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Response Levels Of Learning

      In this book I have taken the idea of the structure of knowledge as the main thing that a teacher or a learner is concerned with, not the stimulus-response model that is much in favor in psychology generally. In fact I spent considerable time in Chapter 12 arguing that the stimulus-response model is not greatly relevant to most academic teaching and learning. However I have maintained that feedback is very important in building a structure of knowledge, and of course feedback consists of something like the stimulus-response model. We ask the learner to respond to a question, a problem, or a situation, and by analyzing his response we assess the state of the growing structure of knowledge in his mind. Also something like the stimulus- response model is very important in drill and review. We must practice what we are learning. I will speak of the "prompt-response" model, rather than the stimulus-response model. The difference is not great. "Stimulus" seems more of a low level prompt, such as a simple visual or aural stimulus. The "prompt" as I am using the term here is on a much higher level in most cases, a cognitive level, such as a question or a problem.

      I will devote a separate chapter to the idea of prompts. In this chapter I will discuss responses. I will be using the term "response" in a very broad sense, not limited to an immediate answer or reply. And I will be concerned with what goes on in the student's mind as well as his or her actions.

      I will divide responses into three levels, the "look-say level", the "ciphering level", and the "reference level". I will start with the look-say level, and to introduce the idea I will use the learning of reading.

      For many years there has been some controversy in the teaching of reading over the issue of "look-say" versus "phonics". I think this is a false issue, as I will explain shortly, but it is a very good place to begin analyzing the different response levels of learning. I will first explain what I understand these terms to mean and then extend these ideas into teaching and learning in general.

      If one were to teach reading purely by the look-say method, then one would show the student a word, tell him what the word is, then show another word and tell what it is, and so on. The sounds of the individual letters would not be taught. The whole word would be the smallest unit of learning. Adequate drill, verbal guidance, and motivation would be provided, but an explanation of how any why letters fit together to form words would not be given. After the student has learned a number of words in this way he can begin to read by himself. However each time he comes to a new word he would have to ask the teacher to name it for him. By this method he would eventually learn a vocabulary of thousands of words, providing he is reasonably intelligent and diligent.

      If one were to teach reading purely by the phonics method, one would first teach the student the sounds of the individual letters and then put these letters together to form words. The student, for example, might first learn the sound of the letter "C", then the sound of "A", and then the sound of "T". He would then be shown how all three sounds fit together to make the word "cat". In this same way all the sounds of the English language could be learned and combined in all possible ways. Alternate sounds, such as the two ways to pronounce "c", and silent letters would also have to be carefully learned. Then a number of rules for combining the various sounds would have to be learned. Once all these sounds and rules are thoroughly learned, the student could figure out new words on his own without the help of a teacher. In this way the student could go on to learn thousands of words, again providing he is reasonably intelligent and diligent.

      Which is the better method, look-say or phonics? Which will give better results in actual practice? These may seem sensible and logical questions, and I think they have kept reading specialists busy for many years, but, as I will try to show, they are the wrong questions. My view is that it is a moot question because any teacher of any competence at all would use both methods together.

      Before proceeding with this discussion on the teaching of reading I will present an example from the field of arithmetic that is similar to this "look-say versus phonics" issue.

      After one learns that decimals mean a particular kind of fraction (a fraction with a denominator of 10, 100, 1000, etc.) then one is able to take a decimal and change it into a common fraction. For example the decimal .24, by definition, means 24/100, which can be reduced to 6/25. This is not too hard to learn for most students. When teaching math at the prison school I found that most of my students could understand the idea without too much trouble, but they would forget it quite easily. Often a student would get hung up in the next chapter because this particular idea had dropped out of mind. I found a way to improve the situation, by the old fashioned method of flash cards. I put about twenty or thirty common decimal-fraction equivalents on 3 x 5 cards. One card, for example, had ".5" on it. The student had to learn that the answer was "1/2". For the card with ".4" the student would have to respond with "two fifths". Each student, as he studied this particular lesson, was required to memorize these answers and be able to call them off in a specified time, one minute as I recall, before proceeding on to the next lesson. As I was using individualized instruction (due to the nature of the prison school) I was able to watch students individually go through this process over a period of many months. I found that by this method the students remembered this bit of learning much better than otherwise. Learning a few decimal equivalents "by heart" did not excuse the student from understanding the rationale, of course. Mathematics has a structure of implication, and students don't really know the topic unless they understand the logical antecedents that determine the answers they memorize. Nor would this process guarantee that forgetting would not occur. However the flash cards proved well worth the time and effort they required from the students.

      After I had taught in this school for a year or so I had a number of sets of flash cards. In addition to the deck I just described I had a deck of about fifty cards on reducing fractions, twenty cards on fractions to decimals, twenty cards on per cents to decimals, twenty cards on per cents to fractions, twenty cards on decimals to percents, and perhaps a few others. I didn't plan out these decks of cards all at one time. I prepared each of them when the need presented itself. Often I would begin by scribbling out a few cards for one particular student who was having trouble, and upon reflection, remember that others had had the same trouble at the same place in the course. After trying out the cards on a few students and observing a benefit I would then make that deck of cards an integral part of the course for all students.

      Now I will apply the ideas of "look-say" and "phonics" to this arithmetic example that I have described. Memorizing that .75 is equal to 3/4 is a type of "look-say" learning. When drilling on the cards I want the students to simply look at the card and say the answer, with little or no thinking in between the prompt and the response. Understanding the reasoning behind these answers corresponds in reading to phonics. This understanding should precede the look-say learning I just described. It may seem odd that learning something by a mindless "look-say" method would be used when understanding is so important. But the two types of learning seem to reinforce each other. I think the explanation is simple enough. The memorization provided supporting structure (as discussed in Chapter Five) as well as practice to increase fluency. Remember that a fragile structure is one of the most common problems for learners. The extra drill and the extra supporting structure greatly alleviate this problem. It might be argued that memorization might drive out understanding. This is a valid consideration, of course, but experience did not show it to be a problem. Indeed the opposite was apparent. Memorization and understanding reinforced each other.

      In general, I think it is true that memorizing will not "drive out" understanding, but memorization can at times displace understanding. Understanding and memorizing are two different things. If the teacher fails to teach for understanding then understanding will not happen, regardless of whether or not anything is memorized. If the teacher fails to teach memorization then memorization will not happen, regardless of how much understanding is present. I taught both understanding and memorization. I made sure, as best I could, that both happened. Memorization can displace understanding only if the teacher allows it by not teaching understanding. Conversely, understanding can displace memorization if the teacher allows it by not teaching memorization. If the teacher will teach both (and if time allows) the students will learn both. In many topics thorough learning requires some of both.

      At this point I will switch from the term "phonics" to the term "ciphering". Phonics implies sound. Phonics is a matter of "figuring out" sounds. It is the "figuring out" part that I am concerned with here. By "ciphering" I simply mean "figuring out", and figuring out is something that must be done to one degree or another in just about any subject. Ciphering is a matter of using conceptual skills to fit ideas together and come up with an answer. Ciphering is structure building, as discussed in Chapter Five. However I will use these two terms a little differently. I use the term "structure building" to refer to a broad-based, extensive, permanent body of knowledge being build up in the student's mind. I will use the term "ciphering" to the process of building up a small, temporary, structure of knowledge that meets an immediate need, that results in an answer to a problem or question.

      Opposed to ciphering is learning that can be immediately recalled, learning that is one mental step from question to answer, learning "by heart", or memorizing. This type of learning I will continue to call "look-say" learning. It is accomplished by brain packing, as I discussed in Chapter Five.

      I am using the terms "response" and "learning" interchangeably. To be accurate I would have to say "learning in which the response is on a look-say level" rather than "look-say learning" However the former is much more clumsy than the latter.

      I have argued that look-say learning reinforces ciphering and ciphering reinforces look-say learning. The two types of learning fit together. Each provides supporting structure for the other. But there is more to it than that. Look-say learning makes possible fluency in a small area of learning. Ciphering does not promote immediate fluency, but it does have broad applicability over a large area of learning. Reading illustrates this very well.

      It might at first seem that phonics would be the logical way to teach reading, for when a child knows phonics he supposedly can figure out any word independently of the teacher. However there are problems. Phonics are slow. Students don't become fluent readers on the basis of phonics. They become fluent readers on a look-say basis. Building up a look-say vocabulary allows the child to read fluently in a limited area. This fluency is very important if the child is to gain satisfaction of accomplishment from his efforts. Learning phonics, a matter of ciphering, cannot provide this fluency, at least not until after several years of effort. Ciphering is hard mental work. An overemphasis on ciphering can make the intensity of effort required of the learner too high. This, as I discussed in Chapter Six, is a factor in the texture of the learning. An inappropriate texture can lead to frustration and decreased motivation.

      But obviously ciphering is needed. If the child is ever to gain independence from the teacher in reading he must learn phonics. Thus learning to read requires a combination of look-say and ciphering, look-say for immediate fluency in a limited area and ciphering for broad applicability and long term independence. Thus, as I mentioned earlier, there should be no controversy about "look-say versus phonics" in the teaching of reading. Both have an important place. Good teachers use both. I would further argue that within broad limits the proportion of the two is not crucial. The teacher's good sense is a far better guide to the optimum proportion for her particular set of students than any preconceived "method" that can be concocted in theory.

      So far I have given examples of look-say and ciphering in reading and arithmetic. In these two subjects the concepts seem to work quite well. However in many other subjects, such as art, history, and literature, these two categories seem to apply less well. However if the categories are subdivided and expanded a bit then they become much more applicable. First I will subdivide the category of look-say learning.

      I will distinguish between "formal look-say" learning and "flexible look-say" learning. Learning the multiplication tables by drill is an example of formal look-say learning. It involves a specifically prescribed set of data and a necessity of attaining a high degree of fluency. Learning facts in history is usually a matter of flexible look-say learning. It involves a loosely defined set of information to be learned and little necessity of attaining a high degree of fluency. When studying the Civil War, for example, one must learn facts. These would include dates, names, events, and so on. But this set of facts (unless the teacher allows the learning to degenerate into a catechism) is rather loosely defined, such as "all the material in chapter three". And on a test the speed of recall, within limits, is not crucial. It is of little importance whether a student takes a half a second or 10 seconds to recall the date of the Emancipation Proclamation. But even with this flexibility many of these facts are on a look-say level. One may use some ciphering to recall the date of the Emancipation Proclamation, but the normal expectation is that one will simply remember it. In both formal look-say learning and flexible look-say learning it is just one mental step from question to answer, from prompt to response.

      In formal look-say learning one accepts the teacher's directions as to which facts are to be learned and the degree of fluency to be attained. In flexible look-say learning there is less direction. Topics which require drill are a matter of formal look-say learning. Topics which require practice, review, or study, are more a matter of flexible look-say learning.

      Other examples of formal look-say learning would be learning the names of the presidents of the United States in order, memorizing spelling rules, learning to instantly identify the silhouettes of enemy aircraft, learning the symbols and oxidation numbers of a list of ions in chemistry, and so on. Other examples of flexible look-say learning would be learning facts about the plot and characters of a play in English literature, learning the postulates and axioms in geometry, learning the color and make of your friend's car so you can find it again in a parking lot, and so on.

      The distinction between formal and flexible look-say learning is not sharp, of course, and sometimes it is impossible to say whether the learning of a given fact is a matter of formal or flexible look-say. Also, the way a topic is handled may change the type of learning. For example if history facts are put in the form of a catechism (that is, a definite set of questions with definite answers to be parroted back verbatim by the students) then it is more a matter of formal look-say learning than flexible look-say learning. As another example, when a teacher is just introducing the idea of multiplication to a class and is not concerned whether any combinations are memorized or not, then any combinations that students happen to memorize would be more a matter of flexible look-say learning than formal look-say learning.

      It should not be assumed that elements of look-say learning are always add-on elements, or that responses of ciphering are found only in subjects with structures of implication. Many implied elements must be put on a look-say response level. They must be instantly available. The implication is important, but the brain must not be burdened with running through that implication continually. In algebra one is expected to memorize the quadratic formula. It is much too useful to do otherwise. Thus it becomes look-say learning. Of course, it must not be blindly memorized. Students are expected to understand its derivation. Indeed they may expect to be asked to derive it as part of a test. The quadratic formula is not an add-on element of algebra. It is very much an implied element, but still it should exist in the student's mind on the look-say level.

      The ciphering level of response (again I will often say "the ciphering level of learning") can also be subdivided. I will use three subdivisions. First there is the "algorithmic ciphering level". In this level one follows a rigid series of steps to arrive at a definite answer. A very large part of elementary arithmetic and algebra is on this level. One learns algorithms for carrying, borrowing, multiplying, dividing, finding square roots, and so on. In typing one learns a definite series of steps to center a title on the page. In electricity one learns a definite series of steps to find the combined resistance of two unequal resistors in parallel. In almost any subject there will be at least a few similar examples.

      Next there is the analytic level of ciphering. This is similar to the algorithmic level except that the steps to be taken are not rigidly prescribed. Rather the process can vary within a certain latitude and still result in the correct answer. However this level of ciphering still has a right and a wrong answer. Both analytic ciphering and algorithmic ciphering are based on reason, but there is a difference. In algorithmic ciphering one does not try to follow the reasoning behind every step each time he uses the algorithm. In many cases the learner may benefit by following the reasoning behind each step when he first learns the algorithm, but thereafter he just applies it. In analytic ciphering the reasoning behind each step is always carefully followed. Reason, in one form or another, is essentially all that analytic ciphering is. A great deal of mathematical and scientific thought is of this type.

      Algorithmic and analytic ciphering blend into one another and are often combined. For example in doing an algebra problem on may use analytic ciphering to set up an equation, and then use algorithmic ciphering to solve the equation. In auto mechanics one may learn the symptoms of a maladjusted carburetor by careful reasoning, but then apply that knowledge in practice in a mechanical way, forgetting much of the rationale.

      A third level of ciphering is "open ciphering". In this level there is no definite series of steps to be taken and no definite answer to be arrived at. Formal debate is a good illustration of this. In arguing one's position one puts ideas together in whatever way seems most advantageous. In subjects such as history open ciphering is very important. If one can put facts together ingeniously then a wide variety of differing, even conflicting, conclusions may be given credit. This is the basic rationale of essay questions in testing. In almost any subject open ciphering has at least some place. In fact I think it could be argued that open ciphering is the way knowledge progresses. It is the daily activity of research scientists, philosophers, detectives, and so on. It is also an important part of such mundane activities as fixing a leaky faucet with limited tools and knowledge, and innumerable other little problems we are all faced with everyday.

      However open ciphering should not be given undue status. "Creativity" is a mantra to some parents, as if it is the highest quality to be found in their precocious offspring. Creativity is open ciphering, and I have just said that open ciphering is what scientists and philosophers do. But, I would argue, it is not what school children ought to be doing all the time. The basics of any subject depend much more on non-open ciphering and look-say learning. These basics have to be learned, and they have to be learned efficiently. Open ciphering on a substrate of massive ignorance may be "creative", but it is also of very low productivity. Open ciphering may be crucial to the research scientist or philosopher, or handyman, but it can be wasteful of time and even frustrating in the classroom.

      Open ciphering is very dependent on intelligence. The lower response levels are less dependent on intelligence and more dependent on effort. As I will argue later, this has important implications for testing.

      I started out in this chapter talking about immediate responses, the look-say level. Then I discussed a less immediate response level, the ciphering level. Next I will discuss a third response level that is even less immediate. This is the reference level. In this level one does not know the answer, nor does one have the means to figure out the answer. Rather one must look up the answer in a reference of one sort or another. This may not sound like a level of learning at all. One can hardly be said to know something if he has to look it up somewhere. However the reference level,. whether we call it "learning" or not, plays a very important part in everyday life and in a wide variety of subjects and topics.

      I will subdivide the reference level according to the accessibility of information. The simplest and most immediate type of reference level is the "catalog reference level". If one wants to know the population of a certain city one can simply go to a road atlas and look it up. If one wants to know the melting point of gold one can simply look it up in a handbook of chemistry. If one wants to know the cosine of an angle of 48 degrees then one can look it up in a mathematical reference.

      Next there is the "explanatory reference level." In this level the information is not so easily categorized or presented as in the catalog level. Information on the explanatory level consists of sentences and paragraphs. On the catalog level, in contrast, the information consists of numbers, letters, or words. A good example of information we expect to have on the explanatory level would be travel information. A travel brochure gives information, but not every travel brochure will cover the same range of information. In one brochure an important topic might be the history of the country. In another brochure the geological features of the area might be of prime importance. In yet another travel brochure the peculiar customs of the inhabitants of the area might be the most important information. All this varied information cannot be reduced to a catalog of numbers or a chart of symbols. It must be presented as an explanation. Yet the information is readily digestible by a very broad group of readers for whom it was written.

      A third reference level is the "textbook reference level". On this level information cannot be reduced to data, nor to readily understood sentences or paragraphs. The information is available only if one is willing to dig it out with great effort. An example of getting information at this level would be a homeowner browsing through a physics text in an effort to get some perspective on whether a given solar heating unit might be adequate for a given application. Another example would be looking through a history text in an effort to find out if two currently warring countries have been traditional enemies. Yet another example would be consulting a book on trigonometry in an effort to estimate the height of a tree by measuring the length of its shadow.

      The difference among these three reference levels is the degree of accessibility of information. On the catalog reference level the information is instantly accessible. On the explanatory reference level it is assumed that some information is needed to support the facts. On the textbook reference level it is assumed that one will need considerable dedication to retrieve any useful information. This makes information so inaccessible that few will even attempt to get to it.

      Response levels differ qualitatively. But they also differ along a continuum of accessibility - how quickly one can get from question to answer. Thus all the levels I have discussed can be put in a series:

formal look-say
flexible look-say
algorithmic ciphering
analytic ciphering
open ciphering
catalog reference
explanatory reference
textbook reference

      Information at the top of this series is instantly available. I will refer to these as "immediate response levels". Information at the bottom of this series is only remotely available. I will refer to these as "distant response levels". Of course the order of this series would not be invariable in all cases. Sometimes, for example, a bit of algorithmic ciphering takes longer than it would take to apply reason to the problem. Sometimes a telephone number is memorized but the person decides it's easier to look it up rather than dredge it out of memory. In this case a catalog reference is chosen over a look-say level. In many other ways the order shown in the above list might be changed. However in a general way I believe the order I have presented is valid.

      With all this in mind, then, the logical next question would be: What learnings should be put on what levels in a given subject? First, it is important to point out that an entire subject is generally not all on the same level. Remember that the look-say and ciphering levels can reinforce each other. The look-say elements of a subject provide a foundation for fluency in a limited area of a subject. They provide a basic framework on to which other learnings can be attached. The ciphering elements of a subject are like the glue that holds the look-say elements in place. They make the subject a subject of implication (as discussed in Chapter Five). The reference level of learning does not reinforce the other types of learning so much, but provides temporary supports in the structure of knowledge. For example when one looks up the logarithm of a number in order to work a problem, that logarithm becomes a temporary, but essential, element in the immediate structure of knowledge.

      Thus a mix of all three levels is most appropriate for most topics or subjects. For example in history the fact that Columbus discovered America in 1492 should be put on a look-say level well before one gets to high school. This historical fact is commonly used as a pillar in the structure of knowledge, a point of departure from which many other facts and ideas will be related. The dates of the founding of every early colony in America will not, at least in elementary school, be put on a look-say level. They would most likely remain on a reference level. History would have some elements on a ciphering level. If a teacher asks the class how many years elapsed from the discovery of America to the date of the Declaration of Independence, the students should be able to cipher it out, but would not be expected to memorize it. The date of Washington's death would probably be left on a reference level, quite accessible to a student writing a report on Washington, but not cluttering up the minds of all students. As another example, in arithmetic the product of six times eight should be on a formal look-say level, but the product of sixteen times thirty-two should be left on a ciphering level, and the cube root of fourteen should remain on a catalog reference level.

      In teaching a subject one must decide what learning should be put on which level. In doing this one can be guided a great deal by traditional practices and the good example of other teachers, but in the final analysis it involves a great deal of subjective judgment. As I mentioned earlier in this chapter I found while teaching general math in the prison school that I needed to put a great many facts on a look-say level. Perhaps another teacher with another class would think many of these facts should be left on a ciphering level. I mentioned in an example in the last paragraph that the date of George Washington's death should be left on a reference level. I was thinking of a high school history course. In a college history course concerned with that specific person or time that date might automatically be put on a look-say level.

      I have mentioned the advantages of each level of learning, fluency for the immediate response levels and broad applicability for the distant response levels. And I have mentioned the disadvantages of each level, lack of fluency for the distant levels and the limited applicability and effort required to attain the immediate response levels. To illustrate some ways in which these various advantages and disadvantages may be balanced I will set up a hypothetical situation and carry it through a number of stages.

      Suppose a person wants to learn physics. Is it really necessary that he spend hundreds of hours learning the subject? Or can he just buy a good textbook of physics, set it on his book shelf, and be done with it? He has, in this way, the entire subject of physics on the textbook reference level. He has the advantage of having expended an absolute minimum of effort. He has the disadvantage of incredibly low fluency. If he ever opens the book to try to find an answer to some question involving physics, he will find that the answer is highly inaccessible. He would not recognize the answer if it were staring him in the face. He would probably not even know when to open the book. Even more importantly, his question would be framed in simplistic terms. He could not frame his question in meaningful terms until he had gained a considerable knowledge of physics.

      The textbook reference level is mostly a hypothetical response level. It is little more than saying that a response of some type is possible because a certain body of knowledge is in existence. It is potential learning, hypothetically possible learning, more than real learning. However it has a certain degree of validity in some situations. A number of years ago I picked up a textbook on soil science at a used book store. As I was interested in gardening at the time I felt that I would be able to get something out of it. As I recall I put in some effort and did indeed learn a little from it, and it did support my knowledge of gardening.

      The textbook reference level is meaningful only when there is a realistic chance of potential learning being changed into actual learning. I did actually learn a little about soil science, and someday I may get interested in it again and learn a little more. Thus it is meaningful. When a knowledge hungry youth first gains access to a large library then the textbook reference level is very meaningful. When a research scientist finds something in another's work that when thoroughly pursued will greatly advance his own work, it is very meaningful.

      But, to continue to hypothetical physics example, suppose that instead of just buying a physics book and setting it on his shelf, the learner browses through it for an hour or so a day for several months, reading and thinking carefully, but neither as intensively or extensively as one would do when taking a physics course. By this means the subject becomes much more accessible to the learner. He begins to build a structure of knowledge. He puts some basic ideas on a flexible look-say level, other ideas on an analytic ciphering level, a few formulas, numbers, and equations on a formal look-say level, and so on. Once this is done the learner has much greater access to the subject than he previously did. If he reads something in a magazine about heat that he doesn't understand to his satisfaction he has some idea where to look in his physics book for clarification. But this does not mean, of course, that his structure of knowledge at this point is identical to the structure of knowledge of one who thoroughly knows the subject. His knowledge at this point is very sketchy. It probably contains many holes, and some important false elements. When trying to apply his knowledge on a specific problem he will do a great deal of thumbing through the book looking for clues of how to approach the problem. Many principles, and a vast multitude of details will remain available on a reference level, and this reference level will remain, for the most part, the textbook reference level.

      Suppose next that the learner is not satisfied just to buy a physics book and get acquainted with it. Suppose he decides to take a college course in the subject. By doing so he studies the subject intensively. He places a great many ideas on immediate response levels. At this approximate stage an important milestone is reached. Enough parts of the subject are at high enough levels of accessibility that the learner is prepared to go on and study more physics. His structure of knowledge, at this point, is far from being as complete as it could be, but it is firm enough to build on.

      I will take this progression yet another step. Suppose now that the learner studies physics until he earns an advanced degree on the subject. At this stage his fluency in the basic topics is very high. He has all the basic facts and ideas, and a vast number of details on a look-say level. His structure of knowledge becomes more a thicket of interlocking ideas than the sparse framework that constitutes the structure of knowledge of the beginner. Problems that the beginner would tackle with a great deal of time and effort would take mere seconds for the learner at this stage. But even at this stage not everything is put on an immediate response level. It would hardly make sense, for example, to memorize the tensile strength of every metal. Such details are still left on the reference level. And one obviously cannot memorize the answers to every possible problem in elementary physics. They must remain on a ciphering level.

      As a last stage in this progression let us suppose that the learner teaches physics for many years at a college. At this stage he has an incredibly dense structure of knowledge. He not only has all the basic and may not-so-basic ideas on a look-say level, but he also has a vast array of details on a very high level of accessibility. His structure of knowledge is a vast, solid edifice. Yet he still has a vast amount of knowledge left on ciphering and reference levels.

      Through each of these five stages the learner continues to expend effort to gain fluency. At no particular stage can one say that the material is "mastered". Nor is there any particular stopping place where one can say that further efforts will produce returns diminished to the point of not being worthwhile. I have described these stages only in an attempt to give some perspective on the matter. Obviously one must aim for some place in the middle ground. The low-effort/low-fluency stage is suitable for some who want only a general idea of the subject. It is not worthwhile for many others because it would not be worth the effort. The high-effort/high-fluency stages are suitable only for specialists in the subject.

      The one milestone worth looking for is the middle stage in which the structure of knowledge in the learner's mind is firm enough to build on. In my hypothetical progression I described this stage as when a person takes a course in the subject. I think that as a general rule this is a stage of fluency that teachers either consciously or unconsciously work for. I further think that for they most part they succeed, at least in a general way. Of course this level of fluency must be determined by the teacher's subjective judgment in any given situation. I have argued that tradition and the good example of others make the best guides until one's own experience and common sense become better guides. Being aware of the different response levels of learning can help in making this subjective judgment.