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Structures of Knowledge


     By "structures of knowledge" I am referring to mental concepts and associations between these concepts. Thus I am referring to what me might call academic knowledge, or "book-learning". There are other types of learning that are also very important, but are not best interpreted as "structures of knowledge." Learning to ride a bike, learning to distinguish by feel between a steel pop can and an aluminum can, learning to carry a tune, and learning to draw, are such examples. They might have components of academic knowledge, but are not primarily structures of knowledge.

     To explain what I mean by a structure of knowledge I will start with a simple example. A father tells his son, "Johnny, there's a candy bar in the grocery sack for you." Johnny then goes directly to the grocery sack, gets the candy bar, and eats it. Johnny has learned something from his father's words. A new bit of knowledge - that there is a candy bar for him in a grocery sack - has entered his head, and this knowledge is immediately made manifest by his actions of getting and eating the candy bar. This type of learning seems instant and effortless. No practice is required on the part of Johnny to learn it. Father spoke and Johnny learned. Is it indeed effortless, or is this misleading? And it seems totally verbal. Is learning the same as the words? Or is there a structure of knowledge apart from the words?

     I will analyze this bit of learning in terms of concepts and associations. Johnny knows what "candy bar" and "grocery sack" are. He knows that "you" refers to himself. These are the three basic concepts involved. The other words establish relations between these concepts. The learning consists of establishing a new configuration of concepts and relations in Johnny's mind. This is a structure of knowledge. Its parts are not new. The parts - grocery sack, candy bar, etc., - were preexisting. But this particular configuration of parts is new. This is a very small structure of knowledge, to be sure, and not a structure that Johnny will need to retain beyond a few seconds. But it is indeed a structure of knowledge.

     Suppose, now, that a concept was missing. Suppose Johnny's father says, "Johnny, there's a candy bar in the poke for you." Johnny then says, "Huh? ? ? ?", excited about the prospect of a candy bar, but wondering where in the world it is. The structure of knowledge is incomplete. Johnny cannot act on that structure of knowledge until he understands the concept of "poke". When he learns it is a slang term for grocery sack then everything fits into place. The structure of knowledge is formed in Johnny's mind and he can act on it.

     Similarly the structure of knowledge can be incomplete if the language used does not make the relations clear. Suppose Johnny's father says, "Johnny, there's a candy bar in the grocery sack for you, my compliments." Johnny is again excited, but not at all sure what his father meant. He understood the part about the candy bar, and he knows where it is. But what was that part about "my . . . . . . ."? It sounds like his father is claiming it, so why did he tell Johnny it was for him? When he learns what his father means by "my compliments" then the relationship between the parts of the structure becomes clear. The structure of knowledge is complete.

     Learning, of the type I am talking about here, consists of building structures of knowledge. A structure of knowledge consists of a configuration of concepts and the relations between these concepts. The relations, of course, are something like concepts in themselves. Thus all learning, of this type, must be built on previous learning. But where does it all begin?

     Few of us are able to remember our first few years of life. To know what it's like to be a baby is probably impossible, but we may conjecture a bit. A baby's world cannot be a verbal world. Rather it must be a world of sights and sounds, and perhaps most importantly, a world of touch and feelings. The brain is a data processing machine. Sooner or later this machine begins to sort out all the input to which it is subjected. Out of complete chaos it begins to put together those sensations which are found together in time or place. We may conjecture, then, that a concept is, or at least begins as, a group of sensations experienced with some consistent relation. This might be considered simple conditioning. I would guess that one of the first concepts formulated by a baby's data processor is "Mama". It is important to note that the baby's concept of "Mama" may be quite different than our concept of "mother". A "mother" is a "person". What reason do we have to think that a baby has a concept of "person". It would seem reasonable that the concept of "person" is abstracted from the concepts of "Mama", "Dada", etc. that would come first.

     Simple concepts arise out of sensory input and a process of conditioning, or at least something like conditioning. Advanced concepts arise out of simple concepts. I would think that generalization and discrimination are the two major processes by which new concepts arise out of old. An example of generalization would be the concept of "person" arising out of the baby's concepts of individual people that he knows. I am assuming, of course, that an infant will develop concepts of individual people before developing the more abstract concept of "person". An example of discrimination would be the concept of "Sunday" arising out of the more general concept of "day". Here, of course, I am assuming that the concept of "day" arises first, and then the concepts of the individual days of the week are formed.

     There may be a temptation to consider generalization and discrimination as verbal processes. This may be true to some extent, but not entirely. I think it is more accurate to say that words are simply one tool, among others, by which concepts can be manipulated and thereby allow generalization and discrimination, and perhaps other processes, to take place.

     Simple concepts must arise nonverbally. When a baby begins to form the concept of his mother he has no words to label that concept. But as time goes on words are attached to concepts. It can also work the other way. Words can come first. A concept can be attached to a word that has existed in the learner's mind purely as an aural sensation. For example a toddler may hear the word "day" and recognize it as a familiar sound months before he begins to understand the concept of day.

     Earliest concepts come from raw sensory data, the sights, sounds, and feelings we are exposed to. By "raw sensory data" I am referring to sensations that are not variations of previous sensations. Without raw sensory data we are limited. "How would you describe color to a blind man?" is a question that most of us have toyed with at one time or another. The answer is easy of course, assuming the person has been blind from birth. You can't describe it! Description or explanation is always a matter of relating the new to the old. A person blind from birth would not have the basic sensory input to make color describable. Nor could you explain sound to the deaf for the same reason. If a person blind or deaf from birth were to suddenly gain his missing sense then he would be faced with totally new and chaotic sensory data, and it would take some time to organize. I think this does happen once in a while.

     Advanced concepts are impossible to form if the simpler concepts from which they are built are missing. A person blind from birth could not form the concepts of "dark green", "pea green", "grass green", etc., which derive from the basic concept of "green". Without the raw sensory input that vision provides there could be no concept of "green".

     There are situations in normal adult life in which we confront new raw sensory data, but they are rare. I have two examples in my personal experience. The first example concerns the songs of birds. I once took an interest in ornithology and began to notice the birds in my own back yard. At this time I was very aware of the field of animal behavior and the idea that behavior can be just as important as body structure in identifying and studying animals. Behavior, of course, includes vocalization. I would see a bird chirping and ask myself if that is the same chirp I remembered from yesterday. I would find to my frustration that I simply didn't know. I couldn't remember the chirp I had heard yesterday, though I had given it close attention. The bird would fly away and I would find that I had again lost the aural sensation just that quickly. The sound was hard to retain (impossible, actually) because it was new raw sensory data to me. I have nothing in my mind to relate it to, no old sensations that could form building blocks to construct this new sensation. Perhaps I have heard the sound many times before, for it was a common backyard bird that produced it, but I never paid any attention to it. I could try to relate it to sounds I knew. I could decide that the sound was something like "chulip", and that is easy enough to remember. But the actual sound produced by the bird was something else again. I could no more remember that actual sound than a person blind from birth could recognize and name colors when suddenly given sight. Had I stuck with bird watching long enough, then surely I would have been able to remember and recognize bird songs. But how long would that take?

     A second example in my recollection, again concerning an aural sensation, is the German pronunciation of the umlaut vowels. Though I studied German for several semesters in college, I do not have these sounds in my mind, nor can I say them. The textbook says "There is no English equivalent for the umlaut vowels." and I believe it! I assume that my professors used the right pronunciation in class, and I would hear the right pronunciation on the tapes in the language lab, but still these sounds are not in my vocabulary. These sounds are indeed new raw material to me, and therefore are very difficult to assimilate.

     If, after several semesters of college German I could not deal with the umlaut vowels, then how long would it take me to deal with bird songs? I really don't know. How long does it take an infant to sort out the sensory input of his world? That I also don't know.

     Often when we think we are learning something brand new it turns out that there is not the least thing in it that is actually new, but only new associations and combinations of the old. For example in a zoology book is a diagram of a worm. The basic parts of the worm are labeled, its stomach, intestine, rhynchocoel, nerve cord, retractor muscle, stylet, brain, etc. The name of the worm is amphiporus. My first task then, if I am studying this topic, is to associate this worm with this name, or at least this particular diagram with this name. I can then associate this name with the particular arrangement of organs shown in the diagram. I can learn that amphiporus has a stomach, an intestine, etc.

     Amphiporus is a new word to me. If learning is a matter of associating the new with the old, then what is "old" here? The letters of the alphabet are old learning to me. This is a new arrangement of letters, to be sure, but the letters are all very familiar. Further, the syllables are not new, and I think syllables can be considered the building blocks of our language more than letters. The first syllable, "am", is not new to me. It forms a part of many words, like "America", "amplify", or "lamb". "Phi" is not new. It forms a part of many words like "fit", "fish", or "amphibian". Further, the combination of sounds very much fits English forms. So the new word "amphiporus" can be easily learned. It is simply a new construct of old building blocks.

     Learning about the parts shown on the diagram of the worm is likewise a matter of making new constructs of old blocks. Stomachs, nerve cords, brains, and so on are old constructs. I do not know what a rhynchocoel is, but with a little effort I can find out. It is some bodily structure, and with my background in biology I could make sense out of an explanation in a biology book. Rhynchocoel can soon be as meaningful as muscle, or heart, or liver.

     Without the simple concepts that one begins learning in infancy one could not learn anything from the amphiporus diagram. A person from a different culture with a different language and even a different alphabet might not at all make the same associations that I would from this diagram. An illiterate person may have the concept of "name", yet have no inkling that this visual configuration below the diagram was supposed to be the name of the worm. As a result he would make no association between the word and the diagram. He would not learn what the diagram was meant to teach. He would not learn because too many concepts are missing.

     I think it is useful to distinguish between "raw sensory data" and "new sensory data" and the amphiporus diagram provides a good illustration of this distinction. The diagram of the worm is sensory data in that it is a picture. But it is made up of many components, and none of these components are entirely new. The diagram taken as a whole is "new sensory data", but not "raw sensory data". It is a new configuration of old building blocks.

     It would be very hard to construct this configuration without the new sensory data of the picture. Theoretically, perhaps, the diagram could be dispensed with in favor of a verbal description. However a verbal description is not an efficient way to provide the sensory input. The best way to provide new sensory data is in the form of sensory data, not a verbal description.

     Similarly a picture of George Washington in a history book provides new sensory data, but not raw sensory data. And similarly the sound of a new tune played on a violin is "new sensory" data. It is not "raw" sensory data, unless the listener has never heard a violin before. New sensory data is an important part of many subjects, from auto mechanics to geography to surveying to zoology, but raw sensory data is rare.

     I will next analyze some small structures of knowledge in order to further illustrate their properties. Remember again that I am talking of structures of academic knowledge. There would be little applicability to habit formation, perception training, or motor skill training.

     Consider this example from calculus. "The derivative of the sum of algebraic quantities is the sum of the derivatives of the separate quantities." The sentence stresses connections. The concepts, presumably, are clear in the learner's mind before he is exposed to this statement. The concepts are "sum", "derivative", and "algebraic quantities". The connections are represented by the words "of", "is", and "separate". These concepts are already built in the learner's mind. The concept of "sum" goes back to elementary school. The concept of "algebraic quantity" comes at the high school level. The concept of "derivative" is the most advanced. It is the beginning point of calculus and is usually learned at the college level. The words used to show the connections between the concepts go back to a preschool level.

     Using these same concepts and almost the same connecting words I can extend this statement a bit and say "The derivative of the sum of algebraic quantities is the sum of the derivatives, but the derivative of the product of algebraic quantities is not the product of the separate derivatives." I have added the concept "product" and a few connecting words such as "not". The result of this can be put into a diagram as in Figure One.

     This diagram is meant only to represent the structure. It is not the structure itself. There are other ways in which it could be represented and still accurately depict the concepts and their relations. The structure must be acquired by the learner, but it need not be visualized in this exact way. Diagrams such as this are sometimes helpful as a teaching aid and sometimes not. The purpose here is to illustrate structure, not to learn calculus.

     By acquiring this bit of information, by sorting out all concepts and connecting them in the right way, the learner has increased his knowledge of calculus. Once this bit of information is in place, it can be applied to calculus problems, and more importantly, it becomes a building block with which to construct other structures of calculus. It becomes part of the foundation onto which other structures can be anchored.

     It must be emphasized, of course, that a person could memorize and be able to reproduce the verbal statement or the diagram exactly without at all understanding it, and this often happens in any class. When such a thing happens there is no reason to say that the information has been learned. The connections may be missing even though the person gives outward evidence that they are there. The concepts may be missing or erroneous even though the person seems to know what he is talking about. One who knows nothing about calculus might be tempted to go to the dictionary for a definition of "derivative". This would do him little good, for the concept can not be formed that quickly and easily. Indeed a student might spend several weeks or even months in a calculus class and still have a concept of "derivative" that is in some way faulty. This is why teaching requires some "digging in" to find out what is really going on in a student's mind, and this is something that only a real live human, not a teaching machine or television set, can do.

     This diagram covers only a very small part of the total structure of calculus. Calculus is only a small part of the total mathematical structure possible for a person to have in his head. And of course all of mathematics is only a small part of all knowledge. It might be possible to draw a diagram that would accurately represent the whole field of calculus, though it would probably not be profitable to do so. The point here is simply that knowledge has a structure. New learning is built on the old.

     Figure Two illustrates a structure of knowledge that is on a much more mundane level. It is a little different type of diagram than Figure One. Each arrow represents an implication. The arrow leads from cause to effect. In most cases it takes several causes to produce one effect. This example shows in more concrete terms the interdependence of the different parts of a structure of knowledge. Every element in this diagram is essential if the whole structure is to have any meaning. Of course statements one through eight must have antecedents that cannot be shown here, but if we take these statements as a starting point then all pieces of the puzzle are in place. Everything makes sense.

     The interdependence of the elements of the matrix can be seen by considering the effect of deleting any of the statements. Without statement one the learner wonders why the other statements are relevant. Why don't they just go to the movies and be done with it? Without statement two the trip would be off, making statement thirteen a mystery. Without statement three the learner doesn't understand the basis of Joe's action. Without number four he is similarly confused. Without statement five the learner doesn't understand the basis of Judy's action. And so it goes for the other statements that make the total structure of implication.

     Figure Two represents a simple bit of knowledge. But consider how far removed it is from the raw sensory input that an infant begins to sort out. Each element of Figure Two is a complex of concepts and relations. In fact each word is a complex bit of knowledge. Then consider how far removed this bit of knowledge is from an advanced technical subject such as calculus. Is there any wonder than education takes years, and is full of opportunities for missteps and failure?

     Both of these diagrams show rather small bits of knowledge. However I think it is worthwhile to notice that their complexity is on a similar level. Which is more complicated, calculus or knowing whether and why we're going to the movies? If I were to replace the words in these two diagrams with nonsense syllables, then it might very well appear that figure two represents the more complex bit of knowledge.

     This is not necessarily misleading. Advanced subjects are not necessarily complex, and subjects commonly learned at a very tender age are not necessarily simple. Is there any reason to believe, for example, that the auto mechanics that a boy learns at age sixteen is more complex than the language he learns by age five? Is the grammar that a child learns in the seventh grade more or less complex than the working knowledge of the social rules and expectations that must be followed in school and on the playground in the third grade? Perhaps it is a moot point, since I have no simple way of measuring the complexity of a subject. However I think it is important to have some awareness of complexity, even if it necessarily must be subjective. And it is certainly important not to assume that advanced subjects are necessarily more complex than less advanced ones.

     I think the size of a body of knowledge is usually more important than its complexity. Is calculus more complex than fractions? Calculus is more advanced than fractions. Calculus takes off with the concept of the derivative. The concept of the derivative rests on algebra. Algebra, in turn, rests on arithmetic. There is no way to understand calculus without first understanding fractions, but that does not mean that calculus is more complex than fractions. Mathematics up to the level of arithmetic is a reasonably large subject, but mathematics up to the level of calculus is a much larger subject. Therefore it takes many more years to learn calculus, but that does not mean it is necessarily more complex.

     Complexity of a subject is usually managed by breaking it down into small segments. We might say that fractions are complex, and that statement has validity. But when we teach fractions we teach only a little bit at a time. Adding fractions, for example, can be summed up in the sentence, "Get a common denominator and then add the numerators." That doesn't sound too complex, and it is not, once it is understood. Compare that to the complexity of understanding why we are going to the movies in Figure two. Adding fractions can be reduced to a flow chart consisting of two boxes, which seems considerably less complex than the diagram of Figure two. But fractions is a big subject compared to the information in Figure two. A considerable amount of learning must take place before the phrase, "Get a common denominator" can make any sense. Therefore fractions can be difficult for many children. Similarly calculus is managed by breaking it up into small parts and learning only one small part at a time. Each small part may or may not be very complex. But calculus takes years to learn because it is a large subject. Learning one small bit is only one step on a long journey. And of course it is a journey that cannot even begin without a good grounding in algebra and arithmetic, which are rather long journeys in themselves.

     Physics would seem to be a complex subject, and indeed parts of it are, but I have been amazed the number of times a seemingly complex topic in physics is reduced to a formula of the form A = BC. A is the simple arithmetic product of B and C. This would seem to reduce to a flow chart of one box.

     Due to the sheer massiveness of most academic subjects there is a constant temptation to overload the learner, to look for shortcuts or panaceas. When this happens the learner may become frustrated and conclude the subject matter is complex. It is a natural conclusion, but not necessarily correct.

     I will next consider in a little more detail the relation of language to knowledge. Words are the handles we have for dealing with concepts and their relations. Words are used by teachers to convey information, and by learners to represent information. But as I have been suggesting, words are not the concepts or relations themselves.

     Consider verbalization as a means to learning in this situation: Jane is the mother of some of the kids in the movie-going scenario of Figure two. She knows exactly how it all worked out that the kids are going to the movie and wants to explain it to her husband, Jim. She could say, "I told the kids they could go, but only if three of them are going together. Johnny wants to go. Judy wants to do the opposite of Joe, cause she's mad at him. Joe doesn't like getting rained on, and there's rain in the forecast, so he's not going, So Judy will go. And Karen wants to see a western, and they show westerns every Tuesday, and today is Tuesday, so Karen wants to go. So they're going." Jane has accurately translated a structure of knowledge into words and delivered those words to Jim. Does it follow that Jim now understands the structure of knowledge?

     Words may be an accurate, or an inaccurate, representation of a structure of knowledge. If a particular string of words is an inaccurate representation - if Jane had misstated something in the above example - then it would seem reasonable that the structure of knowledge could not be conveyed. However would the reverse be true? Does an accurate verbal representation of a structure of knowledge guarantee its transmission to a learner? I think the answer is obvious. No, it does not. Jim may have listened carefully and understood the situation, especially if he is as familiar as Jane with the details of the children's lives. However it is also possible the Joe listened carefully, but did not understand. Learning does not seem so effortless in this situation as it did when Johnny was learning about a candy bar in a grocery sack. It would not be unexpected if Jim is confused and has to ask a number of questions before getting it all straightened out. Jane's accuracy in translating a structure of knowledge into a string of words is only one step in conveying the information. Teaching is more than telling.

     Is verbalization always necessary as a cause of learning? In the example of Johnny learning that there's a candy bar in the grocery sack, is it necessary that this information be verbalized? Suppose his father conveys the message in a less direct method that I first described. Instead of saying, "There's a candy bar in the grocery sack for you." he could say, "Look in the grocery sack, Johnny." Johnny then proceeds to find the candy bar and eat it. Johnny has learned, and words facilitated that learning, but this time the words were not a full representation of that learning.

     Or perhaps Johnny comes into the kitchen where his father is unloading the groceries, observes his father eating a candy bar, and knows from past experience and his father's behavior that there must also be one for him. This case would be an example of gestural language and situational language substituting for verbal language, but with the same learning resulting on Johnny's part.

     Can the knowledge in Johnny's mind exist without words? Or do the words run through Johnny's mind as a precondition for making sense out of the situation? In my view the answer is obvious. No words are needed in Johnny's mind. Words may run through his mind, but, in this last case, they are an effect of the learning, not the cause or substance of it. Similarly words are not necessary when I observe a green pencil on the floor and stoop to pick it up. The knowledge - that there is a green pencil on the floor - came from observation, not words. And words are not necessary when I need the telephone book and remember my wife had it in the bedroom. Observation, memory, and deduction, not words, lead to the learning that the telephone book must be in the bedroom. We engage in a great deal of mundane learning like this everyday with little or no verbalization.

     It is possible to learn without verbalizing what we have learned, and sometimes it is sensible to do so. One idea included in the "discovery method" espoused by "modern mathematics" that came out in the 1960's was that learning, in at least some cases, should be allowed to take place without constant pressure to verbalize. This idea was based on research in which children were given number problems in which they were to discover a key or shortcut. It was found that those children who were forced to verbalize - "The answer is always two less than the first digit", for example - actually had a slightly lower retention rate when tested the next day than those who did not verbalize the short cut they discovered, but only applied it.

     This casts doubt on the idea, "If you can't say it, you don't know it." I don't think we need research to question this idea though. Any number of everyday examples show that we know many things that we can't easily verbalize. Giving directions to a stranger is a good example of this. We may know precisely where the hospital is, but the stranger may not get there on our directions. The parent or teacher who just instinctively knows whether Junior needs sympathy or a good spanking is another good example. Such a parent or teacher may not at all be aware that she can't verbalize how she knows this. But when she is accused by Junior's advocates of unfairness, she becomes painfully aware that she can't verbalize her choice of action to their satisfaction. Still she may not question in her own mind that Junior got what he needed. She can act on her knowledge whether she can verbalize it or not.

     So a good question for educators is just how much verbalization to demand from students. Another good question is how much to depend on what the student does verbalize. When Johnny says, "I know it, but I just can't put it into words", he may be stating the simple truth, or he may be stalling. On the other hand, when a student quotes an answer straight from the text book he may not really understand what he's saying. I think I am safe in saying that the subject matter will determine how much verbalization is necessary. In learning the elementary facts of history just about everything should be verbalized. If the answer is "1492" it won't do Johnny much good to say he just can't put it into words. In the teaching of math, of which I am most familiar, verbalization is often much less important. Knowing whether to add, subtract, multiply, or divide when doing written problems in fifth grade math is very important. But verbalizing the reason for choosing which operation to use may cause more confusion than enlightenment. A working knowledge of the material is the important thing, and verbalization should be simply one means to the end, not an end in itself.

     A structure of knowledge must be built up carefully. Translation of a structure of knowledge into a string of words is a part of teaching, but only a part. Normally a structure of knowledge must be built up with extensive feedback between teacher and learners, which is the subject of the next two chapters.