Notes

INTRODUCTION

1. Piaget is at the center of the concerns of this book. I make a slightly unorthodox interpretation of his theoretical position and a very unorthodox interpretation of the implications of his theory for education. The reader who would like to return to the source needs some guidance because Piaget has written a large number of books, most of which discuss particular aspects of children’s development, assuming that the others have been read as a theoretical preface. The best short book about Piaget is M. Boden’s Piaget (London: Harvester Press, 1979). A good starting place for reading Piaget’s own texts is with H. E. Gruber and J. J. Voneche, eds., The Essential Piaget: An Interpretive Reference and Guide (New York: Basic Books, 1977). My own “short list” of books by Piaget that are most readable and provide the best philosophical overview of his ideas are: The Child’s Conception of the World (New York: Harcourt, Brace and Co., 1929); The Child’s Conception of Physical Causality (New York: Harcourt, Brace and Co., 1932); The Psychology of Intelligence, trans. Malcolm Piercy and D. E. Berlyne (New York: Harcourt, Brace and Co., 1950); The Origins of Intelligence in Children, trans. Margaret Cook (London: Routledge and Kegan Paul); Introduction à l’Epistémologie Génétique (Paris: Presses Universitaires de France, 1950); Insights and Illusions in Philosophy, trans. Wolfe Mays (New York: The World Publishing Co., 1971); The Grasp of Consciousness, trans. Susan Wedgwood (Cambridge: Harvard University Press, 1976). For a critique of the “Piaget Curriculum Developers,” of whom I have said that they are “standing Piaget on his head,” see G. Groen, “The Theoretical Ideas of Piaget and Educational Practice,” The Impact of Research on Education, ed. P. Suppes (Washington D. C.: The National Academy of Education, 1978).

2. LOGO is the name of a philosophy of education in a growing family of computer languages that goes with it. Characteristic features of the LOGO family of languages include procedural definitions with local variables to permit recursion. Thus, in LOGO it is possible to define new commands and functions which then can be used exactly like primitive ones. LOGO is an interpretive language. This means that it can be used interactively. The modern LOGO systems have full list structure, that is to say, the language can operate on lists whose members can themselves be lists, lists of lists, and so forth.

Some versions have elements of parallel processing and of message passing in order to facilitate graphics programming. An example of a powerful use of list structure is the representation of LOGO procedures themselves as lists of lists so that LOGO procedures can construct, modify, and run other LOGO procedures. Thus LOGO is not a “toy,” a language only for children. The examples of simple uses of LOGO in this book do however illustrate some ways in which LOGO is special in that it is designed to provide very early and easy entry routes into programming for beginners with no prior mathematical knowledge. The subset of LOGO containing Turtle commands, the most used “entry route” for beginners, is referred to in this book as “TURTLE TALK” to take account of the fact that other computer languages, for example SMALLTALK and PASCAL, have implemented Turtles on their systems using commands originally developed in the LOGO language. The TURTLE TALK subset of LOGO is easily transportable to other languages.

It should be carefully remembered that LOGO is never conceived as a final product or offered as “the definitive language.” Here I present it as a sample to show that something better is possible.

Precisely because LOGO is not a toy, but a powerful computer language, it requires considerably larger memory than less powerful languages such as BASIC. This has meant that until recently LOGO was only to be implemented on relatively large computers. With the lowering cost of memory, this situation is rapidly changing. As this book goes to press, prototypes of LOGO systems are running on a 48K Apple II system and on a TI 99/4 with extended memory. See S. Papert et al., LOGO: A Language For Learning (Morristown, N.J.: Creative Computing Press, Summer 1980).

3. The history of the Turtle in the LOGO project is as follows. In 1968–1969, the first class of twelve “average” seventh-grade students at the Muzzy Junior High School in Lexington, Massachusetts, worked with LOGO through the whole school year in place of their normal mathematics curriculum. At that time the LOGO system had no graphics. The students wrote programs that could translate English to “Pig Latin,” programs that could play games of strategy, and programs to generate concrete poetry. This was the first confirmation that LOGO was a learnable language for computer “novices.” However, I wanted to see the demonstration extended to fifth graders, third graders, and ultimately to preschool children. It seemed obvious that even if the LOGO language was learnable at these ages, the programming topics would not be. I proposed the Turtle as a programming domain that could be interesting to people at all ages. This expectation has subsequently been borne out by experience, and the Turtle as a learning device has been widely adopted. Pioneer work in using the Turtle to teach very young children was done by Radia Perlman who demonstrated, while she was a student at MIT, that four-year-old children could learn to control mechanical Turtles. Cynthia Solomon used screen Turtles in the first demonstration that first graders could learn to program. At the other end of the age spectrum, it is encouraging to see that Turtle programming is being used at a college level to teach PASCAL. See Kenneth L. Bowles, Problem Solving Using PASCAL (New York: Springer-Verlag, 1977). Controlling Turtles has proven to be an engaging activity for special needs children, for autistic children, and for children with a variety of “learning disorders.” See for example, Paul Goldenberg, Special Technology for Special Children (Baltimore: University Park Press, 1979). Turtles have been incorporated into the SMALLTALK computer system at the Xerox Palo Alto Research Center. See Alan Kay and Adele Goldberg, “Personal Dynamic Media” (Palo Alto, Calif.: Xerox, Palo Alto Research Center, 1976).

4. Touch Sensor Turtle. The simplest touch sensor program in LOGO is as follows:

TO BOUNCE

REPEAT

FORWARD 1

TEST FRONT TOUCH

IFTRUE RIGHT 180

END

Comments

This means repeat all the individual steps

It checks whether it has run into something

If so, it does an about turn

This will make the Turtle turn about when it encounters an object. A more subtle and more instructive program using the Touch Sensor Turtle is as follows:

REPEAT

FORWARD 1

TEST LEFT TOUCH

IFTRUE RIGHT 1

IFFALSE LEFT 1

END

Comments

Check: Is it touching?

It thinks it’s too close and turns away

It thinks it might lose the object so it turns toward

This program will cause the Turtle to circumnavigate an object of any shape, provided that it starts with its left side in contact with the object (and provided that the object and any irregularities in its contour are large compared to the Turtle).

It is a very instructive project for a group of students to develop this (or an equivalent) program from first principles by acting out how they think they would use touch to get around an object and by translating their strategies into Turtle commands.

CHAPTER 1

1. The program FOLLOW (See Introduction, note 4) is a very simple example of how a powerful cybernetic idea (control by negative feedback) can be used to elucidate a biological or psychological phenomenon. Simple as it is, the example helps bridge the gap between physical models of “causal mechanism” and psychological phenomenon such as “purpose.”

Theoretical psychologists have used more complex programs in the same spirit to construct models of practically every known psychological phenomenon. A bold formulation of the spirit of such inquiry is found in Herbert A. Simon, Sciences of the Artificial (Cambridge: MIT Press, 1969).

2. The critics and skeptics referred to here are distillations from years of public and private debates. These attitudes are widely held but, unfortunately, seldom published and therefore seldom discussed with any semblance of rigor. One critic who has set a good example by publishing his views is Joseph Weizenbaum in Computer Power and Human Reason: From Judgment to Calculations (San Francisco: W. H. Freeman, 1976).

Unfortunately Weizenbaum’s book discusses two separate (though related) questions: whether computers harm the way people think and whether computers themselves can think. Most critical reviews of Weizenbaum have focused on the latter question, on which he joins company with Hubert L. Dreyfus, What Computers Can’t Do: A Critique of Artificial Reason (New York: Harper & Row, 1972).

A lively description of some of the principal participants in the debate about whether computers can or cannot think is found in Pamela McCorduck, Machines Who Think (San Francisco: W. H. Freeman, 1979).

There is little published data on whether computers actually affect how people think. This question is being studied presently by S. Turkle.

3. Many versions of BASIC would allow a program to produce a shape like that made by the LOGO program HOUSE. The simplest example would look something like this:

10 PLOT (0,0)

20 PLOT (100,0)

30 PLOT (100,100)

40 PLOT (75,150)

50 PLOT (0,100)

60 PLOT (0,0)

70 END

Writing such a program falls short of the LOGO program as a beginning programming experience in many ways. It demands more of the beginner, in particular, it demands knowledge of cartesian coordinates. This demand would be less serious if the program, once written, could become a powerful tool for other projects. The LOGO programs SQ, TRI, and HOUSE can be used to draw squares, triangles, and houses in any position and orientation on the screen. The BASIC program allows one particular house to be drawn in one position. In order to make a BASIC program that will draw houses in many positions, it is necessary to use algebraic variables as in PLOT (x, y), PLOT (x + 100, y), and so on. As for defining new commands, such as SQ, TRI, and HOUSE, the commonly used versions of BASIC either do not allow this at all or, at best, allow something akin to it to be achieved through the use of advanced technical programming methods. Advocates of BASIC might reply that: (1) these objections refer only to a beginner’s experience and (2) these deficiencies of BASIC could be fixed. The first argument is simply not true: The intellectual and practical primitivity of BASIC extends all along the line up to the most advanced programming. The second misses the point of my complaint. Of course one could turn BASIC into LOGO or SMALLTALK or anything else and still call it “BASIC.” My complaint is that what is being foisted on the education world has not been so “fixed.” Moreover, doing so would be a little like “remodeling” a wooden house to become a skyscraper.

CHAPTER 2

1. “Gedanken experiments” have played an important role in science, particularly in physics. These experiments would encourage more critical attitudes if used more often in thinking about education.

2. There is a joke here. Readers who are not familiar with Noam Chomsky’s recent work may not get it. Noam Chomsky believes that we have a language acquisition device. I don’t: the MAD seems no more improbable than the LAD. See N. Chomsky, Reflections on Language (New York: Pantheon, 1976) for his view of the brain as made up of specialized neurological organs matched to specific intellectual functions. I think that the fundamental question for the future of education is not whether the brain is “a general purpose computer” or a collection of specialized devices, but whether our intellectual functions are reducible in a one-to-one fashion to neurologically given structures.

It seems to be beyond doubt that the brain has numerous inborn “gadgets.” But surely these “gadgets” are much more primitive than is suggested by names like LAD and MAD. I see learning language or learning mathematics as harnessing to this purpose numerous “gadgets” whose original purpose bears no resemblance to the complex intellectual functions they come to serve.

CHAPTER 3

1. Since this book is written for readers who may not know much mathematics, references to specific mathematics are as restrained as possible. The following remarks will flesh out the discussion for mathematically sophisticated readers.

The isomorphism of different Turtle systems is one of many examples of “advanced” mathematical ideas that come up in Turtle geometry in forms that are both concrete and useful. Among these, concepts from “calculus” are especially important.

Example 1: Integration. Turtle geometry prepares the way for the concept of line integral by the frequent occurrence of situations where the Turtle has to integrate some quantity as it goes along. Often the first case encountered by children comes from the need to have the Turtle keep track of how much it has turned or of how far it has moved. An excellent Turtle project is simulating tropisms that would cause an animal to seek such conditions as warmth, or light, or nutrient concentration represented as a field in the form of a numerical function of position. It is natural to think of comparing two algorithms by integrating the field quantity along the Turtle’s path. A simple version is achieved by inserting into a program a single line such as: CALL (:TOTAL + FIELD) “TOTAL”, which means: take the quantity previously called “TOTAL,” add to it the quantity FIELD and call the result “TOTAL.” This version has a “bug” if the steps taken by the Turtle are too large or of variable size. By debugging when such problems are encountered, the student moves in a meaningful progression to a more sophisticated concept of integral.

The early introduction of a simple version of integration along a path illustrates a frequent phenomenon of reversal of what seemed to be “natural” pedagogic ordering. In the traditional curriculum, line integration is an advanced topic to which students come after having been encouraged for several years to think of the definite integral as the area under a curve, a concept that seemed to be more concrete in a mathematical world of pencil and paper technology. But the effect is to develop a misleading image of integration that leaves many students with a sense of being lost when they encounter integrals for which the representation as area under a curve is quite inappropriate.

Example 2: Differential Equation. “Touch Sensor Turtle” (See introduction, note 4) used a method that strikes many children as excitingly powerful. A typical first approach to programming a Turtle to circumnavigate an object is to measure the object and build its dimensions into the program. Thus if the object is a square with side 150 Turtle steps, the program will include the instruction FORWARD 150. Even if it works (which it usually does not) this approach lacks generality. The program cited in the earlier note works by taking tiny steps that depend only on conditions in the Turtle’s immediate vicinity. Instead of the “global” operation FORWARD 150 it uses only “local” operations such as FORWARD 1. In doing so it captures an essential core of the notion of differential equation. I have seen elementary schoolchildren who understand clearly why differential equations are the natural form of laws of motion. Here we see another dramatic pedagogic reversal: The power of the differential equation is understood before the analytic formalism of calculus. Much of what is known about Turtle versions of mathematical ideas is brought together in H. Abelson and A. diSessa, Turtle Geometry: Computation as a Medium for Exploring Mathematics (Cambridge: MIT Press, 1981).

Example 3: Topological Invariant. Let a Turtle crawl around an object “totalizing” its turns as it goes: right turns counting as positive, left turns as negative. The result will be 360° whatever the shape of the object. We shall see that this Total Turtle Trip Theorem is useful as well as wonderful.

2. The phrase “ego-syntonic” is used by Freud. It is a “term used to describe instincts or ideas that are acceptable to the ego: i.e., compatible with the ego’s integrity and with its demands.” (See J. Laplanche and J–B. Pontalis, The Language of Psycho-analysis (New York: Norton, 1973.)

3. G. Polya, How to Solve It (Garden City, N.Y.: Doubleday-Anchor, 1954); Induction and Analogy in Mathematics (Princeton, N.J.: Princeton University Press, 1954); and Patterns of Plausible Inference (Princeton, N.J.: Princeton, 1969).

4. Usual definitions of curvature look more complex but are equivalent to this one. Thus we have another example of an “advanced” concept in graspable form.

5. If turns can be right or left, one direction must be treated as negative. “Boundary of (connected) area” is a simple way of saying “simple closed curve.” If the restriction is lifted, the sum of turns must still be an integral multiple of 360.

CHAPTER 4

1. Here I am picking a little quarrel with Jerry Bruner. But I share much of what he thinks, and this is true not only about language and action but also about the relationship to learning of cultural materials and of teaching.

The systematic difference betweeen us is seen most clearly by comparing our approaches to mathematics education. Bruner, as a psychologist, takes mathematics as a given entity and considers, in his particular rich way, the processes of teaching it and learning it. I try to make a learnable mathematics. I think that something of the same sort separates us in regard to language and to culture and leads us to different paradigms for a “theory of learning.” See J. S. Bruner, Toward a Theory of Instruction (Cambridge: Harvard University Press, 1966) and J. S. Bruner et al., Studies in Cognitive Growth (New York: John Wiley, 1966).

2. The most systematic study is in H. Austin, “A Computational Theory of Physical Skill” (Ph.D. thesis, MIT, 1976).

3. These procedures introduce a further expansion in our image of programming. They are capable of running simultaneously, “in parallel.” An image of programming that fails to include this expansion is quite out of touch with the modern world of computation. And a child who is restricted to serial programming is deprived of a source of practical and of conceptual power. This deprivation is felt as soon as the child tries to introduce motion into a program.

Suppose, for example, that a child wishes to create a movie on the computer screen with three separate moving objects. The “natural” way to do this would be to create a separate procedure for each object and set the three going. “Serial” computer systems force a less logical way to do this. Typically, the motions of each object would be broken up into steps and a procedure created to run a step of each motion in cyclic order.

The example shows two reasons why a computer system for children should allow parallel computation or “multi-processing.” First, from an instrumental point of view, multi-processing makes programming complex systems easier and conceptually clearer. Serial programming breaks up procedural entities that ought to have their own integrity. Second, as a model of learning, serial programming does something worse: It betrays the principle of modularity and precludes truly structured programming. The child ought to be able to construct each motion separately, try it out, debug it, and know that it will work (or almost work) as a part of the larger system.

Multi-processing is more demanding of computational resources than simple serial processing. None of the computers commonly found in schools and homes are powerful enough to allow it. Early LOGO systems were “purely serial.” More recent ones allow restricted forms of multi-processing (such as the WHEN DEMONS described later in this chapter) tailored for purposes of programming dynamic graphics, games, and music. The development of a much less restrictive multi-processing language for children is a major research goal of the MIT LOGO Group at the time of writing this book. In the work we draw heavily on ideas that have been developed in Alan Kay’s SMALLTALK language, on Carl Hewitt’s concepts of “ACTOR” languages, and on the Minsky-Papert “Society Theory of Mind.” But the technical problems inherent in such systems are not fully understood and much more research may be needed before a concensus emerges about the right way (or set of ways) to achieve a really good multi-processing system suitable for children.

CHAPTER 5

1. The most prolific contributor to the development of such systems is Andrea diSessa, who is responsible, among many other things, for the term “Dynaturtle.” H. Abelson and A. diSessa, Turtle Geometry: Computation as a Medium for Exploring Mathematics (Cambridge: MIT Press, 1981).

2. The discussion of the Monkey Problem uses a computational model. However, this model is very far from fitting the notion of computation as algorithmic programming built into most programming languages. Making this model consists of creating a collection of objects and setting up interactions between them. This image of computation, which has come to be known as “object-oriented” or “message-passing” programming, was first developed as a technical method for simulation programs and implemented as a language called SIMULA. Recently it has drawn much broader interest and, in particular, has become a focus of attention in artificial intelligence research where it has been most extensively developed by Carl Hewitt and his students. Alan Kay has for a long time been the most active advocate of object-oriented languages in education.

CHAPTER 6

1. Martin Gardner, Mathematical Carnival (New York: Random House, 1977).

CHAPTER 7

1. For remarks by Piaget on Bourbaki see “Logique et connaissance scientifique,” ed. J. Piaget, Encyclopidie de la Pleide, vol. 22 (Paris: Gallimard, 1967).

2. C. Lévi-Strauss, Structural Anthropology, 2 vols. (New York: Basic Books, 1963–76).

3. Lévi-Strauss uses the word bricolage as a technical term for the tinkering-like process we have been discussing. Bricoleur is the word for someone who engages in bricolage. These concepts have been developed in a computational context in Robert Lawler, “One Child’s Learning: An Intimate Study” (Ph.D. thesis, MIT, 1979).

4. Of course our culture provides everyone with plenty of occasions to practice particular systematic procedures. Its poverty is in materials for thinking about and talking about procedures. When children come to LOGO they often have trouble recognizing a procedure as an entity. Coming to do so is, in my view, analogous to the process of formation of permanent objects in infancy and of all the Piagetionly-conserved entities such as number, weight, and length. In LOGO, procedures are manipulable entities. They can be named, stored away, retrieved, changed, used as building blocks for superprocedures, and analyzed into subprocedures. In this process they are assimilated to schematic or frames of more familiar entities. Thus they acquire the quality of “being entities.” They inherit “concreteness.” They also inherit specific knowledge.

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