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Structures of Knowledge - Part 2


     The methods used to teach any particular subject are at least partially determined by the nature of the subject itself. In Chapter Two I discussed structures of knowledge in general terms. In this chapter I will extend these ideas a bit, especially in reference to academic subjects as they are traditionally taught. I will start by classifying structures of knowledge into two categories.

     Any subject has a structure of accretion, or a structure of implication, or some combination of these two extremes. Either type of structure is built by connecting new bits of information to the old. In a structure of accretion, as the name suggests, the new bits are just added on. Each new bit, which I will can an "add-on element," can exist and make sense by itself. It does not derive from the previously existing structure. An example of an add-on element in the field of astronomy would be the fact that Mars has two small moons. This is a fact that does not logically derive from other facts of astronomy. It is a bit of information that must simply be added on to one's previous knowledge. In a structure of implication, in contrast, one thing implies another. New ideas derive logically from preceding ideas. A new bit of information, which I will call an "implied element," cannot exist by itself. It may not be totally meaningless by itself, but it is best learned as a corollary or consequence of other ideas. An example of an implied element in the field of astronomy would be the fact that the planets are "wanderers" among the "fixed stars". This fact is a consequence of the very elementary facts of astronomy that the stars are immensely far away from us while the planets go around the sun and are our immediate neighbors in space. Therefore they seem to wander around among the fixed stars. This should be understood by the learner, not just memorized.

     In a structure of accretion logic is irrelevant. In a structure of implication logic is essential. In a structure of implication ideas must fit together. In a structure of accretion they do not have to fit together. They are simply lumped together.

     In addition to add-on elements and implied elements there are also "partially implied elements", which make the structure of knowledge intermediate between accretion and implication. A partially implied element is a bit of new information that does not arise logically and inevitably from the old knowledge, but still must fit in with the old. For example, again in the field of astronomy, the fact that Mars takes 687 days to go around the sun is partially implied. A basic knowledge of celestial mechanics leads to the principle that the planets farther from the sun than the earth will take longer to go around the sun, but these principles, at least in elementary astronomy, will not give the actual figure of 687 days. This must simply be learned. Similarly many facts one learns in history are partially implied. For example after one learns a few facts about the Civil War then the fact that the war ended in 1865 is partially implied, but not totally implied. It is a fact that must be consistent with many other historical facts, but unless one is omniscient, it is not a fact that can be determined entirely by logical extension of the other facts.

     It may be evident from these examples that whether a new bit of information is an add-on element, an implied element, or a partially implied element, depends greatly on the previous knowledge that one has. I will return to this point shortly.

     Most mathematical subjects have structures primarily of implication. Each new bit of knowledge is meaningless to someone who does not have the prerequisite concepts and facts. Numbers, for example, mean nothing until one has the some concepts of quantity and counting. Fractions mean nothing without a rather detailed knowledge of whole numbers. Decimals and per cents derive logically as special types of fractions. Algebra rests on a knowledge of arithmetic. Calculus could not exist without algebra. One cannot walk into a college calculus class, having a math background that stopped in the ninth grade, and expect to understand much of what is going on.

     History, in contrast, has primarily a structure of accretion. Each new bit of knowledge can exist without reference to other bits of knowledge. It is quite possible to learn that Columbus discovered America in 1492 though one knows nothing else about history. The fact has some meaning whether or not one is familiar with the prehistory of the fertile crescent, or the history of modern Asia, or the Civil War, or the Crusades, or any of a countless number of other historical subjects. A person could walk into a college history class, having a history background that stopped in the ninth grade, and expect to follow much of what the professor is saying.

     Examples of a pure structure of accretion or a pure structure of implication are very hard to find. Such examples might exist, but only in very limited topics. An example of a pure subject of accretion might be learning to recognize by sight the letters of the alphabet. A young child might learn to recognize a "T" with complete ignorance of what an "A", an "M", or a "Q" look like. However once one goes beyond just recognizing letters and starts to put them together into words then a great deal of implication begins to appear.

     The only example of a subject with a structure of pure implication that I can offer would be topics in very advanced mathematics. High school geometry has primarily a structure of implication, but not entirely. There are plenty of bits of information that are simply added on when needed. This becomes very apparent when a geometry teacher decides to make the course more mathematically rigorous, to define every term only in terms previously defined, to prove everything by basic postulates or theorems. I have seen a few geometry textbooks that try to do this. They succeed logically to quite an extent, but at the price of becoming sterile and academic. I do not think they are suitable for the high school level. I once tried to make an algebra course more mathematically rigorous, but it didn't work out. After a couple of months I became mired in academic abstractions that were hard to understand and my students rebelled. I decided that logical rigor wasn't all the desirable after all. Insisting that everything derive logically from previous concepts and facts can be an elusive goal, and can be counterproductive. Almost any subject, as it is normally taught, has a structure that is made up partly of add-on elements and partly of implied elements, and therefore has a structure intermediate between a pure structure of accretion and a pure structure of implication.

     I think it is worthwhile to categorize subjects by the percentage of accretion and implication in their structures. I know of no way to do this methodically and accurately, but I do have some opinions. I think algebra, as traditionally taught, has a structure of about 70% implication. Geometry has a structure of about 85% implication. Biology has a structure of about 85% accretion. Chemistry and physics, on the high school level, have a structure of about 50% accretion. General science at the elementary school level has a structure of about 90% accretion. History at the elementary school level has a structure of about 95% accretion. At higher levels history may have a structure of only 70% or 80% accretion. Spelling, at any level, has a structure of about 80% accretion.

     I suspect my opinions of these percentages would clash violently with others' opinions. It is easy for those who are knowledgeable in a field to overemphasize the amount of implication in their subject. A historian, for example, would point out that history is not just a collection of isolated facts, but is a system of intricately entwined facts, patterns, principles, and relations. A scientist would be quick to point out that any field of science has certain basic laws and ideas that pervade the whole subject. A mathematician would point out that arithmetic is not a matter of learning the "how to" of a series of isolated types of problems, but is a coherent whole that derives logically from the basic postulates of number theory.

     All this may be true, but I don't think it changes anything. I am considering the structure of subjects as they are learned by the student, not as they are known by the expert. Remember that whether a bit of new information is an add-on element, an implied element, or a partially implied element, depends on the previous knowledge that the student has. The student does not have the same knowledge that the teacher has. For example, a historian might reasonably consider the fact that the Civil War began in 1861 as the logical outcome of many other facts - the election of Lincoln, the secession of South Carolina, the firing on Fort Sumter, and hundreds of other facts that back up these facts. The fifth grader, however, knows very little of all this. If he were expected to learn all these other facts then he would hardly get past 1861 before going on to the sixth grade. And more importantly, all these other facts would be learned as add-on facts, not implied facts. We have to start somewhere. First facts have to be add-on facts. A fifth grader has a very limited knowledge of history. So he must simply add on the fact that the Civil War began in 1861 to his previous structure of knowledge, and then add on lots of other facts that form the basic outline of American history. The result of all this is that the fifth grader's knowledge of history has primarily a structure of accretion.

     Similarly a tenth grade biology student must simply add on the fact that chlorophyll is green to his knowledge of biology, even though an advanced researcher in plant physiology and biochemistry might consider the greenness of chlorophyll as the logical outcome of many other biological and chemical facts. And similarly a fifth grader must simply add to his knowledge the rule that you "invert the divisor and multiply" when dividing fractions, though an advanced mathematician, or even a mathematically inclined high schooler, would know exactly why this rule works.

     Even if one agrees that there is more accretion in most subjects than the experts would estimate, it is still possible to argue that a little implication is more important than a lot of accretion. In one sense this may be true. Many subjects would not exist without a structure of implication underlying them. However this is not of prime concern to the teacher. In saying that a particular subject has a structure primarily of accretion or primarily of implication I am concerned with how the structure is built in the student's mind. By this perspective, for example, third grade arithmetic would have a structure of perhaps 80% accretion because learning the multiplication and division tables consists mainly of memorization. It is quite true that the fact that six times eight is forty-eight is a logical extension of the addition facts and the meaning of multiplication, but one is still expected to know the product of six and eight automatically. Therefore it is treated as an add-on fact. It is memorized.

In some subjects, or parts of subjects, the implication part of the structure is so easy to learn that it may be taken almost for granted. In other subjects, or parts of subjects, the accretion part of the structure is so easy that it may be taken almost for granted. For example, in spelling the implication consists of making use of phonics. If one wishes to learn to spell "diamond" then reason, not just memory, dictates that the first letter must be "d", and the last two letters must be "nd", and there is an "m" somewhere in the middle. However reason can do little more than that. The rest depends on memory. Therefore the knowledge of how to spell "diamond" is a partially implied element. However to a seventh grader the implied part, assuming he knows his phonics reasonably well, would be automatic and therefore would be no problem. Of course it must start with a "d". The problem would be the add-on part, the memorization that there is an "a" after the "i", and that the vowel of the second syllable is "o", and not "a", "u", or "e".

     In some parts of algebra the situation is almost reversed. The accretion part of the structure can almost be taken for granted, but the implication part is difficult to learn. For example, it is purely a matter of convention that on a graph the x-axis is horizontal and the y-axis is vertical. Therefore this bit of knowledge is an add-on element. Yet it is a bit of knowledge that is easy to learn. Many other associated bits of knowledge, which happen to be implied elements, are much harder to learn. Thus algebra has primarily a structure of implication.

     Having talked about structures of implication and structures of accretion it is reasonable to ask what difference it might make. How do these ideas affect methods of teaching and learning? I don't believe I can give a very complete answer to this, but I do have some ideas that I will mention at this point.

     One might first ask if one type of structure is easier to learn than the other. Mathematical and advanced science subjects have reputations for being hard. Is this due their structures being high in implication? Languages have a reputation for being hard. Is this due to their structure being high in accretion? Some subjects are hard for some students, but easy for others. Other subjects work the other way. Is there any way to explain this?

     My answers to these questions are based on the following general principle:


An implied element is hard to learn, but easy to remember. An add-on element is easy to learn, but hard to remember.


     There are many exceptions to this of course, but I believe it holds as a general pattern. For example, to a third grader the meaning of multiplication is an implied element, but the multiplication facts are primarily learned as add-on elements. The meaning of multiplication derives logically from the need to add the same number a number of times, and third graders are expected to understand this. But it takes time. A student can have considerable trouble understanding the idea of multiplication. But once he learns it he doesn't need constant drill on it. It's easy to remember, once it is understood. In contrast, he can quickly learn that three times four is twelve when the teacher tells him, or when he computes it once again on his fingers. But that doesn't mean he can easily remember it. He treats it as an add-on element, one among many add-on elements that he must memorize. He will need a great deal of drill before he can fluently remember all the multiplication combinations.

     An implied element is harder to learn than an add-on element because it must fit with the rest of the structure. The student must put forth the mental effort to carefully fit the new bit of information in place. He must bring to mind relevant parts of the previous structure of knowledge and apply some sort of logic. An add-on element is easy to learn because this mental effort does not have to be forthcoming. The new element does not have to fit in with the old. It is just slapped on as an addition. However an add-on element usually has to compete in memory with a multitude of other add-on elements. When one is learned, another is forgotten, and when the other is learned the first is forgotten. If we could learn only one or two addition combinations they would indeed be easy. But there are a hundred addition combinations in elementary arithmetic. To learn them all is not so easy. The only way to remember all the add-on elements is to put in a great amount of time and effort.

     Consider the example of learning names. If I meet a new person at work and her name is Jane I can easily remember it. I may go home and say "I met a new woman at work today. Her name is Jane." unaware that I had to expend any mental effort at all in learning or remembering. The knowledge of her name is an add-on element. It is easy to learn. It doesn't have to make sense. It just is. It may seem, in this example, that it is also easy to remember. It is easy in isolation, but in hard core learning (like we expect in school) facts don't come in isolation. They come in masses. So I will have to change the example to make it more meaningful. Instead of learning and remembering one name, consider learning a hundred names. That is a common learning task of new teachers at the beginning of the school year. I can attest from experience that it is a big job. It is an important job for a new teacher, so it gets done. But it is a hard job, much harder than simply learning "Jane".

     I will use the term "structure building" to refer to the mental process by which one fits implied elements into the structure of knowledge, and contrast it with the term "brain packing", which I will use to refer to the mental process by which one learns, and attempts to remember, add-on elements. Structure building involves the parts of the brain that have to do with reasoning. It involves memory, but only in a subservient role. Brain packing, in contrast, depends very greatly on power of memory. Reasoning plays less of a role.

     If learning a structure of implication takes different mental processes than learning a subject of accretion, as I have argued above, then it would follow that the teacher's job would be a bit different in each case. Differences are evident in all parts of teaching, from initial presentation of a topic, to practice, to diagnosis, to testing. I will attempt to outline some of these differences.

     Whether a subject has a structure of accretion or a structure of implication is of some concern in the initial presentation of a topic to a class. Coherence is more important when a subject has a structure of implication than when it has a structure of accretion. With a structure of accretion a lack of coherence may be an aggravation to the class, but not a disaster. The students may wonder why the teacher is rambling from one idea to the next, but they can still understand what he is saying and learn the material. With a structure of implication a rambling presentation can result in a complete failure to communicate. It doesn't make sense to the students. I think this is basically why math and science subjects have reputations of being hard. When such subjects are very well taught they are interesting. But when they are taught less well, they cause a great deal of frustration. Subjects that are more on the accretion end of the spectrum can be poorly taught and cause much less frustration.

     Practice in a subject with a structure of implication is a little different than practice in a subject with a structure of accretion. Consider two examples, algebra and spelling. "Drill" applies to both in some sense, but not to the same degree and not in the same way. One studies spelling primarily by brain packing. There is some implication in spelling - learning phonics and learning rules of spelling. But the bulk of work in spelling is simply memory drill. That is not the case in algebra to any great extent. Much of the "drill" in algebra consists of doing problems. When one solves problems one builds up a small dispensable structure along the pattern of the general structure. Then one builds a similar structure with slightly different elements. That is, one does another problem. This is structure building more than brain packing. We might call it "drill" but it is not the memory drill one uses to memorize the multiplication table. Another part of "drill" in algebra is to build the same structure over and over in one's mind. One does this when studying the derivation of the quadratic equation. This is like doing a problem, but the structure is not a small dispensable structure to be thrown away and repeated with slightly different elements. The structure is to be retained. This can degenerate into a task of memorization without understanding. One can, for example, memorize the derivation of the quadratic equation. This is brain packing. But that is not what one is supposed to do. One is supposed to understand the derivation. One is supposed to do structure building, using brain packing only as a auxiliary task when the structure building is not easy.

     Diagnosis of learning problems is somewhat different for the two types of structures. Structure building is susceptible to problems caused by defects in the previous structure of knowledge. Such a defect could be a structural gap, a fragile structure, or a hidden assumption, as I discussed in Chapter Three. Such defects can be totally invisible to the learner. They can be invisible to the teacher unless the teacher has some skill in diagnosing learning difficulties. Brain packing, in contrast, is not susceptible to defects in the previous structure of knowledge. For example if I erroneously think that the planet Neptune has twelve moons, that has no effect on my learning that Mars has two moons. Brain packing, however, is very susceptible to laziness. Brain packing does not take reasoning powers, but it does take effort and persistence.

     When teaching a subject with a structure of implication the teacher must diagnose learning difficulties by detective work. When teaching a subjects with a structure of accretion the teacher must diagnose learning difficulties by methodical survey. Thus when an algebra student flunks a test and asks for help the teacher would have him describe his thinking as he works outs problems in an attempt to find structural defects. When a biology student flunks a test and asks for help the teacher may do a bit of this type of detective work, but basically his job is different. It doesn't take ingenuity to find out what the student doesn't know. It just takes a little time and effort to methodically go over the test and point out the obvious. This methodical survey is something the student can do on his own. The teacher does it mainly to motivate the student. The algebra teacher, if he is successful, will come to the point when he can say. "Aha, here's your problem." The biology teacher, if he is successful, will come to the point when he can say, "Now you know what your job is. It's up to you to get it done."

     The average student can solve his own problems in a subject with a structure of accretion, but needs the help of a teacher to solve his problems in a subject with a structure of implication. Thus poor feedback and diagnosis in a subject with a structure of implication is a disaster, but poor feedback and diagnosis in a subject with a structure of accretion is much less of a disaster. With a structure of implication only intelligence - high intelligence - will compensate for poor teaching. With a structure of accretion effort and persistence will compensate for poor teaching. So again math and science subjects may have the reputation of being hard. They cannot be depended on to yield to effort and persistence.

     And finally, testing is somewhat different for the two types of structures. However I will discuss this more fully in later chapters so I will not attempt to discuss it here.

     I will now leave the idea of implication versus accretion and look at structures of knowledge from a different perspective. Instead of categorizing the structures of various subjects I will categorize the parts of any given structure in accordance to the essentialness of that part to the whole subject. Not all parts of a structure of knowledge are equally essential, and not all parts serve the same function. I will divide the parts of any given structure into:


framework structure

supporting structure

auxiliary structure (burden)



     In explaining these terms I will start from an example in my experience. In my first year of college chemistry I had to learn, among many other things, an equation that is used to determine the size of the molecular crystal unit of a substance. This equation makes use of reflection of X-rays of known wavelength and a bit of trigonometry. I was able to make sense out of this equation without too much trouble and I could do the problems at the end of the chapter that made use of it. Yet on the test a few weeks later I missed it, in spite of the fact that I had reviewed it just the night before. On the test the problem was stated, "Define Bragg's Law". I had no idea who Bragg was or what his law might be. It sounded slightly familiar, but that was all. Later, in going over the test, I discovered that Bragg was the person who figured out the equation for finding the size of the crystal unit of a substance, the equation that I had studied and knew perfectly well.

     So it is not enough to know the chemistry. Sometimes there's something extra added on.

     I use the term "auxiliary structure" to apply to bits of knowledge that must be learned, but that for one reason or another seem to be extraneous to the subject. Nothing essential is changed whether we refer to "Bragg's Law", or "the equation for crystal size determination" or "that funny equation on page 220". The name of the idea is not essential to the idea itself.

     There are many other examples to illustrate the idea of auxiliary structure. In geology it is not enough to understand the phenomena that affect the flow of ground water. We are expected to know that "Darcy's Law" is the name given to this idea. In geometry it is not enough to know the mathematics, we are also expected to know that Euclid wrote the first treatise on geometry in about 300 BC. In biology it is not enough to know about plants and animals. We are also expected to develop a bit of skill in drawing what we observe in the laboratory or under the microscope. In basic training in the army it was not enough to learn to shoot the M-1 rifle. We were also expected to learn that the muzzle velocity is 2800 feet per second (if I remember right).

     The purpose of auxiliary structure is often communication. If I never had to communicate my knowledge of chemistry then I wouldn't have much reason to learn the term "Bragg's law". In fact, language itself can be considered auxiliary structure to all subjects. English is not a part of algebra, but if we are to learn algebra in an English-speaking culture we must know it. Another common purpose to auxiliary structure is adherence to tradition. Learning the muzzle velocity of the M-1 rifle didn't make me more proficient in shooting it. The Army has traditions. They don't need rational reasons. So we learned the muzzle velocity of our rifles. Some auxiliary structure is dictated by simple necessity. Learning to draw what one sees in the microscope is, or at least was for many years, necessary to preserve and communicate what one observes. The distinguishing feature of all auxiliary structure, as I use the term, is that it is considered necessary to learn, yet is not logically a part of the main structure of knowledge, which I will call framework structure.

     Framework structure is the essential learning in a structure of knowledge. In the example of Bragg's Law the framework structure would include a knowledge of electromagnetic radiation, of how wavelength is measured, a knowledge of mathematics, and an idea of how molecules fit together into crystal units. This is the actual chemistry to be learned.

There is another part of structure that is similar to auxiliary structure in that it does not seem to be essential learning. I will call this "supporting structure". I use this term to refer to knowledge that is of some importance, but cannot be considered part of the framework structure. For example the author of a history test might intersperse throughout his book, apart from the actual text, short articles that are of interest to the subject at hand. Thus in a chapter on the founding of the first English colonies in America he might insert a short article on Puritan dress. In doing so he would be providing a bit of supporting structure. The essential features of supporting structure, as I am using the term, are that it is considered helpful in learning the framework structure, but is not considered necessary to learn.

     One might also use the term "supporting structure" to refer to the details of a topic, or the secondary ideas, as opposed to the main ideas. however I tend to think of details and secondary ideas as still a part of framework structure. Supporting structure, as I use the term, is more a matter of filler material, material that is dispensed with if time is short, material that will not appear on a test.

     Supporting structure can be dispensed with, but it is worthwhile because it helps support the framework structure. It either makes the framework structure make sense, or it makes it easier to remember. Auxiliary structure, on the other hand, does not support the framework structure. Rather it must be supported by the framework structure. It is a burden. In fact I often call it "burden". We learn it because, for one reason or another, we must. But it is a load to carry, a burden, not a means of support.

     Auxiliary structure often comes in isolated facts, as add-on elements, as ideas that can be expressed in one sentence, or perhaps as one number. Supporting structure is more often a coherent, though small, system of concepts and relations, a structure of knowledge in other words.

     A structure of knowledge can be compared to a building. The framework consists of the foundation beams, joists, etc., that hold everything in place and prevent the structure from falling down. Supporting structure corresponds to such things as the exterior sheathing and interior paneling that add rigidity to the structure, or the inside walls that add support to the ceiling. Auxiliary structure, or burden, corresponds to such things as insulation, windows, plumbing system, and furnishings. These things are beneficial, even essential, but unlike framework or supporting structure, do not add to the load bearing capacity of the structure. Supporting structure carries its own weight. Burden does not.

     Of course much of this is semantics. What one person calls burden another might call supporting structure. The boundary between the two phenomena is blurred. I think mnemonic devices lie on this boundary between the two. If I meet a Mr. Rose, for example, I might try to remember his name by noting that he has a rosy red nose. Is this bit of information auxiliary structure or supporting structure? It is auxiliary in the sense that it is a bit of isolated and extraneous information, but it is supporting structure in that I willingly learn it to support some other bit of information that I want to retain. The same can be said of the "doughnut rule" that I once learned in calculus. "Doughnut starts with a 'd' so you start with the denominator." This rule helped to remember that the rule that the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator divided by the denominator squared. I found the doughnut rule useful.

     Methods of teaching are not greatly influenced by whether one is teaching framework structure, supporting structure, or burden. However the amount of emphasis that should be given to these parts does vary. This is a matter of prioritization, as I discussed in Chapter Three. Obviously framework structure should be emphasized. Supporting structure may have to be de-emphasized. The students may need to told that a certain topic is of secondary importance and not to spend too much time on it. They may need to be told that a particular bit of information is an illustration, not a principle, or was included purely as a point of interest. Burden often needs to be emphasized simply because it is easy to overlook and not so easy to learn. That was the situation with my learning of Bragg's Law.

     A good example of giving special emphasis to burden comes from my experience in teaching biology. At first I told my students that they were expected to learn the correct spelling of the important biological terms that we meet. This didn't work too well so next I told them I would count off on tests for misspellings. This helped, but they were still careless with the spelling. Finally, about two months in to the school year, I gave them a list of about fifty biological terms and told them we'd have a spelling test on them in a week. I made this spelling test count one half as much as a regular chapter test. This method worked. It finally conveyed the message that they had to give more attention to spelling. They had intuitively realized, I believe, that spelling is burden, not framework structure, and had therefore not given it the attention needed to learn it. I had to add extra emphasis, in the form of a test, to get them to study it.

     There is yet another term I would like to define. That is "overhead". Overhead in learning is just like overhead in business, unavoidable expenses. Overhead in learning is the time and effort, even money, that must be expended in order to gain knowledge. Thus walking to the library is overhead. Whether I live six miles or six steps from the library is irrelevant to the learning that I will gain in the library. Other examples of overhead would be going to the store to buy typing paper, changing a flat tire so I can drive to class, arranging for a baby sitter in order to attend class, arranging a meeting with the teacher to discuss a difficulty or project, getting a library card. All of these things are done in order to contribute to learning, but if the learning could be obtained without doing them we would gladly do so.

     Overhead is normally not learning it itself, though it occasionally can be. When I look up a book in the card catalog of a library, or look up a telephone number, or rummage through a shelf of books trying to find the source of some quotation I want to use, I am engaging in learning. But so long as this learning is only a means to an end, and of no value in itself, then it is overhead.

     Overhead is not a part of a structure of knowledge. I mention it in this context only because some teachers seem unable to separate knowledge from overhead. It would certainly seem sensible that overhead should always be cut to the bone, but this is not always done. I think a prime example of this is the quickness with which many teachers will tell their students, "Look it up!" Unless the learning task at hand is dictionary use or research skills, then "looking it up" is overhead. Consider a ninth grader facing a list of ten unfamiliar vocabulary words. Suppose she is expected to look up each word in a dictionary, learn its meaning, and then use it in a sentence. She is facing a substantial learning task, and a substantial part of that task consists of thumbing through the dictionary. This can be time-consuming. I consider it overhead, pure and simple. It ought to be cut down. The teacher should find a text book in which the vocabulary exercises include the dictionary definitions. If such a text does not exist someone should write one. Learning is too important for time to be used unproductively.

     An even worse situation exists when a fifth grader is writing a composition and asks the teacher, "How do you spell 'discover'?" only to have the teacher say, "Look it up, Johnny!" Sometimes the question is improper, Johnny is more interested in bothering the teacher than doing his homework. But for this example let's assume the students are working quietly on their compositions at their desks and the teacher is circulating around the room giving individual help as needed, and Johnny simply wants the correct spelling. In this situation I think it is shamefully wasteful of Johnny's time to send him to the dictionary where he could easily spend ten minutes before getting the answer that the teacher could have given him immediately. The teacher should minimize this type of overhead just as a businessman should minimize his electric bill.

     This chapter has been very theoretical and abstract. My purpose has been mainly to provide a vocabulary of terms that have to do with structures of knowledge. In the next chapters I will try to use these ideas, and others, in dealing with the practical matters of conveying information.