Modeling the Mind

Introduction

Why I'm writing this

I've been the luckiest of teachers. I have spent twenty years in a supportive environment in which I had the opportunity to make all sorts of experiments in teaching.  In all that time, education seemed to be moving generally in the same directions I was, but with considerably more inertia. Now, means and methods in education are headed back to the past, through a combination of  various factors, mostly political, and none of them based on any theory or research.  It seems that what we want is a student body that can pass multiple choice tests, but little else.  Most of all, it's imporant that kids pass these tests at the youngest age possible. The idea of a student actually understanding something is becoming superfluous to the main goals.

For years, I've been teaching in the public schools of California. Before that, I attended those same schools. In all that time, the schools have been viewed as "problems," "underperforming," etc. Lots of people have tried to "fix" the situation to no avail. I believe I have something to say that most people have missed in all this, mainly because they don't feel that what I've discovered is important. Yet it is important. And I wonder, can I possibly present it clearly enough to convince you?

My main argument is that we need to actaully know our students - and not only their health, their wealth, their families, and their access to society's resources. That's being done already. We need to know  how students actually learn - not how we think they should learn, but how they actually learn.

Over the past sixty or seventy years, there have been hundreds of psychologists, linguists, epistemologists, and other workers who have described the learning process. They have not worked in a vacuum, and their work is a matter of public record. But who reads it? It's not going to make a whit of difference if the only people who know it are hundreds of psychologists, linguists, and epistemologists. Millions of teachers, parents and policy-makers also must know it. Because the fact is, people don't actually learn the way they think they do, particularly the way most Americans think they do. This ignorance, I believe, is one of the major reasons we're constantly falling behind in academic achievement relative to other nations.

As a teacher, I've watched one "instructional reform" after another come and go. And yet, they haven't really come and gone because none of them were really implemented. The truth is that the overwhelming bulk of American education - both public and private - has never seriously deviated from the "basic skills" model that has characterized it for most of the twentieth century. This, by the way, is another major reason that we continue to fall behind in academic achievement relative to other nations. All that those "reforms" ever amounted to was window dressing and new vocabulary for the same old stuff, plus a disruption of whatever education there was.

In their book The Teaching Gap James Stigler and James Hiebert have accurately described the principal reason for this resistance to change - that the teaching system (not just the teachers, but also the students, parents, uncles, aunts, and political leaders) is itself a culture, and thus has its own built-in cultural inertia. This inertia is not caused by laziness, but by lack of imagination - because they only view the situation through certain culturally-determined frames. This is why all those reform efforts accomplished nothing but disruption. The task of education is far more profound and complex than is generally acknowledged.

Hiebert and Stigler escaped this lack of imagination by studying education in foreign countries, where everything really is different, which forced them to adopt a new perspective. If you're interested in learning what they discovered, please read my review of The Teaching Gap which you can find by clicking here.

Hiebert and Stigler have proposed a system for change that I believe will work. The next question, then, is what sort of change we want to make, because it will only work if we're all in it together - millions (or, at least, tens of thousands of us) at a time.

Well, if we all have to work together, what do we all have in common to work with? One thing is classrooms of students, so Stigler and Hiebert have proposed that, within their proposed system for change, we develop and study model lessons, which are motivated and measured by an increase in students' test scores.

But I'd rather look forward to a day when classrooms are just one means of developing minds. I think that day is coming, whether we want it or not. If we place lesson study at the heart of the engine for change, we will certainly improve classrooms, and, I believe, make our classrooms the best in the world. But we're also locking ourselves more deeply into the classroom model. More importantly, we're also locking ourselves into the view that education is a curriculum delivery system, whether it happens in a classroom or not. This is simply our culture's frame, our particular view of education, and our educational culture is precisely what needs to change.

Wouldn't it be better to center any system for change on the student, the mind that's expected to do the learning, whether in a classroom, on the playground, on the job, or in the town meeting? Wouldn't it be better to center our system for change on something that not only allows us to measure progress, but eventually allows us to evaluate the measurement instruments themselves?

And now, dear reader, can you see that what I'm talking about is as far from the common discourse on education as Baghdad is from Brasilia, or Bangkok is from Botswana? And if you can see that, then you will understand the source of my desperation - There's so much arguing about educational window-dressing, and it's lasted for so long, that it's crowded out opportunities to discuss the real situation. The picture I sometimes get is like taking a taxi and noticing that the driver is so focused on the map and on the final destination that he refuses to look out the window at the road. Instead, he steers the wheel by where he thinks he is on the map. If you were in such a car, wouldn't you feel just a little bit desperate? Wouldn't you want to let the driver know that there's a road out there with real turns and real pedestrians? As it is, the driver doesn't believe in either of those things. Or, at least, he doesn't think they're very important.

To my mind, we've got to look at the "road," the student's learning processes, because that's the only way we can reach our educational goals without mowing down a lot of pedestrians in the process.

How can I communicate this message?

How can I tell you what learning really does?

Now, if there really are hundreds of psychologists, linguists, epistemologists, and other arcane-ologists who have already described the mind, what's the problem? Can't we just read what they've written?

We could, but their work has a context and vocabulary that takes a while to get used to, which is time most people don't have. Better, I think would be to give readers some ideas to start with, to ground them in the basic concepts. Then, they can work out the depths and the details in their own situation.

What I'm trying to do reminds me of Rutherford's model of the atom. About a hundred years ago, Rutherford proposed that the atom is not solid, as was previously thought, but mainly open space. It was composed of a tiny nucleus surrounded by orbiting electrons, just like a miniature solar system. Now we know that this model is simplistic to say the least. It's only a crude approximation of what atoms are really like. Yet, this simplistic model is still taught in school because it's easy to grasp, helps one understand simple chemical and atomic reactions, and it's a great starting point for further study, to find out deeper truths about how atoms are made up.

Well, in the area of learning theory, lots of the deeper and detailed truths have already been described, yet there is no simplistic model of the mind to serve as a bridge to understanding them. That's what I'm hoping to accomplish. I don't claim to have discovered all the secrets of the mind, or even very many of them. I simply want to present a basis for discussion and a starting point for more in-depth thinking later on.

Now, Rutherford's model was a picture. Even the simplest views of the mind are going to be more subtle than that. So basically I'm going to describe it through a series of metaphors. I'm going to try and describe the mind as a whole, even though I know it's made up of many different faculties. I figure that these faculties generally work together in the normal mind, and I also suspect that each faculty by itself probably learns in a manner similar to the whole.

And if you want to pursue the story more deeply, you might wish to consult works by Noam Chomsky, Jean Piaget, Stephen Krashen, Konrad Lorenz, Constance Kamii, George Lakoff and a host of others less well known, because those are the kind of people who have already formulated and articulated the principles.

Part One - Here's the Model

Where do you start with a model of the mind?

Three basic observations

Now, I'd like to make some observations, which I hope will provide enough implications to construct a model.

Observation #1. The mind is an organ of the body, and it's like the body's other organs in most ways.

The brain is like other organs of the body. It's composed of smaller components that are themselves like little organs. Most people have heard of different parts of the brain - the cerebrum, the cerebellum, the hypothalamus, etc. etc. What I'm saying, though, is that you can think of the mind (since it's a big part of the brain, after all) as also having components that function like little organs.

These organs of the mind take different forms and functions. Some could be stories (or other such scripts), observations, skills, impressions, feelings, etc. etc. These, in turn have even simpler basic components, which could include basic shapes (such as edges or circles), colors, sounds, and tastes. Not all basic components are tied so tightly to the senses, though. For example, the concept of a phoneme, which is a piece of language, is not tied to hearing itself, but to systems of meaning. The whole system of mind organs is tied together and coordinated by simple logical relationships (such as "same," "opposite," "reverse," etc).

The mind differs from most other organ systems to the extent that the environment shapes its components. This distinction is merely one of degree, of course, since all organs of the body are shaped by the environment. Furthermore, the mind differs from the other systems in that it continues to grow throughout life, just like a tree does. This distinction is also one of degree.

Observation #2. Not everything you do or think is conscious. In fact, most of it isn't.

People make jokes about being "on autopilot" all the time. What they don't realize is how true to the mark this is. It's not simply a matter of thinking and then forgetting. It's a matter of thinking, reasoning, deciding and performing without exerting conscious control. I once read a statistic about how many hundreds of decisions a concert pianist must make each second (like finger placement, arm placement, strength of touch, etc.) in order to perform a piece of music. Rest assured that the pianist is not conscious of all those decisions. If he was, he'd never have time to shape the overall musical message.

This same principal applies even to such subtle activities as socializing. Next time you're in an audience, for example, watch for people sitting together, and see how their sitting postures tend to stay in synch, and perhaps notice yourself doing the same thing.

The question, of course, is this: how much thinking is subconscious and how much is conscious. And, for that matter, is there a clear dividing line between the two? Actually, I don't think there is a clear dividing line, but for purposes of the model, I'm going to assume there is, because it's easier to model it that way.

Sometimes I think that people imagine that they are in charge of themselves the same way a driver is in charge of an automobile. What ever the driver wants, he can make the car perform. However, the true picture is not of a driver in a car, but a rider on a horse. The rider can usually get the horse to do what he wants him to do, but the horse still has a mind of his own, and he's perfectly willing to use it on his own. And in fact, if the journey is long or fraught with danger, then the rider had better listen to that horse, because it will give him useful information.

Observation #3. Learning, then, is at base, a subconscious activity.

In terms of the model, then, one of the most important functions of this "horse" (the subconscious mind) is to learn things. Learning consists of growing more of the "mind organs" mentioned above as well as finding new ways to use them.

The subconscious nature of learning is the kernel of my argument. Most people would think that learning is conscious, and that things, once learned, may recede into the subconscious until such time as they're needed again. What I'm saying is that the conscious mind is like the horse rider above, who pays attention to the horse and gives it focus and encouragement. But it's the horse that actually learns the road and makes the basic decisions about how to negotiate it. In fact, the rider can't even see the road, because the horse is in the way. If he wants to know what the road is like, he'll have to ask the horse.

What about this "subconscious?"

The real story can be hard to accept

This point of view is probably pretty hard for many people to swallow, especially since most people probably don't imagine that so much is accomplished subconsciously. But that doesn't mean I'm wrong. The subconscious nature of the learning process itself might explain why there's such universal ignorance of how learning works, even though we all engage in it. There's a barrier of awareness between our conscious selves and the learning process itself.

Consider, for example, how a baby learns its first language. Most people think the baby learns it simply by imitating his parents. But that can't be right. If it only used imitation, how could children say things they've never heard anyone say before? (And there's a lot more to the argument than just that). Yet most people still believe children simply learn by imitation. So perhaps this one example might encourage the reader to stick with me a little longer in my argument, even if it sounds strange.

At least, nowadays, the idea that a subconscious exists at all is well-known and accepted. That was not always the case. In fact, it was only a hundred years or so ago that Sigmund Freud, the "doctor of dreams" and father of psychoanalysis discovered it and studied it. His gateway to the subconscious was through studying the dreams of his patients, who (by definition) were not conscious while they were dreaming. So it had to be the "subconscious."

Freud, of course, didn't concern himself with thinking and reasoning so much as with emotions and motivations. This is probably because some of the things his patients did, well, no one could understand why they'd want to do them (including the patients, themselves), so he set about the task of finding out why.

Unfortunately, most people still rarely consider that something other than diffuse emotions and lurking motivations might lie in the subconscious mind. Perhaps that's because most people would rather imagine they have the control of a good car driver, rather than that they need to have the empathy of a fine horse rider.

So let's take a look at how this horse thinks. To do so, I'll need to switch metaphors, and also I'd like to start off simple, not with complex learning tasks, but with the way we make observations of the world.

Pattern-matching

How do we perceive the world in the first place?

Now, if we're just looking or listening, most people would consider that a passive activity. And in terms of our conscious mind, perhaps it is. But our subconscious mind is actively at work to accomplish this "passivity." Information doesn't simply flow from the environment into our minds. Our subconscious continually and actively reaches out and takes it and then feeds it into our consciousness.

But even that isn't quite accurate. Because actually, there is no path for information from the environment to directly enter our minds. I and perhaps many teachers have often wished there were. Then we could just tell our students to "open their minds" and we could pour the information in like the coal man filling the basement with coal for the winter. Alas, that can't happen. (Or, as our students might say, "Thank God that can't happen!")

Instead, there is a sophisticated process of pattern-matching taking place.

What do I mean by "pattern matching?" Well, imagine that you're looking for a friend in a crowd. You have an idea of what he looks like, and this idea is like one of the mind's organs that I mentioned before - it's a pattern of thought. Although you can consciously imagine what your friend looks like, this pattern basically exists in your subconscious as a series of simple shapes and spatial positions linked togther logically. You scan the faces of the crowd until there's a match - you recognize him.

My assertion is that all observation works that way, whether seeing, hearing, feeling, or smelling. And it's not just for recognizing "old friends." That's they way we recognize "new friends" as well. In other words, to recognize new information, you have to already have at least the components of that "new" information already in your mind. How does it get there? Well, I'll get back to that later. For now, the reader can see that this is one way that people in certain cultures (not our culture, of course) have long thought that knowledge always arises from within, rather than being gathered from the environment.

One way you can picture this process is through the metaphor of a blind man in a portrait gallery - not a gallery of paintings, but a gallery of valuable sculptured busts. He's allowed to gingerly (so as not to damage them) touch the sculptured faces with his fingertips. He carefully controls the position of his hand and fingers, and thus is able to form a picture of each face in his mind. He actually can only know the position of his hands when they touch the sculpture, but still he can reconstruct the appearance of the face in his mind. And if his sight is restored, he may just recognize it on sight.

His hand positions are like the patterns I talked about before. Each position itself, then is like a mental organ, as are whatever mental structures that correspond to the blind man's fingers. In fact, the hand position itself can be treated like an object, so when the blind man moves on to other sculptures, he can compare them with each other by comparing the first hand position (or pattern) with each new one. In fact, if he remembers all the other patterns that he's seen already, he can compare and contrast them in many different ways.

And then, those comparisons and contrasts can themselves be treated as objects. Pretty soon, this man will have quite a system of catalogued faces, built of mental patterns, which are derived from many sets of finger positions and the relative positions between them. Such a complex structure, then, constitutes learning. And there was no information flowing from the outside in, other than the feedback from the touch of his fingers to the sculpture. And this feedback information was restricted solely to the confirming the patterns of his fingers. If there was a sculptured wart or pony tail that he didn't happen to contact, then, as far as he's concerned, he's still blind to it. It's kind of like the old story of the blind men and the elephant.

We are just like that blind man, even though our eyes see (or we think they do), because real vision works somewhat the same way - not just metaphorically, not just empathetically, but ACTUALLY. One reason I use the illustration of the blind man is that our eyes perpetrate the illusion that what they see is reality. This illusion is so strong that we lend credence to photographs that we would never lend to a painting, even though, in this age of special effects, realistic photographs are not necessarily any more genuine than a kindergartner's sketch.

What you find in the mind that corresponds to the blind man's fingers are sets of logical relations, such as "same," "opposite," "reverse," etc. which make up the comparisons and contrasts. What's understood about the environment, then, is basically an elaborate structure of such logical relations, tested against the environment for accuracy through this process of pattern matching.

So why is it, then, that we can open our eyes and information seems to just flow in?

Well, to begin with, we're not directly looking at the outside world. We're looking at our own subconscious understandings of it. It's our subconscious mind, that, like the blind man, can interpret the evidences our senses into something that makes sense. The subconscious then actively feeds it into our awareness. Information doesn't flow from the outside in. It rises from the depths of our understanding into our conscious mind. We just project that process onto the outside world. This projection is a simplification of what's actually going on in the environment and, as long as the patterns and manipulations we come up with are accurate enough (n.b. - mark well the "if"), it's virtually the same thing.

This, by the way, is also why hallucinations are possible. If information merely flowed into our consciousness from our eyes and ears, there would be no hallucinations, no matter what drug you were on.

What Does This All Mean?

Implications of a model of subconscious, pattern-matching perception

What I've stated so far has a lot of implications worth considering.

First, the patterns that result from making observations, and any consequent sets of patterns that result from trying to understand those observations, are not at all like photographic images or familiar tunes. They are complex constructions that attempt to match the evidence of the senses. They are constructed from logical relations, such as "equal," "not equal," "opposite," "reverse," "included," "excluded," "greater," "less," "rotated," "protruded," as well as some basic shapes like lines and circles, and perhaps some other simple relations.

And this evidence from the sense organs is mainly just a means of confirmation and feedback. You could think of these constructions, then, as not so much a picture or painting of a scene, but a complex, logical, intelligent analysis of a scene.

This means that, later, if you want to remember the scene you were looking at, your subconscious mind doesn't just retrieve a picture to show you. Instead, it uses its analysis to reconstruct the scene and present that to your awareness. Memories, then, are not the retrieval of old sights and sounds (like most people think they are), but the active "recalculation" of them. It's like asking a touch typist to remember where the letter S is on the keyboard. Watch them for a moment as they type "on an air keyboard" for awhile as they reason it out.

A lot of people picture memories like they were items stored in a file cabinet, or like bits and bytes stored in computer memory. They think that the process of recalling them is just that - retrieving them from wherever they're stored. The problem with that picture is that the brain has nothing like a file cabinet or hard drive in it. If you want to think of the mind as a computer, don't imagine it's like the computer on your desktop, with a one or two microprocessors manipulating information that's kept in "memory." There isn't any "memory." Instead, there's nothing but billions of microprocessors.

Second, because memories are actually recalculations of scenes, and not an actual recall of a scene, it's possible to influence them, particularly in the younger mind, when the mind's structures are simpler, and it doesn't have as clear a picture of what's possible and what's necessary. This is why, for example, I have a clear, vivid memory of myself on a lawn next to my dad in Sacramento, two years old, waving at my mother and sister in the window of the hospital where my sister had just been born. My parents tell me that nothing like that ever actually happened, yet I remember it clearly. In reality, this memory was probably built up from overheard conversations, other unrelated experiences, and put together in later years.

Of course, as every Nazi torturer and every cult de-programmer knows, even adults' perceptions and memories can be changed if you apply enough pain and/or deprivation.

This mutability of memory is not always a hindrance to the truth, though. Because memories are built logically, information about the world gained in later years can actually sharpen some memories because the mind understands so much better what is possible and what is necessary.

Third, because your perceptions are actually your subconscious mind actively selecting and testing patterns, the whole process of perception is greatly affected by what you expect to see and by the context in which you see things. It's like the famous character Sherlock Holmes, who always seemed to see clues that no one else noticed. It wasn't that his vision was any more acute than anybody else. He found those obscure clues simply because he was looking for them. He'd already reasoned that they must be there.

Fourth, if we hold our understandings in the subconscious mind, and if the subconscious is anything like the hotbed of motivation and emotion that Freud implied it was, then emotion and motivation can directly affect what we literally see. I still remember the one episode of Seinfeld that I happened to watch a few years ago. In it, the main character was dating a particular young woman, but sometimes she looked beautiful and alluring, and other times ugly and foreboding. In fact, if it wasn't that she always wore the same clothes, you'd think she was two different characters. One of my friends remarked later that she couldn't understand what the heck that was all about. Well, it was about how emotional states can affected the perceptions of the main character, as he wrestled with his feelings about her. To me, it's only a slight exaggeration of what actually happens in our own minds.

Fifth (and perhaps most importantly), if someone can't construct a pattern to match a particular thing, then he remains blind to that thing. Either he does not see it at all, or if he does see it, he perceives it as something different from what it would appear if he had a more sophisticated set of patterns to work with.

This last point is extremely important, because the mind constructs the illusion of a seamless reality. In other words, if you can't understand something, you might never notice it because your mind always presents you with what seems like a complete world view.

These sorts of "blind spots" in the mind remind me very much of the actual blind spot that exists in each of your eyes right now. It's just off the center of each eye's field of vision. There's a small area with no light receptors, because it's where the optic nerve exits through the retina. We're never aware of it because the subconscious fills it in for us, either with information from the other eye, or if the other eye is closed, then it just constructs it out of whatever information comes from the area surrounding the spot. So we perceive a complete view through each eye, even though it isn't actually complete. This is just one simple example of the extent to which our subconscious mind uses the patterns it controls in whatever ways it sees fit.

And if this is how the mind deals with literal blind spots, imagine what it can do to concepts you don't quite understand, or social relations, or causes and effects that aren't what they seem.

So, in a classroom at school, if the students don't have the conceptual equipment (the subconscious patterns) to understand something, you can show them the picture, and they don't see it. You can play them the tune, and they can't follow it or repeat it. You can tell them the answer and they don't grasp it. Furthermore, they treat the lack of information like it was falling on the blind spots in the eye. They fill in the "missing" information with what they do understand so they're often not even aware that they're missing anything. Anyone who has taught or raised children knows this to be the case. And anyone who sets out to plan curriculum better keep it in mind.

Finally, there's an old Buddhist saying that encapsulates this whole model of perception quite well. It goes something to the effect that "when you die, the universe is destroyed." The reason is, of course, that what you perceive about the universe is not actually the universe, but the complex system of patterns that's been built up by your mind. So naturally, when you die, all these patterns die with you, and then, the whole perceived universe is destroyed.

By the way, the reader might have noticed that I keep vacillating between "looking" and "perception." It's my contention that other faculties, such as hearing, touch, smell, etc., work the same way - they're modeled by the subconscious brain, and again, what we perceive is actual the analysis of the situation that the subconscious presents us, rather than a "whole cloth" sampling of what's actually there.

A real-life example of this kind of "blind spot."

One of the nice things about teaching young kids is that these blind spots are rather apparent. One well-known example concerns four-year-olds and their understandings of groups and parts of groups. If you give them, say, six toy dogs and two toy cats (all the same size), you can discuss at length how many there are of each, plus the fact that the dogs and cats are both animals. You can ask them to show you all the dogs, all the cats, and all the animals to make sure they understand exactly which is which. Once you're convinced that they know the difference between cats, dogs and animals, you ask them, "Are there more dogs or more animals?" Typically they'll say "more dogs." Then you say "that what?" and they say "than cats."

Four-year-olds generally can't hold a group and a sub-group in their mind at the same time. Because of this, the question "Are there more dogs or more animals?" makes no sense to them. Therefore they heard the question "Are there more dogs or more cats?" The point is this: they didn't just hear the adult's question unclearly. Of course, like most people they listen to the intent of the question and not the words themselves. Still, it's quite likely they literally heard different words than those that were asked. Their minds filled in this "blind spot" just like your own minds fill in the blind spots in your field of vision.

Adults don't have such obvious blind-spots as four-year-olds. Still, adults can get a flavor of the experience when they try to imitate skilled behavior performed by an other adult. For example, people with no experience in dancing have difficulty seeing, let alone following, even the simplest dance steps when a skilled dancer demonstrates them. Similar experiences may occur when trying to follow and imitate unfamiliar sports skills.

What's important to remember is that we all have such blind spots in our minds. It has to be so, because we perceive (and understand) a seamless world, yet the tools for perceiving that world must, of their nature, leave gaps in our understanding. As we get older, our picture of the world becomes a closer and closer approximation to reality (hopefully), but it can never grasp it completely. Also, because the tools are all subconscious, emotions and health can effect them profoundly. As an elementary school teacher, for example, I often run across this phenomenon when a kid thinks another kid has said something insulting towards them. Sometimes they did, and sometimes they didn't but the supposed recipient of the insult was feeling bad about something else and thus heard what he expected to here. It just shows the importance of asking up front to find out for sure.

What about this mind that constructs itself?

Another point of view that's hard to accept

Now, some people get uncomfortable facing the idea of an internally constructed perception of reality. "If each mind constructed its own perception of reality," they might say, "then everyone would perceive realities that are unique and different. How could anybody agree on anything?" But those people don't reckon with two things that limit the scope of the subconscious's mental constructions, and keep us from falling into a sea of solipsism.

First, there really is a reality beyond our limited perceptions of it, despite what you might have heard back in the sixties and seventies. Maybe what we perceive of it is merely a part of ourselves, but the reality is there nonetheless, so there is a commonality to our experience in it. You won't avoid falling into a pit, for example, just by convincing yourself it's an illusion.

Second, the metaphorical blind man above at least had his fingers to start with. If what I'm saying is true, we must also have some relatively complex patterns to initiate the process, and to deal with the world right from birth. Do such things exist?

Yes. It turns out that many relatively complex patterns are hard-wired into the brain. A good example of this is the perception of perspective. If you imagine railroad tracks receding into the distance, the two tracks sort-of appear to approach each other, the further away they are. Of course, they don't really get closer, it just looks that way. If you make a drawing of it, the two lines of the tracks will draw closer to each other on the paper to show this illusion of perspective.

Experiments with young kittens whose eyes have just opened show that they, too, understand this concept, that they can see depth, because they avoid falling over cliffs, even if the cliff itself and the space in front of it are all underneath one flat level piece of glass, so there's no way they can feel for its presence or detect it any other way than through sight (since for them, there actually is no drop). So if kittens have visual concepts hardwired into their brains, why wouldn't humans also have them?

Third, it's likely that there's some (or a lot of) middle ground between knowledge that's innate and knowledge that is created from contact with the environment. I mean, most people have heard of the phenomenon of "imprinting," where a young bird hatches and learns to recognize whatever being it sees as its mother, whether that being is a bird, a dog, or a person, or whatever. It already knows what a mother is. All it needs from the environment is a particular mother to fit into its definition. Once that identification is made, then it sticks. There are some wonderful films of geese imprinted on the famous scientist Konrad Lorenz . He leads his young goslings through the streets of Vienna and into the river just as a mother goose would.

Well, it seems likely to me that this is just one aspect of a general phenomenon - that there are some mental patterns just waiting to happen, but they need to incorporate certain particulars from the environment to be complete. One good example of this in humans is language. Our birthright includes all the primitive concepts for creating and using language. But in order to actually speak a particular language, we need to gather tools (like particular sounds and word parts, particular forms of syntax, and particular conceptual models that underlie the basic vocabulary) from contact with the environment. This isn't easy to do, by the way, but still easier than inventing a new form of communication that isn't a language as we already know it.

So, again, since we all have the real world in common and many basic complex patterns as a birthright, it's not surprising that two human observers can agree about much of what they perceive. On the other hand, even with the real world on one side and genetically inherited knowledge on the other, and phenomena like imprinting and language acquisition in between, there's still a lot of latitude. Different people really can construct noticeably different views of the world. It's usually not hard for anyone to come up with examples of people whose learned viewpoints are so different from their own (sometimes to the point of seeming to contradict common sense) that they wonder how the other person can possibly think that way. Maybe they think something like "Are they even in the same universe as me?" And of course, that's precisely the point.

Interesting thought: The fact that knowledge is determined partly genetically would be the most unifying factor in the very young (having had little experience in the world) and the way reality curbs one's understanding to the actual instead of the possible would be the most unifying factor in the very old, who have had plenty of time to explore the limits of reality. So when you consider people meeting each other from different cultures, perhaps it's the very young or the very old who can find the most common ground with each other.

So, if that's perception, how does learning work?

What are the two major ways of learning?

There are basically two ways of learning. Most people probably think that learning is just learning and that's all there is to it, but knowledge is like money - the more you have, the more you can make. And just as money can be simply spent for goods, or invested in ways to multiply your means to obtain goods, you can simply learn things, or you can learn how to organize your knowledge in such a way as to multiply what you can learn in the future. (Remember, though, that these two processes both happen subconsciously, beneath one's awareness or direct control)

The famous Swiss epistemologist Jean Piaget was the first to put names to the two ways of learning. I hate his choice of names because they both start with the same letter, "a," so it's hard to keep them apart in my mind. But their use is so well-established, I might as well employ them here, too. He called them "assimilation" and "accommodation." Of course, like everything else in the mind, there's probably no absolute separation between the two. And, indeed, Piaget's conception of them probably varied through his career, and is slightly different from the way I conceive of them.

Assimilation means using pre-existing patterns to understand the world, like the blind man using the same old fingers in new ways. It's the pattern-matching system that I first mentioned a long time ago. In other words, you simply assimilate environmental input into what you already know about and understand. No basic changes in the underlying mental patterns are necessary. They are sufficient to match the environmental input. This process is quite efficient.

So, for example, if you already have the concept of "dog," then it's not much of a stretch to figure out that beagles, poodles, Airedales, Dalmatians, and Saint Bernards are all dogs, too. And, even if some people won't admit it, so are Chihuahuas. If you already have the concept of "team captain," then you already understand something about the functioning of quarterbacks, sergeants, academic department chairmen, and others. And because you expect certain things from a "team captain," you therefore have a greater sensitivity and perception to any subtle differences between such different kinds of captains.

Of course, such expectations also limit your perceptions of events that fall too far outside their scope. Sometimes information that's truly novel slips by or becomes misinterpreted. Here, for example is a story of unknown authorship making the ways through e-mail boxes these days.

A three-year-old went with his dad to see a litter of kittens. On returning home, he breathlessly informed his mother there were 2 boy kittens and 2 girl kittens. "How did you know?" his mom asked "Daddy picked them up and looked underneath," he replied, "I think it's printed on the bottom."

In this case, then, the youngster understood a new situation in terms of the old situation. This process is similar to "covering over" a blind spot, except that there is no blind spot - there's new information that is misinterpreted because it doesn't fit with what the kid could already conceive of.

In general, the perceiver feels quite entertained when he applies the same old patterns, but with slight variations, because its new information, but not so new that you really have to puzzle something out. It's like tickling the fingers of the blind man. That's probably one reason why, for example, there's so much not-quite-repetition in folk songs or children's books, or, for that matter, in the countless Hollywood movies that continually recycle the same dozen or so tired characters and plot elements over and over.

Accommodation, is completely different from assimilation. Accommodation is the production of new and novel patterns. It gets its name from the idea that the mind has to change itself, to accommodate itself to the real world according to the evidence it has, rather than soaking in the same old knowledge that it already knows how to deal with. It's the process that makes it possible to perceive things you literally could not perceive before. It gives you understanding that eluded you for years, and can serve as the gateway to whole new worlds of knowledge that were previously shut away from you.

It's as if the metaphorical blind man was suddenly able to grow new fingers, or, indeed, new hand-parts that are nothing like fingers. It's as if the metaphorical horse of the subconscious suddenly sprouted wings.

And, as you can imagine, it feels nothing like a diverting tickle on the fingers. It's more like childbirth - strenuous, but compelling. And like childbirth, it can be anguishing, frustrating, and exhausting. And it never proceeds as quickly as you might like. Of course, like childbirth, it produces something truly new and wonderful.

How does it work? Here's another metaphor, though in this case, the metaphor is closer to reality than the previous ones were.

Consider the following:

Let's say you glimpse an object that looks like this:

Later you glimpse an object that's like this:

What you're aware of seeing are, of course, the mental patterns that your subconscious mind contains. These patterns often occur in similar contexts, so eventually it occurs to you that they could actually be the same object with different appearances. That's an interesting idea. You reflect on it. You consider yourpattern and your   pattern and try to fit them and stretch them in different ways, to figure how they might actually be the same.

You struggle with this idea. They don't look the same, but on the other hand, you have reason to believe they actually are. You bring in other logical points, such as the idea that an object can appear different if it's seen from a different angle. Perhaps this will clarify the situation. Finally, in a moment of creativity (which usually happens only under sustained and often uncomfortable mental pressures), you fuse (or reconcile) the two models into a new one, that looks like this: Such an object fits all your previous experience, and agrees with the appearances you noticed before, so you keep it.

So now you have a new mental tool or pattern that you can use to assimilate information more accurately than you did before.

It doesn't replace the previous models, like this example might imply, but augments them. It's easy to understand why old and new models can exist together, because there's no actual model in your mind to begin with - just a set of logical relationships, as I've stated before. Adding a few more relationships to a given set of patterns does not have to destroy or even modify those patterns. Instead, what you have is a new pattern, which in turn can be manipulated as an object of the mind.

Now, that's a heck of a lot of thinking to be going on subconsciously - there's strategizing, testing theories, planning experiments, interpreting results, etc. etc. Is it really possible or even likely that so much sophisticated intelligence can be operating, seemingly independently, in our heads beneath our level of awareness? Well, yes, it is.

And there's something else to notice. This process of new construction is based on meaning.

It doesn't take place until you add to your perceptions the meaningful idea that and are really the same object, even though they look different. In other words, their different appearances at different times are linked through some sort of transformation, which is yet another pattern (concept, mental tool) which you constructed like the others, through reason and reflection upon the observations that you made.

Now, it could be that this idea of identity (in other words, that objects tend to maintain their existence from one moment to the next) is something each person constructs for themselves at some point in their development. More likely, though, it's simply a part of the mind that's inherited genetically. Either way, though, it's this bond of meaning and observations that results in new forms knowledge.

Things to notice:

There are a couple of things that the reader should notice here..

First, the process of accommodation results in more tools for the mind to assimilate information. In this sense, then, accommodation underlies all learning, or at least, all new abilities to learn. In this sense, then, if we truly want our students to learn more and know more, this is where we should focus our efforts.

Second, the process of accommodation generally results in ideas (mental tools) that are at once more simple and more powerful than the ideas that had been previously held. The is much simpler than the and the . Also, it helps you understand a lot of other instances you might not have noticed before, like:

Third, the production of these simple ideas rests on a vast array of experience and thought. It's like drawing one simple lesson from a hundred individual and sometimes confusing examples. This is why a "information rich, educational environment" is helpful - having lots of interesting mental stimulation enables the gradual accumulation of simple lessons so that eventually you can put them together into something new. The creation or invention of new mental patterns generally requires holding many bits and pieces of evidence (some of it contradictory) in the mind for a long time until you can reconcile then into a new concept. This takes time.

Also, remember that the new pattern does not replace the old ones, even though it incorporates them. It exists alongside them. For this reason, even though you have a powerful idea, it still remains tied to the situations that produced it. In terms of this metaphor, then, the cylinder shape helps one understand and appreciate the movement and rotations of a cylinder-shaped object through space. But it doesn't necessarily follow that now the person can understand and appreciate the similar movement and rotations of another shape, say, a cube or a pyramid or a cone. The model has to be constructed separately for each of these situations and only then can similarities between them be seen and conclusions drawn about their movements and rotations in space. In other words, more accommodations would be needed.

Fourth, note that the process of accommodation is interactive. But it is not an interaction between the student and teacher, or even between the student and the environment. It's an interaction between one idea and another, where both of these ideas are already a part of a person's understanding.

Indeed, it has to be this way, because a learning person cannot be challenged by a new idea that comes from outside of him, except to the extent that the new idea resonates with something already inside his mind.

There is some information that comes from the environment, of course. But it's only simple things, such as whether events happen in sequence or simultaneously, or if you run into something while walking, etc. All other information that seems to flow from the environment actually flows into your conscious mind from out of your subconscious, from the pattern-matching process that I outlined above. To put it at bit crudely, then, the fact that you got hit over the head comes from the environment. When you look to see exactly what hit you, that comes from your subconscious.

In the metaphor above, the person already possessed the ideas of the circle and the rectangle. When coincidence and timing forced him to consider that they might be the same shape in different guises, he used this consideration to link the two ideas, wielding them one against each other.

The person knew that the circle had to be the same identical object as the rectangle. But they didn't look the same. That's a contradiction. It's at once compelling and annoying, something that can drive you crazy if you don't solve it. In this case, the only way to resolve the contradiction was to invent (construct) the cylinder shape, which can look like either a circle or rectangle, depending on one's viewpoint.

This point is particularly important because of the use of "concrete objects" (also known as "manipulatives," "models" or simply "toys") in education. These can include various counters, and also more sophisticated objects, such as counters glued into groups of ten or hundred to demonstrate "place value"in arithmetic, etc. Such objects are, of course, very useful, and in fact should be used in education. But the reason they can be effective is slightly different from the reason many people think.

If you take, for example, the common picture of a small child playing with blocks, (or, these days, Legos), many people might think that learning occurs as the child internalizes the shapes of the blocks and the ways they go together. This picture, however, is misleading. What's actually going on is that the child is recapitulating the shapes of the blocks in his mind. In other words, he's constructing mental models of the blocks based on evidence from the environment, such as if the block fits somewhere or not, how it lines up with other blocks, etc.

Again, these mental models themselves are made up of simple shapes and logical relations, such as "it fits" or it doesn't fit," etc. This, then, is why a child plays with such objects, instead of just scanning them in, like a computer might.

Of course, by elementary school age, most kids have a pretty good understanding of the objects themselves, at least insofar as how to manipulate them or fit them into the right place. In elementary school, then, physical objects are usually not used for themselves, but to represent something else. For example, they may be used to represent quantity, through their size or by counting them.

However, numbers do not exist in the environment the same way physical objects do. They exist only inside the mind of each individual. You can prove this to yourself by asking yourself how you could demonstrate "three." You can't. You can only demonstrate three of something. You can't demonstrate "three" without the "something."

Another way of looking at it is if numbers existed in the environment, it should be as easy to teach an animal to count as it is to teach him to fetch the paper or jump a hurdle or manipulate other environmental objects. Most animals can't count, even though many can learn to manipulate objects at least as well as a young child. That's because numbers just don't exist in animal's brains. And for those few that can conceive of numbers, they can't count very high.

What's my point then? Young children need to interact with something to develop concepts of number (patterns of thought that they can use mathematically) through the process of accommodation. But because numbers don't exist in the environment, children can't get any feedback whatsoever from the environment itself. The only possible way to develop these concepts is through problem solving of some sort or another, where one concept or pattern is wielded against another in the hope of sharpening both.

This consideration certainly puts the idea of "manipulatives" as demonstrators of number concept in a different light, doesn't it? In reality, "manipulatives" can only exhibit what's already in the mind of the beholder, unless they're being used as some sort of record-keeper in solving problems.

Such problem solving doesn't have to be formal, such as the math problems in a textbook. Much of it (and, probably most of it) is achieved rather informally. I remember the comments of one famous mathematician who enthusiastically recounted his first math breakthrough as a preschooler; he was amazed to find that if he lined up some objects and counted them from left to right, it turned out to be the same number as when he counted from right to left. Thus, through one instance of accommodation, was born a lifelong devotion to mathematics.

And, by the way, he achieved this accommodation because he was playing undisturbed. If an adult had told him to count each way, his mind would have been focused on following the adult's directions and it would have probably taken him longer to reach the same conclusion.

This is the sort of "problem solving" I'm talking about, where very basic, basic things are learned (much more basic than what are called "basic skills" these days.) What I'm not talking about is anything like using physical objects to demonstrate borrowing and carrying in arithmetic. Even though such skills are labeled "basic skills," they are in reality far from that. When properly understood, they are sophisticated concepts that can be developed only by years of focused problem solving. When not properly understood they are only a recipe to be repeated mechanically to get an answer that doesn't mean anything to anyone.

Fifth, there's a lot of effort and discomfort involved in accomplishing an accommodation. It's obviously quite a leap to go from 2-D objects like a and a to a 3-D . Long periods of reflection and struggle are needed to construct any truly novel mental pattern. And since all new constructions are at least somewhat bound up with emotions and whatever else might be found in the subconscious, there could be psychological impediments, especially if you have to redefine ideas that might have served you well for years.

There are certain things you might learn that involve a lot of this accommodation type of progress. One of these is mathematics (and I mean real mathematics, not just arithmetic and blindly finding answers by following prescribed methods). Another is learning a new language - real language learning, and not just studying about the language, which is all most high schools ever get around to. Anyone who has actually learned to speak another language knows the fatigue that results as your brain almost seems to rearrange itself in order to accommodate the new concepts behind the language.

This means, then, that learners tend not to make accommodations when assimilations will do. This is why, for example, most foreign language instruction involves learning about the language rather than learning the language. It's much more quick and expedient to learn about something a concept than to learn the concept itself. This is also why college math departments like to wait until a mind is "more mature" in the subject before they provide instructions in advanced (yet also elegant and simple in their way) subjects such as linear algebra and 3D geometry.

The effort required to bring a new mental pattern to fruition has been termed "cognitive dissonance" by Piaget and others. It's called "dissonance" because it feels like an uncomfortable clash of ideas to the person undergoing it. In some sense, the emergence of the new pattern is a response to this pressure, and signals a new calm and equilibrium, at least until the next challenge comes along.

Sixth, accommodation can only happen when all the components are in place. This means that a rich environment, with many varied small playful, problem-solving-type experiences will nurture the process

Even in a rich environment, the learner must be engaged and accepting of new ideas for accommodation to function. If he cannot accept one of the required components to produce new understanding, then new understanding won't develop. Instead, he'll fall back on the same old mental patterns and assimilate the situation, distorting it in the process to fit his established world view. Scott Adams recently drew and published a Dilbert cartoon that illustrates this idea neatly:

This cartoon doesn't just show how failure to accept a premise can lead to a wrong conclusion. It shows how failure to accept an idea can lead to (or reinforce) a faulty world view.

It also shows that, from a teacher's point of view, you may be teaching one lesson while the student learns a completely different one. Kids are always learning. You can't stop them from learning. However, they aren't always learning the lessons we think they are learning, or think they should learn, because they have their own learning agendas. And, short of torture, they really do have primary control over their own minds.

Before finishing this section, I'd like to just list a few more examples of the kind of inventions kids have to make, to give you a more complete flavor about what we're talking about. Piaget called them "operations," but that's not a word we need to remember. "Concepts" or "patterns" will do just fine.

• the realization that when you count objects in one order, you get the same final count as when you count them in another order
• the fact that changing the shape of a lump of clay does not change the amount of clay in the lump.
• Similarly, the fact that the number of objects in a group remains the same, even if you spread them out so it looks like there are more.
• The idea that the walking a straight line between two points is the shortest path.
• Understanding that a beagle is also a dog, or a white horse is also a horse, or a classroom's students are included in a school's students.
• the ability to sort objects by their properties (such alphabetical order and size place).
• the imagination to predict what you'll see when you view a scene from another standpoint.
• the fact that the water surface inside a filled glass stays parallel to the table even when you tilt the glass.
• The fact that a man walking west on an east bound train can be stationary with respect to the ground.

Other Areas of Learning

Learning is to perceive is just one aspect of learningThere are other areas outside of the classroom that might also exhibit the process of accommodation I've described. In fact, I would assert that all learning works in ways analogous to what I've described so far.

So, for example, instead of inventing a 3D shape as in the example above:

• a musician might invent a scale out of familiar sets of notes, or a new way that one chord can lead to another.
• An athlete might invent a movement by modifying or combining movements that are already mastered.
• A politician might use input from his town meetings to conceive of a message with many ramifications to increase his support.
• A writer might take another look at his story line and realize a new metaphor or theme might be more appropriate.
• A historian might use myriad bits of historical evidence to limn the direction society is headed in.
• A cook might come up with a new recipe for an old dish.
• A potter might discover a way to feel the changing thickness and texture of the pot on his wheel through touch.
• A physicist might conceive of a new model for the atom!
• Someone might invent a word to describe a detailed new process, using parts of words they already know.

All these are constructions resulting from reflective thinking.

At this point, the reader might say, "Wait a minute! All those examples are cases of inventing something for oneself. Isn't there more to learning than coming up with novel inventions all on your own?

And my answer is that, at base, no. Other than the concepts that we're born with, or ones that wait for particular tools or triggers from the environment, it's necessary for invention to happen before anything truly novel can be understood, or even perceived. Again, this is because there is no path for information to flow directly into a person from the outside. Instead, a system of pattern-matching in the subconscious mind is continually modeling and testing its models. The results of that subconscious mind are fed to our awareness and we perceive them as the outside world.

More examples

This whole process of accommodation, because it underlies all other learning, is critically important, so here are three more real-world examples:

Example 1

A little kid notices that every time he waves his hand, it produces some air (what we could call wind), seemingly out of empty space. But when he stops waving, there is no air (so can you imagine how big a hand somebody must have to produce a gale?). But as he learns more about the world, it no longer makes sense that a hand movement can produce something out of nothing. So now there's a contradiction. Finally he comes up with the idea that he's not producing something from nothing, even though it seems so. Instead, the air is always there, filling the space around him, even though he's not aware of it. It's not produced by the hand. Hand movements merely make it palpable.

Example 2

It's also like a sociologist who has been studying the obligatory control of information by Communist dictators in Cuba or China. One day, he talks to some friends in the computer industry and finds out they control information carefully, too. But they aren't Communists. Then he notices ads saturating radio and TV broadcasts, attempting to crowd out competing products or political candidates. But they aren't Communists or dictators, either. As many more such contradictory examples occur to him, he comes round to the idea (he accommodates, in other words) that it's not a matter of Communists or dictatorships, but that positions of authority or power tend to control information, partly for practical reasons, and partly because they simply can.

Example 3

There's a famous story which illustrates how subconscious the whole process really is. It's the story of how Auguste Kekule discovered the chemical form of benzene, an oily liquid often found in gasoline. It was well known at the time that each benzene molecule consisted of six carbon atoms and six hydrogen atoms. All other such carbon compounds then known had molecules consisting of a row or branching pattern of carbon atoms, with hydrogen atoms attached in several places. But such a structure could not explain the chemical properties of benzene. In other words, there was a contradiction. Here's what happened next, as expressed by some uncredited text I found on the internet:

Kekule was to say later that he must have dozed off at this point. In his dream the black balls of carbon turned into black imps with forked tails that began racing around the room and would soon be upsetting the apparatus of the laboratory. He was ready to run the rascals out. Then, almost suddenly, the confusion died away as each imp grabbed the tail of the one ahead of him, the six forming a whirling circle. One hand of each imp held a tail, the other a white handkerchief, and they waved to him as the group whirled by. He said that he came awake with a start, realizing that the imps were acting out the formula for benzene. As his hand grabbed the sketching pencil, the imps were back to black balls again and the handkerchiefs had changed to hydrogen atoms. How simple the arrangement turned out to be. "The carbon atoms of benzene form a ring."

And here's what it looks like (more or less):

Actually, a later scientist, Linus Pauling, had an additional insight, because even with this ring arrangement, not all the properties were explained. He realized that the bonds between the carbon atoms were not double and single alternating, but sort of a 1½ bond, like this:

What about the part of the mind that is aware and conscious?

Consciousness and it's leadership role -

To have come this far, and not even broken into the conscious mind yet!

And really, that's the order it should be, because consciousness arises from the subconscious mind.

In the beginning of this essay, I compared the relationship of consciousness to the subconscious as like a rider on a horse. You can also compare it to a gardener or vintner working a field. He lives off the fruits that grow there, and makes his shelter from them as well. In other words, the conscious mind uses and adapts to the information fed to it by the subconscious. The gardener cannot actually produce the growth in his crops, he can certainly direct their development within certain limits. He also has free reign in the field to explore and collect things to the extent that he's able.

Of course, mind is mind, whatever faculty it makes up, so the gardener is really made of the same stuff as the garden. So instead of an ordinary gardener, imagine something like the "tree-herding" Ents in J. R. R. Tolkien's Lord of the Rings, who have tended their trees so long that they have grown to be like them, so much like them that it's sometimes hard to tell the Ent from the tree.

There's a well-known Taoist tale that illustrates how this relationship works. It's the tale of the ever-sharp knife:

There once was a butcher whose knife never dulled, even after years of use. As time went on, this knife's fame spread far and wide. People came from everywhere to just to view it, and find out how it was made, thinking maybe they could buy one or construct one for themselves. But if that's what they wanted, they were sorely disappointed.

It was actually just an ordinary knife. The secret, as the butcher explained, was how he wielded it. His skill came from the fact that he could see what others could not. His eyes were so sharp that he could make out the tiniest of spaces that separated the joints between bones. His hand was so skilled that he always hit those spaces precisely. Consequently, the knife never really hit anything solid, and so it never went dull.

To be sure, when he had first begun to butcher, he could not have accomplished this feat. His first few knives needed sharpening all the time. But through years of patient practice, he had honed his skills and his perception to the point that it was now rather easy. Perhaps, he suggested, if his visitors were willing to do the same, they could achieve the same results.

In this example, the butcher began with the ordinary abilities of movement and perception that most non-infants have. As he attempted his first cuts, some failed and some succeeded. This simple feedback allowed him to better his aim, but it also allowed him to construct better mental models of the material he was working with. Once these mental models were in place, the pattern-matching process of perception became more accurate. To further develop his skill, he kept looking closer and closer. What he was looking at, of course, wasn't the material itself, but the mental models that his subconscious had constructed through the process of accommodation. This process of looking allowed his conscious mind to construct its own models (again, through a process of accommodation) that reflected the models already in the subconscious.

Once he had these models consciously in place, then, he could direct his learning even more accurately. Further accommodations were constructed in the subconscious, which, in turn, allowed further constructions in consciousness, and so on. Through this cycle of directing, learning, and becoming aware, directing, learning, and becoming aware, not only did his hand coordination improve, but his ability to see (in this situation) was magnified, and thus his understanding became more powerful..

In a nutshell, then, the process goes like this:

The conscious mind directs learning (in other words, it sets the overall agenda and seeks out learning situations). The subconscious mind accomplishes the learning. Finally, the conscious mind learns the skills and the details from the subconscious. Now it can direct further learning with more precision.

Anyone who has learned something complex involving perception has experienced this. Obviously, this takes place in sports. Dart throwers see the target more accurately. Hurdlers see the hurdles more accurately, first basemen see the ball more accurately, etc. In all these cases, vision sharpens, but not all vision - only the visual skills you need to play the sport well.

I myself have had some experience with such limitations on skill learning. Since I'm a musician more than a sportster, I can pretty easily pick up on hand positions, particularly on instruments I know, but even on other instruments or in other realms of hand-use. I can often get sound out of new or foreign instruments that stymie the efforts of others. It's not the same, though, when I'm shown any sort of fancy footwork, whether in sport or in dance. In that case, even slow movements are at times too fast for me to follow. On the other hand, another area I'm adept in is computer use, and I often have to slow down my clicks and drags so that non-computer users can follow what I'm doing.

Vision isn't the only faculty that works this way. Anyone who has learned a foreign language to fluency knows what it's like when, after long experience with a new language, one can suddenly hear and follow words in quick conversation. It's as if, after working so hard for so long, conscious thought can finally ride upon the subconscious structure beneath, and thoughts actually seem to speed up.

At that point, the language learner can simply listen to the sounds of the words and not strain to decipher them. As he listens, he can feel his subconscious struggling to feed meanings to the words. Usually it uses assimilation, but on rarer occasions, accommodation occurs in order to provide deeper meanings to old familiar words. That's why, by the way, at this point in coming to fluency, language learners need a lot of sleep!

And, of course, it's not just perception of the real world that works that way. When mathematicians talk about being trained in mathematics, their conversation sounds a lot like an athlete being trained in a sport. And, just as the dart thrower sharpens his view of the target, the mathematician sharpens his conception of mathematical processes, which allows him to go to the next, usually "simpler," level.

Now, when the subconscious is having trouble processing something, it summons the conscious mind inside to direct its handling of memories. The language learner above can feel this when all of a sudden he hears words that are not very familiar to him. At that point, listening speed slows down as one's consciousness is called in to figure out the words.

And, when the mind finds itself dealing with contradictions or other facts that are hard to accept, it also calls in the conscious mind, which provides the coordinations necessary for accommodations to occur. The Dilbert cartoon above shows this idea quite well. As the boss's subconscious tussles with some rather uncomfortable truths, it metaphorically screams for the conscious mind's help in focusing on the problem. The "spoink" is when the boss consciously feels the tussle that's beneath the surface of his awareness, drawing his conscious mind into the discussion. In that case, of course, accommodation was begun, but not completed.

One more example - musical style and musical taste.

Another place where I experienced the "speeding up" of the conscious mind as it helped the subconscious grow was in music. When I was young, I loved jazz tunes, but couldn't follow the improvised solos. It just sounded like the musicians were playing random scales or something. It didn't seem very musical. But, I continued listening and thinking about what I was hearing. Gradually my attention focused on particular motifs, particular musical conventions, particular ways soloists weaved ideas together, and gradually it became clear. I could follow the solos perfectly well and "see" into the minds and hearts of the people making them.

I think most people experience something similar with music. Sometime during childhood or adolescence they become enamored of a particular style. They learn the tunes, the lyrics, the dance, and they totally immerse themselves in it. Conscious minds and subconscious minds work together to further develop their listening skills until they simply listen with pleasure. No further noticeable effort is required (which means, then, they are mostly assimilating instead of accommodating)

Of course, their expert listening skills are mainly applicable to the style they love, and consequently less applicable, the further you get from that style. From their own point of view, other styles might seem simplistic, unskilled, out of tune, or even just plain annoying - not because they really are, but because the listener can't hear the subtleties (or the soul) that is there. It's like the stereotypical snob who loves rock and roll, but hates country music.

I remember once hearing a skilled performance of Chinese Opera by a local amateur group. Many of the Americans (and also some of the Chinese) in the audience seemed to treat the experience as a tortuous duty to get through. The only reason they put up with it was out of respect for the millions (of mostly Chinese) who enjoy it, and out of respect for those particular performers, who loved and cared enough about the music to spend all their spare time perfecting it. So that's the kind of thing I'm talking about. And perhaps those dissatisfied audience members did reflect that there must really be something there, since those performers loved it so much.

And the fact that understanding of musical style is based in the subconscious in turn accounts for the fact that most people see musical styles merely in terms of what appeals to them. From their point of view, it's simply a matter of what they happen to like or not like, and not what it really is, a matter of what they can and cannot understand. (Of course, I'm referring to the style itself, and not to particular lyrics, which may have their own appeal or repulsion separate from the style)

What does all this mean? (Part Three)

Implications of a conscious mind that focuses attention and coordinates learning

1. Well, one thing it means is that, though the conscious mind is in control of learning, in some ways it really isn't, because it can only direct, and not accomplish, learning. Furthermore, what it does learn is learned from the subconscious, and not directly from the environment.

What a strange system! Wouldn't it be better if brains simply learned information and skills consciously, and then, once learned, just stored them for later in subconscious memory. After all, that's probably how most people think learning happens. But that's just wishful thinking. Alas, none of us had the privilege of designing our own brains. Such a wish is like wishing that a horse rider could grow his own horse-legs and hooves, or that a gardener could grow vegetables off his own toes or boxwood from his own ribs. It might be kind of handy, but it just ain't gonna happen.

2. Is there any advantage to designing a mind so it learns in such an indirect way? Of course, there is, otherwise I wouldn't bring up the question. The key is the idea of multitasking. As I mentioned earlier, the brain is like a computer where each part is a complete microprocessor. So instead of information storage, you have a trillion little living calculators (brain cells or neurons) that are constantly sending signals of one kind or another to each other. These signals constitute the patterns that typify thinking. The neurons grow and change in order to establish and to maintain these patterns.

This is, of course, beyond anything the conscious mind can keep track of. But each of these little patterns seems to have its own "life," its own inputs and outputs, and, in a certain restricted sense, its own thoughts, all going on at the same time as all the others. Of course, it can learn and reinforce its learning somewhat independently of other thoughts.

I think that this general process also operates on a larger scale. Who, for example, has not had a heavy concern gnawing away at them in the background, affecting one's mood wether or not it was consciously being contemplated? Similarly, who has not anticipated a wonderful and exciting moment to the point that the excitement bled over into everything they did, even when they didn't think they were thinking about it? In terms of learning, I have often noticed that if I back away from a subject for a period of days, and then return to it, it's almost as if my mind has been mulling it over in the background "while I was gone," and now it seems clearer than ever.

So it may even be that this multiprocessing community of thoughts needs freedom from consciousness from time to time in order to grow efficiently and, indeed, to hone its thoughts to clarity.

To take the idea a little further, it also may be that we all harbor the seeds of multiple-personalities within our minds, which become especially noticeable in the behavior of people afflicted with certain mental illnesses or in the art of certain great novelists.

3. As you learn something, there's a really fine balance going on between the concerns of the conscious mind directing things and the subconscious learning them and teaching them to the conscious mind..

The conscious mind is like the mayor of a small town who has big plans for the town's future. He is elected by the townspeople (the subconscious) and has a mandate to set the direction and goals. He's in a position to appreciate the big picture, but when he wants to build a road, he can't do it himself. He hires a construction agency. When he wants to create an arts program, he can't do it himself. He hires a director. When he wants to field a baseball team, he doesn't run it himself, he hires a coach. And if he wants to promote healthy relations between different segments of the population, he doesn't do it by himself He appoints a committee. And if he's smart, he'll listen to what those people are telling him, and as he learns more about his town from the people who work for him and the people who elected him, his own knowledge of the town grows and he's better able to chart the town's future direction.

On the other hand, no matter how knowledgeable he thinks he becomes, he cannot actually be more knowledgeable than the populace he leads, he cannot lead them where they cannot imagine going, and he cannot motivate them to do things that their present skills will not allow.

The Development of the Mind

You mean children aren't really just little adults?

It's patently obvious that adults possess more knowledge than kids. Not only that, but they understand more subtle jokes (at least, some of them do).

The child's mind, of course, is not merely childish, but childlike. And it's not simply information-deficient. Anyone who has worked with kids can see this. But beyond this simple intuition, how exactly do kids' minds function differently than adult ones? It's not simply a matter of superior adults vs. inferior juniors. Kids have talents and abilities that most adults have lost. For example, they attend to (and produce) language sounds that adults simply don't perceive.

But the differences go beyond such perceptual abilities. Kids also have a different set of mental tools for thinking, for problem solving. They really think differently than adults do. It's hard sometimes to convince adults that this difference is really the case, because adults don't usually remember themselves thinking qualitatively differently when they were younger. However, memories aren't simple recordings - they are logical reconstructions, produced in the here and now. What people think they remember is always interpreted (constructed) through their present understandings. Still, the mind is the mind. So it seems reasonable that basic human learning processes probably won't change much with age.

Still, simple accumulation of mental tools through the process of accommodation does make a significant difference. With greater numbers and variety of tools you can simply understand a greater number and variety of things.

Of course, there are a couple obvious limitations to this process of accumulation. Since the thinking mind possesses many different faculties, such as visualization, language, time sense, etc., different parts of the mind are likely to develop at different times, just like an adolescent's feet and hands have a growth spurt before the rest of them does. Language is one faculty that develops pretty fully relatively early, for example. And, of course, these different faculties interact with one another during development. So, even though the language and syntax abilities develop early, they aren't always used appropriately until the mind's time sense and logical abilities can order the world well enough to fit a language's potential to express those things. (So, for instance, kids in elementary school often have trouble with verb tense, and younger kids often misuse words such as "because" and "if" and others that relate ideas to each other.

It's also pretty obvious that some concepts need to be learned before others. For example, it seems reasonable that you can't understand what multiplication is before you understand what addition is. Unfortunately, even with such seemingly obvious examples as addition vs. multiplication much is still uncertain about what exact paths people take to reach more advanced understandings. In fact, the quest for such paths is a sort of "holy grail" of educational psychology, one which a young grad student could easily study right through to his retirement.

The reason such paths are so hard to determine is that such seemingly simple concepts such as addition and multiplication themselves rest upon a myriad of more basic understandings. And it's these basic understandings that produce the growth. In other words, multiplication is not simply deduced from addition, even though multiplication can be explained as "adding the same number again and again" to those who have already constructed both of those concepts.

And because there are so many of these more basic understandings underlying something as "simple" as addition, there are many ways to reach a given new understanding from a given group of starting points. And because these paths are longer than you might think, you could say, for example, that addition and multiplication are both developing for years before they're explicitly used in school. Compared to this long incubation, their final emergence in school is almost simultaneous, and so multiplication really doesn't derive from addition, even though it does derive from it in standard arithmetical explanations.

Besides these points about order and accumulation of knowledge, which simply follow from what I've said before, there are some critical points about development that I haven't stressed at all. But now's the time.

Firstly (in fact, fundamentally), the mind, like every other part of the body, has an inborn drive to develop and grow. In other words, kids are thinking, reasoning, problem-solving creatures right from the very beginning. You can no more keep someone from learning (or wanting to learn) than you can keep them from eating. The only question is "What's for dinner?" The mind is continually and actively reaching out to grab interesting tidbits and to digest a hearty meal.

To me, this fact is so obvious that it goes without saying. I know, however, that my viewpoint is not shared by everyone. However, I'll address whence this difference of opinion arises in the second half of this paper.

Secondly, the mind's internal coordination of ideas grows ever more profound and far-reaching with time. In other words, you first acquire various mental tools, (mental patterns), and then you coordinate one pattern with another through logical relationships. These coordinating patterns themselves can be later coordinated one to the other.

One simple example of this process is the mind's use of opposition.

To begin with, there is no opposition, because opposition is a relation between two concepts, and not (to begin with, anyway) a concept itself. So, for example, an infant's mind begins building itself up merely in positive terms. In other words, it builds mental patterns and tools that are useful in getting what it wants. Everything about an infant or young toddler is positive in this sense. Even what we might think of as "opposing, rejective, or negative behavior," such as spitting out food or crying, is not really an opposition or negation. It's a positive, assertive, rejection of something or response to pain. In other words, infants and young kids are at a time in life when they just "go for it," no matter what they do.

But gradually, one's mind, one's understanding can coordinate crying with laughter, spitting out with taking in, the positive thing with its negated opposite. These coordinations obviously have to be mental abstractions, if only from the fact that these opposites rarely occur at the same time. You usually don't laugh and cry at the same time, nor spit out and swallow. So you have to reason out any comparisons between them.

Furthermore, the mind can find many sorts of oppositions or negations. The opposite of something positive could be something negative, but it could also be something reversed, upside down, compensated, shattered, inside-out, skewed, or assigned to a different category.

By the way, another fruitful field of research (meaning no one's figured it out yet) is to discover exactly what sort of logic is used to build up this extensive internal coordination. The example I gave was based on the use of negative and positive (which could actually be inside/outside, forward/reverse, etc). But there are other types of logic, both binary (like negative/positive) and multi-dimensional (such as "fuzzy logic") that could be used as the foundation for a logical scheme. One theory that I've read suggested that the mind's logic might be built up from the binary of possibility/necessity, for example. Another way might be from "induced/derived." Another might be from "shared/private." At this point, I don't think anybody knows for sure. Indeed, more than one type of logic might interact to produce the mind's internal coordination of ideas.

Thirdly, these mental coordinations don't spring suddenly into existence. Although there may be a short window of time when they seem to coalesce, they are actually only teased slowly out from the original patterns.

And, since they occur so as to make patterns within patterns, or ideas within ideas, the same sort of process should be functioning at different levels of this resulting hierarchy of ideas. So a simple illustration should be representative of all, and here it is:

A few pages back I used the metaphor of first seeing a and then seeing a and eventually realizing (constructing) that it's a 3-D seen from different angles. One of my points was that this new construction does not supercede the old ones, but co-exists with them as merely a more efficient way of viewing old material. In other words, it remains somewhat bound to the original concepts.

So, let's say the same kid sees a and then he sees a in similar situations to when he saw the and the in the first example. He doesn't automatically put them together into some sort of 3-D arrangement, say a , just because he did it that way before. He would only do that if he already had produced a generalization about the way the world works that specifies that similar observations always go together that same way. Of course, it's still possible that he'd put them together that way. He may even put them together more quickly than the first time, since he had already had the practice of doing so. But even if he gets that far, he still probably won't see a connection between what he did in the first instance and what he did in the second.

This is because making these generalizations, and particularly, becoming conscious of such generalizations, only happens after many small instances have been done the same way, much like a musician who only masters playing a G major scale after having practiced it over and over at many times and in many rhythms and speeds.

Only after many such observations and problem solving will he finally be able to see a new situation, say a and a , and put them together to make a .

At this point, you might say, well, can't you just teach them the generalization and then you could dispense with all this experimentation? Unfortunately, however, "understanding" is a pattern-matching process. If you try to "teach" the kid a concept he doesn't already possess, then he will have no way of dealing with it directly. He'll have to resort to the process of assimilation, which means distorting or perhaps even ignoring the "teaching" message as it was actually meant.

The only way of teaching the concept would be to put him in a situation where he must problem-solve and reason about his observations such that the best solution is to construct the generalization. Obviously, such a process takes time. It's also not particularly easy to structure such learning situations.

Shifts in Thinking - the "Upper Grade Kid"

Fourthly, and critically, there are times in a kid's life where mental coordinations become so profound that a shift in thinking styles becomes possible.

This sort of paradigm shift doesn't just occur with mental development. It happens whenever tools and connections develop to a certain volume, even in the real world. Take the Internet for example. It got started in the 1960's, but it was really only a research tool for scientists and generals. But then, bit by bit (or maybe, byte by byte), more and more computers linked up to it. More networking schemes were added to connect them. Finally everything was ready and the key software was invented to turn it all into the World Wide Web. This web software would not have been so transforming back when only scientists and generals used it. Now it's no longer an adjunct to life. It's a "given." We depend on it for basic communication. And I can't imagine many scientists and generals of the 1960's would have believed that we'd all be shopping for books and tools on it, and using it to keep in touch with our dear old formerly computer-phobic family members.

There are lots of such "critical mass" processes (including nuclear reactors themselves). If you look at ancient history, for example, Enough People + Division of Labor = Civilization. In city planning, Enough Cars + Enough Roads = Suburbs and Outlet Stores. In biology, Sophisticated Lab Techniques + Enough Knowledge of DNA and Viruses = Genetic Engineering and Manufacturing. Enough TV Sets + Enough Broadcast Stations = Mass Media. In humanities, Enough Decorations on Objects + Enough People with Time on Their Hands = Art for Art's Sake. It's like once you get to a certain level, a whole new paradigm shift is possible.

What sort of paradigm shift are we talking about in the development of the mind? We're basically talking about what happens when, after several years of growth, the ever-burgeoning mass of logical coordinations begins to hit the level of conscious understanding, so that now kids can consciously begin to think along such logical lines. In terms of what I wrote above, it's like Enough Formerly Subconscious Thinking about the Real World + Control and Direction by the Conscious Mind = Flexible and Logical Understandings That Kids Can Consciously Apply to Problems. From the adult's point of view, you can talk and reason with them in ways that just didn't make sense to them before.

What we're talking about is not a single shift to a new kind of mind. Instead, it's a series of shifts in thinking that generally begin around the age of seven, sometime during what corresponds to second grade in traditional American schooling. They dominate the mind's growth by accommodation for the next several years. Of course, some kids begin these shifts sooner and some later. The variation in starting age probably depends on biological factors and maybe nutritional factors, the same way kids of similar chronological ages experience growth spurts at different times from each other. Because the shifts are knowledge based, it may also be that lack of experience in the world may delay them slightly.

Probably most kids have begun them by the time they're nine years old. I like to think of kids who have embarked upon this shift of thinking as being "upper grade kids," named after the upper three grades (4, 5, 6) in an elementary school setting It's only in late third grade that you can be fairly certain that just about all the students have started this process. And after grade six, the students begin moving into a different series of shifts (as we shall see later) so grade six makes a convenient upper age limit.

I used the word "shift" to describe this process because each shift seems to happen relatively suddenly, over a period of weeks or months. This seeming suddenness and comprehensiveness of change is, at least in part, because of the involvement of the conscious mind, which can focus attention on goals. Thus, once the process starts, the conscious mind notices it, focuses attention on it, and expeditiously completes the process.

What kinds of reasoning tasks are we talking about? Really, not much different than the logical connections that make up all conceptual knowledge, except that now they are available to the will and the understanding of the child. Mind you, the child is not yet consciously aware of these logical connections being logical connections. From their point of view, it's simply the self-evident way the world works, as they (consciously, at least) simply continue to "go for it" in a positive sense.

A typical example of what I'm talking about is shown by the well-known "ball of clay" experiment, which illustrates the logical relationship of "compensation." In this experiment you start with two balls of clay, exactly alike. And if the child doesn't agree they're exactly alike you let them make adjustments until they're sure. Then you roll one of them out into a hot-dog shape or a pancake shape, leaving the other as a sphere. Then you basically ask the child which one is bigger or if they're still the same size (and there are many ways to ask this question, all of which produce the same results), to see if the child believes there's still the same amount of "stuff" as before.

Typically, a child who has not reached this shift in thinking says the clay is no longer contains the same amount of material, and will choose one or the other as bigger. Those undergoing the shift will insist they're still the same amount, either because you could roll it back into the same shape again, or because it's bigger in one direction and narrower in another direction, so the two changes compensate each other. This is evidence that the logical relationships are reaching their conscious minds. On the other hand, these kids aren't aware that the logical relationships are logical relationships. They just know that the world works that way.

Those who have passed through the shift (the "upper grade kids") will often just say that the clay has to be the same size because it's the same clay as before. What's interesting about that is that the pre-shift child also knew it was the same clay as before, but didn't think that was a good enough reason to claim it still contained as much "stuff." But now, the older child, through building up concepts logically, simply justifies his opinion by "stating the obvious." To him, it's the self-evident way that the world works.

This same child, or an adult for that matter, could get really frustrated trying to explain this self-evident property of the world to the younger child, who, would seem to get it and not get it at the same time, because the premise (it's the same clay as before) doesn't imply the same conclusion (it's just as big as before). The younger child could be simply "taught" to remember it's the same material as before and therefore it's the same size. But in that case, it has no more meaning than if you taught him it's the same because the clay's name is Bill or because the clay is blue.

After understanding this "conservation" of the amount of clay, this older child will later go on to realize similar things about weight a couple years later. A couple years after that, he'll realize similar things about volume. This apparent delay in simply generalizing a concept of "conservation" occurs because increasingly abstract ideas, such as weight and volume, need further logical bolstering before a child can see (construct) the truth of them. Someone like me can group these particular three shifts together because I can see that they're all similar situations. A child of that age would not necessarily see that they're similar.

This kind of thinking, where the logical relations constructed in the subconscious are taken as guides for conscious thinking, is sometimes called "concrete thought," or "concrete thinking," or, as Piaget used to say, "concrete operations," (from the mathematical use of the word "operations," which means adding, subtracting, multiplying, dividing, etc.)

The term "concrete" is an appropriate label because this kind of thinking is well-suited to manipulating physical (i.e. concrete) objects to solve problems. Indeed, if kids of this age group have physical objects available to them, they can often solve problems far more complicated than most adults would think possible.

Here are some more examples of the kinds of things kids can accomplish after they pass through this shift.

They can put a row of objects in size place. Younger kids cannot do this, and to the extent they succeed it's usually only with a pair or three of objects.

They can organize objects into groups based on properties (a classification, in other words). Younger kids arrange objects into groups based on what seems to look nice together. Or, if they're able to separate objects into categories, they are inconsistent about it (which you can determine by interviewing them, not simply by your own examination of the groups).

They can organize their classification hierarchically, so they understand that some groups contain other groups, and are in turn contained by larger groups. Younger kids just arrange groups disjunctively, separately from each other. Even if you place one group inside another, objects are understood as still being in one group or the other.

They can imagine (or predict) what a scene would look like from a different point of view. By this, I mean that they can imagine what a group of objects would look like if they saw it from the other side of the room. This is an ability which increases in sophistication over years, probably because it is actually composed of several shifts in thinking

They can see understand that the level of water in a bottle is horizontal no matter how you tip the bottle. This means that they can use the concepts of vertical and horizontal to structure their view of the world.

They can understand that lengths don't change and thus you can compare and measure them.

They can understand that they will always be the older than their younger siblings by the same amount.

They can understand that clocks don't speed up or slow down and that the same time measured by different methods is still the same amount of time.

They can understand (usually by around age 9 at the earliest) that a point on a plane can be accurately placed using a vertical and a horizontal measurement. Thus they can begin coordinate graphing)

They can understand that if you cut up a piece of paper and arrange the pieces into another shape, it still has the same surface area.

They can begin to predict the shape of a shadow cast by a three dimensional object, but only for the simplest of shapes.

Shifts in upon Shifts upon Shifts - adolescent thinking becomes systematic

After several years of bringing one logical relationship after another within the awareness of the conscious mind, another qualitatively different shift in thinking begins to occur. The conscious mind has been using relationships between ideas to guide its thinking. Now it becomes aware of the relationships themselves and begins to logically relate one relationship to another.

This generally begins happening around age 11 at the earliest. For some kids, it doesn't happen until age 14 or later. For this reason, it's useful to think of the shift beginning in early adolescence or in the pre-teen years. Of course, this type of thinking is highly dependent on the subject matter and how much you think about it. So in some domains of knowledge, it won't occur at all.

When relations are related in this way, specific content (such as specific examples) are no longer part of the relation. So the only way for it all to make sense is for the set of relationships themselves to be complete. This is probably one reason why it takes so long to develop (besides the fact that the accumulation of enough relationships to make it feasible takes a long time).

And because the relationships operate without specific content, it's possible to figure things out and add the examples later. In other words, it's possible to reason something out "abstractly," and then see where you can apply the principles you've derived. Because the relationships make a complete set, the mind can now better determine what all the possibilities of a given situation are, and to notice when one possibility isn't there or if it changes.

In other words, the mind can now think systematically in terms of hypotheses, exploring all the possibilities of a situation, holding all the factors of a process separate, yet coordinated from one another. So, for example, if a learner was studying the effects of adding bleach and/or fabric softener to a given laundry detergent, he would naturally figure out all the possible combinations of detergent, bleach, and fabric softener so as to isolate the contributions of each, as well as isolate each kind of interaction that they may have with one another. A child who hasn't entered this shift simply won't do this. They may try one or two combinations of things, but in general, they're inclined to either try them each separately, or try all of them together in order to "make the most difference." Certainly they will never form a system to explore all combinations and possibilities.

This is the sort of thinking that characterizes great problem-solvers in many fields, such as figuring out why your car runs rough, why your computer always hangs at the same spot, why the paint sticks on the front of your house, but peels on the back, why the Christmas tree lights won't light, what the animal you're tracking is like, and what it really meant when someone said what they said.

Besides combinations, other logical relations are possible. One key idea that develops out of relating relations is the concept of a continuously measurable line (or plane or space). Up until early adolescence, distances are really understood as being intervals, so measuring them is like putting milestones along the way and then counting the stones or the resulting line segments. In order to be continuously measurable, measurable to any given fractional part, then, it's as if there were an infinite number of possible spots to measure to, all squeezed together on the line. Of course that would result in an infinite number of line segments of length zero. Now, infinite numbers of spots are not something the mind can directly conceive. Lengths whose lengths are zero are also not conceivable. It only makes sense because the mind at this point can take the relationship of numbers of spots to lengths of segments and take that to it's logical extreme. In other words, it coordinates the two related ideas of number of spots versus length of segments.

Remember, of course, that all this is happening subconsciously, beneath the direct control of the learner, though the conscious mind's ability to focus attention and coordinate ideas is probably involved in some way.

Now, until you come up with the idea of a continuously measurable interval, you can't properly conceive of the measurement relations between different-dimensional objects - for example the numerical relationships between the lengths of a rectangle's sides (1-dimensional) and the coverage of its surface area (2-dimensional). It is only at this point that students can properly understand, for example, that the length of a rectangle (one dimensional) times its length (also one dimensional) gives its surface area (two dimensional) in square units. Or, if you had a rectangular prism (brick-shaped object) with such a rectangle as its base, then the area of the base (two dimensional) times the height of the prism (one dimensional) gives its volume (three dimensional) in cubic units.

For most adults, it's hard to believe that such sophisticated thinking is necessary for such seemingly straightforward problem solving, especially since many kids younger than eleven are successful in calculating the areas of rectangles, and a few can even do the volume calculations. But they do so because they are following directions, and not because they understand what the directions mean. For adults who have already constructed the more sophisticated understanding, then, it's simply the self-evident way the world works, so it's hard for them to conceive that kids don't perceive what's so patently obvious. And, again, it was all constructed subconsciously, so the adults don't have any memory of having constructed it.

There are many other mathematical ideas that come to coordination during this second shift. One example is the idea of proportion, which is the relation between two ratios. Since ratios are themselves relationships, then, proportion is a relationship between relationships.

A related idea from physical sciences is that of density, which is the relationship between weight and volume, two abstract ideas. It's always an interesting experiment to see, for example, if kids can figure out which objects sink in water and which ones float. Only after this second shift can they understand the actual reason.

After the adolescent shift, students can take a much more thorough and systematic view of causes and effects. Whereas, for "upper grade" students, it's a difficult enough achievement just to understand the normal ideas of motivation and the normal course of cause-and-effect, adolescents can even consider such abstractions as teleological causes, which can occur after the effects they produce.

What does all this mean? (Part Four)

Implications of a mind that changes with age

The most important implication of this process of mental development is that kids of different ages should not be taught using the same methods. Or, to amend that a little bit, kids have to reach a certain developmental state before particular methods make any educational sense to them. On the other hand, methods from lower grades will always find a home in the upper grades and high school whenever similar learning situations occur.

One simple way to look at it is that there are three populations of learners in Kindergarten through twelfth grade, and that three different methodological bodies of knowledge should be developed to address their needs. The first group is from preschool through first grade. The second group is from mid-third grade through sixth grade. The third group is from ninth grade through twelfth grade. In this schema, then, grades two (and possibly three), as well as grades seven and eight, are transition grades, where classrooms have a greater variety of educational needs.

Only the third and last group can be mainly taught under the paradigm of learning basic principles and then applying them. The first two groups should only be taught through problem solving. And, of course, problem solving needs to remain part of the methods for the third group as well.

Only the second group should be taught to master what are usually thought of as "basic skills" in math. Of course, the third group should also be taught to master such basic skills if it's still needed, but most skills should be mastered if they start learning them in third grade. For the younger group, their problem solving should not require any awareness on their part that they had learned a skill, but simply that they were successful or not in reaching a goal or getting something in the real world to work. Thus, they can learn to add and subtract better than they currently do (and I mean "better" in ways that standardized tests usually don't measure) simply by playing the right games.

Something similar should also be true for reading and writing. In many ways, of course, language skills can be taught much earlier than third grade, since the language faculty (meaning the grammatical manipulation of meaning) generally develops much earlier. This early maturation may be because, to a greater extent than most learning, language is simply the unfolding of genetically pre-determined processes that need little environmental feedback to develop, other than the paring away of structures not used in particular languages, and the acquisition of the particular words. Of course, just because skills of listening and speaking develop early, it doesn't mean that kids can automatically generalize what they've learned to reading and writing. Still, they have that base available as a resource when reading and writing skills develop.

However, I suppose it all might depend on what subconscious structures an individual chooses to underlie their understanding or processing of language, so that for some individuals, third grade might not be "too late" to begin learning basic literacy.

It's important to keep one's thinking about these three groups of students distinct, at least as ideals or theoreticals. In the past, educators have noticed that upper grade kids can do amazing things and solve amazing problems if they have concrete (physical, real) objects to work with. They imagined, then, that one could introduce new learning concepts to these kids first with physical objects, and, as the kids learned to work with these objects, they could abstract the meaning and then solve the problems without the "crutch" of using physical objects. They figured that concrete thought was just a step on the road to more sophisticated, abstract thought, which doesn't require physical objects to act upon.

Well, it is a step on that road, but most kids don't arrive there until they're eleven years old or later. The only group that this method works well with is the third, adolescent, group (and also with adults, of course). It works with them, I suppose, because only they have fully developed mental "objects" available to their conscious thought processes - by these I mean the constructed relationships between different objects in different times and places.

Does this mean, then, that all that effort trying to teach conceptual thought through physical objects in elementary school is a total waste of time? Well, no, not if the physical objects are used in a problem-solving setting, where kids use them to solve real problems. This builds their conceptual thought in that they can produce ever-more sophisticated ways of dealing with objects and indeed can go beyond their own conscious understanding, in a certain sense.

But if the physical objects are just held up at the front of the class as demonstrations, then, yes, it's mostly a waste of time. They'll remember the objects fine, of course. What they won't get is what the physical objects have to do with the mysterious "concepts" that the teacher keeps talking about.

In fact, one of the main jobs of a teacher in the elementary grades is to serve as the main person in the room who can help kids draw connections between one concept and another, as the kids construct them first in one area and then in another; to help kids see that the principles that they just thought up "work the same way" as other principles that they thought up earlier in a different context or curricular area..

Another implication is that many things we think of as "cultural customs" have some sort of root in mental development. This age of seven is pretty significant in many cultures. For example, in Sweden, that's when kids first start attending school. In a way, that's been traditionally true here, as well. I mean, first grade starts at six, but first grade used to be more or less what kindergarten has become now - an introduction. Real learning started in second grade. Similarly, the Roman Catholic Church doesn't allow kids to take communion until they reach the age of seven, on the grounds that only then can they reason well enough to understand what they are doing.

The ages of 12-14 also mark a significant time in many traditional cultures - when kids become adults. Partly this is because they grow up and mature physically. But at the same time, they can finally enter into a more sophisticated, adult way of thinking about things. It's also an age that traditionally marks readiness for high school, which, at one time, was what we thought of as "higher education."

Well, okay, but so what?

What difference does it make if we learn the way the model states we do?

At the beginning of this paper, I expressed my desperation with the situation of education in America, a situation with much broader roots that any particular educational institution, such as public schools or private schools. It's roots sink much deeper than the teaching (or education) profession itself. They spread throughout our communities, and are grounded in a broad consensus about the nature of education and learning that overwhelms and smothers the imagination needed to change things for the better, to make students more successful and make the institutions and classrooms that they inhabit more human.

To this end, then, I've produced this model of the mind from my studies of educational psychology and the evidence from my own experimental subjects, also known as my students. It's my hope that I've articulated it clearly enough to begin a discussion of how the mind works and what teachers should do to deal with the educational issues it implies.

Along the way I've stated many of the most obvious educational implications, as well as the most obvious applications. But I'm not done. My sense of desperation does not come from a failure to understand how to lift students out of the apparently desperate education they're presently getting. My sense of desperation comes from the fact that well meaning, but uninformed, people, in great numbers, are systematically and thoroughly squelching any hope that the necessary changes can be made.

I mean, if you're an elementary school teacher (or to some extent, if you're a middle school teacher), and this model of the mind is even half-way valid, then those thirty-odd kids you see in front of you may inhabit the same classroom as you, but they also dwell in a completely different kind of universe. And even worse - you're the alien. You can't reach them by babbling on in your alien speech, or waving your alien appendages, although it may be possible for you to fool yourself that you can, if you use enough graphs, charts, and multiple-choice tests. And therein lies the danger.

So, the second half of this paper will address the situation we have now, and the effects it has on kids' minds, assuming that the model I've presented has some validity, I'm hoping that if enough people understand the consequences of our present way of thinking about education, then many will take up the challenge and search out ways to do better.

So if the first half of my paper is titled "Here's the Model," the second half will be titled "So What?"

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This is where I am in my draft as of November 25, 2000. I probably wouldn't post it now, except to get my former draft off the web site.  I've got quite a bit left to say about this, but there's no time. Contact me through this site if you have any questions.

By the way, if anyone is interested in seeing a survey of the kinds of learning theoryies that are out there, you could certainly do much worse than reading Theories of Developmental Psychology by Patricia H Miller, published by W. H. Freeman and Co. copyright 1983 ISBN 0-7167-1432-9 It basically gives clear summaries of many of the most common theories you'll find about how humans development.  It's great reading for a basic background in the subject, and would be a wonderful resource to be kept in the principal's office or teacher's room of an elementary school.

I noticed the third edition (1993) which is ISBN 0-7167-2309-3 is available at Amazon.com. It's now out of print, but cheap copies can be had  from Amazon's site. It would make a great addition to a school district library. Click here to see it on Amazon.com

A great companion volume to Patricia Miller's book would be Models of Teaching by Bruce Joyce and Marsha Weil, published by Prentice Hall Just as Miller's book summarizes the major views of human development, this book summarizes the basic kinds of teaching styles that are out there.  It's highly recommended for any teacher, novice or experienced, who wants to be challenged by teaching styles and methods that are different from what they currently use. It really is a well of ideas about how to run a classroom or a school. Again, it would be a wonderful resource to be kept in the principal's office or teacher's room of an elementary school.

The Sixth edition was published in 1999 with ISBN 0-2053-1038-9  It's pretty pricey, but used copies are available. It's certainly worth having in a school district library. Click Here to see it on Amazon.com.

There's an excellent book for understanding more about constructivism, Jean Piaget's theories, and especially about how this all applies in the real-world classroom,  It's by Ed Labinowicz, and it's called The Piaget Primer. It's also not cheap, but well worth the price, particularly if you've read other books about Piaget and found them a bit difficult to navigate. It's published by Modern Curriculum Press, and its ISBN is 0-2010-4090-5.  To see it on Amazon.com,