task of making an organization that will work. And all this is what is perhaps most disastrously missing from the traditional school experience especially of the sciences, most especially of mathematics.

Let us focus on one particular component of this knowledge: the art and techniques of experience of “debugging”. The school experience of mathematics is dominated by the normative attitude implied by “right answer vs. wrong answer”. The mathematician’s experience of mathematics is dominated by the purposefulconstructive attitude implied by the struggle to “make it work.” He abandons an idea not because it happened to go wrong, but because he has understood that it is unfixable. Dwelling on what went wrong becomes a source of power rather than a piece of masochism (as it would appear to most fifth graders in traditional math classes). We contend that ours is the only clearly defined proposal for producing this shift of attitude in the elementary school. To do this we change the context of mathematical work from “workbook exercises” to using mathematical ideas to dominate a powerful technology.

The spirit of what is being said can be enriched by some quick references to points of contact with people we consider as intellectual allies. Using a phrase from Illich we would say we are fashioning the computer into a convivial tool, and science in general into a convivial mental tool.

Simon’s concept “The Sciences of the Artificial” has also greatly influenced us. These sciences are important both in their application and in their inherent simplicity and intelligibility. They should be at least as explicitly represented in the lives of children as “Natural Science”. Simple but completely functioning models and “minitheories” help immensely a child know “where he is at” in exploring the complex network of ideas about real, natural systems.

There is an obvious flavor of Dewey in our thinking. Learning is best when embedded in living experience. Dewey’s followers fall into hollow romanticism for lack of the technical means to embed the learning of complex modern knowledge in meaningful experiences.

Edith Biggs, Dienes, Gattegno and the blocks, rods and sticks we see in the infant schools are steps in the direction we like; we are trying to take giant leaps in the direction they have defined. Piaget is too close for brief comment except to say that we see much of the “Piaget and Education” community as standing him on his head by emphasizing the negative aspect of his work, namely his demonstration that children of certain ages have surprising “deficiencies” (which some say should be “remedied” by the schools). Much more important in our view is the demonstration that normal kids “remedy” these “deficiencies” all by themselves, without formal teaching. We’d rather see more knowledge, acquired in the way children (successfully) acquire “conservations”, than see “conservations” foisted on children in the way schools (usually unsuccessfully) try to teach mathematics.

2.2 New Conceptualizations of Knowledge

Under this heading comes the real challenge for the future. Disappointingly few of those who quote, or even adopt, our work see this, or view the steps taken so far as pointing to it. We are quoted (and praised or criticized) as making “turtles”, giving children computer graphics and designing programming languages. Well we are pleased to have done that. But the real tasks were more fundamental and more mathematical than technological. A typical question was: how could children make computer controlled displays do anything? For example, would they have to use Cartesian coordinates? Our turning point came when we decided to stop scratching around in the mathematician’s cupboards looking for already elaborated geometries that might do the job. Instead, we decided to make our own. And after a number of false starts and many ideas from many people in various corners of M.I.T., there gradually took form a new piece of mathematics, now called Turtle Geometry.

Of course, nothing is really new. Once it is made, we see many points in common with established geometry. But this does not undermine our thesis that we have stumbled on a new paradigm for research in education. Most such research accepts a given body of knowledge and worries about how to deliver it into the heads of the children. We say: no, what you want is to create new, more suitable knowledge. This leads into research that looks more like — indeed actually is — mathematical research.

More generally: we said we want to give the child a mathematical experience like an engineer’s and we cited the new technologies as providing a material a child can engineer. But to carry this idea into practice we were faced with the problem of giving a child access to necessary knowledge of mathematics, physics, control theory, programming, etc., etc. The problem is: what knowledge is really necessary, and can it be formulated in learnable sequences for our purposes? Here we find ourselves in virgin territory. What knowledge, what intellectual structures do you really need to be an engineer, to dominate and manipulate the physical world? There is at least one aspect of the answer to this question about which we feel firm: the traditional knowledgeset followed by all the high schools and engineering schools may be sufficient (for some students) but it is neither necessary nor optimal. If it were necessary our enterprise would be hopeless!

Two of the more compelling examples of what we have been able to achieve so far in this direction are:

• Turtle Geometry: which provides a conceptual frame for manipulating geometric objects without the algebra needed for doing the same job in the Cartesian frame.
• Our Qualitative Physics Projects: (which interlocks with Turtle Geometry) which has shown how to develop enough mechanics for a usable subset of control theory and planetary theory without anything resembling the familiar theories of differential equations or integration. We do not mean anything as trivial as replacing conceptual understanding of integration by “number crunching”! Physicists who glance at §3.3 will see why we claim that our mechanics is as theoretical and abstract as the classical ones; it is merely different, conceptually clearer and much more accessible.

How far one can go remains to be seen. We see the fragments of progress we have made as compelling evidence for our thesis that much of science can be reconceptualized to become vastly more accessible. But scarcely anyone ever tries when the existing conceptualizations are perfectly satisfactory for working scientists.

Some readers might protest that this is unfair to the physicists and mathematicians who worked so hard on the curriculum reform movements (PSSC, SMSG, etc.). We are not trying to devalue the intellectual quality of their work. But we are trying to draw a distinction between making local changes in the exposition of, say, traditional physics, and globally changing the structure and conceptual foundations of the subject. To a first approximation, the PSSC physics book defines the same concepts, states the same “laws” and uses the same mathematical formalisms and theorems as the traditional books. Whether it does this very much better than they did is not the point at issue here. We are trying to define the conceptually different enterprise of defining other principles, other concepts, other theorems… to arrive by a different route at the same ultimate conceptual and instrumental mastery of the physical world.

The project of re-conceptualizing areas of knowledge applies also to some usually regarded as extra-scientific. Music is one which several members of our research team (particularly Jeanne Bamberger) are pursuing. In principle vast new horizons can be opened for any child who likes music by having access to a computer with a music generator. In particular he is no longer prevented by lack of dexterity from exploration of composition and other musical experiments. The computer becomes an obedient orchestra and will play any piece the child can describe. But in what formalism will he describe it?

As Turtle Geometry gives a child a grasp on movement in space (which he may use in geometric or physical applications) so we need to develop ways to think about relations in time and in “tonal space” to give him a similar control over music. Work in this area started in our laboratory later than work on geometry, and has not reached the same level of development. But we see it as exceedingly important for a number of reasons of which we mention only two. The first is the importance of music for many children. The second is the importance of being able to think clearly about time for many purposes other than music. It is curious that the study of the temporal is so superficially represented in thinking about Education despite its importance and the difficulties students manifestly have when dealing with dynamic problems.

There is a minor paradox which we must draw to the attention of any reviewer who might have missed it. If you want to make mathematics for children you work terribly hard to make the mathematics as simple and transparent as possible. You would like to make it almost unnoticeable, so that children would learn it like we all learned our mother tongue, without even knowing there was anything difficult. Fine. But then people (like reviewers) pick it up without noticing it; and they say: what on earth have you been doing? In our case (as we have already noted) no one quite says that; but people often praise us for machines and don’t notice what we really are proud to have done. Even when they talk about the machines, they fail to notice what is mathematically interesting about, say, the turtle, namely the independence of forward and rotational motions.

2.3 A Cognitive Theory About and FOR Children

Much of our work strikes educators (of very different theoretical persuasions) as valuable “in its own right” without reference to an explicitly spelled-out cognitive theory. We have shown that children, even mathematically recalcitrant ones, learn geometry very well through working on computer graphics. Since everyone believes that it is desirable for children to learn geometry, there is from one point of view, nothing more to be said in justification except to ask whether the results are generalizable. But from another point of view there is a great deal to be said about the relation of these experiments to theoretical questions. The purpose of this section is to explain the latter point of view and indicate what we mean to do about it.

The Goose and the Golden Eggs

Turtle Geometry can be (has been!) judged and found acceptable by educators of many theoretical persuasions. But they didn’t discover it. We did and we see its discovery as having been guided not merely by technological and mathematical thinking, but also by a general cognitive theory, which has gradually emerged in the Artificial Intelligence Laboratory. We are very unwilling to let the goose starve while we enjoy the golden eggs.

Artificial Intelligence and the Philosophy of Education

The cognitive theory underlying our work draws on ideas from the Piagetian tradition of thinking about children and from those aspects of Artificial Intelligence concerned with thinking about thinking in general.

Recommended literature for detailed discussion:

• S. Papert, “Teaching Children Thinking”
• M. Minsky & S. Papert, “Artificial Intelligence” (A Progress Report available from the A.I. Lab)
• T. Winograd, Understanding Natural Language, Academic Press, 1972.
• T. Winograd & S. Papert, Lecture Notes in Process Models for Psychology, Rotterdam University Press, 1973.
• P. Winston, “Learning Structural Descriptions from Examples” (Ph.D. Thesis)
• M. Minsky & S. Papert, “Thinking About Thinking”
• G. Sussman, “A Theory of Skill Acquisition” (Ph.D. Thesis)
• I. Goldstein, “An Intelligent Model for LOGO” (Ph.D. Thesis) Does Knowing Impede Doing?

There is a popular trend in cognitive theory and educational practice with which we are in most direct conflict: the prevalent idea that “verbalized” knowledge is undesirable for elementary mathematics; instead, it is said, children should “discover” concepts through “non-verbal intuitive” processes. Manifestations of this trend are found in the writings of J. S. Bruner (for example his doctrine of the “impotence” of “words and diagrams” in the acquisition of “enactive knowledge”), of M. Polanyi, of H. Furth and many other currently influential authors.

The existence and importance of “non-symbolic” knowledge has received considerable attention in the Artificial Intelligence Laboratory quite apart from our interest in education. If real it would obviously deeply affect the general enterprise of representing knowledge in computers: if true it implies the need either to seek “non-symbolic” representations in machines or to recognize that certain intellectual functions could only be achieved in machines (if at all!) through processes fundamentally different from those operative in human intelligence. But everything we know about machine intelligence runs counter to both those suggestions.

From these studies has emerged a more sophisticated concept of what it is to be “symbolic” or “verbal” and a specific theory of a methodological trap which caught Bruner, Furth and others. Stated simplistically this theory is: Bruner’s language quite possibly is impotent for the purpose, say, of telling someone how to ride a bicycle. But it does not follow that language as such is impotent for this purpose. Indeed a major contribution of the Information Sciences (including control theory, computer science, A.I., etc.) is to have made more powerful languages for describing complex processes. Thus we ask the following question of anyone who concludes that “you can’t tell someone how to do so-and-so”:

Do you mean you failed to tell him in the available versions of ordinary natural language or do you mean that he could not learn a technical language in which he could be told? If anyone is brave enough to take the second option we ask further: “can you explain to us how you managed to consider all possible languages?”

Translated into Educational terms these ideas lead to the enterprise of developing a language in which to talk to children about cognitive matters about which one is normally silent.

Time Constraints of Experiments on Learning

The idea of constructing special languages to express ideas and procedures usually classified as “intuitive” leads to an important methodological point about the form of experiments in cognitive psychology and education.

Consider a typical experiment on a question like: Does Verbal Instruction Help or Hinder Learning a Physical Skill such as Juggling? The standard methods for such an experiment take about as much time as is needed to acquire measurable proficiency in juggling — at most a few hours.

Now consider our more complex question about whether (and how much) verbal instruction can help subjects who have acquired a technical language and a whole set of concepts and intellectual skills. If the experiment includes learning the technical language it might need very much more time. In fact experiments in our laboratory use children who have had many months of experience in programming and talking about computers and computational processes.

The difference is relevant also to experiments on the difficulty of accelerating (or changing the order of) “Piagetian stages” through teaching. The typical traditional experiment involves at most several hours of exposure to an experimental condition. Our informal experiments cast very serious doubts on conclusions (especially negative ones) drawn from all reported experiments on verbalization and on the invariance of Piagetian stages. A different time scale for the experiment, and a very different experimental setting seems to produce qualitatively different results. Over the next few years we will give major attention to translating these impressionistic observations into rigorous experiment.

Cognitive Theory as a SUBJECT in Elementary School

We have a double interest in working towards simpler and more explicit statements of our cognitive theories. One is the ordinary scientific interest in doing so to confront it more clearly with experimental reality. The other comes from a specially reflexive sub-thesis of our theory, namely the contention that a child’s cognitive development greatly benefits from the child knowing about cognitive processes.

What does it mean for a fifth grader to study cognitive theory? Certainly we do not mean: send him off to read Piaget or Bruner or Newell and Simon! Even if they wrote more plainly, what they say is too purely theoretical.

We are interested in a more applied, more practically oriented version of cognitive theory.

The simplest image of this is nothing more than a more articulate and more sophisticated ability to discuss such activities as learning skills, memorization, solving problems. Most children are so poor in their ability to do so that very little positive knowledge can make a big difference. For example, if a child really believes (as some do!) that the best way to memorize material is “to make your mind a blank and say it over and over” then any slight knowledge of active mnemonic skills will put him in a better position to memorize. Similarly for the even greater number of children who know nothing explicitly about heuristic knowledge, very elementary heuristic skills make a noticeable difference.

The educational problem is: how to involve the child in such knowledge. Lectures are surely hopeless! We have explored a two-pronged approach, using two kinds of “experience”:

The Concept of a Learning Lab

Schematically the idea is to give children many experiences in learning “very learnable and discussable” skills; in teaching these skills to other children; in experimenting with different ways of teaching and learning. A favorite subset of skills is the CIRCUS ARTS (Juggling, Bongo Boards, Tight Rope, Unicycle, Circus Ball, etc.). These have these advantages:

• almost everyone likes to learn them;
• once one has understood a suitable “technology” of learning, they are quickly learned;
• once one has a proper descriptive language it is easy to analyze, discuss, compare notes, transfer, etc. And it soon becomes evident that doing so ACTUALLY HELPS ONE TO LEARN;
• they have enough in common so that a student who had to be taught one of them didactically, can transfer this knowledge and so increase his chances of mastering the next through self-generated learning strategies;
• they also have enough in common to allow a child to make simple mini-theories to cover more than one.

The Computer as a Model Pupil

It is obviously possible to conduct “learning labs” without computers. Nevertheless we believe that the interaction of a “learning lab” experience and a suitable computer experience has very particular value. The point is that “teaching the computer” objectivizes the process; it develops the concept of formal description; and it plants ideas such as “bug”, “sub-procedure”, “state”, “control variable” (“input” in our jargon), etc. which enormously help the analytic aspects of the work in the learning lab.

2.4 Studying Heuristics

We assume the reader will accept that Polya is right in principle about the value of studying heuristic knowledge explicitly. But much work is needed to make this idea practically useful at the Elementary level or more useful at Advanced Levels. In particular:

1. Polya is at his best in discussing relatively advanced and even somewhat esoteric topics (for example the harder problems in Euclidean Geometry). So the least we need is to “do a Polya” on more elementary topics — and more commonly intuitive ones.
2. Some problem domains exemplify heuristic principles much better than others. So the best strategy to learn heuristics for problem domain A might be to first learn problem domain B, study heuristics there, and then transfer them to A. If B is worth while in itself the gain is all the greater. And for quite fundamental reasons we believe that topics in computational mathematics are superb training grounds for heuristic thinking.
3. Polya’s particular set of heuristic principles is, of course, far from complete. The tradition of work on “Heuristic Programming” or “Artificial Intelligence” has pursued more deeply than he in certain directions, which happen to be especially suited to writing programs. Since our laboratory is a leader in this area of work, it is not surprising that we should choose as an area of special interest to develop more systematic and fundamental formulations of heuristic knowledge.

2.5 Relationship of Our Work to Schools

We classify the ways our work relates to schools into two broad categories and an intermediate one:

• Reformist: Producing new materials and methods which can be incorporated into traditional schools without fundamentally changing their nature.
• Revolutionary: Producing a total alternative to the “school” as it is known today.
• Intermediates: Developing new forms of “supplementary” learning centers in the spirit of Science Camps, Oppenheimer’s Exploratorium and so on.

Although the “revolutionary” goals figure largely in the thinking of most members of our group, our practical work with children has necessarily been confined to the other two categories. Most belongs to the first category. The pattern we have used most is exposing children in a public elementary school to our materials for about two hours a week over a half or whole school year. The most interesting deviation from this pattern was a spectacularly successful project in which children were given free access to computers, devices and counsellors over a three week period last summer. From the “revolutionary” point of view we see all this as a necessary preparatory step towards more radical experiments in the global redesign of learning environments for children.

2.5.1 Material Suitable for Use Within the Traditional School

Our most elaborated module is a “course” designed to be used somewhere between the fifth and the ninth grades. Its educational objectives include:

1. The fundamental ideas and skills of programming; fluency in LOGO; experience in planning, developing and debugging a programming project.
2. Elements of Turtle Geometry (which includes a large part of the geometric knowledge normally taught at these grade levels and much more).
3. Various other formal and heuristic mathematical ideas including: use of variables, functions, recursive definitions, etc. on the formal side; and some explicit Polya-like heuristic principles.

We hope by next year to have a book on Turtle Geometry ready for publication. Modules on other topics are less developed but advancing rapidly (Music, Linguistics, Physics, Biology, Heuristic Programming, Perceptual Psychology, Circus Arts).

2.5.2 New Concepts of “School”: New Methodologies of Education Research

We believe that the major impediment to the emergence of a truly modern theory and practice of Elementary Education is the methodology of experiment which makes small changes to a large and complex ongoing system. The consequence is the self-reinforcing syndrome of gloom and pessimism one might call “Jencksinism”.

We propose a different paradigm for research on Education:

1. Take a theory of Education.
2. Develop the consequences of this theory far enough to design what it projects as a really good set of conditions for the intellectual growth of children.
3. Implement these conditions on a minimal viable scale (a community of children on the scale of a small school).
4. Equip your experimental “school” with all the resources of people, technology and ideas required by the design. IGNORE THE PROBLEMS OF COST PER STUDENT AND OF PERSUADING PRINCIPALS, TEACHERS, SCHOOL COMMITTEES AND EVEN COLLEAGUES.
5. Run your experiment for the time required by your theory (2-3 years). Then:

◦ SUCCESS: The results are so qualitatively different from what would normally be expected that no sane observer says: “how do you measure that?”. In this case the next problem is to study why the experiment worked, whether it can be generalized, what can be learned from it.

◦ FAILURE: If under these ideal conditions the results are so poor that the statisticians want to test them for significance you declare the experiment a failure, try to understand why it did not work, perhaps try another.

2.5.3 Intellectual Centers Parallel to Schools

We are particularly interested in a type of “Resource Center” at which children can spend larger slices of time than is usual in “museums” (e.g. all day for three weeks; a full day once a week for half a year). An experiment conducted in Exeter, England as part of the U.S. National Presentation at the International Congress on Mathematical Education achieved a degree of involvement and sophistication far exceeding what we have seen under forty minute school periods.

2.5.4 Opening Avenues for the Severely Physically Handicapped

A computer controlled music generator opens new horizons for children to experiment with music and especially to compose it. Severely handicapped people cannot overcome lack of dexterity physically. For them availability and mastery of a computer with musical and graphic capabilities could produce an even greater improvement in the quality of life. We intend to deepen contacts with institutions teaching handicapped children and train teachers in using terminal devices (such as those developed by Kathafian’s Cybernetics Organization).

2.5.5 Remedial Mathematics for Adults

With small modifications our fifth grade courses seem to be as enjoyable and as instructive to adults (college students in academic trouble, future teachers, open-admissions university students) as to children.

2.5.6 Paradigm for Experiments on Developmental Psychology

Previous psychologists have sought to observe children; the point, however, is to change them. As long as psychology will not take effective steps to change them it will never become a truly experimental science. We propose to use “curriculum design” as an experimental technique for psychology.

2.6 A New Kind of Professional for Research on Education

Developing a program of graduate studies in Education research in close collaboration with M.I.T.’s Education Division.

2.7 Mundane Aspects of Computers

There are people who say that our work is admirable but too expensive to be used for the vast majority of children. They are diametrically wrong. It is expensive only because it is not used for the vast majority of children. Mass production of a standardized computer really could bring the cost down to a very economical level: If every child had a computer, computers would be cheap enough for every child to have a computer.

SECTION 3: CONCRETE EXAMPLES

Concrete Examples to Illustrate the General Concepts and Goals Mentioned in Section 2.

3.1 Example 1: Alan Kay’s Concept of the Dynabook

The Dynabook is (in the intention of its designer, Dr. Alan Kay at Xerox PARC) a computer about the size and shape of an average book and inexpensive enough for mass use. Most of one side is a display panel with printquality text and graphics. It is self-contained, portable, and exceeds a PDP-11 in capacity.

In a world where every child has a Dynabook, learning mathematics becomes embedded in an active, personal, interactive social context. A child uses it to compose music, simulate space navigation, or run interactive physics models. Modern technology allows us to realize Dewey’s vision: children learning complex modern knowledge by direct, meaningful participation and playful imitation of real adult scientific activity.

3.2 Example 2: The “LOGO Turtle Lab”

The LOGO Turtle Lab utilizes a time-shared PDP-11/45 equipped with graphics CRT displays, plotters, music boxes, phoneme generators, floor turtles, light turtles, and motors. Children learn geometric concepts by programming commands rather than doing coordinate algebra.

Contrast descriptions of a circle:

• Cartesian: y = b + √(R² – (x – a)²)
• Turtle (LOGO):

and watch what you do” lead children to discover geometry naturally.

3.3 The Power of the Idea of Powerful Ideas

Two key examples of powerful scientific ideas introduced at an elementary level are State and Local.

State in Turtle Geometry

The state of a turtle consists of position and heading. State operators are FORWARD (changes position) and RIGHT (changes heading).

Total Turtle Trip Theorem (T3): In any closed path where the turtle returns to its starting state, the total angle turned is an integer multiple of 360°.

To draw an equilateral triangle:

TO TRI

  FORWARD 100

  RIGHT 360/3

  FORWARD 100

  RIGHT 360/3

  FORWARD 100

  RIGHT 360/3

END

Debugging using State: When combining BOX and TRI to draw a house, students learn to insert state fixes (e.g. RIGHT 90 , LEFT 60 ) between subprocedures.

Pledge’s Algorithm for Maze Solving: Jonathan Pledge (age 12) created a maze-escaping algorithm. By keeping track of total rotation (TOTROT) rather than heading modulo 360°, the turtle avoids “turtle traps” (ushaped obstacles). In mathematical terms, Pledge introduced a “shadow turtle” in a covering space where state includes accumulated rotation :TOTROT = 0.

Local vs. Global Action

Comparing an inadequate wall-following procedure ( CIRCUMNAVIGATE , which assumes exact global alignment) with a robust local control procedure ( CRAWL , which continually tests touch sensors and makes local angle adjustments). Local action provides the intuitive foundation for calculus and physical laws.

3.4 Ramifications of Turtle Geometry into Physics and Biology

Biology Applications

• Tree Growth: Why do trees grow straight up on mountain slopes rather than perpendicular to the ground? Vertical growth requires a simple local gravity sense (“gravity is everywhere”), whereas growing perpendicular to sloped ground requires complex non-local sensing.
• Animal Aggregation: Modeling wood lice (humidity preference) and planaria (darkness preference) via orthokinesis (velocity modification) and klinokinesis (turning rate modification) driven by local gradient sensing.
• Build-An-Animal Kit: Cybernetic models with jointed rods, motors, and feedback sensors.

Physics & Balance

Studying a two-rod motorized joint balancing problem. Contrast strategies:

1. Position Strategy: Turn motor until lean angle is zero (fails due to overshoot bug).
2. Velocity Strategy: Stop motor when angular velocity is zero (still overshoots).
3. Acceleration Strategy: Adjust motor when angular acceleration changes (achieves dynamic balance).

Qualitative Expression of Newton’s Laws

In Turtle terms, a particle state is (position, velocity). The only active operator is Force, which acts as an accelerator changing velocity without instantaneously altering position.

Elementary Planetary Theory & Velocity Space

Instead of proving orbits are ellipses via differential equations, we use qualitative Turtle geometry:

• Prove orbit closure using angular momentum (Kepler’s equal area law) and energy conservation.
• Turtle Theorem in Velocity Space: All gravitational orbits (1/r² force) form perfect circles in velocity space because impulse vector kicks (Δv = F · Δt) have constant magnitude when divided into equal-angle sectors.

SECTION 4: SPECIFIC RESEARCH PLANS

4.1 Computer Controlled Devices

• Goal 1: Set up a dedicated hardware workshop at M.I.T. for building computer-controlled devices.
• Goal 2: Construct a modular Build-An-Animal kit starting with a two-rod rotary joint module.
• Goal 3: Develop user terminals tailored for pre-literate children.
• Goal 4: Develop terminals tailored for individuals with severe motor disabilities.
• Goal 5: Design portable hand-held terminals.
• Goal 6: Design analog input/output terminals.
• Goal 7: Acquire portable high-resolution display systems for field experiments.

4.2 Knowledge for Children

• Goal 8: Individual teaching work with pre-literate children.
• Goal 9: Small group teaching with mid-elementary students from diverse urban backgrounds (Cambridge schools).
• Goal 10: Remedial mathematics teaching for older students and adults.
• Goal 11: Undergraduate instruction at M.I.T.
• Goal 12: Experiments in clear written and visual expression of computational ideas.
• Goal 13: Active collaboration with programming language creators (PLANNER, CONNIVER, LISP).
• Goal 14: Two new seminars on “Intuitive Mathematical Sciences” and “Heuristics” offered through M.I.T.’s Education Division.
• Goal 15 & 15A: Integrated Music-Math-Science projects and conceptual studies of musical time structures.

4.3 Contexts for Work With Children

• Goal 16: Establish a remote school outpost facility in a Cambridge urban school.
• Goal 17: Plan and design a comprehensive experimental Demonstration School / Learning Center environment.

4.4 Dissemination and Publication

• Publish scholarly papers, books, and educational films demonstrating children at work in LOGO environments.
• Distribute standard LISP-LOGO implementations and hardware specifications to outside universities and research centers (in collaboration with BBN).