Mindstorms: Children, Computers, and Powerful Ideas

Author: Seymour Papert


The canonical work on designing programming systems for learning, and perhaps the greatest book ever written on learning in general, is Seymour Papert's "Mindstorms".

Learnable Programming, Bret Victor


The use of technology in the field of Education has taken place in one form or another essentially since formal education was created thousands of years ago. Today, we often hear how our current celebrity technology, computers, have "revolutionized" the education system. Seymour Papert challenges this perspective in "Mindstorms: Children, Computers, and Powerful Ideas." Papert backs up his opinions with insights he gained in over a decade of research on the use of computers in education at the MIT Media Lab. He presents his perspective of how we have failed to leverage computers as the learning tools they have the potential to become. Dr. Papert also shares his perspective on the problems we face in education and how we missed an opportunity to really change the system. Although he feels we missed an opportunity early-on in the life of the computer, he leaves the reader with some thoughts on how we can correct the path we are on.

The book was originally published in 1980, which sounds extremely dated. However, I feel practically everything in it is still relevant well over 30-years later. After reading the book I can't help but see the way we use computers to teach concepts as being anything but primitive at best (Worst case? "Harmful").

I really can't recommend reading this highly enough. If you are interested in computers, education, or just "learning" in general (how humans do it, et cetera), you'll likely love this book. If you have young kids, I think it will change how you evaluate the "learning tools" your children use on a computer.

I will even go so far as to say that I think this should be required reading at university-level Computer Science and Education programs.

Walking away from books like this make me quite frustrated; feeling like we have really failed to advance a solid foundation that was in place 30-40 years ago. This is a similar sentiment I get when reviewing the "computer science" work (vs. education accompanied by computers) accomplised in the 1960-1980 timeframe.

Chapter Overview

Chapter 1: Computers and Computer Cultures

Papert lays the foundation for the rest of the book: Computers have the potential to change not only the way we educate, but also _what we are able_ to teach to children. Many people disagreed and others were using computers to essentially replace the methods used prior to computers due to historical biases, both of which would only delay significant changes. However, the cost of computers was approaching a point which would allow ubiquitous access. This fact, combined with the freedom individuals would have by owning their own computers, would create an environment which Papert believed even the broken, bureaucratic public education system would not be able to stop people's ability/desire to lead the education revolution from their own homes/devices.

The question to ask about a program is not whether it is right or wrong, but whether it is fixable.

[page 23]

I believe that the computer as a writing instrument offers children an opportunity to become more like adults, indeed like advanced professionals, in their relationship to their intellectual products and to themselves. In doing so, it comes into the head-on collision with the many aspects of school whose effect, if not whose intention, is to "infantilize" the child.

[page 31]

But making good choices is not always easy, in part because past choices can often haunt us.

[page 32, On how difficult it is/will be do make revolutionary changes to the existing system]

BASIC is to computation what QWERTY is to typing.

[page 34, Referring to BASIC being an inferior language used for reasons which are (were) no longer relevant.]

It took years before the designers of automobiles accepted they were cars, not "horseless carriages," and the precursors of modern motion pictures were plays acted as if before a live audience but actually in front of a camera.

[page 36]

Chapter 2: Mathphobia: The Fear of Learning

Papert describes the fear we have created around Math as a society (really, the "sciences", in general). He then outlines his opinion of why we are collectively "afraid" of it (my summary: "poor teaching methods, outdated tools, and lack of environmental reinforcement"). Dr. Papert also briefly mentions the "Turtle Geometry" project, an attempt at providing "appropriable" mathematics for children. He wraps up the chapter with the three principles of appropriable mathematics:

  1. Continuity: The mathematics must be continuous with well-established personal knowledge from which it can inherit a sense of warmth and value as well as "cognitive" competence.
  2. Power: It must empower the learner to perform personally meaningful projects that could not be done without it.
  3. Cultural resonance: The topic must make sense in terms of a larger social context ("…it will not truly make sense to children unless it is accepted by adults too").

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

[page 40]

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.

[page 43]

The invention of the automobile and the airplane did not come from a detailed study of how their predecessors, such as horse-drawn carriages, worked or did not work. Yet this is the model for contemporary educational research. […] It is analogous to improving the axle of the horse-drawn cart. But the real question, one might say, is whether we can invent the "educational automobile."

[page 44]

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.

[page 51]

Chapter 3: Turtle Geometry: A Mathematics Made for Learning

In Chapter 3, Papert describes the Turtle Geometry environment/tools and discusses how the design intentionally promoted mathetics. He also highlights that he felt it was a tangible example of George Polya's principles of problem solving in action. Dr. Papert then describes how Turtle geometry's "syntonic characteristics" allow for critical concepts such as variables and recursion to naturally present themselves to learners. This organic discovery tends to give individuals a unique appreciation for the power and utility of these tools[1].

The chapter concludes with a hypothetical, yet realistic conversation between two children attempting to accomplish the goal of drawing a flower in LOGO. The story shows how several of the concepts outlined in the chapter are encountered within the Turtle geometry environment: how learners naturally reuse familiar concepts (one of Polya's principles), how "debugging" is a natural (and encouraged!) part of learning/understanding, the skill of learning to "build up" complex things by (re)using several smaller/simpler things (abstraction), and inversely, how to break complex challenges into smaller/simpler tasks.

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)

[page 60]

Turtle geometry is learnable because it is syntonic And it is an aid to learning other things because it encourages the conscious, deliberate use of problem-solving and mathetic strategies.

[page 63]

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.

[page 69]

[After learners notice that the difference between a triangle and a square is only the number of turns, not the total degrees traveled, known as the "Turtle Trip Theorem"]

"Thus the child's encounter with this theorem is different in several ways from memorizing its Euclidean counterpart: "The sum of the internal angles of a triangle is 180 degrees."

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.

[page 76]

Chapter 4: Languages for Computers and for People

In Chapter 4, Papert presents the importance of "formalized terminology" (defined language/terms around an area of study) in learning, and challenges the idea that it is only useful in "cognitive" learning scenarios (vs. physical). He discusses how formalized terminology and languages open the door to very powerful ways of expressing ideas and learning new concepts. Dr. Papert uses the formalization of analytic geometry by Descartes opening the door to several new fields of geometry as an example of how powerful this can be when it comes to sharing new concepts[2]. He goes on to explain how a formalized terminology also aids in teaching physical skills, using "juggling" as an example (instead of "teaching" beginners by saying "keep practicing"). This topic brings up the power of abstraction (breaking large, complex problems into smaller, more manageable ones), and how it applies to both "cognitive skills" (such as computer programming or mathematics) as well as physical skills (such as juggling or walking on stilts). He also provides an example which raises the question of whether learning how to create formalized terminologies in a cognitive setting could help people learn physical skills faster. According to research by Howard Austin, the answer to this question appears to be a resounding "yes"[3].

Another central theme to this chapter is the importance of learning the skill of "debugging." Papert talks about how traditional educational settings tend to discourage this skill by focusing on whether an answer is either "right" or "wrong." His stance is that educating within the LOGO environment tends to promote a much more constructive attitude of "How can we fix it?" when "bugs" (incorrect behaviors) are included into a solution. Papert feels this shift in attitude is significant, as it promotes and values discovery and understanding versus the "right/wrong" dichotomy which paralyzes individuals in their learning process.

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

[page 96]

People are capable of learning like rats in mazes. But the process is slow and primitive. We can learn more, and more quickly, by taking conscious control of the learning process, articulating and analyzing our behavior.

[page 113]

The fact that computational procedures enhance learning does not mean that all repetitive processes can be magically removed from learning or that the time needed to learn juggling can be reduced to almost nothing. It always takes time to trap and eliminate bugs. It always takes time to learn necessary component skills. What can be eliminated are wasteful and inefficient methods. Learning enough juggling skill to keep three balls going takes many hours when the learning follows a poor learning strategy. When a good one is adopted the time is greatly reduced, often to as little as twenty or thirty minutes.

[page 113]

Discovery cannot be a setup; invention cannot be scheduled.

[page 115]

The LOGO environment is special because it provides numerous problems that elementary schoolchildren can understand with a kind of completeness that is rare in ordinary life.

[page 115]

…the success of a mathematical theory served across more than an instrumental role: It served as an affirmation of the power of ideas and the power of the mind.

[page 119]

Chapter 5: Microworlds: Incubators for Knowledge

The following excerpts outline this chapter very well:

Two important mathetic principles in this chapter. First, relate what is new and to be learned to something you already know. Second, take what is new and make it your own: Make something new is, play with it, build with it.

[page 120]

The theme of this chapter is how computational ideas can serve as material for thinking about Newton's laws.

[page 121]

Papert felt that one of the reasons Newtonian physics was so difficult for people to grasp was because the traditional approach to teaching the topic does not include these mathetic principles. Another reason he proposes is the long list of mathematic prerequisites required prior to even being introduced to the general concepts, which, as previously mentioned, are generally taught using outdated/inappropriate methods.

These shortcomings are addressed by taking the "Turtle geometry" system and enhancing it to be a "dynamic system", creating "Dynaturtles." Dynaturtles are governed by two principles which closely resemble the first two of Newton's Laws of Motion very. Associated these principles with the familiar "turtle environment" allowed learners to easily grasp the concepts and apply the mathetic principles mentioned to the physics domain (the third principle only applies when multiple particles, or "turtles," are present).

Dr. Papert provides examples of how "microworlds" can be created which only allow one dimension of Newtonian physics to be modified (eg: velocity, acceleration), making it easier for learners to understand and associate to other things they know from life and what they learned already via Turtle geometry. With these simple (and inaccurate) models, it is easier to make the leap to full-blown Newtonian physics its more complex laws of motion.

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."

[page 124]

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.

Our educational system rejects the "false theories" of children, thereby rejecting the way children really learn. And it also rejects discoveries that point to the importance of the false-theory learning path. Piaget has shown that children hold false theories as a necessary path for the process of learning to think. The unorthodox theories of young children are not deficiencies or cognitive gaps, the serve as ways of flexing cognitive muscles, of developing and working through the necessary skills needed for more orthodox theorizing.

[page 132-133]

So, rather than stifling the children's creativity, the solution is to create an intellectual environment less dominated than the school's by the criteria of true and false.

[page 133]

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.

[page 134]

Chapter 6: Powerful Ideas in Mind-Size Bites

Children (people) have intuitions about the world. Sometimes these intuitions are valid, sometimes they are not. Unfortunately when they are false we often invalidate the idea with very concrete things, such as an equation which explains some unintuitive phenomena in physics. Instead, we should try to change the child's (person's) intuition and treat the equations/facts as secondary knowledge.

An example of this is approach is illustrated via a hypothetical conversation between Aristotle and Galileo about the speed of falling objects. The conversation shows how it would (may) have been possible for Galileo to challenge Aristotle's perspective of gravity purely by logic (without equations or measurements). The takeaway is that discussions at this level of abstraction are extremely powerful, as people do not get tied up in the minutia of things like timing, measurements, or equations...yet they can completely shift ways of thinking. Building on these intuitions affords an individual to "connect the dots" with the details, such as equations and measurements, in a "natural" way. The individual truly understands the subject matter when this connection forms.

Programming (particularly with LOGO, of course) promotes procedural-thinking. Procedural thinking is incredibly important because it teaches you how to break down problems into simple components and "debug" them when something is wrong. This ability is obviously valuable in the world of "computer programming," but it applies in essentially every aspect of life. Surrounding kids with opportunities to hone their ability to do this is nothing short of fantastic.

Critics of computers claim(ed) that if you spend too much time working on computers you will start to "think like a computer." However, thinking like a computer is just one way of thinking (albeit an important/valuable one) and having the ability to leverage this way of approaching problems (procedurally) is a great skill and one that we should not fear to teach.

Yet the Turtle is different—it allows children to be deliberate and conscious in bringing a kind of learning with which they are comfortable and familiar to bear on math and physics. […] The Turtle in all its forms (floor Turtles, screen Turtles and Dynaturtles) is able to play this role so well because it is both an engaging anthropomorphizable object and a powerful mathematical idea.

[page 137]

[On how getting to know a topic or area of study is similar to meeting new people]

Good learners are able to pick out those [ideas] who are powerful and congenial. Others are are less skillful need help from teachers and friends. But we must not forget that while good teachers play the role of mutual friends who can provided introductions, the actual job of getting to know an idea or a person cannot be done by a third party. Everyone must acquire skill at getting to know and a personal style for doing it.

Here we use an example from physics to focus the image of a domain of knowledge as a community of powerful ideas, and in doing so take a step toward an epistemology of powerful ideas. Turtle microworlds illustrate some general strategies for helping a newcomer begin to make friends in such a community. A first strategy is to ensure that the learner has a model for this kind of learning; working with Turtles is a good one. This strategy does not require that all knowledge be "Turtle-ized" or "reduced" to computational terms. The idea is that early experience with Turtles is a good way to "get to know" what it is like to learn a formal subject by "getting to know" its powerful ideas. […] Our discussion in Chapter 4 suggested that theoretical physics may be a good carrier for an important kind of meta-knowledge. If so, this would have important consequences for our cultural view of its role in the lives of children. We might come to see it as a subject suitable for early acquisition not simply because it explicates the world of things but because it does so in a way that places children in better command of their own learning processes. (emphasis added)

[page 137-138]

The physics that had a bad influence on social sciences stressed a positivistic philosphy of science. I am talking about a kind of physics that places us in firm and sharp opposition to the positivistic view of science as a set of true assertions of fact and of "law." The propositional content of science is certainly very important, but it constitutes only a part of a physiscist's body of knowledge. It is not the part that developed first historically, it is not a part that can be understood first in the learning process, and it is, of course, not the part I am proposing here as a model for reflection about our own thinking. We shall be interested in knowledge that is more qualitative, less completely specified, and seldom stated in propositional form. If students are given such equations as ƒ = ma, E = IR, or PV = RT as the primary models of the knowledge that constitutes physics, they are placed in a position where nothing in their own heads is likely to be recognized as "physics."

[page 138]

[A fictitious conversation earlier in the book] has something to teach us about one of the most destructive blocks to learning: the use of formal reasoning to put down intuitions.

[page 144]

Chapter 7: LOGO's Roots: Piaget and AI

Papert opinions on learning and approaches to education were heavily inspired by Piaget, yet he presents Piaget's work in a different way than most. He feels a big reason for this is because prior to the widespread availability of computers, it was hard to broadly apply Piagetian learning principles. In Chapter 7, Dr. Papert steps back "to reinterpret Piaget…[and] to develop the theories of learning and understanding that inform our design of educational situations."

Learning to ride a bicycle is used to illustrate the idea of "studying learning by focusing on the structure of what is learned" (see the quotes for this Chapter for some excerpts). He discusses how the building blocks known as "mother structures" within the Bourbaki style of mathematics, and how the approach was similar to the Microworlds he presented earlier in the book. On page 160, Dr. Papert gives three traits that make the Bourbaki mother structures learnable:

  1. "…each [mother structure] represents a coherent activity in the child's life that could in principle be learned and made sense of independent of the others.
  2. "…the knowledge structure of each has a kind of internal simplicity that Piaget has elaborated in his theory of groupements…"
  3. "…although these mother structures are independent, the fact that they are learned in parallel and that they share a common formalism are clues that they are mutually supportive; the learning of each facilitates the learning of the others.

Papert also describes where he and Piaget differ in their perspectives, "My perspective is more interventionist. My goals are education, not just understanding. So, in my own thinking I have placed a greater emphasis on two dimensions implicit but not elaborated in Piaget's own work: an interest in intellectual structures that could develop as apposed to those that actually at present do develop in the child, and the design of learning environments that are resonant with them." The Turtle is given as an example of these two interests being applied.

Chapter 7 ends with Papert describing how children pick up conservation skills before combinatorial almost universally[4]. Many have suggested that the reason for this is because the brain must mature to a certain point (approximately 5 more years of development) before this is possible to learn. Piaget disagrees and feels that the reason has more to do with the fact that there were far fewer opportunities to encounter and interact with combinatorial situations in the "pre-computer" world. By introducing kids to computers (and programming), many combinatorial questions move from being advanced computations to simple procedures. Taking this a step further, if society removed the self-imposed shackles of the education curriculum from a pre-computer world, Papert says, "Children may learn to be systematic before they learn to be quantitative!"

Piaget has described himself as an epistemologist. What does he mean by that? When he talks about the developing child, he is really talking as much about the development of knowledge. This statement leads us to a contrast between epistemological and psychological ways of understanding learning. In the psychological perspective, the focus is on the laws that govern the learner rather than on what is being learned. Behaviorists study reinforcement schedules, motivation theorists study drive, gestalt theorists study good form. For Piaget, the separation between the learning process and what is being learned is a mistake. To understand how a child learns number, we have to study number.

[page 158]

The bicycle without a rider balances perfectly well. With a novice rider it will fall. This is because the novice has the wrong intuitions about balancing and freezes the position of the bicycle so that its own corrective mechanism cannot work freely. Thus learning to ride does not mean learning to balance, it means learning not to unbalance, learning not to interfere.

[page 159]

Only when mathematics becomes sufficiently advanced is it able to discover its own origins.

[page 163]

The important question is not whether the brain or computer is discrete but whether knowledge is modularizable.

[page 171]

Research in artificial intelligence is gradually giving us a surer sense of the range of problems that can be meaningfully solved on modular agents, each of them simple-minded in its own way, many of them in conflict with one another. The conflicts are regulated and kept in check rather than "resolved" through the intervention of special agents no less simple-minded than the original ones. Their way of reconciling differences does not involve forcing the system into a logically consistent mold.

[page 173]

Chapter 8: Images of the Learning Society

As the title suggests, Dr. Papert takes a step back in Chapter 8 provides a broad view of his perspective on what a "Learning Society" could be. He highlights how difficult creating the types of environments he's outlined in this book would be. He stresses that the reasons for this difficulty are not due to the economic or technical constraints of the past). Instead, the biggest challenges have to do with overcoming the public opinion of what learning and education are.

He gives two examples which dovetail nicely with the chapter topic: samba schools in Brazil and a shift in downhill skiing teaching methods which occurred in the 1960's.

The tie-in with the samba schools of Brazil was that, like the Turtles discussed in this book, the learning process is not highly structured and involves both experts and beginners working together constantly. Learning is very natural and practically constant, even without orchestrating it completely. However, the samba environment differs from the Turtle environment because samba is actually part of society in Brazil, whereas teaching in the way outlined in this book is far from the norm for us.

He also used the transition to teaching skiing via shorter skis which took place in the 1960's as an example of how new technology allowed an age-old method of learning (to ski) to be completely turned on its head in less than a decade. (Note: This transition is often referred to as "the move to GLM.".)

The following excerpt from the last page of the book summarizes both the chapter, and the book quite well:

The computer by itself cannot change the existing institutional assumptions that separate scientist from educator, technologist from humanist. Nor can it change assumptions about whether science for the people is a matter of packaging and delivery or a proper area of serious research. To do any of these things will require deliberate action of a kind that could, in principle, have happened in the past, before computers existed. But it did not happen.

[page 189]

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

[page 183]

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.

[page 184]

The price of the education community's reactive posture will be educational mediocrity and social rigidity. And experimenting with incremental changes will not even put us in a position to understand where the technology is leading.

[page 186]

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.

[page 186]

The computer by itself cannot change the existing institutional assumptions that separate scientist from educator, technologist from humanist. Nor can it change assumptions about whether science for the people is a matter of packaging and delivery or a proper area of serious research. To do any of these things will require deliberate action of a kind that could, in principle, have happened in the past, before computers existed. But it did not happen.

[page 189]


  1. Using computers to present flash cards of math/reading problems and standardized assessment are incredibly common, yet awful ways to use a computer in education. Learning by experience with the topics is far superior. Let them play, provide feedback loops, and encourage "debugging."

  2. "Debugging" (troubleshooting) is a critical thinking skill which is practically ignored by society.

  3. "Externalizing" something you are doing on the computer is a valuable way to "debug" and/or "design" things within the computer ("playing Turtle" is an example from this book).

    For example, when kids associate "the turtle" (on the screen) to their own physical body instead of a digital cursor makes it easier for them to "act out" what "the turtle needs to do," as if they were the turtle.

  4. Language matters.

    The language you use when interacting with a computer is important to how you think about and approach concepts. This is an important thing to consider when developing something for children to interface with, particularly when in the nascent stages of learning a concept. Limit syntax as much as possible without reducing the depth of what is possible.

    For example, you can start a person off in the LOGO environment basically saying that "The turtle understands 3 commands: forward, left, right. But, you can teach him new commands if you like"...which takes us to...

  5. Abstraction is a critical thinking skill.

    Learning to take a set of complex tasks and wrap them in a higher-level concept that is easy to understand and re-use is invaluable in both computing and in problem-solving in general. This is a concept you naturally develop when programming, and one which is hard to learn "naturally" in many other settings.

    An example of this would be taking the idea of "going forward some number of steps, then turning right 90-degrees...and repeating that four times to make a square" and wrapping it up in a new "command" (function) named "square" which takes an argument of how many steps forward to "walk" (ie: "square 20"). When this happens, the person no longer thinks about the "implementation details" of a square unless they find a bug later. Instead, they think only of a square and how large it should be. If they need to refresh their memory or fix a bug they have found in their implementation, this is easy: they refer to the "square" command they defined and fix it there. This is what Bret Victor refers to as "moving up and down the ladder of abstraction."

  6. Concrete & Formal thinking: Can computers impact this relationship? [5] [6]

    Papert disagreed with the details of Piaget's definitions of these two stages of cognitive development, but not to a degree he wished to point out at great lengths. However, he felt the computer was capable of "concretizing" cognitive skills which were traditionally thought to be only possible when an individual was in the "formal" stage of cognitive development. He gives two examples of situations he felt proved his point in the book: combinatorial thinking (figuring out possible states in a system) and self-referential thinking (thinking about thinking itself). Often in the book Papert refers to "turning children into epistemologists." Doing so seemed to be something he considered an ideal, and something the computer could absolutely have an influence on.

  7. Imprecise models are important in learning.

    Although we tend to shy away from "inaccurate models" of new concepts, they are actually critically important to the learning process.

    An example of how this happens naturally is how children learn to speak: they get it wrong over and over, saying partial words only their parents understand. They realize it isn't quite right when other people do not understand them, and they keep refining how they speak until the message is consistently understood. In this process we praise and encourage improperly pronunciated "dads" and "moms" (and every other word), yet we seem to forget this approach when a child is learning more "serious" topics such as math or physics.

Related links


Syntonic learning

"The Turtle circle incident illustrates syntonic learning.This term is borrowed from clinical psychology and can be contrasted to the dissociated learning already discussed. Sometimes the term is used with qualifiers that refer to kinds of syntonicity. For example, the Turtle circle is body syntonic in that the circle is firmly related to the children's sense and knowledge about their own bodies. Or it is ego syntonic in that it is coherent with children's sense of themselves as people with intentions, goals, desires, likes, and dislikes. A child who draws a Turtle circle wants to draw a Turtle circle wants to draw the circle; doing it produces pride and excitement." (page 63)

"Here we see a cultural syntonicity: The Turtle connects the idea of angle to navigation, activity firmly and positively rooted in the extraschool culture of many children." (page 68, pointing out how the use of angles comes up in navigation, boating, and flight)

Analytic geometry

One of the most striking examples of the power of descriptive language is the genesis of analytic geometry, which played so decisive a role in the development of modern science.

Legend has it that Descartes invented analytic geometry while lying in a bed late one morning observing a fly on the ceiling. We can imagine what his thinking might have been. The fly moving hither and thither traced a path as real as the circles and ellipses of Euclidean mathematics, but one that defied description in Euclidean language. Descartes then saw a way to describe it by saying how far it was from the walls. Points in space could be described by pairs of numbers; a path could be described by an equation or relationship that holds true for those number pairs whose points lie on the path. The potency of symbols took a leap forward when Descartes realized how to use an algebraic language to talk about space, and a spatial language to talk about algebraic phenomena. Descartes's method of coordinate geometry born from this insight provided tools that science has since used to describe the paths of flies and planets and the "paths" of the more abstract objects, the stuff of pure mathematics.

A Computational View of the Skill of Juggling

"Variants of this teaching strategy [from pages 110-111] have been used by many LOGO teachers and studied in detail by one of them, Howard Austin, who took the analysis of juggling as the topic of his PH.D. thesis. There is no doubt that the strategy is very effective and little doubt as to the cause: The use of programming concepts as a descriptive language facilitates debugging." [page 111]

Combinatorial thinking
A simple example of this type of problem would be to provide someone with a group of differently colored beads and to ask them how many combinations of colors could they make.
Concrete thinking
"This stage, which follows the preoperational stage, occurs between the ages of 7 and 11 years and is characterized by the appropriate use of logic."
Formal thinking
"Intelligence is demonstrated through the logical use of symbols related to abstract concepts." (achieved in adolescence and into adulthood)