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Arranging Structures


     I have spoken extensively of knowledge as having a structure. One thing implies another, ideas lead to other ideas. Little bits of knowledge fit together to form bigger bits, and the bigger bits fit together to form even bigger bits, until eventually vast mountains of knowledge are built. Such structures of knowledge may be visualized as physical structures with posts, beams, girders, supports and so on. By this perspective it may seem odd to speak of "arranging" structures. It would seem that the arrangement of a structure would be a matter of the nature of the structure itself, not a matter of how one might wish it to be. However as teachers we are as much concerned with transmitting knowledge as we are with having knowledge, and, as I will try to show, this results in our being very much concerned with the arrangement of structures of knowledge.

     A structure of knowledge cannot be transmitted wholesale from one mind to another by mental telepathy. Rather, knowledge is transmitted through the medium of language. Ideas must be translated into words by one mind and those words translated back into ideas by the other mind, and this must be done a little bit at a time. Translating ideas into words is a common part of our everyday life and is often done fluently and automatically, but not always. Sometimes it is done haltingly and inefficiently. And it is usually done only with small structures of knowledge. For example when I go into a hardware store to buy a "thingamajig" I may translate quite a bit of information into words before the clerk figures out what I need. I may not do this very efficiently or coherently, but that matters little if in a few minutes I am on my way home with my thingamajig. Efficiency matters little in this situation because the total amount of knowledge that is to be transmitted is small. Similarly one may pass on a great deal of neighborhood gossip, effortlessly and automatically translating information into words. But again only small, loosely connected structures of knowledge are involved, and coherence and efficiency are not of critical importance.

     When larger structures of knowledge are to be transmitted then efficiency and coherence become much more important. One may do a fine job in transmitting neighborhood gossip, but be totally at a loss when faced with preparing a ten-page report or a ten-minute speech. "But where do I begin?" one plaintively asks. Where indeed does one begin? And what does one say next? And after that?

     Words must necessarily come in a sequence, like the links of a chain. But a structure of knowledge is not like the links of a chain. Rather it is like a tree, or a mountain, or a net, or, as I have said before, a building. Some part of that tree or mountain or net or building must be translated into words first, some part next, and so on and on. The larger and more complex such a body of knowledge becomes, the more difficult it is to translate into a chain of words. Thus we have composition and speech courses in school. "Language arts" in one form or another is a very important part of the curriculum at almost any level of schooling. Yet so difficult is the task of communication that even after studying his native language for years in school the average person is hardly prepared even to sit down and write a letter to the editor of his local paper.

     It certainly can be argued, then, that a considerable amount of what a teacher does consists of what we learn in speech and composition courses, and therefore such courses should be emphasized in the training of teachers. However that is not the point to which I am heading. Speech and composition courses are concerned with translating small structures of knowledge into formal language. Teachers are concerned with translating larger structures of knowledge into informal language. A ten-minute speech or a ten-page report may seem like a large undertaking compared to the linguistic efforts we put forth in our everyday conversation, but still only relatively small structures of knowledge are involved. A subject such as arithmetic or chemistry or world history has a structure tremendously larger than the structure of knowledge that can be reduced to ten pages or ten minutes.

     Preparing a polished formal presentation, as I discussed in Chapter One, is at most only a small part of what most teachers actually do everyday in the classroom. It represents the performance perspective of teaching, which I have argued is inappropriate for most teaching situations. Most of the talking that teachers do everyday is a matter of extemporaneously responding to the words and actions of the students. It has more in common, on the surface at least , with the talking that goes on in the hardware store or over the back fence than with the talking that goes on in a speech class or at a lecture in an auditorium.

     Speech in everyday life is interactive. It is two way. It is flexible and open-ended. When I am asking for a thingamajig at the hardware store the clerk gives me numerous prompts to help her understand what I am asking for. When people gossip over the back fence they give each other numerous prompts that keep the conversation flowing. When preparing a speech or writing a paper we don't have these prompts. This makes it much harder than conversation. When a teacher talks to her class she should get feedback as she talks. This may not be identical to the prompting we get when gossiping over the back fence, but it is important. That is the whole point of the management perspective of teaching. Thus teaching is much more a matter of informal speaking than of formal writing.

     This does not mean, of course, that a teacher need not plan his lessons carefully. He must plan his lessons, but on a large scale. He must plan his topics, his chapters, his units, but he need not plan every word or sentence. Both a teacher and a speech writer are concerned with taking a mass of information and translating in into a chain. But a speech writer is concerned only with producing a chain of words, sentences, and paragraphs. A teacher is primarily concerned with producing a chain of topics, and with superimposing assignments, feedback, activities, and tests, over that chain. The individual words and sentences need not be planned, just as I need not plan every word and sentence when I am asking for a thingamajig at the hardware store.

     Teaching can be thought of as a process of tackling a mountain of information by systematically taking a bit at a time, translating that bit of information into words, delivering those words, checking to see if it is successfully transmitted, and then repeating the process with another bit of information, and so on and on until the subject is complete or the school year is over. By this perspective the analogy of a structure of knowledge to a building must be changed a bit. For the purposes of this chapter a structure of knowledge is analogous not to a fully erected building, but to a building that must be disassembled and loaded onto a freight train for transport to a new site. The new site is the learner's mind, and the learner's mind must reassemble the building. That is, it must build up a structure of knowledge. The analogy is not perfect. Brick and mortar are not easily disassembled. But if we picture a building with a steel frame that can be disassembled after interior paneling, exterior sheathing, and plumbing and electrical systems are removed, the analogy is fairly apt. The structure is reduced to the form of a chain, each link being a loaded railroad car, and the links of the chain can be formed and rearranged in many different ways. Some of these ways will be good ways. Other ways may present problems.

     Thus, by this perspective, the arrangement of parts of a structure of knowledge is definitely not predetermined by the nature of the subject itself. Rather the parts can be arranged for transport in many different ways. Some of these arrangements will prove to be successful. They will result in efficient and effective learning by the students, with resulting satisfaction of accomplishment and respect for learning. Other arrangements of the subject matter will prove to be less successful. Learning will be inefficient and frustrating. I will use the above analogy, of a building disassembled and loaded on a train for transport, to illustrate a number of ideas in this chapter, ideas of how best to arrange the parts of a structure of knowledge.

     Perhaps the first requirement in packaging a structure of knowledge is to decide what must be included and what must be excluded. Superficially, this may seem to be a question that does not have to be asked. If I am to teach algebra then I will include everything that is algebra. If I am to teach American history from the Civil War to the present than I will include everything that has happened in America from the Civil War to the present. I think simply asking the question makes it plain that it is not that simple. I cannot possibly include every thing that has happened in America since the Civil War in a course. Much will have to be left out. I will have to choose the most important parts and exclude all the res. Also I cannot possibly include any topic that can be considered algebra in an algebra course. There is no end to the subject, even on an elementary level. We can only decide what the most important topics are that can be made to fit into the time available.

     One important factor in choosing topics would be the formal expectations of one’s employer. If you teach in a big department and the department says to include the statehood of Hawaii in a particular history course then of course you will do so. Even if there is no formally stated requirement to cover a particular topic one may be strongly influenced by requirements that one informally perceives.

     Assuming one has a reasonable degree of freedom in selecting topics then one should give careful thought about what to include and what to exclude.

     Coherence is the first requirement in packaging a structure of knowledge. In Chapter Four I gave some examples of incoherence and tried to explain why coherence is very important. I said that a textbook would not be very coherent if the ideas needed to understand Chapter Two are not presented until Chapter Six. In the building-on- a-train analogy coherence is represented by having each part of the building arrive at the right time. If the building is loaded coherently the workers will find that each time they are ready for another piece of the building, that piece arrives on the next car. If the building is loaded on the train incoherently the workers will find that often when they are ready for another piece only frustration and confusion arrive on the next car. They find that they must unload and store pieces of the building that cannot be used until later, but they must also make do without pieces that are needed but have not yet arrived. The result of all this is a construction site cramped with stored parts and cluttered with incomplete projects. In teaching and learning this corresponds to a student's mind cramped with information that he doesn't know what to do with (other than remember it for a test), groping for bits of missing information, and confused about what to expect in the current shipment of information. The student may sit quietly and takes notes conscientiously, but that does not mean that all is well. His mind is a construction site just as much as any piece of real estate, and that construction site should be neat and organized, not cramped and cluttered.

     How can we make a course of study coherent? What guides can be used to know that the information we present at any one time will make sense to the learners? The answer to this lies ultimately in the teacher's common sense, his knowledge of the subject, and his ability to interpret and respond to the students' words and actions. Thus experience is a big factor. However experience is not the only factor, and certainly not every experienced teacher teaches coherently. There are at least two guides to coherence that are commonly used and are usually beneficial. One is the guidance of tradition, and the second is the guidance of a good, carefully chosen, text.

     I would certainly not argue that just because something is traditional then it is best or right. However I would argue that in a good many cases something that is best or right therefore becomes traditional. And I would argue that unless one has a good reason not to, it is only sensible to follow tradition. For example if I suddenly found myself in the situation of teaching grammar, which I haven't studied seriously since I was in the eighth grade, then I would follow tradition and start out by introducing nouns, then verbs, then adjectives, and so on. After I had taught grammar a while I might make many departures from what I perceive to be traditional approaches, but only for what I felt were good reasons that would not detract from coherence.

     A second guide to coherence, and I believe the most important one in most cases, is to choose a good textbook and stick to it. I once had a colleague who maintained that "Anyone who doesn't follow the book is a fool." I don't totally agree with that, of course, but I understand his point. As I mentioned in Chapter Four, the author of almost any text goes to great lengths to make his book coherent. For the most part they succeed. They succeed to a far greater extent than a beginning teacher can expect to by assembling a variety of "resources" and throwing them out to the class. It is disheartening to me to see a teacher, whether in high school or college, decide that the readings must follow his lectures and not the other way around. He therefore assigns a bit of one chapter and a bit of another, and then a bit of another text, and so on. In doing so he throws away a great deal of coherence. Occasionally - perhaps more than occasionally, for teachers tend to be a dedicated lot - a teacher will scramble the text yet still put in the effort needed to keep coherence. This surely seems a waste of time and effort though. It's like taking a building that is already carefully loaded on a train for transport, unloading and reloading the cars, scrambling up the cars and then carefully arranging them again (and providing new instructions to the assembly crew). The new arrangement may be just as good as the old, but it is unlikely to be much better, and it is very seldom enough better to justify the time and effort involved.

     However I certainly do not advocate slavish devotion to the letter of a text. Some textbooks, in spite their authors' efforts, are simply not very coherent. This could be because it is carelessly written, but more often the authors are more concerned with following some educational fad than in providing a coherent exposition of the subject matter. Several times in my experience I have faced the situation of making do with such an incoherent text. In one case the seventh and eighth grade math books were so incoherent as to be almost worthless to me. I ended up teaching almost totally from duplicated work sheets. This is not at all hard to do in arithmetic, but it is hardly a solution in a course such as history or science. When I started to teach biology I faced a similar situation. The textbook was incoherent because the authors attempted to be "modern". I'm not sure, as I look back on it, just what was modern about it. At any rate I managed to switch to another text after about a week.

     Another reason for not slavishly following a textbook has to do with the needs of the immediate class. A text may be very coherent and suitable for a typical class, but not for the class presently before you. So far I have been talking about coherence with the unstated assumption that information presented is information learned. That is obviously not always the case. As I discussed in Chapter Three, the "assumption of fluency" is a common, but serious, error. Fluency comes from practice, and thus practice is provided for in most textbooks. However the author of a text cannot know his audience intimately and cannot know just how much practice should be provided. Even if he tries to provide more exercises, problems and questions than he thinks any class will ever need, he can still easily misjudge. Incoherence can result just as easily by the forgetting of a key bit of information as by never having learned it. Thus a teacher may need to modify an otherwise good text at times by putting in extra assignments, or by reviewing previous material when needed.

     Coherence in a subject with a structure of implication is somewhat different than coherence in a subject with a structure of accretion. In a structure of implication coherence is often a matter of "all or nothing". The topic being studied either makes sense or it doesn't. The parts must fit together, and if parts are missing or faulty they cannot fit together. In a structure of accretion coherence is more a matter of having an orderly succession of topics and orderly transitions between the topics. However this does not mean that coherence is not important in a structure of accretion. If a student must flit from one topic to another with no rhyme or reason, even though he may have no trouble understanding the material, at least in an immediate level, his motivation and interest will be less than they would otherwise be. Even more importantly he may miss some important ideas. He may learn the facts of the subject, but the facts may not add up to a unified structure of knowledge in his mind.

     In a subject with a structure of implication it may be tempting to think of coherence only in terms of logic, but this can be a mistake. Logic leads to logical extremes, and logical extremes do not always add up to efficient learning. For example in arithmetic one might logically go from the addition of simple fractions to the addition of mixed numbers, to the addition of compound fractions. However experience indicates that addition of compound fractions is not an appropriate next step. It is quite enough for students to be introduced to the idea of compound fractions (by a compound fraction I mean a fraction in which the numerator and/or the denominator is itself a fraction or a mixed number). When I first began teaching math I had a tendency to go to such logical extremes. I thought I was just being sensible in such cases, or imaginative perhaps. My students thought I was leading them off on wild goose chases. I have seen some high school algebra and geometry texts that were so rigorously logical that I would consider them totally unsuitable for any high school students other than an occasional genius who likes to do puzzles. Coherence, even in subjects with structures almost entirely of implication, is at least partly subjective. One must often put topics together just because they "seem to go" together.

     Another concept that relates to choosing topics is what can be called the minimal and maximal teachable structures. No topic, it might be argued, can be totally exhausted. You can always find, if not additional concepts, at least additional applications, tangents, or related concepts that could be considered part of a given topic. The minimal teachable structure would be the set of related concepts that cannot be further reduced without some negative consequence. In math this means teaching a bit of some topic, but leaving out some important concept that ties the whole thing together. In any subject it can mean teaching so little of a topic that students will forget it entirely within a short time.

     A maximal teachable structure would mean a set of related concepts that cannot be augmented without somehow obtaining diminishing returns. When one exceeds the maximal teachable structure the students begin to feel that the teacher is wasting their time, pounding away at the subject with no benefit. The best motivation occurs when neither the bounds of the minimal nor maximal teachable structure are exceeded.

     I have used the building-on-a-train analogy to discuss coherence in a course of study. Next I will use the same analogy to discuss texture. In chapter Four I discussed texture and defined it as how closely or openly feedback is spaced. In this chapter I will define texture a little more broadly than I did in Chapter Four. I will define it as the size of chunks in which information is presented. I find I use the term "chunk theory" in my own mind to denote texture. The "theory" part of it is simply the idea that chunks of knowledge should be neither too big nor too little. If a chunk of information is too big, then it should be cut down. If a chunk of information is too little, then it should be built up. “Chunk theory” is a useful term, but I think the term “texture” is the better term.

     As I discussed in Chapter Four the texture should be neither too coarse nor too fine. I will speak of the "optimum texture" as a medium texture. I have no recipe for deciding just how much learning should be put into one package, but some general principles seem obvious. Younger students need a finer texture. Older students can take a coarser texture. Less capable students need a finer texture. More capable students can benefit from a coarser texture.

     Consider this illustration using the building-on-a-train analogy. The construction workers at the new site have a six ton crane available, but are faced with the arrival of a twenty ton piece of the building loaded on a flatcar. They might manage by some means or another to unload it, but how would they ever put it in place? Or, at the other extreme, suppose the workers have a six ton crane, but the building is packaged in pieces of only a few hundred pounds each. The six ton crane could to the job, but it would be exceedingly cumbersome and inefficient. If only a six ton crane is available at the new construction site then surely this fact should be taken into account by the packagers of the building. The big pieces should be broken down to chunks no more than six tons, and all the little pieces should be assembled together in packages of at least two or three tons.

     Translated into teaching and learning this analogy means that information should be presented to the learner in appropriately sized packages. What would constitute an "appropriate sized package" of information? I will discuss several factors that influence, either objectively or subjectively, the size of a chunk of learning.

     The length of the standard lesson time is one of these factors. To teach a mass of information in a series of five minute lessons is to have a very close texture. To teach the same mass of information in a series of four-hour lessons is to have quite a different texture, a very coarse texture. For some reason the one-hour lesson has come to be traditional. It certainly seems to be a workable tradition, but it is not the only possible way in which things can be done. When I was in college preparing to teach I did my practice teaching in the summer, and instead of meeting an hour a day for 36 weeks the class met four hours a day for nine weeks. At the beginning of the summer I wondered if this arrangement would be hard on the students. I had visions of the entire course degenerating because four hours of geometry in one sitting is just too much for high school students. There were some troubles in the course, to be sure, but so far as I could tell these troubles had nothing to do with the arrangement of time. The students seemed just as capable of thinking mathematically at 12:00 noon as they were at 8:00 a.m. However I don't think this means that it doesn't matter how long lessons are. I expect lessons are traditionally one hour long for good reason, and basically the reason is to have a texture that is neither too coarse nor too fine. For that matter our four-hour sessions of geometry were broken up into four fifty-minute sessions with ten minute breaks in between. Therefore I hypothesize that for high school and college students the traditional one-hour lesson represents some approximation of optimum texture.

     Another factor that influences texture is the intensity of effort required from the learner. When high intensity of effort is required the chunk of learning is, or at least feels, bigger. Therefore if a particular topic requires high intensity of effort, it makes sense to make that topic shorter, in order to keep a medium texture. And conversely, if low intensity of effort is required then longer time, or a larger mass of information, may be assembled in one chunk to maintain a medium texture. For example in teaching spelling the teacher may keep the students struggling on hard words for a short length of time, or more routinely memorizing easier words for a longer length of time. The total amount of information to be learned may be about the same in each case. The learning experience in each case is not necessarily the same. The two learning experiences may be different, but from the standpoint of texture both may be satisfactory. Similarly an arithmetic teacher may keep his students struggling for a short length of time on a few hard problems or he may have them spend more time on easier problems. Again the total amount of brainwork may be approximately the same. The learning experiences are not the same, but they may both have approximately the same texture.

     Intensity of effort of a given learning task should be considered quite apart from considerations of texture. If the intensity of effort required is too high students will eventually rebel in one way or another, for high intensity can be perceived as frustrating to the students, even painful, just as too much stimulation can be perceived as painful. A low intensity of effort, on the other hand, is easier on the students. They are not pushed to perform. However this does not mean that low intensity is better than high intensity. If the intensity of effort required is too low motivation may decrease. Athletes, as an analogy, like to run, not just walk. Minds also like to run. Requiring only a low intensity of effort can be like too low a level of stimulation. It can be boring.

     I have identified four factors that determine texture, length of time between opportunities for feedback, amount of information in each lesson, length of time of lessons, and intensity of effort required. There may also be other factors that I have not identified. It might be possible to come up with a mathematical formula that would take into account these factors and give a numerical rating of texture on a scale of coarse to fine. However I think it is also possible, and probably just as productive, to judge texture totally subjectively. Good teachers do this automatically whenever they give careful thought to the next day's lesson. They may not verbalize their ideas on texture, but they adjust the texture to fit their students, and that is the important thing.

     When the teacher has the parts of a structure of knowledge arranged in a good order and packaged in the right sized chunks then he is well on the way to having a good course of study. However there is yet another concern that I will discuss here - orientation. The learner must know where he is in the subject matter at any given time, and hopefully where he has been and where he is going. In the building-on-a-train analogy consider the following situation. A boxcar containing parts of the heating system arrives at the new building site. These parts are well packaged. They come at the right time and are in the right sized chunks. But, unfortunately, the workers don't know whether they are dealing with the heating system for the east wing of the building, or the west wing. If they start to assemble it on the wrong wing they may do a great deal of work before they find that it's not going to fit. If they are on the ball they will try to figure out which wing it goes in before they start working on it, but this also involves time and effort. The best solution is for the packagers of the building to send along instructions. Then the workers at the new construction site will have no problem of orientation.

     In teaching and learning the above scenario corresponds to something like the following example: A history teacher is lecturing about the economic system of a country, explaining how it was affected by and contributed to the War of 1812. The students listen carefully and take notes, but they have a problem. They don't know whether the teacher is talking about the economic system of the United States or of England. They can continue to take in the information, and it will continue to make at least some sense to them, but until they get oriented, by finding out which country the teacher is talking about, they cannot really assimilate the information they are receiving.

     A lack of orientation can be considered a structural gap. Some essential bit of information is missing which prevents other information from completely making sense. However a lack of orientation is a special type of structural gap. It arises only in the process of communication. It arises because the structure of knowledge must be broken down and then reassembled. To return to the building-on-a-train analogy, no problem of orientation could arise if the building were not to be moved. So long as the heating system of the east wing of the building sits undisturbed in the east wing there is no confusion. But when the heating system is disassembled and loaded for shipment then there is the possibility of a great deal of confusion. Correspondingly when a structure of knowledge sits undisturbed in the teacher's mind there is no problem. But when that structure of knowledge is broken into parts, translated into words, and thrown out to the students, there is the possibility of a great deal of confusion.

     I will give an example of a problem of orientation that caused me confusion for many years. Like most actual problems of orientation it is not so transparent as the imaginary example of the history teacher I gave above. In high school I took physics, and I think I learned it reasonably well. However I was always confused with velocity and acceleration problems. The formulas and equations would make some sense to me, and I could use them to work problems,, but I always felt that something was unclear. Some time ago, under what prompting I cannot remember, I got out a physics book and proceeded to get it straightened out. My problem, as it turned out, had to do with the concept of velocity. There is not one, but at least four, concepts of velocity. The first concept is constant velocity, where no acceleration is involved. The other three concepts of velocity involve acceleration. There is initial velocity, final velocity, and average velocity. Any given formula with the term "v" would refer to one of these concepts, but only one.

     On the surface this might seem to be a failure to discriminate rather than a problem of orientation. This might be part of the problem. However I think orientation is also involved, for I did discriminate among the four related concepts. Had I not, then the formulas, equations, and exercises would have made no sense to me whatsoever. I think what happened was that the teacher explained the concepts to us quite carefully, but was not always careful to tell us everyday just which of the four concepts we were dealing with at the moment. Thus any particular formula, equation, problem, or example would make sense, but after we were through with the chapter the whole mass of information just didn't quite "click". It didn't click because I never knew quite where to put some bits of information in relation to the other bits of information. In other words I had a problem of orientation.

     To once again return to the building-on-a-train analogy, consider a workman who works on the heating systems of both the east wing and the west wing of the building. While working he makes mental notes of details that cannot be completed at the moment for one reason or another. Weeks later those mental notes cause him a great deal of confusion. Weeks later, for example, a needed replacement part comes in and he finds it's not needed after all. Or he spends half a day making routine checks and then discovers in his paperwork that he's already made and documented those checks. The explanation, of course is that he didn't pay enough attention to which wing of the building was involved. When he was working he obviously was oriented. He knew whether he was working on the east wing or the west wing. However he failed to give that distinction the emphasis that was needed. His orientation was adequate for the moment, even for the day or the week, but it was not adequate for the longer term. Similarly my orientation of the several concepts of velocity was adequate for the moment, but not for the long term.

     A similar problem of orientation occurred the first year I taught algebra. One of my better students was miffed after a test because she thought she should not have missed some problems. She was a conscientious student and had carefully learned everything she thought she was supposed to know. Unfortunately she applied some rules for working with polynomials to equations. Polynomials and equations don't follow the same rules. She knew that. But polynomials and equations are superficially similar and she mixed them up. After the test I vaguely realized that had I been a bit more on the ball a week before I would have pointed out something to the effect of, "Equations were in the last chapter. Polynomials are in this chapter. So don't mix them up." My miffed student probably realized about the same thing.

     I remember this example because of the feeling tone involved. This particular student was not one to complain without good reason. I wonder how many other problems were missed on tests that year that could have been prevented by providing a little more orientation every now and then?

     I think there is something to the idea that a teacher does a better job when the subject doesn't come too easily to him, that a person who has trouble understanding the subject himself is in a better position to help others who have trouble understanding it. I think part of the reason for this is that to one for whom the subject is easy, the orientation is always obvious. To my physics teacher it was always obvious whether we were concerned with average velocity or final velocity. To me it was always obvious whether we were concerned with polynomials or equations. And to the hypothetical history teacher from my first example on orientation it was quite obvious whether he was talking about the economy of England or the United States. A teacher for whom a subject does not come so easily, in contrast, is better prepared to give orientation cues when needed, for orientation is not always so obvious to him.

     The way to prevent problems of orientation is to label each chunk of information as it is given, to send along instructions as to how this bit of information fits in with previously learned information, where it is located in the emerging structure of knowledge. Of course it is not always clear just when and how to provide these labels and instructions. Here again experience is a great help. However there are a few general rules about giving orientation cues that I think are worth mentioning here.

     First, each new topic should be oriented as it is introduced. Whenever the subject changes the students should be told just how and where the new topic is to be connected to the old. Much of this type of orientation is done automatically. It is built into our language as we speak, at least if we are capable of speaking coherently. When one says, "Let's first look at . . ." or, "On the other hand. . ." or "Remember we already established that . . ., he is giving orientation cues. Unfortunately teachers often neglect to give orientation cues for larger milestones in the subject. One may automatically give orientation cues for each new paragraph, but fail to give any orientation cues at all for a new chapter or unit. Or he may give inadequate cues, cues that fail to impart the importance of the transition. In a book the super-large type in which a chapter title is printed is an orientation cue, but large type cannot be directly translated into spoken language. Therefore extra effort must be put forth in orienting such large divisions of subject matter. One must become conscious of the spoken orientation cues one gives to students and make sure they are adequate.

     Second, I think it is sensible to insert orientation cues whenever circumstances cause a break in the continuity of the course. For example, a student's question during the middle of a class period might lead the teacher far astray from what he had intended to talk about. When this happens he should announce very clearly when he is returning to the main track. As another example, whenever the students are away from the subject for a time, such as for a school vacation, or even a weekend, it may be necessary to remind them just where they are in the subject matter when they return.

     A third rule is to provide a surplus of orientation cues. Provide enough orientation, and then provide more. Give cues when plainly needed, and give cues when not so plainly needed. In algebra the name of the chapter may be "polynomials," which certainly seems like adequate notice that we are working with polynomials, not equations, but it still does not hurt to remind student of the difference. In biology the name of the chapter may be "insects", and that may seem like adequate notice that the subject is not vertebrates, but it still doesn't hurt to remind the students now and then that this chapter has nothing to do with vertebrates. In history the topic for the day's lecture may be plainly stated in the course syllabus, "England's economy as a cause of the War of 1812", but it is still sensible to tell the students that you are going to be talking about England's economy, not the United States'.

     Orientation problems, like so many learning problems, can be deeply hidden. The student who is disoriented in some way may or may not be able to state his problem succinctly and ask for help. He may very well have only a vague feeling that something is not quite right with the structure of knowledge that he is building. However if a teacher is conscientious about keeping his students oriented as much as possible, then students will get in the habit of expecting to be oriented. Then when they are disoriented they are in a better position to ask for help. To return to the building-on-a-train analogy again, if workers at the new construction site are in the habit of having the arriving parts well labeled then they will be quick to complain when a load arrives without such labeling.

     The thesis of this chapter is simply that the process of communication of ideas requires that the parts of a structure of knowledge be arranged in such a way as to facilitate that transmission. In the next chapter I will discuss some general strategies for doing this.