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When developing classroom plans and teaching methods, many teachers use collaborative learning techniques.

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THE 2500-YEAR-OLD PYTHAGOREAN THEOREM

Pythagoras of Samos was a Greek philosopher responsible for important developments in mathematics, astronomy and the theory of music. He left Samos because of the tyrant who ruled there and went to southern Italy about 532 BC. He founded a philosophical and religious school in Croton that had many followers.

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Computing In Learning

Computing power has become more available and affordable than ever before. Satellite transmission can beam instructional material to sites thousands of miles away. Computer graphics can create "virtual environments" in which the user sees and interacts with an artificial three-dimensional world.

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EFFECTS OF CHILD ABUSE
All children are born to grow, to develop, to live, to love, and to articulate their needs and feelings for their self-protection. Although growing up can be difficult, most children and young people receive the love and care they need to develop into healthy, happy young adults.

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globe. Japan, the United States, and European
countries are entering now a major competition to make computers that will capture the world market in the coming decades. Will these computers serve the needs of people or will they undermine what we hold most dear? Thoughtful men, in the smaller countries of Europe as well as other where’s, fear the computer revolution as a carrier of intellectual and cultural colonialism. If the world is more than a marketplace, we need to think deeply and act vigorously to advance the adaptation of intelligent technologies to forms which will be culturally congenial.

The richness of humanity is the diversity of its cultures, but now as never before the destructive power of modern technology makes it necessary that we all recognize we are many peoples of one world. Complementing the rich cultural diversity of our traditions, the growth of a common, scientific knowledge inspires the hope that we may achieve and share a secondary culture of ideas. Computers, which can help represent explicitly the best ideas of modern science, can aid in the diffusion of such powerful ideas to create a popular, secondary, scientific culture.

The central representations of modern science are "ways of looking at the world." They are equally useful to children and adults. Simplified computer models of the everyday world, computer microworlds, can help people understand and learn; they provide a toy to tinker with, from which to learn a scientific view of "what's what" and "how it all fits together.” More advanced computer facilities provide tools for more advanced work. Computer microworlds are popular in a specific sense: they do not train anybody to do any job, even though playing with them provides a sufficient orientation for a more purposeful training to follow. In this specific sense, they are suitable for the introduction of inexperienced people to the possibilities of modern technology.

Microworlds and Learning

The central problem of humane education is how to instruct while respecting the self-constructive character of mind. Teachers face a dilemma in motivating children to do schoolwork that is not intrinsically interesting. Either the child must be induced to undertake the work by promise of some reward or he must be compelled to do the work under threat of punishment. In neither case does the child focus his attention on the material to be learned. The problems are someone else's problems. The work is seen as a bad thing because it is either an obstacle blocking the way to a reward or a cause of the threatened punishment.

Psychologists know that - however much insights do occur - learning is often a gradual process, one of familiarization, of stumbling into puzzles and resolving them by proposing simple hypotheses in which a new problem is seen as like others already understood and performing experiments to test the latest "theory."

Computer-based microworlds can be seen as sets of programs designed to provide virtual, streamlined experiences, play worlds with agents and processes one can get to know and understand. Properly designed microworlds embody a lucid representation of the major objects and relations of some domain of experience as understood by experts in the area. This is where the knowledge of the culture is made available, in the very terms in which the microworld is defined.

Children can absorb that knowledge because the microworld is focused not on problems to be done, but on "neat phenomena" - these show the power made available by knowledge about the domain. If there are neat phenomena, then the challenge to the knowledgeable expert is to formulate so crisp a presentation of the elements of the domain that even a child can grasp its essence. The value of the computer is in building the simplest model which an expert can imagine as an acceptable entry point to his own richer knowledge.

If there are no neat phenomena that a child can appreciate, he can make no use of knowledge of the domain. He should not be expected to learn about it until he is personally engaged with other tasks which will make the specific knowledge worth learning as an aid in achieving some other personal objective.

Designing Computer-Based Microworlds

Designing computer applications for education might be called cognitive engineering, for its objective is to shape children's minds. Such a goal must carry with it a commitment to cognitive science, the study of how knowledge functions and changes in the mind. In light of the profound influence of computers in the schools, designing educational applications without such a commitment would be irresponsible.

I believe that Jean Piaget, the Swiss student of knowledge, formulated the general solution to the problem of how intelligence develops. Although the field of cognitive science has advanced beyond Piaget's innovative theories by revising and extending them, his insights into the nature of learning continue to influence teaching methods. The union of computer microworlds and Piagetian theory is the subject of this article.

Piaget and Education

Central to the work of Piaget is constructivism, the view that the mind incorporates a natural growth of knowledge and that the mind's structure and organization are shaped by interactions among the mind's parts. In The Science of Education and the Psychology of the Child (Viking Press), Piaget challenges educators to answer two questions: How does instruction affect what is in the mind ? and What remains in the mind from the process of instruction long after the time of instruction has passed ? In the same work, Piaget disputes both the effectiveness and the ethical correctness of many of the practices of modern education:

"If we desire to form individuals capable of inventive thought and of helping the society of tomorrow to achieve progress, then it is clear that an education which is an active discovery of reality is superior to one that consists merely in providing the young with ready-made wills to will with and ready-made truths to know with. "

The Dilemma of Instruction

Given Piaget's view that learning is a primary, natural function of the healthy mind, we might consider instruction in any narrow sense unnecessary. Children (and older students of life as well) learn the lessons of the world, effectively if not cheerfully, because reality is the medium through which important objectives are reached. Nevertheless, in certain situations children often rebel against the lessons society says they must learn. Thus the educator's ideal of inspiring and nurturing the love of learning frequently is reduced to motivating indifferent or reluctant students to learn what full functioning in our society requires.

Teachers face a dilemma when they try to move children to do school-work that is not intrinsically interesting. Children must be induced to undertake the work either by promise of reward or threat of punishment, and in neither case do they focus on the material to be learned. In this sense the work is construed as a bad thing, an obstacle blocking the way to reward or a reason for punishment. Kurt Lewin explores this dilemma in The Psychological Situations of Reward and Punishment. (in A Dynamic Theory of Personality, McGraw-Hill, 1935). The ideas of Piaget and Lewin have led me to state the central problem of education thus: "How can we instruct while respecting the self-constructive character of mind ?"

Computer-Based Microworlds

In Mindstorms: Children, Computers, and Powerful Ideas, (Basic Books, 1980) Seymour Papert proposes computer-based microworlds as a general solution to the problem of motivation. One argument for Papert's proposal runs as follows: learning is often a gradual process of familiarization, of stumbling into puzzlements, and resolving them by proposing and testing simple hypotheses in which new problems resemble others already understood. Microworlds are in essence "task domains" or "problem spaces" designed for virtual, streamlined experience. These worlds encompass objects and processes that we can get to know and understand. The appropriation of the knowledge embodied in those experiences is made possible because the microworld does not focus on "problems" to be done but on "neat phenomena"--phenomena that are inherently interesting to observe and interact with.

With neat phenomena, the challenge to the educator is to formulate so clear a presentation of their elements that even a child can grasp their essence. A well-designed computer microworld embodies the simplest model that an expert can imagine as an acceptable entry point to richer knowledge. If a microworld lacks neat phenomena, it provides no accessible power to justify the child's involvement. We can hardly expect children to learn from such experiences until they are personally engaged in other tasks that make the specific knowledge worthwhile as a tool for achieving some objective. This amounts to an appropriate shifting of accountability from students (who have always been criticized for not liking what they must learn) to teachers, those who believe that their values and ideas are worth perpetuating.

A Constructive Alternative

The broader availability of technical training and its vigorous application have created an explosion in the quantity of knowledge available. Too often the rate of knowledge growth outpaces the ability of teachers to absorb and communicate what is known, and skills developed through schooling are obsolete or irrelevant before students are in a position to apply them. Beyond the arena of instruction waits another problem, pervasive but less well recognized: commonsense knowledge is becoming harder to acquire. We are all served by tools more complex than we understand. The objects we depend on "contain no user serviceable parts"; if they did, few of us would have the experience to know what to do with them. A second source of impenetrability is the increasing complication of life by extrinsic rules, even at the simplest level. As an everyday example, consider how the imposition of percentage sales taxes makes simple addition of small purchases harder. The decreasing accessibility of common sense knowledge makes the instructional contribution to cognitive development even more critical than it has been in the past.

The reforms of yesterday have been significant and effective, even if not entirely adequate to cope with the problems of today. The long during lectures of the past are now sometimes circumvented by work with concrete materials, thanks to the followers of Montessori. Teachers do now ask "How can we present material in ways more congenial to the developmental level of the pupils ?" Piaget's effort to focus on the activity of the pupil remains a positive and potent force in education. Following him we ask the different question, "How can we instruct while respecting the self-constructive character of mind ?" Granting the good will of teachers and the effectiveness of past reforms, instruction is still typically frontal lecture in form. Teachers still teach. The children still "get teached", whether they learn or not. The reasons may derive from institutional efficiencies, as Bauersfeld has suggested (in an invited lecture at Purdue University, 1987), but I argue here that today's problems in education have their roots within the views of mind and learning from which we generate curriculum. We need today a constructivist alternative to pupils' "getting teached", a genetic vision of knowledge and its growth that is at least as explicit and well articulated as the standard view, specifying

• components of knowledge
* interrelation of knowledge components
* functioning of knowledge components
* the emergence of behavior from the interactions of components
* processes of development (observed and ideal)
* methods of evaluation derived from the view but recognizing the legitimate interest of society in evaluation of the process and results.
* actions and options available to educators.

A Constructive Alternative to "Getting Teached"

A view of knowledge in which skills are seen as decomposable into a series of procedures (sub-skills) leads to a focus on the logic of prerequisites in curriculum. This may be an error, as suggested by this simple example. In the elementary grades, addition instruction is often sequenced by the magnitude of addends. Teachers have told me "We don't add any sums higher than 12 in the first grade." Yet it is only logical to focus on the addition of single digit sums within a specific scheme of representation. Children may know very well that 15 cents and 15 cents make thirty cents or that four quarters make a buck. Different aspects of knowledge are more or less salient according to the scheme of representation involved. A focus on the varieties of representations and their relation to human learning (drawing on studies in Artificial Intelligence and the Cognitive sciences) will provide better guidance for generating curriculum. Curriculum design requires a logic of psychological genesis, not the structured logic of the domain expert.

Local Clusters of Empirical Knowledge

The way to knowledge lies through the fields of ignorance. There is no single path. People typically develop familiarity with a domain through a limited experience of its various possibilities (we can say they experience some aspects of the domain as a microworld). The development of disparate bodies of knowledge (call them microviews) based upon the experience of microworlds is guided by personal action and social interaction. Only later, if ever, do people achieve a unified comprehension. Integration of knowledge derives from later experiences of which two kinds (at least) appear possible:
* rational reconstruction of prior knowledge through reflection
* discovery of an especially apt, concrete model

Such an especially apt model, even though encountered after other experiences, can through its simplicity and good fit help explain the nature and operation of objects and processes in more complex and less obvious situations. I call such a concrete model a (temporally) post-cadent logical ancestor. This kind of cognitive reorganization derived from the internal construction of a post-cadent ancestor would be expected to dominate the learning of young children, inasmuch as rational reconstruction may remain uncommon except in circumstances where formal capabilities are well developed. When there exists a cluster of cognitive structures derived from varieties of experiences relating to the same kind of knowledge (as experiences with counting , and coins, and measure may all relate to processes of addition) a post-cadent ancestor would be more generally named the nucleus of a microview cluster and the process of internalizing such a nucleus would be cluster nucleation. What would make a nuclear microview central ? Aptness as a crisp representation of the relevant knowledge is a first answer. Simplicity would make the microview easier to remember in detail and easier to think with. A cluster nucleus would present a more "thinkable model" for a domain than would less lucid representations. It would dominate its cluster-members by its efficiency in thought. The nuclear model finally would come to serve as a ground of explanation for their processes as well, and this is the relation which gives it the power to advance reformulation of earlier experiences and the integration of disparate experiences into a coherent, augmented area of knowledge. If we express this in terms of a geographic metaphor, the nucleus of a cluster could be seen as a regional capital or the central city of a metropolitan area, in communication with its earlier established empirical domains as suburbs. Such a geographical metaphor can help us discuss how such clusters can relate to one another.

Structured semantic networks

On the ground, cities and suburbs are connected by local roads and rail lines. Major cities today also are interconnected in more immediate ways, as by air or express trains. If we imagine that these cluster nuclei, these mental capitals, are interconnected, we have a graphic image for knowledge in the mind as a structured semantic network. If we adopt such a vision for guidance in thinking about knowledge within individuals, the critical questions to be asked are first, how does such a network grow and second, what are the interrelations of the development of such personal knowledge with the social context ?

Genesis in the Cognitive Network: psychological issues

Why should it be the case that different schemes of representation are important in education ? One reason is the Galton phenomenon: some people think predominantly with words, some with images, some with kinesthetic feelings; most of us think in mixed modes which alternate and interact variously in different domains of experience. Another reason is that multiple representations covering different aspects of a "common domain" permit the expression of a more robust and adaptable set of responses to externally presented problems than any single formulation would permit. The nucleation of such a cluster by the dominance of one single mode (because its best model is somehow more fit than others) would integrate and strengthen its structure. One conjecture for the larger scale organization of mind is that until such nucleation is achieved, a cluster's interconnection to and integration with the broader cognitive network would not be stable. A corollary of this conjecture is that networks of "nuclei" alone would not be a desirable outcome of instruction because they could not by themselves form a robust, flexible, and coherently integrated network.

Genesis in the Cognitive Network: sociological issues

The way to knowledge lies through the fields of ignorance. Why should one begin the journey ? How can one value what neither has been experienced nor understood ? Abstract or future advantages may be important to some adults, but human engagement is the primary motive for children, and it remains primary for many adults as well. Conventions are shared knowledge about what we do together and how we do it. We all accept conventional knowledge and, in bits and pieces come to understand the function and logic in detail of the various things we do. A child may, for example, be quite happy to take turns at beginning games of tic tac toe with a playmate while lacking completely any sense of the relative advantage adhering to that first move. There may be many different ways of gradually coming to understand the rationale for accepted practice. Instruction may be one. Observing others and the consequences of their action is another. Working out encountered problems may lead to a purely internal discovery. The essential situation is, however, that society provides frameworks of interaction with discoverable meanings; the engagement with these frameworks leads to their later exploration and comprehension in detail.

Educational Implications and Directions for Research

This vision of mind as a structured semantic network offers the hope of solving some problems. If the focus on multiple representations of knowledge in different modes of experience argues that a broader base of empirical experience is needed for stability of learning, it also presents a concrete proposal about what makes for such stability. Further, the conjecture that cluster nucleation is a prerequisite for stable integration of local domains of knowledge into the larger cognitive network provides us guidance as educators. The hypothesis itself is a formulation by which we can judge whether any student should or should not be able to integrate new learnings into what is already known. If, on the other hand, it requires a deeper penetration of what a pupil knows to provide guidance for his activities, that is a sign of the increasing depth of our appreciation of cognitive development, however much in fact it increases the practical burdens of instruction.

The question of sequence in curriculum comes now in a new light. Doubtless there is a sense in which knowledge of one level is prerequisite for later learning. However, within a common domain of experience, prerequisite knowledge sequences may be largely irrelevant if empirically rooted microviews are built up through everyday experience- induced associations and through the adoption of conventional knowledge from social interaction. Within the terminology of Vygotsky, one might say that prerequisites are unimportant within the zone of proximal development. What is a prerequisite for stable understanding is a breadth of empirical information which can be rationalized and comprehended through the development of a post-cedent logical ancestor in cluster nucleation.

If we focus closely on what sort of thing a nuclear microview might be, the primary characterization is that it should be a "thinkable model", one fit to the domain and efficient enough that it can serve as the ground of thought experiments -- those very reflections which will bring about the rationalization of the local cluster and create its stability. One implication for education is that we should try to catalog and articulate known models which might be able to play such a role. With respect to empirical knowledge and the experiences on which it is based, one may inquire of disparate microworlds about the extent and limitations to their variety, and also explore how they are interrelated with each other and nuclear models.

Aiming to build a structured semantic networks in pupils' minds requires the development of a more structured view of knowledge. Some models are more important than others. With this proposal, we have a principle for judging which are more important and why they are more important -- without diminishing at all the requirement for sufficient breadth of experience that potential nuclear models can have some base of material to function with and reorganize. Finally, given that one may imagine a flexible and extensive scheme for representing human knowledge as a structured semantic network, would it be possible to reorganize our view of what is known -- say the contents of encyclopedias -- into a form which would be compatible with this vision of human knowledge ? Could one imagine pupil workstations in which the students would have representations of knowledge (their own and the worlds') which would help them compare their knowledge network for a domain with other possible organizations and extensions ? If such is possible, should the potential be explored ? Could it be exploited as an method of evaluation ? How else would it be exploited ?

References:

1. Ferguson, I., with Martin, E., & Kaufman, B. (1990). `The Schemer's Guide'. Fort Lauderdale, FL: Schemers Inc.

2. Yazdani, M. "Computational Story Writing," in Computers and Writing, Williams and Holt (Eds.) Norwood, NJ. Ablex, 1989.
 

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