A Logo Vision from Argentina by Horacio C. Reggini

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Capítulo relativo a la Argentina publicado en el libro “Logo Philosophy and Implementation”, editado por Logo Computers Systems Inc., Québec, Canada, 1999.

This is my experience on Logo in Argentina, involving judgments of its value and development, an appreciation of its beauty and techniques and an interpretation of the present situation.


No author can escape from his singular first memories on a subject. I learnt about computers many years ago, in 1959, at the IBM Watson Scientific Computing Laboratory in New York on the occasion of having a scholarship at Columbia University. On one hand the subject seemed tedious as the only possible way of programming the computer was through machine language. On the other hand through this experience I became fully aware of John von Neumann’s genial ideas which he clearly explained in his wonderful book The Computer and the Brain (1958): “… since the orders that exercise the entire control are in memory, a higher degree of flexibility is achieved than in any previous mode of control. Indeed the machine, under the control of its orders, can extract numbers (or orders) from the memory, process them (as numbers!), and return them to the memory (to the same or to other location); i.e., it can change the contents of the memory -indeed this is its normal modus operandi-. Hence it can, in particular, change the orders (since these are in the memory!) -the very orders that control its action-. Thus all sort of sophisticated order-systems become possible, which keep successively modifying themselves and hence also the computational processes that are likewise under their control”. The preceding concepts enthusiastically enraptured me and I decided that I was going to be part of the new computer era.


Back in Argentina in 1960, I became a professor in several universities and carried out different projects in the computer field. Among these projects I set up the first Computer Center for Undergraduates at the School of Engineering in Buenos Aires University on the basis of an IBM-1620.


In 1963, as a member of the MIT Interamerican Civil Engineering Program I started to travel frequently to Boston and Cambridge. I was lucky enough to be a witness of one of the most wondrous phases of the advance of computer ideas: the MAC Project at MIT where I met Marvin Minsky, J. C. R. Licklider, Robert Fano, Ivan Sutherland and many other founders of the new world. Lately I met Seymour Papert and my passion and advocation to Logo was ignited.


My awareness of the power of computers pushed me to get involved with the MAC Project in 1964. I was able to log in MIT IBM-7094 time-sharing system from Buenos Aires, Argentina, -thousands of miles away- via a radio teletype link. This experience deeply influenced me and was the onset of my research and commitment to the role science and technology can play in its relation with society and education.


In the seventies, visiting Papert’s Logo projects like the Brooklin Public Schools in Boston and Computers in City Public Schools sponsored by the New York Academy of Sciences, my concern was focused on how the information technologies could help kids find new and better ways of learning. In 1980 I was also present at the pioneering Logo experience at the Lamplighter School in Dallas, Texas. I was delighted to see the happiness of kids and teachers when writing and executing Logo procedures on a lot of Texas Instruments TI-99 Home Computers spread along the building. The Logo software then available had still certain drawbacks: the turtle could not draw too much as it “run out of ink” and decimal numbers were not accepted. At the same time it had wonderful colored “sprites” and many people who were expecting to run Logo in a personal computer adhered to it promptly.


Immediately, I agreed with the Texas Instrument Argentine branch to prepare a Spanish Logo version for their TI-99 Home Computer. Soon the Spanish Logo became available in Argentina and neighboring countries, particularly in Uruguay. At that moment, I considered it essential to have Logo’s ideas printed in Spanish; so, I set up Ediciones Galápago, which published in 1981 the translation of Mindstorms by Seymour Papert, under the title Desafío a la mente. Computadoras y educación. Ediciones Galápago also translated and published The Second Self -El Segundo Yo- by Sherry Turkle in 1984, The Society of Mind -La sociedad de la mente. La inteligencia humana a la luz de la inteligencia artificial- by Marvin Minsky in 1986, The Media Lab -El laboratorio de medios. Inventando el futuro en MIT- by Stewart Brand in 1988.


In 1982, I wrote Alas para la mente. Logo: un lenguaje de computadoras y un estilo de pensar -the corresponding title in English would be Wings for the Mind. Logo: a Computer Language and a Style of Thinking- which has been extensively used in Spanish speaking countries. As an example of the spirit of the book, I include a dialogue among children that simulate “walking” on the computer screen, showing their involvement with the situation, the enrichment of their original idea while advancing in the project and their feelings of achievement, all key factors in a genuine learning process (Annex 1). Alas para la mente was translated into French: Logo, Des ailes pour l’esprit, Cedic/Fernand Nathan, 1983, and into Italian: Logo, Ali per la mente: il linguaggio di programmazione ideato per l’educazione e il gioco creativo, Arnoldo Mondadori Editore S.p.a., Milano, 1984.


Counting with the availability of Desafío a la mente by Papert, and my own book Alas para la mente, together with the fact that personal computer were becoming popular in Argentina, I organized a great number of workshops, courses and seminars for teachers all over the country. Three enthusiastic assistants helped me at that time: Teresa Carabelli, Mercedes Torino and Paula Bontá -presently Director of Design at LCSI in Quebec. My involvement with Logo ideas led me to frequently publish articles on national and local newspapers, and attend interviews for well known local magazines and TV programs, trying to help spread the Logo spirit. Teachers and professionals from the educational field showed great enthusiasm and eagerness to participate in the new digital world. Most of what has been done with Logo in Argentina grew out from the interest, investment and dedication of small groups of persons. They were frequently headmasters and teachers often accompanied by parents from family school associations.


There was one school in particular, the Instituto Bayard (Bayard Institute), that led the way. With the support of the headmaster and owner of the school, Annelise Henriksen de De Forteza, I could train all her teachers and, soon, all the kids from the school started to develop Logo projects. The Instituto Bayard became a Logo “lighthouse” for many educators who were looking for positive changes in learning.


In 1982, a group of people inspired by the new winds brought by Logo to Argentina decided to join efforts and created the Asociación Amigos de Logo (Friends of Logo Association). This was a non-profit organization which main goal was to plant the seeds and promote the growth of Logo ideas. Asociación Amigos de Logo grew thanks to the effort and hard work of its members. We rarely got any support from outside. The first big step was the organization of the Primer Congreso Internacional Logo. Las Computadoras en la Educación y la Cultura -First International Logo Congress. Computers in Education and Culture- which was held on September 16/18, 1983, at the premises of the Instituto Bayard in Buenos Aires. (Annex 2).


None of us, not even the most optimistic, would have imagined the success and the response it would achieve. One thousand participants attended from Argentina, Uruguay, Brazil, Colombia, Peru, Mexico, U.S.A., France and Spain, and it was sponsored by the I.B.I. Intergovernmental Bureau for Informatics (Rome), the UNESCO (Paris), Subsecretaría de Informática, Ministerio de Educación, Secretaría de Cultura, and Secretaría de Educación de la Ciudad de Buenos Aires. There is something I would like to emphasize about this event: it was not a mere cluster of magistral presentations, but a real exchange of experiences and ideas. It is worth noticing the sincerity of several lecturers who, understanding that the Logo modality was in full process of evolution, did not hesitate, when answering the requests of the attendants, to say “We haven’t thought about that yet”, or “It had not occurred to us”. Many children also participated actively in the meeting showing their own Logo microworlds, adding fresh and spontaneous remarks that promoted deep thoughts in the adults. “This meeting has been an extraordinary blend of a seminar together with a weekend party” said Julián Marcelo, who represented the I.B.I., at the closing ceremony.


Unfortunately, who should be our key speaker, Seymour Papert could not attend. Robert Mohl from MIT was highly demonstrative of the latest progress of Logo at worldwide level. The Primer Congreso Internacional Logo. Las Computadoras en la Educación y la Cultura 1983 of Buenos Aires strengthened the situation of Logo, not only in Argentina, but also in all Latin American. Many of the attendants of the Buenos Aires participated later at the Logo-84/85/86 Conferences at MIT.


New Logo versions and new personal computer models became available in Argentina like LCSI Apple Logo and LCSI IBM Logo. Texas Instrument Argentina, taking into account the success of its Spanish Logo version for TI-99, asked Harvard Associates of Boston and my team to develop a Spanish Logo software for its TI-PC model. We also prepared, in 1986, the Spanish Logo vocabulary and documentation for a MSX computer -assembled by Telématica S.A. in Argentina- following the MSX-Logo developed by LCSI in Canada. This computer became quite popular and was used by lots of schools. Lately, LCSI commissioned us a translation and adaptation into Spanish of its LogoWriter software.


During 1984 and 1985, I worked in the development of 3D-Logo and published the book Ideas y formas:. Explorando el espacio con Logo (Ideas and Forms. Learning to Build in 3-D Space with Logo). It deals with the virtual construction of three-dimensional objects following the powerful and elegant Logo intrinsic geometrical approach: the shape of any object can be described by defining the necessary movements of a turtle to trace its edges. Given an initial position of the turtle -location and orientation-, with regards to a viewpoint in front of the screen, 3D-Logo renders the corresponding image. This book was translated into French, Logo dans l’espace, ACT-Informatique, Cedic/Nathan, Paris, 1986, and into Italian, Idee et Forme. Explorando lo spazio con il Logo, Edizione Sisco Sistemi Cognitivi, Roma, 1987. The construction of 3D-shapes captured the imagination of many children and adolescents, to the point that it was quite natural to see all kinds of 3D-objects on the computer screens of many schools.


I also carried out a research on the geometrical generation of polyhedra using intrinsic geometry implementing my 3D-procedures in ObjectLogo -Paradigm Software Inc.- for the Apple Macintosh. Results, including the determination of the number and the structure of all possible different paths of traveling along all the faces of a regular polyhedron without passing more than once by the same face, were published in my paper Regular Polyhedra: Random Generation, Hamiltonian Paths and Single Chain Nets (Annex 3).


In 1988, I wrote Computadoras ¿creatividad o automatismo? Reflexiones sobre tecnologia (Computers, Creativity or Automatism? Thoughts on Technology). It brought together a collection of articles and conferences on Logo, stressing the premise that only a wise use of new information technologies can help us to become freer and more creative persons. The book also included an analysis of the ideas of Karl Popper and Seymour Papert (Annex 4).


Logo grew in Argentina through groups of people who believed in Logo more as a philosophy than a mere computer language. I can recount on hundred of wonderful and extraordinarily rewarding small experiences all over the country. I can even give the names of Logo kids who clearly realized about the meaning and importance of computers and followed their careers at MIT, or some that have successfully found a place in the information society, but, unfortunately, I can not recount on valid experiences supported by local or national governments. Most official projects on the field have dealt with the idea of using doubtful “educational software” and trusted on the computer companies for support and training. The present Logo situation is more or less the same as it was years ago. There are many schools that permanently seek and work hard to find better ways to understand Logo and make wonderful use of it; others, as in God’s flock, scarcely realize about the philosophy underneath Logo. One school, among the first ones, is the Colegio Las Cumbres in Buenos Aires. There, Mónica B. Coni -in charge of staff development under the slogan Digital Learning– is continuously training teachers and giving innovative cues for imaginative Logo applications.


I want to end with a comment about one of the essential natures of Logo, not very clearly understood in educational areas. Logo was enthusiastically received in Argentina as a big innovation but -as it has also happened in other countries- the force and dynamics of the beginnings decayed with the years. I believe this circumstance can be explained by different reasons. The main reason had to do with a way to look on Logo just as a mere technological auxiliary tool to be applied principally to what has been done up to the present. That vision overlooked or denied the delicate Logo approach that implies selfsconscience -to be oneself-, to be in control -to be the owner of oneself or to be oneself- and selfdecision -to act from oneself or by oneself-. These human attitudes require “personal reflection” and demand “thinking about thinking”, but -unfortunately- these traits are not the ones that prevail in influential groups and are simply not encouraged enough. We are in a situation in which some of the Logo approach virtues are, at the same time, a barrier to overcome as the society in general does not comply with them. Better Logo versions and explanations of its meaning, the ascending power and spreading of persons that had the experience and pleasure of thinking using Logo in their childhood, the evolution of society to better human life conditions, will gradually produce the necessary changes for its due appreciation and success.


Annex 1

Alas para la mente. Logo: un lenguaje de computadoras y un estilo de pensar, by Horacio C. Reggini, Ediciones Galápago, Buenos Aires, 1982. Winds for the mind. Logo: a computer language and a style of thinking.

From the prologue

This book is addressed to all those people who want to know about the new roads laid down by computers as mind auxiliaries; it attempts more specifically to introduce a different way to using computers and to contribute to establishing new links with personal thinking. The election of the title Alas para la mente (Wings for the Mind) was a difficult aspect to solve. The first titles that I considered were the following:


– The computer: a prodigious instrument of the mind

– The blooming of computer in the field of intelligence

– Computers: an intellectual revolution


All the above sentences contain the ideas that animate the book but I want to avoid the word “computer”. I have used it many times in articles and conferences. Moreover, and this is what really counts, that term and the other one “computation” so often used, immediately insinuate complex and esoteric matters. I believe it should not be that way. In the past, persons who professed esoteric doctrines wanted their knowledge to be shared only by a few ones. This situation must not be repeated nowadays because of cultural as well as social reasons. Scientific progress brings us to the world of computers. To restrict or to propagate the knowledge of science  and technology are two alternative paths which are up to us. Alas para la mente (Wings for the mind) chooses the second one: it aims to demystify the allegedly mysterious and complicated computer and make it accessible and simple to everybody. These and other beliefs have also influenced the book so that it continually analyzes educational issues and often refers to children with respect and love.


Reflection, critical thinking and dialogue are essential aims in education. Young people live interested in real problems, and they want to fly up to them. But sometimes their wings are not powerful enough, like those of Icarus who, in escaping from imprisonment, fell into the sea when the wax of his wings melted as he flew near the Sun. This book intends to provide real and effective “wings” to thinking. The “wings” this book dreams of are meant to be powerful and amiable ones, capable of taking us where we choose to go: far away in the analysis of external phenomena or deep inside ourselves in the study of our own way of thinking.


The book describes and uses the wonderful Logo language created by Seymour Papert. Logo or Logo-like languages will be used in all future computers. Logo is powerful and natural tool for procedural description. It can be understood by everybody and it is an excellent vehicle for exploring multiple phenomena. A lot of achievements in the past, carried out times where the present science and technology did not exist, proved the effectiveness and power of imagination. For taking real advantage from a computer it is necessary to follow the intuitive path as well as the logic-mathematical one. The Logo language eliminates the severe barriers that prevent most people from using computers creatively on the basis of essential knowledge and resources, following the sage proverb that says that nothing must be done with many elements if it can be done with a few.


The book is mainly related to the ideas by Papert in Mindstorms. I often refer to children in the way I am going to do immediately. But, I want to make it clear that the same concepts and points of view also apply, with more similitude than we may imagine to adults. What happens is that we adults believe we know everything and so we do not want to learn anymore. I wish we adults were more like children so as to enjoy more the beauties of life, to laugh or sing and to listen without mistrust or prejudice. Moreover, I firmly believe that a more just and worthy society will be built by children tomorrow on condition that adults today generously provide them with the possibility of achieving an authentically better education.


Some of the basic premises of the book-closely related to Seymour Papert thoughts and the Logo approach are the following:

– The computer becomes an outstanding learning tool.

– The computer acts a way of making “learning mathematics” natural an enjoyable.

– The process of learning how to think and the understanding of the way they think, allow children to grow up intellectually.

– Instinctively, almost without help, children develop inductive and deductive reasoning.

– Since children find their learning meaningful, they get interested and so learning goes on naturally.

– Children learn to see numbers from a different point of view: since numbers begin to be comprehensible, they start to be appropriable; since numbers are useful to carry out their purpose, they become personally meaningful and even, necessary. Number are alien objects no more.

– A very special kind of affective relation is quickly established between person and computer which resembles more a dialogue of friends than a prepared lesson. In this way learning becomes an irresistible temptation to everybody, and so we can see groups immediately interacting and trading ideas, as soon as they face the computer.

– Reality becomes a representable object. They can divide it into approachable and comprehensible parts and so they begin to understand it, and what is more: they feel prepared to interact with a new and friendly world.

– The computer is personally meaningful to the user.

– Both body syntonic and ego syntonic learning develop.

– Children are allowed to see what they think. They see their thoughts come to life on the screen.

– The aesthetic dimension is continually placed in the forefront.


All the comments above involve self-conscience, self-control and self-decision, which are principal objectives of development and acquisition of personality:

– Self-conscience means to be in oneself, to assert oneself.

– Self-control means to be the owner of oneself or to be oneself.

– Self-decision means to act from oneself or by oneself.


The book intends to be understood by children, adolescents and adults. It also wants to be useful to specialists, often used to considering computers from other points of view. I do hope that someone will venture and enjoy the intellectual flight proposed by this book.

Chapter 26. Fifth Scene

Five scenes with dialogues between children are inserted in the book Alas para la Mente. Logo: un lenguaje de computadoras y un estilo de pensar, which the author uses either to give examples of the ideas stated in previous chapters or just to explain a new subject. The content of these dialogues, besides its reference to formal aspects of Logo language, reveals, as is the intention of the whole book, the characteristics of a genuine learning process.


What follows is a translation from Spanish into English of Chapter 26, entitled Fifth Scene. It is a short story of a group of children using a computer. The conversation furthers the general habit of relating mathematics to personal knowledge. Observe how Claire looks at her feet; …see how all are involved with the situation; …the variable RHYTHM (the use of a symbol to name an undefined value) is presented as a means of “walking faster or slower”. The group works out different alternatives, enriching the original idea and making the first model more complete. The scene ends with everybody happily looking at what is happening on the screen, “something” that they, by themselves, planned and achieved with self-decision.


The scene

Ximena rushes into the room because she has had an idea and she wants to make sure it is not to be lost. Everybody loves her a lot, so they save what they are doing in the diskette and say:


– Logo is all yours, so long as you give us an explanation about what you’re thinking.


Ximena sits down, and begins touching keys. Then she explains:


– My idea is to make the right foot and the left foot appear on the screen and tell them to walk upwards without leaving footprints.


Peter likes the subject and says:


– Of course you’ve thought of an agent for each foot. Each one will have to “carry” different “shapes”, which will be very much like real feet.


To begin, Ximena writes MAKESHAPE 11, and a grid of 16 x 16 little squares appears on the screen. Touching keys with signals  she makes some squares become black and so a form similar to the left foot is obtained.




SHAPE 11                                                              SHAPE 12


When she writes MAKESHAPE 12, another grid, completely clean, appears on the screen; she touches the “drawing keys” once more and makes a figure representing the right foot. Everybody takes part in the drawing. Claire takes her shoes off to look at her feet and to compare them with those Ximena is drawing on the screen.


Now, Ximena writes the procedure FEET:



CALL [1 2] “PAIR


SETXY [-8 (-5)]



SETXY [8  5]







– I use the order TELL to communicate with each agent. It’s like telling them to “wake up” or to “listen” to me. I order agent 1 to carry SHAPE 11 and agent 2, SHAPE 12, and I place them near the center of the screen. When I need to tell the same thing to both agents 1 and 2, I address to PAIR, which is the “list” formed by their numbers. So, I tell them both to dye yellow and to point upwards.

















Claire keeps her breath when Ximena writes WALK, and bursts into enthusiastic shouts when the feet begin to WALK on the screen. Peter claps and Ximena seems moved.


Jules, who leads the group, suggests that they make the feet walk more or less quickly, for which it is necessary to change WALK into a “procedure with an input”.


Ximena changes the procedure in this way:












and she tells the feet to make a “quick walk”:




Peter, sparkling, remarks:


– Well, I’m leaving now. I’m so pleased for all we’ve done today.



The feet, in the meantime, walk for ever on the screen, at the rhythm that Ximena, Claire, Peter and Jules have decided by themselves…






Annex 2


First International Logo Congress: Computers in Education and Culture. 1983.



Opening speech (by the President of the Congress, Horacio C. Reggini, September 16, 1983, Buenos Aires, Argentina).


“Dear ladies, gentlemen and children:


In behalf of the Asociación Amigos de Logo, I welcome you to this Congress. Many people, from neighbor countries and from other ones of the world have come to be with us in this occasion, as well as numerous delegations and visitors from other cities of our country.


Some time ago, when we were seeking a place to hold the Congress, we regarded theaters, auditoriums and other places often used. But we wanted a different congress. We fancied about trees, and plants, and flowers. We used to have our Association meetings in the director office, very close to this place. And, as it often happens, the nearest, most loved and familiar things are those that we often disregard, those which we do not notice. Only sometimes, when they are missing, we take them into account. One day we noticed this patio, the patio of a school, which has a tree, plants and flowers and where there is always joy, play and love for life. This patio thus became a symbol for this Congress. I hope its spirit is among us these days.


We are here because we have some points of coincidence. To some, these coincidence are strictly scientific: the study and the deeper knowledge of a new kind of computing, Logo, which may imply a different use of computers in many fields, professional as well as commercial and industrial. For others, the Logo mode is intimately aimed to open new ways in education. So, Logo acquires social and cultural characteristics that make a mere technological innovation become a potential factor of authentic progress.


We are confident that the lectures, the papers, the children works and the fruitful exchange of ideas will be valuable contributions to the Logo International Community that is growing day after day. This congress is a challenge to all of us and undoubtedly will help achieve a more convenient use of computers and an improvement in education.


In behalf of the Association I thank you all for your presence, and all those who, in one way or another, have lent us a hand. Specially we want to thank the children who have devoted many of their hours preparing their creative works for us to see them.”



On Logo as an educational philosophy


“The purpose of Logo is to contribute to the complete development of human being, which is the goal of all authentic education. Through Logo, computers can improve the conditions for such learning, while still respecting the learners linguistic and cultural identity. I will discussed these ideas and how Logo is sometimes erroneously introduced in schools as a technique; just as another computer language, instead of a philosophy with implications for the very structure of the educational system.


From a very early age the child draws from the material around him to develop coherent theories about the world; “naive theories” through which she explains and understands the real world. The knowledge she acquires is limited by the limitations of her environment. She lacks the material to turn some concepts into concrete ideas.


The computer -a new Aladdin’s lamp, with its capacity to become any object through a kind of metamorphosis- provides the child with those models she does not find in the real world. Because the machine can turn formal thinking into concrete thinking, computers can help the child learn this type of knowledge more quickly, and so stimulate her intellectual development.


What usually happens, however, is that once the child enters school her previous theories about the world clash with the information supplied by her teachers. The child doesn’t understand why her theories are incorrect or incomplete, nor why they are discarded. Students learn lessons by rote without understanding what they are saying. Latter, they reject what they have been forced to learn. In addition, the system of reward and punishment produces and intellectual paralysis in the learner, who does not dare to hold views different from those of her teacher for fear of committing a mistake and receiving a low mark.


An effective learning process requires certain strategies in order to resolve these conflicts. In an ideal education system:


1. The child reflects on what she knows and expresses it in a coherent, and orderly way so that she may be understood and also be able to defend her ideas. Thus, she will gain self-confidence and an awareness of the power of her ideas.


2. The child participates in a well-developed, qualitative approach to  knowledge.


3. The child is not afraid of making mistakes -it is not important whether what one does is right or wrong, but whether mistakes can be corrected and improved.


4. The child is encouraged to be creative and allowed to personally appreciate the world. Each person interprets things from her own viewpoint.


5. The child is offered the opportunity of being like an adult in relation to her intellectual product.


Putting such an educational system into practice is extremely difficult in ordinary classes. And in this sense, computers -provided with a suitable language-, can help.


A Logo procedure -teaching the machine how to do something- is the formalization of a piece of knowledge. This formalization can be tested, executed and verified. For the computer, a procedure is a sequence of intelligible and executable phrases. For the computer user, the procedure is the expression of the understanding of a concept. As that understanding can change, a procedure is also subject to change, improvement and permanent revision. In Logo environments, the students discusses their work with their teachers and classmates. This verbalization is vitally important. For the student, building a procedure is not only naming a series of instructions: it is an initial passage into abstractions and the subsequent concrete manipulation of an idea. With Logo, students learn to analyze problems, develop abstractions, formalize solutions to problems, and break down the solutions into a series of solutions to sub-problems.


Students also acquire the habit of verifying general solutions in particular cases: of considering errors not as disasters, but as temporary obstacles which must be overcome; and of developing constructive self-criticism. Frequently, success in learning largely depends not only in intellectual factors, but also an emotional ones. Whether a child likes an area of study depends on her ability or inability to assimilate this special type of knowledge into her own set of models. These intellectual models, different for each person, are acquired in the course of a lifetime. It is easy to learn what one likes or understands, but hard to learn what is unpleasant or incomprehensible. What a person learns depends on the models available to him.


Because the computer is capable of adopting innumerable forms and of supplying widely varying models to suit individual interests, it helps to overcome such barriers to learning. Also with the computer, knowledge acquires a recognizable personal purpose: The child can do things with it. The steps of learning are inverted; the computer allows students to come into contact with the practical uses of knowledge, before coming into contact with its formal enunciation.


Education, when properly understood, must always try to help the child to use her freedom in the most authentic way, a responsible freedom. From an early age the child must progressively exercise her right to choose -under her parents and teachers help- from among the different options offered to her, and to assume the responsibility for and the consequences for her actions, celebrating her accomplishments and correcting her errors. It is this sense of freedom that Logo environments hope to achieve. The child feels free in the formulation of her work, in the steps she will follow to reach her goal and according to her own rhythm: she freely organizes her microworld, but she assumes responsibility for what happens within it.


Working with Logo, children may form a mental image and then design ways to create their image. Or a child may start working with no definite goal, and she discovers something new with each step she takes. Her activity resembles scientific research and artistic creation, as the artist often looks in amazement at the result of her own work. Thus, learning becomes a personal adventure. Also with Logo, children conquer a world which they create and own, and into which they project not only intellectual ideas, but also their own desires. They discover the pleasure of creation without external demands nor absolute truths. Freedom and creativity are not suppressed by fear of punishment. Mistakes are not bad and it is possible to learn from them. Thus, freedom is used with responsibility.


In Greek philosophy, the teacher is the one who guides, who helps the spirit to discover essential truths. The teacher does not show the truths but instead encourages the learner to discover them. Same concept applies to Logo”.



Annex 3


Ideas y formas. Explorando el espacio con Logo, by Horacio C. Reggini, Ediciones Galápago, Buenos Aires, 1985. Ideas and Forms. Learning to Build in 3-D Space with Logo

From the prologue

This book aims to bridge the gap between the traditional ways of describing and representing three-dimensional objects, and those ways offered by modern computers to accomplish that objective. Examples are given and projects are developed to stimulate interest and encourage the exploration of the structure and composition of geometrical shapes. By teaching the computer to produce three-dimensional objects, the reader may achieve greater understanding and comprehension of the elegance and complexity of shapes in space. A capacity to appreciate and understand the beauty of everyday life is important not only us a source of aesthetic pleasure, but also as a medium of knowledge. To know is to delve into things to discover their meaning. The study of shapes is one of the ways of unraveling mysterious natural designs and designing new ones.


The book deals with the creation and manipulation of spatial figures with Logo. The computer is not considered an object of study in itself or used to teach a specific problem or to make a step by step demonstration of a given exercise; instead, it is a tool or chisel that allows people to behave like experts in a field in which they are not, making it possible for them to play creatively with the chosen subjects. The computer is used here as a medium for human expression, intellectual or artistic, offering the user the chance to experience the emotion and joy of the creative act.


The knowledge required to use the Logo language is minimal: only the capacity to define new words or procedures. The commands that make possible the construction of three-dimensional figures are simple and are shown gradually.


The Logo implementation of primitive commands to enable us to use the turtle in space is based on the idea of organizing a microcosm in which the knowledge and the design of spatial objects could become part of natural and egosyntonic process. Through these commands it is possible to draw easily a three-dimensional object upon the computer screen. The image is generated by the movements of the turtle in the space; the representation of the image in the plane of the screen -through a central conic perspective- is performed automatically by the system.


Corporal syntonicity is one of the powerful ideas that makes possible the drawing or graphics with the Logo turtle. This idea of corporal syntonicity allows us to use corporal knowledge to describe any object. To draw a spatial object it is not necessary to know the laws of representation, only the movements that should be made to describe it. In other words, the description of a succession of movements results in a drawing on the screen.


Moreover, the incorporation of a spatial dimension in the Logo microcosm implies an important qualitative jump in the conception of the turtle as “an-object-to-think-with”, as it movements, not only limited to displacements and rotations on a plane surface, reach a more realistic approximation to true corporal movements.


There are many similarities between the methods of technological production, scientific research and artistic creation: fundamentally, all consist of creating order from chaos. The production of a work of art and the concretion of any scientific project requires intense dedication. Both are highly complex tasks demanding resourcefulness, planning, knowledge and debugging. Michelangelo studied anatomy at night, and the result of those studies can be seen in the perfectly molded lines of his Moses or his David. In the same way, professionals, researchers, and artists, spend much of their lives on tests, explorations and studies. The contents of this book signal a road; the rest is in the hands of those who may want to go along it fully.




Logo: Not just for kids

Note by Margaret Grammer, published in magazine Argentine News, ps. 21-23, Buenos Aires, December 20, 1985.


For those who know Logo, the computer language, the image of a Logo environment is of children playing with designs on a computer intersecting spirals, squares, or more complicated plane drawings. But for those who know Horacio Reggini, the image is different. Some of Reggini’s “Logo Kids” are architects and engineers and the designs they play with are skyscrapers of Buenos Aires: past, present and future. Indeed, in Reggini’s hands, Logo has escaped the two-dimensional plane and moved into space. Three-dimensional Logo may change the way we use computers and, with time, it may even change our way of life.


Logo as a computer language has been understood as a tool for education. Easy to learn, understand, and use, Logo is taught in schools around the world. But Logo, says Reggini, is not just for kids. “With Logo, you can tackle any kind of problem,” he told Argentine News recently. “Anything you can do with another computer language, you can do with Logo…” The difference, says Reggini, is that with Logo it’s better. “It’s easier and more natural.” What makes it so attractive for educators and for children is that from the very beginning, Logo user can create procedures that are interesting and satisfying.


In the sixties, Reggini taught and wrote about other computer languages, but when he met Seymour Papert, who created Logo with other researchers in the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology, it was the beginning of a long relationship. “Logo was the first computer language adapted to the natural conditions of intellectual development”, explained Reggini in a publication of the Friends of Logo Association (Asociación Amigos del Logo), of which he is president. “Today, Logo can also be considered a philosophical approach to education, which adds an important social component to its powerful capabilities in the computer field.”


“Reggini’s first book on Logo, Wings for the Mind (Alas para la mente, 1982) presented the Logo system in Spanish for the first time. Translated into French and Italian, Wings for the Mind is a step-by-step introduction that shows the fundamentals of Logo. With it, Argentine children use Logo for computers graphics, driving a tiny, triangular “turtle” around a computer screen, and writing procedures for increasingly complex geometrical designs. When Reggini introduced several new procedures, building on the existing software, those designs could become three-dimensional and, especially important, be moved in space.


For many years, Reggini has been interested in visual perception and in the construction of spatial shapes. He explains in the prologue to his new book, Ideas and Forms (Ideas y formas: Explorando el espacio con Logo): “From a practical and professional aspect, I have been involved in designing resistant structural elements and supervising their construction.”


His book published in September 1985, provides the framework for three-dimensional Logo. In it, Reggini uses his own designs as examples. He shows buildings from above and below, and from different viewpoints around it. He draws spectacles and watches (Reggini’s own, he says), chairs and even his bicycle in three-dimensional Logo. Reggini’s first models were simple. He was testing the idea. But it wasn’t so long before he and his team were writing data for the Catalinas complex -a set of buildings on the Buenos Aires waterfront that includes the Sheraton Hotel, the IBM building, and the UIA building. Reggini’s office on the Leandro N. Alem Av. overlooks the complex, and he and his engineering partner, Hilario Fernández Long, originally designed the structures for several of the buildings. In those days, says Reggini, a perspective drawing could take days, at least, to complete. His Logo perspective took just a few minutes.


Reggini’s work surprised even Logo’s creator. “Horacio has created a new microworld for Logo which must rank among the two or three best that we know” said Papert recently, queried specially for Argentine News. “The real brilliance is that he took an idea, namely the three-dimensional turtle that’s been around for a long time as an abstract piece of mathematics, and turned into a living piece of mathematics that can be used in the same spirit as the classical (Logo) turtle.”


Reggini brought architects and engineers to his office. Soon they were drawing buildings and city blocks. One architect, when she heard that no other Logo researchers were as advanced in three-dimensional Logo, wondered about other Logo users: “If they don’t use 3-D Logo, what do they do with it?” she asked. Most recently, Reggini’s Logo team has been working on a three-dimensional drawing of Piazza San Marcos in Venice. They drew the plaza, and turned it, looking at it from all sides. But they’re not stopping there: “We’re working on a way to see it from inside the plaza,” says one of the team. “Next time we’ll have coffee in the Piazza San Marcos…”


For Reggini, the applications for three-dimensional Logo go beyond engineering or architecture. “You can apply three-dimensional geometrical Logo to any design of any kind… The computer becomes a tool, he says -the means to an end, and not the end in itself. The user must be more interested in the idea he or she wants to develop than in the machine or the computer language… I feel that with computers, you can also experience the joys of being, of doing, of realizing that you are thinking. And you are interacting with your thoughts and putting form to them”.


In addition, says Reggini, those using computers could begin to see themselves as artist or artisans: “in the same way that the artist or the artisan, starting from inspiration, elaborates his idea both in his mind and in the material until obtaining the final shape, so (three-dimensional Logo) describes a similar process in which the computer serves a tool and Logo as a mean of expression.” The artisan, working with tools, creates objects by hand for daily use, he explains. The tool used are not complicated, nor do they required special knowledge to use, except for the skills, judgment, and aesthetics of the artisan. “The artisan may change his mind, because of the work in progress… the material itself may guide his hand, his mind.” Reggini sees Logo as just such an uncomplicated tool, easily mastered, that offers a new means of expression to the “artisans” of our day.


“The process a person follows to define a form with Logo resembles the work of an artisan”, says Reggini: “The integration with the machine is intense and straightforward, with trial and error, and consequent modifications throughout.” Logo offers a tool that designers of all kind can control completely. On receiving an award from the Argentine Academy of Exact, Physical, and Nature Sciences, Reggini said: “The creative engineer will be able to experiment with his ideas unhampered by current restrictions. He will once again be the absolute master of his work, as were the great masters of the Renaissance. Leonardo da Vinci mixed his own paints and Michelangelo made his own chisels. Today, with our sculpturing  done on a computer we have a full range of new instruments and tools. Total control could eventually lead to a new kind of engineering.”


“It bothers me when people say computers tend to develop a more analytical or mechanical way of thinking”, says Reggini. The computer has entered every area of our lives, besides its more common use in science, business, and industry, he explains. But as a vehicle for human expression, either technological or artistic, the computer also offers the opportunity to experience ideas and emotions. “Art, architecture, and engineering bring together, in some way, intelligence with intuition, basic pillars of all design. Technical resources do not limit human activity: on the contrary, they can widen human intuition, reason and decision greatly if they are conveniently used. When I see children with Logo, they learn, they learn new things, a way of being. They choose subjects that are significant to them and they make mistakes and they correct their mistakes… and that is not just to develop an analytical mind. They are feeling that they are living.”


Several children in Buenos Aires schools are already using three-dimensional Logo. “We are watching them, trying to see what they are doing and learning from them… Adults tend to do old things with new tools, applying what they already know… But children will find new uses for the computer. This is important, so we must have the courage and the generosity to let them do it.”




Creación y representación de formas tridimensionales. Creation and Representation of Three-Dimensional Shapes.


An Inaugural Lecture by Horacio C. Reggini delivered at the Academia Nacional de Ciencias Exactas, Físicas y Naturales (Argentine Academy of Mathematics, Physics and Natural Sciences) on November 27, 1987. Anales, Tomo XL, 1988.




The first part of the presentation deals with the geometrical description and  computer generation of three-dimensional shapes. The importance of intrinsic geometry and modular techniques is evident in the construction of a set of basic designs: diverse and intricate forms can be created from simple building blocks, their complexity arising from the combination and interaction of a large number of small elements and processes. Using a three-dimensional Logo system developed by the author, several architectural examples are reproduced and displayed from various viewpoints. The most important aspect of three-dimensional Logo is its use as a rigorous and intuitive language for describing and generating objects. The same system may be applied to robot guidance. Thus computers become a versatile means of expression, providing computer users with an opportunity to feel the emotion and joy of the creative act.


The second part of the presentation deals with the two-dimensional representation of three-dimensional shapes. It has been noticed that classical rules of perspective -which govern orthodox drawings and photographic images- are not always satisfactory. Artists have dealt with this problem in different ways. In this presentation, a perspective system based upon curved projection rays is introduced. This new system reduces certain distortions and rigidities observed by some people in classical perspective. The curvature of the rays is controlled by an image index, which can be selected at will by the user in another three-dimensional Logo system also developed by the author. Examples of images generated with different indexes are shown. The possibility of adding such an option to new visual devices, such as television,  is raised, whereby the users can choose among alternatives according to taste. This opportunity to select the proper index would be possible in the near future thanks to modern techniques of computerized virtual reality and digital image processing. It is concluded that these ideas are rooted in the belief that it is essential to give people adequate resources for thinking and acting freely.




Regular polyhedra: random generation, Hamiltoniam paths and single chain nets


Paper by Horacio C. Reggini, Monografía nº 6, Academia Nacional de Ciencias Exactas, Físicas y Naturales, Buenos Aires, 1991.



Computer simulations are carried out using the Logo language in relation with the number of trials necessary to create each regular polyhedron by a random method. Statistical results (distributions, averages and standard deviations) are calculated and analyzed, relating obtained values to face paths and plane nets of each polyhedron.

All the possible different paths of traveling along all the faces of a regular polyhedron, given two initial faces, without passing more than once by the same face -called “Hamiltonian face  paths”- are determined: 2 for the tetrahedron, 10 for the cube, 6 for the octahedron, 1264 for the dodecahedron and 54 for the icosahedron. The number of paths that arrive at a neighbour face of the departure face are respectively: 2, 8, 4, 512 and 20. The number of paths that arrive at the opposite face  to the departure are: none, 2, 2, 144 and 2.

If it is possible to travel around without re-entering a face, this means that the polyhedron can be cut along its edges so as to make it into a single chain of polygons. Thus, Hamiltonian paths are associated with corresponding single chain plane nets that by folding along common edges generate the regular polyhedra. It is deduced that the total numbers of the different single chain nets are: 2 for the tetrahedron, 8 for the cube, 5 for the octahedron, 680 for the dodecahedron and 36 for the icosahedron. If mirror-image nets are not counted the corresponding numbers become: 1, 4, 3, 340 and 18 respectively. Drawings of the nets are included.

Finally, the problem of the face circuits is related to the dual problem of the vertex circuits (1, 1, 3, 2 and 33 different routes for each regular polyhedron, or 1, 1, 2, 1 and 17, if mirror-image circuits are not counted) studied by Hamilton a century ago. Divisions of the polyhedra into two segments by respective vertex are shown.


Note: Other articles on 3D-Logo by Horacio C. Reggini:

1. Exploring Three-Dimensional Space with Logo, Logo-85 Conference, Plenary Session IV, M.I.T., Cambridge, MA, July 1985.

2. Explorando formas espaciales con Logo, Summa, Architecture Argentine Journal, Nos. 221-222, ene.-feb. 1986, Buenos Aires.

3. Generation of space shapes with a computer, The Argentine Engineering Institution Bulletin, 370, dic. 1986, Buenos Aires.

4. Exploring 3-dimensional space with Logo, Micromath, Basil Blackwell Ltd., Vol. 2, N° 1, Spring 1986, London.

5. Towards an Artisanal Use of Computers. Their application to the design and study of three-dimensional forms, Logo-86 Proceedings of The Third International Logo Conference, M.I.T., Cambridge, MA, July 1986.

6. Exploring 3-Dimensional Space with Logo, Logo Memo 102, The Media Lab, M.I.T., Cambridge, MA, July 1986.

7. 3-Dimensional Logo Commands and Procedures, Logo Memo 103, The Media Lab, M.I.T., Cambridge, MA, July 1986.

8. Projects with 3-Dimensional Logo, Logo Memo 104, The Media Lab, M.I.T., Cambridge, MA, July 1986.

9. Geometría computacional: generación al azar de poliedros, caminos hamiltonianos y redes unifilares, Anales de las Conferencias del II Encuentro Hispanoamericano de Historia de las Ciencias, Real Academia de Ciencias Exactas, Físicas y Naturales de España y Academia Nacional de Ciencias Exactas, Físicas y Naturales de la República Argentina, Buenos Aires, 10/16 de septiembre de 1990.

10. Three-dimensional Logo, Journal of the International Council for Computers in Education, Special Interested Group for Logo-Users Educators, April 1990, Vol. 8, No. 8.

11. Tree graph representation of Hamiltonian paths, Computers & Mathematics, vol. 27, n. 5, ps. 101-114, Pergamon, Elsevier Science Ltd., Oxford, 1994. Paths along faces of a polyhedron can be assimilated to paths along the branches of a tree graph. This paper shows the “pruned” trees corresponding to all Hamiltonian paths of regular polyhedra.



Annex 4


Computadoras: ¿creatividad o automatismo?, by Horacio C. Reggini, Ediciones Galápago, Buenos Aires, 1988. Computers: Creativity or Automatism?

From the prologue

More than three decades ago I first had the chance to make contact with the machine that, without fear of overstating, can be called the discovery of this century: the computer. The first computers, at that time, were not only extremely expensive but almost inaccessible. A few years later, I became close to the MAC Project at the Massachusetts Institute of Technology which since the early ’60s, has researched how technology can enhance human beings intellectually and creatively. “MAC” comes from “Machine Aided Cognition” which was the general goal of the project, and also from “Multiple Access Computer” which meant its operative mode. Since then, I have remained tied to computers as a consulting engineer, a professor and a researcher.


During this time, the initial surge of the new technology has not declined and enthusiasm remains high all over the world. Today, new discoveries, like fiber optics and parallelism techniques in new computer design allow us to foresee technological marvels that go beyond yesterday’s fantasies. The future, wich once seemed so far away, becomes reality every day.


I have always believed that computers should be placed in people’s hands as a means for growing, no matter what their activities are. But it was only in the ’70s and -much more intensively- after 1980, with the invention of small personal computers, that informatics could escape from the narrow limits imposed by the setting of the laboratory or large organizations. Today, millions of people around the world have a fluent dialogue with their computers, which permits them to work, create and study independently and autonomously.


However, real interaction with computers is just beginning. Technological humanity has just stammered its first words and it is still too early for evaluations. What role will computers play in the next decades? We must search together for the answer. Those who succumb to “computer magic”, that unthinking enchantment by the technological novelty, anticipate golden years when technology will provide us with thousands of material benefits.


On the other hand, those who reject, also unthinkingly, anything new, warn of the dangers of a coming “cybernetic slavery” that will end up turning us into robots and stripping us of our more genuine values. I believe that only a thorough understanding of their significance and an ample freedom of application will allow us to find new and more human uses for computers, so their enormous technological potential does not turn us into mere passive receivers of information.


The road divides and we must choose: creativity or automatism? This book responds to the first premise as I am convinced it is scientists’ responsibility to contribute to the most harmonious introduction of the new technologies into society. I also believe that the more we learn about technology, the deeper we can perceive the essence of our humanity. The articles, lectures, presentations and interviews, etc., that comprise this book have one common thesis: only a wise use of the new technological tools can make us more complete, creative and freer people. This thesis is applied to the different areas I have been interested for years: computers in general, computers in education, artificial intelligence, the design and construction of forms in space, the psychology of learning, and the development of scientific knowledge. It is also the basic concept which gives unity to the book and to the variety of areas dealt with. I have revised and made minor changes in some of the chapters to avoid unnecessary repetition and to make the marginal superpositions among them more acceptable; although, in some cases, I have allowed some repetitions so that each chapter may stand independently.


I believe the ideas concerning computers are new and important and of vital significance for the process of human evolution. That is why, more than ever before, a sensible integration of the new technology is necessary at every level, while avoiding the fragmentation of culture and without leading towards undesirable institutions or life styles. In my contribution to all this, I have chosen two paths: on one hand, to influence education, trying to make of computers tools to enable more creative, less dogmatic personal growth; on the other hand, to investigate, develop and diffuse through mass media, proper methods and means to apply computers in various areas of daily life.


These pages express my thoughts as well as many of the alternatives that mark out the road yet to be traveled.




From the epilogue

Shall we advocate creativity or automatism? It would be an erroneous and artificial classification to imply that the two words are mutually exclusive. Instead, I intend to indicate two possible alternatives as well as two complementary aptitudes. We tend to use the word creativity to refer to the production of artistic or new ideas, and to use automatism to mean a sequence of actions produced so mechanically that they can be carried out without interfering with any other activity. I believe that both processes are vital and necessary. In that sense, the title of this book could have been creativity and automatism. Automatism as a result of previously acquired action scripts -which required creativity when first used- is beneficial, as it sets free other parts of the mind. Automatism in the use of computers must not be neglected, but it will be insufficient if it is not combined with sparks of creativity which allow us to change or modify our routines, giving birth to previously unknown goals or imagining new destinies.




Chapter 12. Popper and Papert


“I must not search for my dignity in space, but in the government of my thoughts… through my thoughts I embrace the world”, Blaise Pascal wrote more than three centuries ago. Even then, Pascal considered intelligence as the only way to freedom, and education as the only way to intelligence. But, like any human gesture, education and learning theories suffer the influence of the cultural and philosophical ideas of the age. Today, perhaps more than ever before, education holds a privileged position but… what sort of education?


Creative learning


In the light of new ideas, influential thinkers of our century, coming from different currents of thought, coincide in characterizing learning as a process of acquisition and practice of new methodologies, skills or aptitudes necessary to face new situations. They formulate a new emergent paradigm in education based on lack of dogmatism, respect for individual freedom and critical thinking.


Among those who stand out for their contributions to the reform of educational theories, are Karl Popper and Seymour Papert. The former, born July 28, 1902, in Austria, is considered one of the most important philosophers of science, whose ideas about methodological fallibility, conjectures and refutations, and the evolutionary theory of knowledge, among others, constitute a valuable contribution to universal thinking. The latter, born February 29, 1928, in South Africa, is a mathematician and epistemologist widely known, for over a decade, for his research and development into the use of computers in the classroom. His activities, following Piaget’s ideas, his contributions to artificial intelligence and the creation and worldwide distribution of the Logo computer modality have earned him a privileged position among modern educational reformers.


Both have merited authoritative articles published previously by this medium (1,2); and some time ago, I also referred to the educational approach of Logo modality (3,4). On this occasion, however, I attempt to point out some similarities and coincidences between Papert’s ideas, embodied in his unique computation system, and Popper’s ideas regarding education implicit in his extensive philosophical work.



The error as a source for understanding


For a long time it was believed that education is essentially a transmission of knowledge from teacher to student in which knowledge is received from the outside; we accept it as long as it can be adequately justified and we increase it through the inductive method, beginning with individual cases to arrive at a universal rule. This was the position held by such outstanding philosophers as Locke or Hume. However, already in this century, Karl Popper, among other authors, was concerned in his philosophical research with the growth of knowledge, both in the individual and society, and criticized the theories that considered the child as a blank page and the mind as little more than a file of impressions gathered by the senses. Refuting one by one the logical foundations of that position, Popper established the hypothesis that knowledge is only increased by means of conjectures and refutations, and learning only takes place through trial and error, where this method becomes a source of knowledge: “To learn how not to commit errors, we must learn from our errors” (5).


For Popper, “the learning process consists mainly in correcting expectations that are not satisfied” (6) and these expectations are just what generate the process of trial and error.


In “Mindstorms”, Seymour Papert describes the fundamental ideas of his educational philosophy. Much of his work is devoted to evaluating the role of error, which is not considered an imperfection that must be eliminated but a functional element that participates in the process of achieving an objective. He asserts: “Many children are held back in their learning because they have a model of learning in which you have either ‘got it’ or ‘got it wrong’. But when you learn to program a computer you almost never get it right the first time. Learning to be a master programmer is learning to become highly skilled at isolating and correcting ‘bugs’, the parts that keep the program from working. The question to ask about the program is not whether it is right or wrong, but if it is fixable. If this way of looking at intellectual products were generalized to how the larger culture thinks about knowledge and its acquisition, we all might be less intimidated by our fears of ‘being wrong'” (7). Later he explains that much of the work with Logo is devoted to strategies of debugging, such as developing autonomous subprocedures so that errors can be easily isolated and corrected. The priceless by-product of learning these skills is a new attitude about errors. “Errors benefit us because they lead us to study what happened, to understand what was wrong, and, through understanding, to fix it” (8).


Learning through trial and error shapes a behaviour that works. The debugging process –search for and correction of errors– in Logo procedures plays, for Papert, a fundamental role in the understanding of any subject: “School teaches that errors are bad; the last thing one wants to do is to pore over them, dwell on them, or think about them. The debugging philosophy suggests an opposite attitude. Experience with computer programming leads children more effectively than any other activity to ‘believe’ in debugging” (9).



An evolutionist theory of learning


This concept of error as a tool for the growth of knowledge harmonizes with Popper’s idea about the selection of ideas. Just as in the world of living organisms, evolution takes place through natural selection of the fittest, Popper believes that, in the world of ideas, theories evolve through trial, where incorrect theories are eliminated. This concept can also be applied to education. H. J. Perkinson writes: “Karl Popper has done more than destroy the transmission theory of education. He has also given a new theory of evolutionary epistemology, upon which it is possible to build a new theory of education, a Darwinian theory of education. This theory could perhaps be called learning from our mistakes. According to this theory, the learner is active, not passive, not a receptor of knowledge; a seeker of order, not needing motivation or control in order to learn. The learner learns from making mistakes” (10).



Natural learning: active, autonomous and critical


“The best learning takes place when the learner takes charge” (11), writes Papert. The Logo creator believes children are active builders of their own intellectual structures and “innately gifted learners”, acquiring long before they go to school a vast quantity of knowledge by a process he calls Piagetian learning: “learning without curriculum… supporting children as they build their own intellectual structures with materials drawn from the surrounding culture. In this model, educational intervention means changing the culture, planting new constructive elements in it and eliminating noxious ones” (12). “Piagetian learning is typically deeply embedded in other activities. For example, the infant does not have periods set aside for ‘learning talking’. This model of learning stands in opposition to dissociated learning, learning that takes place in relative separation from other kinds of activities, mental and physical.” (13).


Papert uses the term constructionism in reference to two aspects of education. From the constructivist theories of psychology he adopts the thesis that considers learning as a reconstruction rather than a consequence of the transmission of knowledge. From educational experience, he draws the evidence that learning is particularly effective when it is immersed in an activity in which the student feels he is building something significant and meaningful (for example, an artistic piece, a machine that functions, a research project, or a computer program). In this case the student builds a meaningful entity instead of acquiring facts and data devoid of a context in which they can be immediately used and understood (14). Besides, Papert includes within constructionism social and affective aspects that go beyond mere cognition. The fact that the child is dedicated to building something, and the particular circumstance that he is doing something he feels and in which he believes, adds new dimensions to the learning process. He believes that certain learning takes place early and spontaneously, because the child as builder finds the materials with which to build. Others do not develop because of the relative poverty of the culture in those materials that would make the corresponding concepts simple, and concrete, and not because of their complexity or formality. Therefore, his insistence on giving children adequate access to learning environments or microworlds. “The educator must be anthropologist. The educator as anthropologist must work to understand which cultural materials are relevant to intellectual development”, Papert says (15).


Popper defines the human being as an active seeker of knowledge so much so that he remembers his first years of school with the following words: “I will always be grateful to my first teacher, Emma Goldberger, who taught me how to read, write and the basic rules of arithmetic. These three subjects, I believe, are the only essential things that must be taught to a child; and some children still do not need to be taught anything at all so as to learn them. Everything else is a matter of atmosphere and continuous learning by thinking and reading” (16).


Logo, on the other hand, is a synonym of self-education: each person establishes his or her own goals, recognizes his or her mistakes, analyzes them and learns from them. Papert writes, with reference to the use of Logo: “…when a child learns to program, the process of learning is transformed. It becomes more active and self-directed. In particular, the knowledge is acquired for a recognizable personal purpose. The child does something with it. The new knowledge is a source of power and is experienced as such from the moment it begins to form in the child’s mind” (17).


Karl Popper refers to this topic in a dialogue with John Eccles on the occasion of the preparation of their book “The Self and its Brain”, when they spent time together in Villa Serbelloni, Bellagio, Italy. There, in lovely and stimulating surroundings, beside Como Lake, in the morning of September 20, 1974, he comments: “I think that it is terribly important that, throughout our whole life, we should avoid being mere passive receptors of information. There is a special danger in childhood: that our schools may treat children like the gondola kitten*. This was especially true when children had to sit in a confined way so as to reduce the possibility for the children to move, so they shouldn’t disturb other children and, especially, the teacher. In other words, our children were once like gondola kittens. While it would not matter as much if people our age spent their time staring at television screens, I think it is undesirable for television or teaching machines to be used as a means of instruction such that children play a passive role: they just sit and learn. I don’t deny that television has its good side if used sparingly, but a growing young person should be stimulated to face problems and then to try to solve these problems, and should be helped in solving these problems only if help is needed. He should not be indoctrinated, and should not be fed with answers when no questions have been asked: when problems don’t come from within” (18).




* The most elegant and delightful example of the role of activity in visual learning is provided by the experiments of Held and Hein (19). Kittens from the same litter spend several hours a day in a contraption which allows one kitten fairly complete freedom to explore its environment actively, just as a normal kitten. The other is suspended passively in a gondola that, by a simple mechanical arrangement, is moved in all directions by the exploring kitten so that the gondola passenger is subjected to the same play of visual imagery as the active kitten, but none of this activity is initiated by the passenger. Its visual world is provided for it just as it is for us on a TV screen. When not in this contraption both kittens are kept with their mother in darkness. After some weeks, tests show that the active kitten has learned to utilize its visual fields for giving it a valid picture of the external world for the purpose of movement just as well as a normal kitten, whereas the gondola passenger has learned nothing.

The spirit of scientific method in learning environments


The fact of learning has much in common with building a scientific theory (20). Papert emphasizes this idea in reference to learning a physical skill. He considers programming as a source of descriptive mechanisms, that is to say as a means of strengthening language. When a child learns to program with Logo, he uses the bricolage or tinkering method, a technique often used in scientific research or in artistic creation. It is a road with a beginning but which holds many surprises (21, 22, 23).


Popper describes the human being as a fallible creator of knowledge through conjectures and refutations. Consequently education should emphasize the process of problem solving rather than obtaining results. Popper believes it is impossible that knowledge is derived only from observation. Observation, he asserts, comes after expectations. And this applies to the baby who tries to grab everything he sees as much as to the scientist who creates new theories about the universe: both proceed by trial and error, correcting their theories in the light of unfulfilled expectations. Popper says: “If I perceive a sofa, I immediately think one can sit on it. This is part of perception. That is why perception essentially surpasses the content of the signs” (24). “The process of bringing, by means of mutation and selection, the cognitive structures to the real world is not so much a process of adaptation as a process of active construction of theories and of testing them through the difficult process of examination” (25). Papert adopts a similar criterion for the application of Logo and explains: “…when a student, or anyone, sticks with the drive to make an actual connection, an actual program actually run (in a computer or in life), then rigor grows. Discipline flips from the external and oppressing ‘get it right’ to the internal and intellectual ‘make it work'” (26). When programming with Logo children get used to the idea that, in most cases, it is better to find a quick, simple, economic, solution to a problem, and then debug it until it works properly, than to insist on a definite, slow, and complex solution, apparently without mistakes. And he adds: “What an individual can learn, and how he learns it, depends on what models he has available” (27).


Thus, the classroom –just as the laboratory– provides a context where theories can be built and refuted, a place where errors can be committed… and where one can learn from them. The teacher is not a judge but a person who provides the proper environment where the child can recognize errors and correct them. Autonomy is a goal of learning that can only be achieved when learning itself becomes an exercise of autonomy.



The child as protagonist of the learning process


These theories about the learning process impose changes in the teacher-student relationship. In the new learning settings teachers learn and students can create complex theories. Karl Popper considers that a new ethics is necessary based on a new kind of fallible, uncertain knowledge, as he believes that errors are unavoidable. And he explains: “Any new discovery, or new concept, falsifies a theory established up to that moment; a theory that does not produce it, that can not produce it. The theory, for example, that one can not speak with someone in a distant place, therefore telephones can not exist, is constantly verified till the telephone was invented. The same is true for the theory that says that people can not fly. Thus, this means that any theory that has been built based on the fact that anything like that does not exist, has been up to that moment constantly confirmed” (28). That is why, the importance given to the development of critical thinking oriented to the discovery of error; a concept shared by advanced researchers like Marvin Minsky who asserts: “To understand how something works, it helps to know how it can fail” (29).


Although Karl Popper never dealt specifically with education, H. J. Perkinson shows “his system of conjectures and refutations may be fashioned into a model of education –an approach which views the teacher not as someone who injects predigested knowledge into the passive minds of students, but as one who elicits student responses, creates environments in which ideas can be criticized freely and mistakes made without humiliation: in short, a classroom in which the student is respected as a solver of problems and originator of ideas” (30).


For his part, Seymour Papert, as was pointed out by Cynthia Solomon, a researcher who carried out a comparative evaluation of the use of computers in education, follows Piaget in his assertion that people have different theories to explain the world, and that those theories keep changing as long as they learn, and, what is most important, that they learn while doing (31). Papert’s ideas rest upon the educational theories that sustain that learning environments must be focussed on the student and not on the teacher.



Two educational utopias


So far, I have pointed out some coincidences that arise from an analysis of Popper and Papert’s ideas. I don’t intend to say this is all but, I want to emphasize some premises that have gained increasingly wide acceptance in different educational circles. Both thinkers propose less authoritative education, based on human fallibility and on the search for freedom. And it is in their dreams where they coincide once more. Popper writes: “In 1917, I understood very clearly what I had been feeling within since long ago: In our famous high schools… we were wasting our time miserably, although our teachers were very learned and endeavoured to make of our schools the best in the world. The fact that part of their teaching was extremely dull –hours and hours of agony without hope– was not new for me. At school one was exposed to discovery when one’s own thoughts were engaged with something disconnected from the lesson; one was forced to pay attention… I dreamed of creating a school where the young could learn without weariness and where they were stimulated to present problems and discuss them; a school where one should not have to listen to undesirable answers to unasked questions”(32).


Papert writes: “…the classroom is an artificial and inefficient learning environment that society has been forced to invent because its informal environments fail in certain essential learning domains… I believe that the computer presence will enable us to modify the learning environment outside the classroom so that much -if not all- the knowledge schools presently try to teach with such pain and expense and such limited success, will be learned in the same way the child learns to talk, painlessly, successfully, and without instruction” (33).


These concepts could be regarded as utopias to many people. We instead consider them as necessary beacons for today’s education.




1. Zanotti, L. J., “Papert: un desafío que debe aceptarse”, I.I.E., Revista del Instituto de Investigaciones Educativas, Nº 43, Buenos Aires, noviembre de 1983.

2. Darós, W. R., “Concepción Popperiana del aprendizaje”, I.I.E., Revista del Instituto de Investigaciones Educativas, Nº 61, Buenos Aires, abril de 1988.

3. Reggini, H. C., “Irrupción de las computadoras en la educación”, I.I.E., Revista del Instituto de Investigaciones Educativas, Nº 35, Buenos Aires, abril de 1982.

4. Reggini, H. C., “Revisión del aprender y del enseñar” I.I.E., Revista del Instituto de Investigaciones Educativas, Nº 43, Buenos Aires, noviembre de 1983.

5. Popper, K. R., “Sociedad abierta, universo abierto (Conversaciones con Franz Kreuzer)”, Edit. Tecnos, Madrid, 1985, p. 156.

6. Berkson, W. and Wettersten, J., “Learning from Error, Karl Popper’s Psychology of Learning”, Open Court Publishing Company, Illinois, E.E.U.U., 1984, p. 8.

7. Papert, S., “Mindstorms, Children, Computers, and Powerful Ideas”. Basic Books, Inc., Publishers. New York, 1980, p. 23.

8. Ibid. p. 114.

9. Ibid.

10. Perkinson, H. J., “Education and Learning from Our Mistakes”, en “In Pursuit of Truth, Essays on the Philosophy of Karl Popper on the Occasion of his 80th Birthday”, Edited by Paul Levinson, Humanities Press, Atlantic, Highlands, N. J., 1982, p.141.

11. Papert, S., Op. cit., p. 214.

12. Ibid., p. 32.

13. Ibid., p. 48.

14. Papert, S. “Constructionism: A new Opportunity for Elementary Science Education”, NSF Proposal, M.I.T., November 1986.

15. Papert, S., “Mindstorms, Children, Computers, and Powerful Ideas”, p. 47.

16. Popper, K. R., “Búsqueda sin término: una autobiografía intelectual”, Edit. Tecnos, Madrid, 1985, pág. 43.

17. Papert, S., “Mindstorms, Children, Computers, and Powerful Ideas”, p. 21.

18. Popper, K. R., And Eccles, J. C., “The Self and Its Brain”. Springer International, Berlin, 1977, p. 435.

19. Held, R. and Hein A., “Movement-produced Stimulation in the Development of Visually Guided Behaviour”, Journal of Comparative and Physiological Psichology, 56, 1963, ps. 872-6.

20. Papert, S., “Mindstorms, Children, Computers, and Powerful Ideas”, p. 96.

21. Ibid., p. 173.

22. Reggini, H. C., “Alas para la mente”, Ediciones Galápago, Buenos Aires, 1982, p. 91.

23. Reggini, H. C., “Ideas y formas”, Ediciones Galápago, Buenos Aires, 1985, p. 15.

24. Popper, K. R., “Sociedad abierta, universo abierto (Conversaciones con Franz Kreuzer)”, p. 92.

25. Ibid., p. 99.

26. Brand, S., “El laboratorio de medios. Inventando el futuro en el M.I.T.”, Ediciones Galápago, Buenos Aires, 1988, p. 126

27. Papert, S., “Mindstorms, Children, Computers, and Powerful Ideas”, p. VII

28. Popper, K. R., “Sociedad abierta, universo abierto (Conversaciones con Franz Kreuzer)”, pág. 40.

29. Minsky, M., “La sociedad de la mente”, Ediciones Galápago, Buenos Aires, 1987, Cap. 14.2, p. 142.

30. Levinson, P., “In Pursuit of Truth, Essays on the Philosophy of Karl Popper on the Occasion of his 80th Birthday”, p. 9.

31. Solomon, C., “Computer Environments for Children. A Reflection on Theories of Learning and Education”, The M.I.T. Press, Cambridge, MA, 1986.

32. Popper, K. R., “Búsqueda sin término: una autografía intelectual”, ps. 43, 54.

33. Papert, S., “Mindstorms, Children, Computers, and Powerful Ideas”, p. 9.








Buenos Aires, Argentina, May 1998.