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A coherent structure of knowledge is an organized system that will stick together. An incoherent structure of knowledge is a mass of loosely connected facts, concepts, skills, habits, and perspectives, that do not stick together. There are many ways that a subject can be presented and be coherent. There are also many - very many - ways that a subject can be presented and not be coherent. It is the teacher's job to make sure that the subject is presented coherently.
"Presentation", in this context, of course, has a broad meaning. I am speaking of much more than a single day's lecture. I am using the management perspective of teaching, not the performance perspective. "Presentation" in this perspective means the order of topics to be presented, the assignments given, the provision for feedback and testing, the texture of the course, and all the other topics I have discussed in previous chapters.
Of course it does not follow that just because a subject is presented coherently that every student will build a coherent structure of knowledge in his or her mind. Nor does it follow that an incoherent presentation will necessarily produce an incoherent structure of knowledge in every student's mind. Constructing a solid, coherent structure of knowledge depends on a great deal of diligence, effort, and ability on the part of the student. If this is lacking an incoherent structure of knowledge may result from even the best presentation. We don't expect all students to make A's in every class, after all. And we don't demand complete mastery of one topic or course as the prerequisite for going on the next topic or course. And, since students apply their efforts and intelligence to what the teacher presents, it is common for the more able students to make a great deal of sense out of even an incoherent presentation. Students are certainly not limited to learning only what the teacher knows and can convey. If that were the case there could be no human progress. So the connection between process and product - between the teacher's good or poor teaching and the student's good or poor learning - is not perfect. Nevertheless it is certainly reasonable to assume that a better teaching job will, on the whole, produce better learning. A coherent presentation, therefore, is very important to good teaching.
In this chapter I will discuss a number of methods, approaches, and perspectives of teaching with a view to their coherence or lack of coherence.
A method of gaining coherence attempted by many teachers is to have some "unifying theme" that will bring together all the disparate parts of the structure of knowledge. Often this comes perfectly naturally in a subject. In geometry, for example, formal logic serves very successfully as a unifying theme. In fact, without formal logic, it would not be geometry as we know it. In history, at least in the beginning courses, the progression of time is a logical and intuitively satisfying unifying theme.
However in many situations a unifying theme is identified by the teacher at the beginning of the course, but it does not work out so well. Rather than arising naturally out of the subject matter it seems forced. It requires extra effort on the part of the teacher and learners to take it into account and it doesn't really seem to unify. For example, in a college course in organic chemistry that I took some years ago the professor told us that we should keep two basic questions in mind for whatever topic we were studying: "What is it made of?" and, "What is it good for?" These two questions were to be a unifying theme for the course. When we studied the simplest hydrocarbons we would be concerned with these two questions. When we studied more complex hydrocarbons we would again ask these same two questions. When we studied alcohols, carboxylic acids, aromatic hydrocarbons, amines, and amino acids we would again ask these two basic questions. Unfortunately things didn't quite work out the way we were led to expect. These questions were of some importance, but so were many other question and ideas. Stereochemistry, for example, which has to do with the shape of molecules, was a very important part of the course, but did not relate closely to the two questions that were supposed to be the unifying theme. Similarly the analysis of chemical reactions into the categories of addition, substitution, and elimination reactions were very important in the course, but again had very little do with the unifying theme.
In an American history course I took in college the unifying theme was supposed to be the question, "Why?". For each fact or idea we learned we were supposed to ask "Why?" and this was supposed to lead to all kinds of fascinating and productive endeavors. Of course this didn't work out exactly as planned. "Why?" is a very important question in practically any field of knowledge, but it is hardly all that a history student can concern himself with. He must also concern himself with learning a large mass of "What?", and "When?" and "Who?" and a lot of ideas and concepts that will not fit neatly into any of these categories. The question "Why?" will serve to bring together some of these facts, concepts, and ideas, but not all.
In both of these courses the "unifying themes" did little harm, except to occasionally divert attention from more important matters, but they did very little good. Is this because a course does not need a unifying theme, or that unifying themes do not exist, or that the professors had chosen the wrong theme in each case? My view is that unifying themes do exist, that they are important, but that usually the unifying theme of the course is identical with the name of the course. Thus the unifying theme of a course in organic chemistry is nothing other than "organic chemistry", and the unifying theme in a course in American history is nothing other than "American history". This may seem obvious, redundant, or to be begging the question, but I think it is important. The only theme that runs through every topic in organic chemistry is "organic chemistry", and the only theme that runs through every topic in American history is "American history."
These "unifying themes" identified by the professors were only subthemes, and not necessarily the most important subthemes. In organic chemistry other important subthemes that might have been identified could be: "the effect of molecular shape on chemical properties", "the incredible carbon atom", "big molecules from little ones", or "How does molecular bonding translate into chemical properties?". In American history other possible subthemes that might have been mentioned are: "Ideologies as historical forces", "Economic determinism", Environmental determinism", "Individual will versus collective drift", or "the English influence on American history and culture". All of these subthemes are about as important as the "unifying themes" stressed by the professors. But none of them will serve to bring together every topic covered in the course.
It is useful to identify subthemes as they arise in the course. This is automatically done by good teachers, and by good textbooks. But it is not useful to take a subtheme and elevate it to a main theme. It is misleading, and it does not add coherence to a course. Coherence, as I mentioned in the last chapter, comes from carefully choosing topics and carefully presenting them, from packaging information in the right size chunks, from being sure that facts needed for chapter two are not first presented in chapter six, from planning an orderly succession of topics and making sure the students know just where in the emerging structure they are at any given time, from not assuming fluency where fluency cannot be reasonably expected, from getting enough feedback from students to discover and correct mistakes, and so on. In other words coherence comes automatically from good teaching.
In both of these examples the professors identified what they perceived as a main theme that already existed. They didn't choose an extraneous theme and impose it on the subject. But in other cases an extraneous theme may intentionally be chosen. I came across one biology textbook, apparently made for the junior high level, which seemed to have a "field trip" pattern of organization. For example a chapter might start out on the seashore, describing the plant and animal life that one would encounter there. After a few pages it would move into the forest beside the shore and spend a few more pages describing that environment. Then it would take off on some convenient tangent and discuss the chemistry of life, then perhaps taxonomy, then go into some depth on some particular species, then would describe some particular line of research, and so on. Each new topic would have some connection to the preceding topic, but often this connection would be very tenuous or superficial, just as on a nature walk one comes across a wide variety of phenomena that just happen to be in approximately the same place at the same time.
This type of unifying theme, if one may call it that, has the "advantage" of seeming natural. It presents information in about the same way one encounters phenomena in "real life" or non-school life. However I do not think this is at all appropriate as a way of organizing biological information for junior high school students. The purpose of education should be to bring order out of chaos, to explain phenomena, not simply to reflect them, or to catalog them. Thus different topics must be organized so that they are connected by relations that will endure, not by relations that are superficial. An example would be studying sharks and dolphins together because they have similar shapes and both live in the ocean. This might be suitable for second or third graders, but hardly for any level of education above that. Dolphins are mammals and sharks are fish. On the college level dolphins would be included in a course in mammalogy, along with gorillas and tigers. Sharks would be included in a course on fish, along with other fish. On the high school level dolphins and sharks would be in separate chapters. The connection between dolphins and sharks is very superficial compared to the dissimilarities.
With this "field trip" pattern of organization relations are transient or superficial. Such an approach may be quite appropriate for very young children, because children must be exposed to a wide variety of phenomena before they can begin to systematically put some order into their world. But surely at the junior high level it is not an appropriate means of organizing information. The "field trip" pattern of organization is not a "unifying theme". It is not even a subtheme. It is a totally extraneous theme.
This same book made use of another extraneous theme. A number of topic headings were "What's new in . . . ?" or "What have scientists recently discovered about . . . .?" Apparently the assumption is that young people are automatically interested in anything that is new. I do not think this is true. Even if it were true it makes a poor theme around which to organize a subject matter. And even if it were true it would be overshadowed by a larger truth - young people are interested in what they are successful at, and good teaching will make them successful at building a coherent structure of knowledge. An extraneous theme will not lead to a coherent structure of knowledge, and thus, in the long run, will not lead to either satisfaction or interest.
Next I will discuss what may be called the "activity method" of teaching, and relate it to the concept of "integration" or "differentiation" of subject matter. In this method an activity is chosen, and this activity is supposed to be the means of a great deal of learning. For example a class might choose a project such as baking cookies. In following the recipe, so it is claimed, they would be getting practice in reading and would be learning to follow directions. In measuring out the ingredients they would be learning arithmetic. In working together they would be learning social skills. In discussing their project they would touch on history, sociology, etc, etc, etc. The net result of all this is supposed to be a "learning experience". Each bit of information, each skill, concept, or idea, is held to the other bits of information by its relation to the activity. Many subjects are integrated into this activity.
The activity method makes an ideal of what I call "super-integration". This is the idea that many subjects can be combined together. "Integration" of subject matter is to be contrasted with "differentiation" When disparate topics are brought together they are integrated. When they are separated they are differentiated. As a general rule topics are integrated to a great extent in the lower levels of education, and are differentiated to a great extent in the upper levels of education. In kindergarten, for example, we have "social studies". This is a very broad subject. After a few years we have history differentiated from geography. In high school we may differentiate sociology, economics, and political science from history and geography. In college each of these subjects may be differentiated into dozens of courses. At each successively higher level of education we differentiate the subjects more and more. As another example, in elementary school we simply have "science", not biology, chemistry, and physics. In college we differentiate the broad category of science into many subjects. Chemistry may be differentiated into dozens of different courses with different titles. Biology is differentiated into botany, zoology, and microbiology, and each of these is differentiated into dozens of courses.
Much concern about integration versus differentiation of subjects is mundane. Should history and geography be integrated together in the fourth grade into "social studies"? Or should they be differentiated into two different subjects? Should spelling be integrated with grammar in the eighth grade, or should they be differentiated? Should ninth graders take a general science course, integrating a number of topics as is done in the lower grades, or should they take a more specific subject such as earth science, space science, biology, or chemistry? Should college freshmen take "American Civilization" integrating history and political science, or should they just take "American history"? These are issues to be decided by subjective judgments. They might be hotly argued issues at times, but are still within the general rule that education proceeds from integrating everything at the lower levels to differentiating everything at the upper levels.
My criticism of the activity method is that it attempts to integrate far too much and therefore it simply doesn't work. Not much knowledge is actually gained, because structures of knowledge cannot be built up without considerable differentiation of topics. Differentiation of topics is needed because concentration is needed.
Consider the example of learning to add fractions. By the activity method, using the example of baking cookies, presumably the students will have occasion to add fractions in following the recipe. Perhaps the recipe gives quantities for a regular batch of cookies, and quantities for a extra large batch of cookies, and they teacher tells the class that they need to combine the two recipes to make enough for the whole class. There is the "teachable moment" concept that is supposed to apply here. Since the students are in a situation where their cookie baking is dependent upon adding fractions, then, by this "teachable moment" concept, they will be motivated to learn to add fractions. Obviously there is something to this idea, but I think much less than educators would like. The idea that in this situation the students will really want to learn to add fractions is questionable, and also the idea that they will actually do so is questionable.
Just when the teacher is trying to turn their attention to fractions they may be eager to dump the ingredients in a bowl and start mixing. The teacher may be trying to tell them that they can't mix the ingredients until they know how much, and to know how much they must deal with fractions, but that does not mean that they will respond as we would like. They should want to learn fractions, of course, but "should" does not always translate into "will". If the fractions are seen as something in the way of their project they might actually be motivated to avoid the learning. This is especially true if they feel that the measurement is not crucial, that they could bake the cookies just as well by estimating quantities. An avoidance of the learning could be strengthened by the group nature of the project. If five students are involved in the activity, then as many as four of them might be just biding their time while one of them, a de facto leader, works with the fractions enough so that the teacher will let them go ahead with the baking. "Does that make sense?" the teacher asks the group when an answer is arrived at, and all eagerly nod their head in the affirmative. So they combine the ingredients and start mixing, but a perceptive teacher should wonder just how much was really learned.
Adding fractions is normally the result of a considerable amount of hard-core learning. Students do assignments, listen to explanations, do more assignments, listen to corrections, do more assignments, take tests, do review assignments, and so on. Can doing a few incidental problems while baking cookies compare with this?
It might be argued that the few incidental problems connected with today's activity of baking cookies must be added to a few more incidental problems in tomorrow's activity, and so on day after day. However I would argue that the total result can not add up to much, for a number of reasons. First of all the activities are too demanding of time. The "activity" is generally very time consuming, leaving little time left for learning. Also, the chopped up nature of the incidental problems militate against their cumulative effect. A few problems in adding fractions while baking cookies added to a few problems in adding fractions while doing another activity, results in a very incoherent presentation of the topic of adding fractions. We differentiate subjects so we can concentrate on them. Learning to add fractions takes concentrated effort. This concentration is lost when a few problems here and a few problems there are embedded in a mass of other concerns. And, of course, the goals of completing the activities militate against a cumulative effect that is supposed to result in learning. Can one concentrate on learning to add fractions when the concerns about getting the cookies done before the bell rings seem more important?
Hard core learning takes concentration of effort. Otherwise learning either doesn't take place or it is quickly lost and does not provide a foundation for future learning. I will have much more to say on this in a chapter on recitation strategies. The point here is that the "activity method", at least as I have heard it described, cannot provide this concentration of effort.
Consider super-integration in light of the building-on-a-train analogy. Integrating every subject into every hour of lesson is like integrating a bit of every part of the building into each railroad car. That would not be very efficient. The new construction site would be crowded with parts stored for future assembly. Perhaps the analogy is not very good. There may be many ways in which packaging the parts of a building on a train would be considerably different than packaging a structure of knowledge for communication to students. But in the central idea that a structure of knowledge must be built up in the student's mind, and therefore the elements of that structure must be provided in a coherent manner, I think the analogy holds.
The activity method can be interpreted as having "relevance" as a unifying theme. It can be argued that whatever is relevant to the child's world will be of great interest to him or her, and therefore highly motivating. I think there is a level of education in which this theme of immediate relevance is quite appropriate. I don't know of any other way to teach four year olds. As I have almost no experience teaching below the junior high level I do not know just where this theme of relevance is appropriate and where it should give way to more enduring themes. However I believe that by the fourth or fifth grade most subjects should be taught by following the theme of the subject, not the theme of relevance to the child's world. Relevance is not unimportant of course. It can be used to great advantage by a skilled teacher. It is used to great advantage by even unskilled teachers. However is still an extraneous theme. It detracts from coherence if carried too far. To return to the building-on-a-train analogy, it is like letting a five year old's whims direct the packaging of the building. It is not realistic to expect good results.
The activity method is similar to the "project method", and shares many of its disadvantages. The project method is of sufficient importance that I will devote a separate chapter to it.
The last "method" of teaching I would like to discuss from the viewpoint of coherence is the "spiral method". The idea of the spiral approach is that the learner goes through a series of topics, then goes through the same series of topics at a deeper level, and then again at still a deeper level, and so on. In elementary school the spiral approach occurs quite naturally in many subjects with each year of instruction making up successive loops in the spiral. In arithmetic, for example, the third grade course of instruction may start out with counting, then go on to simple addition, then subtraction, and then touch on multiplication. The fourth grade would again start with counting and addition, repeat the work on subtraction, and then go a little deeper into multiplication and get into division. The fifth grade would repeat all this but go deeper into division and start into fractions. Even at the eighth or ninth grade level the course may start out with the same simple beginning and repeat virtually all that has come before.
The alternative to the spiral method, which we might call the "straight-through method", is to never repeat a topic once it has been thoroughly covered. There are few subjects in elementary school where this would be an appropriate approach, but it is commonly used in higher levels of education. For example there would be little spiraling in a high school course in economics. Once the class completes a chapter on banking they would not expect to return to it again, except perhaps briefly for reference or review. Once they completed a chapter on the law of supply and demand they would not expect to cover it again. And once the entire course is completed the students will not expect to repeat the course next year.
The main advantage of the spiral method is that it provides for review. This is especially important in lower levels of education because young students take very little responsibility for their own education. A great deal of review is a way of insuring that ideas and skills are retained. Another advantage, if it may be called that, of the spiral method is that the natural way in which a child learns about the world from the earliest age contains a great deal of spiraling. He learns a little about one thing, then a little about another, then a little about other topics, and then comes back to learn a little more about the first topic. A third advantage, again if it may be called that, of the spiral method is that most subjects naturally spiral of their own accord to at least some extent. Advanced topics tend to return to simple topics, and simple topics lead to advanced topics. Therefore it is impossible not to spiral to some degree.
There is a natural progression from a great deal of spiraling at the lower levels of education to very little spiraling at the higher levels of education. This is similar to the other natural progressions that I have discussed.
An important question one must consider about the spiral method is how "deep" each cut in the subject matter should be. There are two reasons why each cut should be not too shallow. First, if one barely touches a topic before going on, that topic is easily forgotten, meaning that the loop was wasted. This is what I was saying about the "activity method" and its failure to provide concentration of effort. Second, a certain amount of effort must be expended in getting into a topic each time one returns to it. The teacher must reintroduce the topic. He must orient the students to where they are in the structure of knowledge. If only a small bit of new knowledge is presented in each "cut" then a large percentage of time is spent in simply getting oriented. This is an inefficient use of time and effort.
An example of not cutting deep enough on each loop comes from my own experience in elementary school. I well remember wondering why we seemed to learn the very same grammar year after year. We learned about nouns and verbs in the sixth grade, and that seemed like it was just a repetition of what we had learned already in the fifth grade. So why, I wondered, where we going through the same old stuff again in the seventh grade? At that time, of course, I didn't think in terms of the spiral method, but looking back on it I realize that my complaint was simply that we were not cutting very deeply with each successive loop. In fact I wonder if some years we didn't fail to cut as deeply as in the previous year. If one cuts no deeper on each loop one is circling, not spiraling, and that can be frustrating. A little circling can be beneficial as practice, but too much circling is just busywork.
It was explained to me, a number of times as I recall, that the reason we had to study the same old grammar year after year was simply that the students keep forgetting it. I never quite accepted that explanation when I was young, and I still don't. I make no claims to knowing anything about the teaching of grammar, but I do have a hypothesis about the yearly repetition that so irked me. My hypothesis is that a vicious cycle was in operation. Teachers made very shallow cuts into the subject of grammar each year because the students seemed to forget so much of it, but they forgot so much of it simply because each year's cut was so shallow. The reasoning that produced this situation seems sensible enough. If students have so much trouble remembering a little grammar, then does it not follow that they would have a lot more trouble remembering a larger mass of grammar? The fault of this argument is that it fails to take into account the demoralizing effect of the inefficiency that results from making too shallow cuts in the subject matter. Why should we bother to learn and remember this year's grammar if it seems like last year's, and we expect to face it again next year?
Fortunately my eighth grade English teacher broke out of this pattern. She gave us a very deep cut in grammar, and I learned more grammar that year than I ever have before or since. The course, at least the grammar part of it, had a straight-through feeling for me, and I liked it.
I have dwelled on the danger of cutting too shallowly on each loop of the spiral, but it would seem reasonable there is also a danger in the other direction. If one cuts too deeply in each loop one switches to a straight-through mode, and one may never get around to another loop in the spiral. If there are valid reasons for spiraling, and in the lower grades at least there usually are, then one loses these advantages by cutting too deeply. One might reason, for example, that students learn to reduce fractions by spending about six hours of effort on it in the sixth grade, the same in the seventh grade, and yet again in the eighth grade. Why not just spend eighteen hours on it in the sixth grade and be done with it? I expect this wouldn't work very well, though I would not try at this point to explain exactly why. The point is, as is so often the case, that one must strike a balance between two extremes.
It may seem that this has been a very negative chapter, that I have spoken disparagingly of the activity method, of unifying themes, of the spiral method, and of every other effort that conscientious teachers have made in order to get their subject across. My aim, however, is not so much to emphasize the faults of all these efforts as to emphasize the value of sticking to a careful and coherent presentation of the subject. In most cases, as I have said before, the best way to do this is simply to choose a good textbook and stick to it. Projects, themes, relevance, and other approaches to organizing material should be given consideration, but should always be subservient to the natural coherence of the subject itself.