Great Throughts Treasury

"A second key mathematical concept whose understanding is facilitated by the Turtle is the idea of a variable: the idea of using a symbol to name an unknown entity. [?] In principle you could describe [a spiral] by a very long program that would specify precisely how much the Turtle should turn on each step. This is tedious. A better method uses the concept of a symbolic naming through a variable, one of the most powerful mathematical ideas ever invented."

"A programming language is like a natural, human language in that it favors certain metaphors, images, and ways of thinking. The language used strongly colors the computer culture. It would seem to follow that educators interested in using computers and sensitive to cultural influences would pay particular attention to the choice of language. But nothing of the sort has happened. On the contrary, educators... have accepted certain programming languages in much the same way as they accepted the QWERTY keyboard. An informative example is the way in which the programming language BASIC has established itself as the obvious language to use in teaching children how to program computers... Today, and in fact for several years now, the cost of computer memory has fallen to the point where any remaining economic advantages of using BASIC are insignificant. Yet in most high schools, the language remains almost synonymous with programming, despite the existence of other computer languages that are demonstrably easier to learn and are richer in the intellectual benefits that can come from learning them. The situation is paradoxical. The computer revolution has scarcely begun, but is already breeding its own conservatism."

"A more subtle difference is in the fact that some of them leave the Turtle in its original state. Programs written in this clean style are much easier to understand and use in a variety of contexts. And in noticing this difference children learn two kinds of lessons. They learn a general "mathetic principle," making components to favor modularity. And they learn to use the very powerful idea of "state." (emphasis added)"

"An unknown but certainly significant proportion of the population has almost completely given up on learning. These people seldom, if ever engage in deliberate learning and see themselves as neither competent at it nor likely to enjoy it. The social and personal cost is enormous... Deficiency becomes identity: "I can't learn French, I don't have an ear for languages;" "I could never be a businessman, I don't have a head for figures;"... These beliefs are often repeated ritualistically, like superstitions... Although these negative self-images can be overcome, in the life of and individual they are extremely robust and powerfully self-reinforcing. If people believe firmly enough that they cannot do math, they will usually succeed in preventing themselves from doing whatever they recognize as math. The consequences of such self-sabotage is personal failure, and each failure reinforces the original belief. And such beliefs may be most insidious when held not only by individuals, but by our entire culture."

"Adults have been brainwashed into thinking that they can't really learn about computers without being taught, so it's more difficult for them to feel comfortable with a computer. Deep down, I think they're afraid of learning about computers."

"As in a good art class, the child is learning technical knowledge as a *means* to get to a creative and personally defined end. There will be a product."

"But the interesting cases are those where the conflict remains obstinately in place however much we ponder the problem. These are the cases where we are tempted to conclude that intuition cannot be trusted. In these situations we need to improve our intuition, to debug it, but the pressure on us is to abandon intuition and rely on equations instead."

"Children begin their lives as eager and competent learners. They have to learn to have trouble with learning in general and mathematics in particular."

"Discovery cannot be a setup; invention cannot be scheduled."

"BASIC is to computation what QWERTY is to typing. Many teachers have learned BASIC, many books have been written about it, many computers have been built in such a way that BASIC is "hardwired" into them. In the case of the typewriter, we noted how people invent "rationalizations" to justify the status quo. In the case of BASIC, the phenomenon has gone much further, to the point where it resembles ideology formation. Complex arguments are invented to justify features of BASIC that were originally included because the primitive technology demanded them or because alternatives were not well enough known at the time the language was designed."

"Do away with curriculum. Do away with segregation by age. And do away with the idea that there should be uniformity of all schools and of what people learn."

"Children do not follow a learning path that goes from one "true position" to another, more advanced "true position." Their natural learning paths include "false theories" that teach as much about theory building as true ones. But in school false theories are no longer tolerated."

"Corporate America needs to get its act together to see that the education system is changed so it produces what it needs. The educational system that teaches kids to be passive recipients of knowledge worked when most workers were sitting in assembly lines."

"Educators who have said, "We don't like that, so we'll continue to teach as if it's not happening," are just aggravating the gap between what happens in schools and what happens in the real world. Because of their personalities, or for cultural reasons, some kids might better express themselves through moving images and sound."

"Even with the most stupid video games, kids learn more about learning than they ever did before, because they want to learn codes and moves before other kids figure them out. They're motivated to seek out someone or search the Net for help. A student who makes a video game has to solve mathematical problems to make special effects happen on the screen."

"Every maker of video games knows something that the makers of curriculum don't seem to understand. You'll never see a video game being advertised as being easy. Kids who do not like school will tell you it's not because it's too hard. It's because it's--boring"

"First (at least in the context of LOGO computers), the Total Turtle Theorem is more powerful: The child can actually use it. Second, it is more general: It applies to squares and curves as well as to triangles. Third, it is more intelligible: Its proof is easy to grasp. And it is more personal: You can "walk it through," and it is a model for the general habit of relating mathematics to personal knowledge."

"For what is important when we give children a theorem to use is not that they should memorize it. What matters most is that by growing up with a few very powerful theorems one comes to appreciate how certain ideas can be used as tools to think with over a lifetime. One learns to enjoy and to respect the power of powerful ideas. One learns that the most powerful idea of all is the idea of powerful ideas."

"Generally in life, knowledge is acquired to be used. But school learning more often fits Freire's apt metaphor: knowledge is treated like money, to be put away in a bank for the future."

"I am talking about a revolution in ideas that is no more reducible to technologies than physics and molecular biology are reducible to the printing press. In my vision, technology has two roles. One is heuristic: The computer presence has catalyzed the emergence of ideas. The other is instrumental: The computer will carry ideas into a world larger than the research centers where they have incubated up to now."

"I have asked many teachers and parents what they thought mathematics to be and why it was important to learn it. Few held a view of mathematics that was sufficiently coherent to justify devoting several thousand hours of a child's life to learning it, and children sense this. When a teacher tells a student that the reason for those many hours of arithmetic is to be able to check the change at the supermarket, the teacher is simply not believed. Children see such "reasons" as one more example of adult double talk. The same effect is produced when children are told school math is "fun" when they are pretty sure that teachers who say so spend their leisure hours on anything except this allegedly fun-filled activity. Nor does it help to tell them that they need math to become scientists---most children don't have such a plan. The children can see perfectly well that the teacher does not enjoy math any more than they do and that the reason for doing it is simply that it has been inscribed into the curriculum. All of this erodes children's confidence in the adult world and the process of education. And I think it introduces a deep element of dishonestly into the education relationship."

"I am convinced that the best learning takes place when the learner takes charge."

"I don't like MTV, and I don't like the culture that goes with it. It's OK in very small doses, maybe. Nevertheless, it's a social reality and has influenced how kids perceive things around them, the pace of life and the way people do things."

"I prefer software where kids build something and run into problems they have to solve."

"I do think that we'd do better if we just offered all the bureaucrats in the Department of Education very attractive early retirements. But whether you want to abolish the department is another matter. Maybe there's room for recruiting a lot of visionary people who would do very good things: develop new techniques, new ideas, foster innovative models, disseminate those ideas."

"I tell adults about the experiences of more than a hundred teachers I've interviewed. They tell me that allowing the child to help them learn helped them become better teachers. That's because they no longer had to pretend they were the experts - not only about computers but about other things."

"I see "school math" as a social construction, a kind of QWERTY. A set of historical accidents (which shall be discussed in a moment) determined the choice of certain mathematical topics as the mathematical baggage that citizens should carry. Like the QWERTY arrangement of typewriter keys, school math did make some sense in a certain historical context. But like QWERTY, it has dug itself in so well that people take it for granted and invent rationalizations for it long after the demise of the historical conditions that made sense of it."

"I think Disney is a very interesting company, and it does have an extremely powerful means for projecting a certain way of thinking."

"I think we should allow for schools within schools, where 100 out of 500 kids may be organized by the way they work and what they do, and what they do often is more progressive. I would like to see a lot of kids of different ages, maybe even some adults, work together on a project."

"I think schools generally do an effective and terribly damaging job of teaching children to be infantile, dependent, intellectually dishonest, passive and disrespectful to their own developmental capacities."

"I was really looking at computers as a way to understand the mind. But at M.I.T., my mind was blown by having a whole computer to yourself as long as you liked. I felt a surge of intellectual power through access to this computer, and I started thinking about what this could mean for kids and the way they learn. That's when we developed the computer programming language for kids, Logo."

"In a classical joke a child stays behind after school to ask a personal question. Teacher, what did I learn today? The surprised teacher asks, Why do you ask that? and the child replies, Daddy always asks me and I never know what to say."

"If children really want to learn something, and have the opportunity to learn it in use, they do so even if the teaching is poor. For example many learn difficult video games with no professional teaching at all!"

"In my vision of this field its professionals will need special combinations of competences. Apart from a foundation in scientific knowledge and technological skill they will need high degrees of psychological sensitivity and 'artistic' imagination. For the ones who will make the greatest social contribution will be those who know how to mold the computer into forms which people will love to use and in ways which will lead them on to enrichment and enhancement."

"In my vision the computer acts as a transitional object to mediate relationships that are ultimately between person and person."

"Imagine that children were forced to spend an hour a day drawing dance steps on squared paper and had to pass tests in these "dance facts" before they were allowed to dance physically. Would we not expect the world to be full of "dancophones"? Would we say that those who made it to the dance floor and music and the greatest "aptitude for dance"? In my view, it is no more appropriate to draw conclusions about mathematical aptitude from children's unwillingness to spend many hundreds of hours dong sums."

"If a kid really is retarded and can only come up to a certain level, he will still have more success if what he learns is connected with something important to him."

"In many schools today, the phrase "computer-aided instruction" means making the computer teach the child. One might say the computer is being used to program the child. In my vision, the child programs the computer and, in doing so, both acquires a sense of master over a piece of the most modern and powerful technology and establishes an intimate contact with some of the deepest ideas from science, from mathematics, and from the art of intellectual model building."

"In my vision, space-age objects, in the form of small computers, will cross these cultural barriers to enter the private worlds of children everywhere. They will do so not as mere physical objects. This book is about how computers can be carriers of powerful ideas and of the seeds of cultural change, how they can help people form new relationships with knowledge that cut across the traditional lines separating humanities from sciences and knowledge of the self from both of these. It is about using computers to challenge current beliefs about who can understand what and at what age. It is about using computers to question standard assumptions in developmental psychology and in the psychology of aptitudes and attitudes. It is about whether personal computers and the cultures in which they are used will continue to be the creatures of "engineers" alone or whether we can construct intellectual environments in which people who today think of themselves as "humanists" will feel part of, not alienated from, the process of constructing computational cultures."

"Indeed, a central part of Newton's great contribution was the invention of a formalism, a mathematics suited to [conceptualizing and "capturing" Newtonian physics]. He called it "fluxions"; present-day students call it "differential calculus.""

"It is not uncommon for intelligent adults to turn into passive observers of their own incompetence in anything but the most elementary mathematics. Individuals may see the direct consequences of this intellectual paralysis in terms of limiting job possibilities. But the indirect, secondary consequences are even more serious. One of the main lessons learned by most people in math class is a sense of having rigid limitations. They learn a balkanized image of human knowledge which they come to see as a patchwork of territories separated by impassable iron curtains."

"It's not what you know about the computer that's important, but your ability to do things with it. By studying French in an academic setting, you get to know a lot about it, but typically, you can't express yourself well or have an interesting conversation with it."

"Juggling and writing an essay seem to have little in common if one looks at the product. But the process of learning both skills have much in common. By creating an intellectual environment in skills and interests something to talk about."

"My basic idea is that programming is the most powerful medium of developing the sophisticated and rigorous thinking needed for mathematics, for grammar, for physics, for statistics, for all the hard subjects.... In short, I believe more than ever that programming should be a key part of the intellectual development of people growing up."

"Many children who grow up in our cities are surrounded by the artifacts of science but have good reason to see them as belonging to "the others"; in many cases they are perceived as belonging to the social enemy."

"No knowledge is entirely reducible to words, and no knowledge is entirely ineffable."

"Nothing bothers me more than when people criticize my criticism of school by telling me that schools are not just places to learn math and spelling, they are places where children learn a vaguely defined thing called socialization. I know. I think schools generally do an effective and terribly damaging job of teaching children to be infantile, dependent, intellectually dishonest, passive and disrespectful to their own developmental capacities."

"Nothing enrages me more than when people criticize my criticism of school by telling me that schools are not just places to learn math and spelling, they are places where children learn a vaguely defined thing called socialization. I know. I think schools generally do an effective and terribly damaging job of teaching children to be infantile, dependent, intellectually dishonest, passive and disrespectful to their own developmental capacities."

"Nothing could be more absurd than an experiment in which computers are placed in a classroom where nothing else is changed."

"Now more people are doing work that requires individual decision-making and problem-solving, and we need an educational system that will help develop those skills."