Seymour Papert

Epistemological Pluralism: Styles and Voices within the Computer Culture

The prevailing image of the computer represents it as a logical machine and computer programming as a technical, mathematical activity. Both the popular and technical culture have constructed computation as the ultimate embodiment of the abstract and formal. Yet the computers intellectual personality has another side: our research finds diversity in the practice of computing that is denied by its social construction. When we looked closely at programmers in action we saw formal and abstract approaches; but we also saw highly successful programmers in relationships with their material that are more reminiscent of a painter than a logician. They use concrete and personal approaches to knowledge that are far from the cultural stereotypes of formal mathematics.1

Software Design as a Learning Environment.

Abstract This article describes a learning research called the Instructional Software Design Project (ISDP), and offers a Constructionist vision of the use of computers in education. In a Logo‐based learning environment in a Boston inner‐city public school, a fourth‐grade class was engaged during one semester in the design and production of educational software to teach fractions. Quantitative and qualitative research techniques were used to assess their learning of mathematics, programming, and design, and their performance was compared with that of two control classes. All three classes followed the regular mathematics curriculum, including a two‐month unit on fractions. Pre‐ and post‐tests were administered to the experimental and control groups. The evaluation revealed greater mastery of both Logo and fractions as well as acquisition of greater metacognitive skills by the experimental class than by either control class. Selected results from several case studies, as well as an overall evaluation are p...

Teaching Children to be Mathematicians vs. Teaching About Mathematics

Summary The important difference between the work of a child in an elementary mathematics class and that of a mathematician is not in the subject matter (old fashioned numbers versus groups or categories or whatever) but in the fact that the mathematician is creatively engaged in the pursuit of a personally meaningful project. In this respect a childs work in an art class is often close to that of a grown‐up artist. The paper presents the results of some mathematical research guided by the goal of producing mathematical concepts and topics to close this gap. The prime example used here is ‘Turtle Geometry’, which is concerned with programming a moving point to generate geometric forms. By embodying the moving point as a ‘cybernetic turtle’ controlled by an actual computer, the constructive aspects of the theory come out sufficiently to capture the minds and imaginations of almost all the elementary school children with whom we have worked—including some at the lowest levels of previous mathematical perfo...

A Critique of Technocentrism in Thinking About the School of the Future

Publisher Summary The presence of computers and other new technologies in learning will play a determining role in the way that both technology and culture evolve in the coming generation. The future of computer could be made in many different forms. It will be determined not by the nature of the technology, but by a host of decisions of individual human beings. Thinking of the future as an information age certainly focuses on some exciting new developments. There is more access to more information than there has ever been before. But, there is also a dangerous side to it from an educators point of view—the danger of seeing the most important aspect of education as the provider of information or even the provider of access to information. The chapter presents a critique of technocentrism in thinking about the school of the future. The author has coined the word technocentrism from Piagets use of the word egocentrism. Technocentrism is the fallacy of referring all questions to the technology. Questions regarding whether technology have this or that effect, whether using computers to teach mathematics increases childrens skill at arithmetic, or whether it will encourage children to be lazy about adding numbers because calculators can do it, reflect technocentric thinking. The issues about how to use the computer in education reflect deeper issues of educational theory and philosophy.

Artificial Intelligence, Language, and the Study of Knowledge*,†

This paper studies the relationship of Artificial Intelligence to the study of language and the representation of the underlying knowledge which supports the comprehension process. It develops the view that intelligence is basedon the ability to use large amountsof diverse kinds of knowledge in procedural ways, rather than on the possession of a few general and uniform principles. The paper also provides a unifying thread to a variety of recent approaches to natural language comprehension. We conclude with a brief discussion of how Artificial Intelligence may have a radical impact on education if the principles which it utilizes to explore the representation and use of knowledge are made available to the student to use in his own learning experiences.

Programming-Languages as a Conceptual Framework for Teaching Mathematics

Formal mathematical methods remain, for most high school students, mysterious, artificial and not a part of their regular intuitive thinking. The authors develop some themes that could lead to a radically new approach. According to this thesis, the teaching of programming languages as a regular part of academic progress can contribute effectively to reduce formal barriers. This education can also be used to enable pupils to access an accurate understanding of some key mathematical concepts. In the field of heuristic knowledge for technical problem solving, experience of programming is no less valuable: it lends itself to promote a discussion of relations between formal procedures and the comprehension of intuitive problem solving and provides examples for the development of heuristic precepts (formulating a plan, subdividing the complexities, etc.). The knowledge gained in programming can also be used for the discussion of concepts and problems of classical mathematics. Finally, it can also facilitate the expansion of mathematical culture to topics in biological and physical sciences, linguistics, etc. The authors describe a programming language called ‘Logo’ adapted to objectify an enduring framework of mathematical experimentation.

Unrecognizable Sets of Numbers

When is a set <italic>A</italic> of positive integers, represented as binary numbers, “regular” in the sense that it is a set of sequences that can be recognized by a finite-state machine? Let π <subscrpt><italic>A</italic></subscrpt>(<italic>n</italic>) be the number of members of <italic>A</italic> less than the integer <italic>n</italic>. It is shown that the asymptotic behavior of π <subscrpt>A</subscrpt>(<italic>n</italic>) is subject to severe restraints if <italic>A</italic> is regular. These constraints are violated by many important natural numerical sets whose distribution functions can be calculated, at least asymptotically. These include the set <italic>P</italic> of prime numbers for which π <subscrpt><italic>P</italic></subscrpt>(<italic>n</italic>) @@@@ <italic>n</italic>/log <italic>n</italic> for large <italic>n</italic>, the set of integers <italic>A</italic>(<italic>k</italic>) of the form <italic>n<supscrpt>k</supscrpt></italic> for which π <subscrpt><italic>A</italic>(<italic>k</italic>)</subscrpt><italic>n</italic>) @@@@ <italic>n<supscrpt>P/k</supscrpt></italic>, and many others. The technique cannot, however, yield a decision procedure for regularity since for every infinite regular set <italic>A</italic> there is a nonregular set <italic>A′</italic> for which | π <subscrpt><italic>A</italic></subscrpt>(<italic>n</italic>) — π <subscrpt><italic>A′</italic></subscrpt>(<italic>n</italic>) | ≤ 1, so that the asymptotic behaviors of the two distribution functions are essentially identical.

A case study of a young child doing turtle graphics in LOGO

This paper explores some important issues with regard to using computers in education. It probes into the question of what programming ideas and projects will engage young children. In particular, a seven year old childs involvement in turtle graphics is presented as a case study.

Programming-languages as a conceptual framework for teaching mathematics

This is a report of research and teaching d~r~cted toward the development of a new mathematics curriculum in which presentation depends fundamentally on the use of computers and programming. The work was centered mainly on a mathematics teaching experiment with seventh grade children utilizing a programming language, LOGO, specifically designed for the teaching of mathematics. We also conducted an investigation of the use of LOGO in teaching much younger children-a group of second and third graders. After a brief exposition of the LOGO language, the two teaching activities are described in some detail, including many examples of the classroom and laboratory materials used. The report begins with a discussion of the reasons why the learning and teaching of mathematics are so difficult, and states the underlying issues that have dictated the kind of approach undertaken here. Following the descriptive material on the teaching experiments is a discussion of the results including some evaluations of the years work and of the project. A detailed description of the LOGO programming language and system is appended. taught the group of second and third grade children. We did not begin the teaching with a large body of previously developed classroom materials. These had to be created concurrently with the teaching as the INTERFACE v4, #2 APRIL 1970

The Turtle's Long Slow Trip: Macro-Educological Perspectives on Microworlds

In the literature on microworlds, the writing that is closest to the “macroeducological” intent of this essay is Celia Hoyles’ (1993) Microworlds/ Schoolworlds: The Transformation of an Innovation. By loose analogy with the usage of the terms macro-economics and micro-economics my neologism recognizes as a field worthy of serious theoretical attention the study of phenomena such as microworlds on the level of the functioning of the system of education or, as I should rather say, the learning environment. I use the opposing term “micro-educological” to encompass the kind of work most of us who care about microworlds do most of the time; typically work focused on the learning process or on the invention and study of specific means of learning. These macro and micro domains are not intended to be exclusive; for example, I shall be looking at epistemological questions that straggle between them and serve as a basis for a unified approach, which differentiates what I have in mind from typical writing by sociologists and historians of education. Hoyles is on macro-educological territory when she describes a systematic “trivializing” tendency in School’s1 adoption of the microworld idea. For example, the idea of a computational world rich enough for children to make mathematical discoveries is turned into a set of virtual manipulative materials to exercise mathematical manipulations taught in a traditional “instructionist” mode. On a general level, I join her intention to explain the phenomenon, as rooted in incompatibilities of fundamental cognitive values. Yet, I am less willing than she