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6.4 The Implications Piaget is an interactionist. His theory clearly places the inside-out process of growing as primary and the outside-in process of conditioning as secondary. The child is unfolding naturally from the inside-out but will not do so unless that process is "fed" by appropriate information from the outside-in.2 The "food" is, however, best provided as a smorgasbord than as the traditional set menu. When presented with an all-you-can-eat buffet, children will demonstrate a healthy appetite and will select a balanced diet. Teachers need not talk down to them because they are small nor force-feed them because they are reluctant to eat. However, they do need to know how best to serve the smorgasbord. Teachers communicating with young children are like anthropologists in their own culture dealing with people who are more different from themselves than adults in other cultures. For example, information must be presented to younger children in concrete form. Many children get turned off mathematics because it is presented too abstractly too soon. This vision is, perhaps, best presented, somewhat whimsically, by telling a story of three Wise Men bringing gifts to a child - or to anyone who has ever been a child. The first Wise Man is Jean Piaget. His gift is the theory described above. The second Wise Man is Seymour Papert. His gift is a language. A student of Jean Piaget, he and his colleagues at the Massachusetts Institute of Technology (MIT) developed a computer language called Logo, which would enable computers to facilitate human development, as described by Piaget. Logo played a central role in two brilliant books on education [PAPERT 1980, 1993]. The third wise man is Guy Montpetit. His gift is the distribution to the general public of a disk containing the Logo language. A student of both Piaget and Papert, he returned home to Montreal to found Logo Computer Systems Inc. (LCSI) to manufacture and distribute software using the Logo language. Papert first "taught" his language to a mechanical "Turtle" attached to a keypad. A child (of whatever age) would type FORWARD 40 and the Turtle would take 40 turtle steps forward, leaving a trace of its trip with a pen on a paper on the floor. The child may then type RIGHT 90 and the Turtle would turn 90 degrees to the right. Using four basic words in the Logo language (FORWARD, BACK, RIGHT, LEFT), the Turtle can be instructed to draw any shape on the paper. Let us imagine that a child has commanded the Turtle to draw a triangle, then a square, then a pentagon, and so on. The child may already have discovered the Total Turtle Trip Theorem - that is, regardless of the shape of the figure, the turtle must turn a total of 360 degrees to get back to the position in which it started. This is only one of many insights into geometry the child could get using only those four simple words. Let us say that the child now wishes to teach the Turtle how to draw a circle. The Logo teacher may encourage the child to "think Turtle". This could help the child realize that the Turtle will have to go forward a little, turn a little, many times. Few children would have the patience to key in FORWARD 1, RIGHT 1 360 times. No child would be willing to repeat this chore every time he/she wanted the Turtle to draw a circle. This brings us to the third Wise Man.
Montpetit replaced the mechanical Turtle on the floor by a triangle on a screen and the keypad by a keyboard. A disk, supplied by LCSI, could be considered as the "mind" of the Turtle. It mediates between you at the keyboard and the Turtle-triangle on the screen. It solves the two problems mentioned above. The first problem of repeating FORWARD 1 RIGHT 1 360 tedious times is solved by simply typing REPEAT 360 (FORWARD 1 RIGHT 1). REPEAT is another Logo word which the Turtle "understands" because it is programmed into the disk. The second problem of repeating this command every time you want a circle is solved by teaching the Turtle a new word. This is done as follows: One of the best ways to understand the relatively unfamiliar concept of a computer language (Logo, BASIC, etc.) is by analogy with the more familiar concept of a natural language (English, French, etc.). A natural language, the linguists tell us, consists of a hierarchy of units and sets of rules for combining acceptable units at one level to create meaningful units at the next level. What's that? Linguists don't talk too clearly. Figure 5-1 may help make this more concrete. A natural language consists of phonemes (roughly corresponding to letters) which can be combined, according to the rules of vocabulary, to yield morphemes (roughly corresponding to words) which can be combined, according to the rules of grammar, to yield sentences which can be combined, according to the rules of logic, to yield discourses. The structure of a computer language is essentially the same. Thus, learning a language, whether natural or artificial, consists largely of learning those acceptable units and the appropriate rules of vocabulary, grammar, and logic to combine them into units at the next level. Though they have essentially the same structure, there are some interesting differences between Logo and English. Logo was developed mechanically from the outside in, whereas English grew organically from the inside out. It was consciously developed for a specific purpose - to talk to computers, or, more accurately, to talk to computer programs or, more whimsically, to talk to Turtles or to Turtles which had been transformed into triangles.
It is not possible to talk to computers, computer program, or Turtle-triangles (at least, not yet). You can't yet write to them either (you can, if you like, but you won't get an answer). You can, however, type to them and they can type back. The set of characters (corresponding to the phonemes in a natural language) are all on the keyboard. The Logo language is designed so that you can talk-type on the keyboard to the Turtle, represented by a triangle on the screen, and the Turtle can talk-type back to you on the screen.
The Turtle has a basic vocabulary of Logo words, which it can understand. You could think of them as the instincts of the Turtle. However, you can teach the Turtle new words (as we saw above in teaching the Turtle the meaning of CIRCLE). You can, of course, invent new words in English. However, they will be of no value in communication, except with those people to whom you teach their meaning.
Sentences in English can be roughly classified as statements (followed by a period), questions (followed by a question mark) and commands (followed by an exclamation point). When you talk Logo to the Turtle, you tend to use commands and the Turtle tends to use questions. It is like a conversation between a Sergeant-Major and a Private.
When talk-typing to the Turtle, you must talk very precisely. The Turtle is not very smart. It will not understand you unless you type exactly what you want. You can also talk very concisely. Most Logo words have a shorthand version - e.g. FORWARD can be written FD and RIGHT can be written RT. You can, of course, do the same in English (e.g. TELEVISION can be written TV and MOdulator-DEModulator can be written MODEM). However, since it takes longer to type than to talk, this practice is more common in Logo than in English. Logo differs, in a number of ways, from various traditional computer languages (FORTRAN, COBOL, BASIC, etc.). Those languages were created at a time when computer memory was very expensive. They are therefore designed to conform to the requirements of the machine rather than of the person. BASIC is a brave attempt to consider the needs of the person by simplifying FORTRAN so that it is easy to learn. It is not, however, easy to use. Nor is it powerful enough for complex programs. Thus, one must learn BASIC to do simple programming and, then, learn FORTRAN to do complex programming. Logo, on the other hand, was specifically designed to help children learn to learn. It is simple yet powerful. Thus, it has a very low threshold and a very high ceiling. Turtle Geometry, an undergraduate textbook, starts with a first few fumbling turtle steps and ends with Einstein's General Theory of Relativity [ABELSON & DISESSA]. At first, Logo could be used only in laboratories with very large computers. However, now that home computers are becoming smarter and smarter and cheaper and cheaper, Logo can be made available to the general public. A language like Logo, which is based on child development, is easier to learn and to use than languages like FORTRAN, which are based on engineering. It is designed for people rather than for machines. There is, however, another reason why it is better to "talk" Logo rather than FORTRAN. Benjamin Lee Whorf has argued that the language we speak determines the way we think [WHORF]. Though this theory is very controversial, there is considerable evidence that language determines thought, in some aspects and to some extent. Perhaps, the computer language we use also determines the way we think.3 A language designed for people would, therefore, have a more congenial effect on thought than a language designed for machines. Many much more sophisticated computer languages have emerged since LOGO. However, it is important to preserve its emphasis on the concrete. It's obviously important for children in their concrete operations stage. Much of math phobia is due to the introduction of abstract mathematics during this stage and thus perpetually turning people off mathematics or requiring them to perform mathematical operations by rote without genuine understanding of its principles. I once taught a gifted class in Grade 8 mathematics (I was the only one in the class who wasn't gifted!). Since, in elementary school, they had little difficulty with the math curriculum, teachers had enriched the curriculum by teaching them some algebra. To solve equations, they would recite a little mantra: "Put it over the other side and change the sign". Thus if X + 3 = 8, then X = 8-3 = 5. However, X/3 = 8 would confuse them and √X = 8 would totally baffle them. I had to teach them the basic principle in concrete terms: An equation is a balance and therefore, in order to maintain that balance, you have to do the same thing to both sides. X + 3 = 8 X/3 = 8 √X = 8Many complained about the "extra" step but those who replaced their abstract mantra with this principle found that the concrete principle applied in all cases and guided them through very abstract algebra. I met Guy Montpetit at a dinner party:. "Why don't you come to LCSI and help us with the documentation for the Atari Logo.".I found myself at a meeting with Seymour Papert on one side and Marvin Minsky (probably the world's foremost expert on artificial intelligence) on the other. I had the following conversation with myself: "What am I doing here? You're being paid 300 bucks to listen to Seymour Papert talking to Marvin Minsky. Relax. Enjoy. Learn." One of the ideas that emerged from that meeting was Logo Lego. Imagine building a truck and a crane with your Lego set and being able to program them with the Logo language. You program the truck to back into a particular position and the crane to pick up an object and place it in the bed of the truck. It misses. Back to the drawing board. Imagine what abstract principles a child (or anyone who has ever been a child) could learn in getting that concrete load into that concrete truck using that concrete crane.
2 When he presented that unfolding process at Cornell University during my time as a graduate student, the first question he was asked was "how do you speed it up?" He took his pipe out of his mouth and said "Ah, the American question!" Apparently this is the first question he was always asked when talking to a North American audience. His answer was, essentially, that one should let the process unfold and not rush it. 3 My thanks to my friend, Gordon Sheppard, for this insight. |