Chap 4 Correction Of Learning Problems

Correction of Learning Problems

In the last chapter I discussed diagnosis as if the correction were obvious once an accurate diagnosis is made. To quite an extent this is true. If you find a structural gap you close it by providing the missing bit of information. If you find a fragile structure you strengthen it by assigning more practice. If you suspect a hidden assumption you try to expose it and correct it. But sometimes there is more to it than that. In this chapter I will go a little deeper into the steps that can be taken to correct learning difficulties. In doing so I will discuss, or at least mention, a number of topics what will be more fully developed in later chapters. These topics are:

branching

coherence

levels of fluency

texture

mental habits

definitional versus operational concept formation

The two basic forms of correction are explanation and practice. When a student doesn't understand something you try to explain it to him. If the problem is not a matter of understanding, but a matter of fluency, then you assign more practice. Often both explanation and practice are called for. I will discuss both explanation and practice at some length. I will also discuss a middle ground between explanation and practice.

I would like to approach explanation from the perspective of a planned course of study. A course of study consists, among other things, of a series of topics that are presented to the learners. Explanation may be seen, by this perspective, as "branching". Branching means directing the student's attention to something other than the current, or expected next, topic. One may branch forward, or branch out, but most commonly one branches back, back to a topic that appeared earlier in the course.

The following example illustrates a mundane, routine example of explanation by branching back. This is a real example from my experience at the prison school, one among many that occurred everyday and that would all be equally appropriate to cite.

Student: "I'm stuck on this problem. "Find the interest on $6400 at 4 1/2 % interest for 45 days." Okay, I used the short cut like it shows here on this page. I mark off two decimal places to get $64. That would be the interest for 60 days at 6%. Now I cut it down. 45 days is 3/4 of 60 days, so I take 3/4 of $64. That's $48. Now I cut it down again, because 4 1/2 % is less than 6%, but what fraction . . . . how do I figure that . . . . ?"

Rude: "Well, you put 4 1/2 over 6. That's a compound fraction. Do you know what to do with a compound fraction?"

S: "Uh, . . . . I remember something about that . . . ."

R: "Here it is on this lesson back here. Do you remember what you did back here?"

S: "Oh, . . . let me see here . . . . "

In short my correction was "Branch back to lesson . . . " The key word was "compound fraction" The moment I said this things clicked in his head. This was all this particular student needed to get going again. he had a gap in his structure of knowledge, and I could easily fill it in. Or, if you prefer, he had a fragile structure that he could strengthen and use once I pointed it out to him. The part of the structure dealing with compound fractions was so fragile that it seemed to be entirely missing. He knew about compound fractions. At least he had previously completed a lesson on compound fractions. But needed a bit of reminding to use that knowledge. Whether one considers it a structural gap or a fragile structure, the cure was to branch back to a previous lesson.

This example was rather clear cut because the student was more intelligent and verbal than most. He could lay his problem right on the line. Also it occurred near the end of the time that I taught in the prison school and I had things going as I wanted them. I knew exactly what lesson to refer him to and I knew the lesson would make sense to him. Many similar examples would be much less clear cut, but would boil down the same thing - "Branch back to . . .".

The idea of branching back is dependent on having something to branch back to. If the missing or troublesome bit of knowledge has never been covered in the first place then one cannot branch back to it. This brings up the idea of coherence.

A structure of knowledge, or a course of study, is coherent if each bit of knowledge builds upon the previous knowledge, if all the elements that are needed for an idea are taught before the idea is introduced, if an explanation relates the unknown to the known, not to other unknowns. For example in a coherent course of study I can explain that per cent means hundredths and therefore a per cent can be changed to a decimal by removing the per cent sign and moving the decimal point two places to the left. This will make sense to the student because he has already been introduced to decimals and understands them adequately. This does not mean that understanding will be instant and without effort, of course. The learner must put forth mental effort or no learning will take place. It does mean, however, that if this mental effort is forthcoming then understanding will be forthcoming, because all the elements needed to form the new structure of knowledge are pre-existing and need only be assembled into the new structure. If the course of study is not coherent then all the mental effort in the world will not produce learning because not all the required elements are there. You cannot understand why you move the decimal point two places to the left if you don't understand the meaning of decimals. Similarly in a history course it would be incoherent to try to explain that slavery was one cause of the Civil War if the students are unaware of even the most basic facts about slavery in the united States at that time. Or in a science course it would be incoherent to try to explain water pressure to students who had no understanding of weight, force, or area.

Coherence can be defined a little more broadly than simply having the topics in the right order, or having topics arranged in accordance with strict logic. I will have much more to say about this in Chapters Six and Seven.

In a highly coherent and well administered course one has no trouble branching back. The teacher knows intimately how each bit of knowledge relates to other bits of knowledge, and where each bit of knowledge is to be found. The students, of course, will not have such intimate knowledge of where all the details are to be found, but the ideas will fit together for them. Things will "click." Such coherence does not come automatically or effortlessly. It comes only by a teacher's conscientious effort. An acceptable level of coherence can generally be achieved with a little planning and common sense. However good teachers will go beyond that. They will refine their courses considerably with experience. Whenever a student has a problem, which normally occurs many times a day, the teacher must ask himself or herself how that problem can be avoided for other students. Did the student learn something superficially? Should he have had more practice on that something? Was something explained poorly by the teacher or by the book? Can the problem be avoided for other students by adding an extra lesson in the workbook, or by requiring more practice on some concept? How can the same thing be done in another or better way? In this way experienced teachers can become highly efficient. The students benefit not only in learning more, but in enjoying it more also.

The author of every textbook, with perhaps a few exceptions, is very much concerned with putting coherence into the book. You will seldom find a text that carelessly explains the ideas of chapter two in terms of the concepts in chapter six. However I do not think the same can be said of teachers. It is quite common, unfortunately, for a teacher to subordinate coherence to other considerations. In an effort to provide a "variety of resources" one may try to teach a course by throwing together two texts, three supplementary books, a few lectures, a few more film strips and movies, and a guest speaker or two, and expecting the students to profit by the result. Such a mishmash of "resources", if not very carefully managed, results in an incoherent mishmash of knowledge in the students' minds. A considerable amount of learning may have taken place, but the different bits of knowledge are not all neatly tied together as they should be. There are inconsistencies that the students haven't resolved, and isolated bits of knowledge that seem irrelevant, and masses of knowledge that are not worth the effort expended to acquire them, a lack of good prioritization, and so on. Further, learning has taken place inefficiently. The learner put in considerably more time and effort than was really needed to learn what he did. Thus, though students may respond at times to such a course of study for a number of superficial reasons, they will not get much real satisfaction of accomplishment from their efforts, or respect for learning. I will go into this idea of coherence in more detail in later chapters. The important point here is that the idea of branching back depends on coherence. There must be something to branch back to.

From a strictly logical point of view it might be argued that one can only branch back - one cannot branch out, or branch forward. If a bit of knowledge is needed, but is not previously covered, then it is logically the next step. When one takes the next step he is progressing, not branching. However there are times when explanation is needed that does not seem to be a matter of "branching back," but is more a matter of "branching out". One student may need more information or explanation than others.

For example, in teaching general math in the prison school I found that most students could understand fractions with the aid of a few drawings to show halves, thirds, and quarters. But for some of the less able students I found that I had to have something more concrete. Therefore I kept a box containing fractional parts of circles that could be used for demonstrating fractional relations. I had these parts color-coded. The half circles were orange colored, the thirds were green, the fourths yellow, the fifths red, and so on. When a student couldn't seem to make sense out of fractions by the use of drawing and verbal explanation I would pull out this box of circles. I could then show very concretely that one third equals two sixths, or that a half is bigger than a third, and so on. In other words, I could branch out when needed.

Another way of branching out that I used consisted of having a "help book". This was simply a loose leaf notebook in which I put examples, additional explanation, hints, etc., for various lessons which proved difficult for many students. Thus very often when a student had a problem and I could see that it was the same problem other students had I could just say, "look in the help book, Jones." This was a matter of branching out because the contents of the help book were not a part of the regular course of study, but were available when needed.

Branching is very important in some forms of "programmed instruction," which came out a number of years ago. The basic idea of programmed instruction is that the student should be reinforced immediately for each small step that he successfully completes. Normally this reinforcement is accomplished by requiring the student to answer a question after every sentence or two of the text. He checks his answer immediately after he writes it down, and if his answer is correct he continues to the next step. Each correct answer is a reinforcement that motivates him to continue working and learning. Branching comes into the picture where the student misses a question. In the simplest type of program, a linear program, the student would simply branch back to the previous step. However in a more sophisticated form of programmed instruction, a branching program, he would branch out to a subprogram. The subprogram would be designed to explain to the student what he had done wrong and to get him back on the right track.

On the surface it would appear that a good branching program would approach the ideal of the management perspective of teaching, that it would provide feedback, that it would adjust to the problems of the individual learner. However programmed instruction has not lived up to the glowing promises that accompanied its introduction. I think part of the problem is that too many branches would be required in order to make a comprehensive program. One of the first things I learned when I first started teaching was that there are an unbelievable number of ways in which a student can go off on the wrong track and get hopelessly stuck. A real live teacher can take each problem as it occurs and get the student back on the right track. However the author of a programmed text would have to anticipate thousands of potential problems and write out endless subprograms if he wants to anticipate every possible problem. Thus a simple course in arithmetic could come out thousands of pages long, and there would still be a few students who would find new ways to get stuck.

It does not require a "programmed " text to provide some branching written into the book. Whenever a text includes a phrase such as "se e section 42," or, "as discussed in the last chapter," or "refer to the explanation on page 16", it is in effect providing a branch which the student may or may not follow depending on his needs at the moment. Footnotes serve the same purpose. Again, such measures cannot totally substitute for a real live teacher simply because only a very few of the many possible problems that students will actually encounter can be anticipated.

The bulk of branching, in spite of subprograms, help books, footnotes, and so on, still consists of verbal explanation on an individual basis. No matter how much coherence a teacher may achieve in his course, and no matter how extensive his efforts to anticipate possible problems, students will find new ways to have trouble, and will need the teacher's help. This is why I stress the management perspective of teaching over the performance perspective.

I will now turn to the second principle form of correction - more practice - and I will discuss the idea of practice in general. In chapter one I touched on the idea of practice as being necessary for learning, because teaching is more than telling. Then in Chapter Three I discussed the ideas of fragile structures and the assumption of fluency. However I did not go into much detail about practice. When is practice needed? What kind of practice? How much? How can you tell how much practice is too much and how much is not enough? How can you tell if practice is effective?

Hopefully a teacher has some good answers to these questions before he or she enters the classroom for the first time.

The purpose of practice is to firm up a fragile structure. But how firm can a structure be? Or how firm should a structure be? It is sometimes said that the student should "master" one topic before he goes on to the next. Is this to be interpreted as a requirement of nothing less than total and complete fluency?

Fluency comes by degrees. To illustrate various degrees of fluency consider the following example. A seventh grade class in general math is to learn to reduce fractions. Though they have been introduced to fractions as early as the fourth grade, the concepts are still new to most students. Increasing degrees of fluency in reducing fractions are represented by:

1. As the teacher explains it, it "clicks" in the student's mind.

2. The student has done twelve problems of a twenty problem assignment when the bell rings to end the class period.

3. The class has completed and discussed two daily homework assignments on reducing fractions and is ready to go on to the next topic which is the addition of fractions.

4. The class learns to do the opposite of reducing. They learn to take fractions to higher terms so that fractions with unlike denominators can be added and subtracted.

5. The class takes the chapter test on fractions.

6. The class continues on through the chapters on decimals, percents, business problems, formulas, areas and volumes, and so on. In each of these topics fractions frequently occur that must be reduced.

7. The students go through the eighth grade and repeat a considerable amount of what they learned in the seventh grade.

At level one a part of a structure of knowledge is built. Associations are made that were not previously present in the students' mind (or have been forgotten). Would these associations remain if no further work were done beyond this point? Could the teacher immediately drop the subject once the students understand it and go on to the next topic? Is the structure of knowledge firm? No, the structure is obviously not firm. It is so fragile that if the students stopped there they would have just wasted time. The more intelligent students might retain the idea for a while, perhaps even for several days, but the average student would not. The idea made sense, but with no practice the idea would be immediately dropped from most students' minds.

At level two the structure of knowledge is considerably firmer in the students' minds. If the topic were suddenly dropped at this point the knowledge might remain in many of the students' minds for some time. However even if understanding persisted for a while the usefulness of that understanding would be almost nil. At this stage the student does not automatically think "3/4" when he sees "6/8". The knowledge is far from being an integral part of the students' life and thought.

At level three the concept is still not automatic, but it is ingrained strongly enough that the class can continue to the next topic. Perhaps some students will automatically think "reduce it" when they see the fraction "6/8", but most will not. However at this stage the idea has become very real to the students. It makes sense to them, and they recognize it as something that they are expected to learn, as something they will see again. This is not to say that they have much appreciation of the mathematical importance of the concept. Rather they simply realize that when two full class periods are spent on a topic then they are expected to learn it.

Each higher level of fluency makes the structure of knowledge firmer in the students' minds. At some point the average student automatically thinks "2/3" when he sees "4/6", and automatically looks for the possibility of reducing when he gets a fraction for an answer on a problem. Each successively higher level of fluency provides a firmer foundation for the continued learning of mathematics.

The early levels of fluency come very quickly. The later levels of fluency come successively more slowly. The more practice one does the more the returns are diminished. This law of diminishing returns is why I reject the idea that one must "master" one topic before going on to the next. Mastery, in an absolute sense, is impossible. Rather than mastery, one must simply aim for a degree of fluency that will allow the student to comfortably take the next step.

If one attains only a low degree of fluency in one step then the next step is difficult. If very many steps are taken with low fluency then the student has seriously compromised his potential to continue in the subject. I have seen this happen many times with college students. A student may scrape through an introductory chemistry course, for example, with average grades, but with little fluency. Concepts that are "crammed" into the brain the day before a test are often lost the day after. Thus a passing grade may mean only that fragile structures of knowledge were momentarily erected only to come tumbling down after they have served their purpose. In such a case the student has no desire to take another course in chemistry, has no appreciation for chemistry, and within a year or so has absolutely no knowledge of chemistry. The student's time and energy would have been much more beneficially spent if he had learned a considerably smaller body of knowledge, but learned it to a higher degree of fluently. In college this state of affairs is tolerated for a number of reasons. Colleges are more interested in producing an elite of specialists than a mass of generalists, and those who get a poor foundation in a subject can simply drop out of that subject after one course. However such a state of affairs is certainly not appropriate for high school or elementary school and should not be tolerated.

Nor should one aspire to an excessively high level of fluency. The result is a great deal of wasted effort and stagnation. I experienced this myself when I took a Latin course in high school. The first half of the course went by routinely and I rather enjoyed the subject. But then we got a different teacher and our progress stopped cold. The new teacher kept us on one lesson interminably. She apparently had decided that we were to "master" that lesson before proceeding. After about a week on the same lesson we began to feel a bit stagnant. After two weeks on the same lesson our homework was done very perfunctorily, if at all. I don't remember just how long we finally spent on that one lesson. I presume we finally continued, but we didn't get much done the rest of the year. This stagnation considerably weakened my interest and accomplishment in Latin.

Thus there are problems in aiming for either too much or too little fluency in any given topic. The solution is to aim for the middle ground. One must acquire enough fluency to provide a foundation for future learning, but one should not aim for total and complete fluency. Just where this middle ground lies is something one cannot determine out of context. However from the standpoint of correction of learning difficulties the principles involved are fairly clear: Fluency does not come automatically. I comes only with practice. Under normal circumstances more practice produces more fluency, but in decreasing degrees. Too little practice results in an inadequate foundation on which to build further learning. Too much practice is a waste of time and effort and produces frustration.

This next example of correction represents a middle ground between explanation and practice. The correction is more situational than verbal. As part of the placement diagnosis for a new student in my classes at the prison school I would check out their fluency on arithmetic combinations. I would have the new student call off the answers to the flash cards for addition, subtraction, multiplication, and division. When a student was slow on these, taking more than two and a half minutes to go through an entire deck of one hundred cards, I would have him work on them to speed them up to an automatic response level. And of course I would try to show him how to study them. This help would go something like this:

Rude: "Denton, you know these cards, but you're still pretty slow on them. It'll slow you down through the whole book unless you can speed them up a little. I'll show you how to work on them. What's this?" (showing the first card)

Denton: "Uh . . . I know it . . . it's fifty. . ah, . . . nine times six . . . fifty-six!"

R: "Nope, fifty-four."

D: "Sure, fifty-four."

R: "And what's this?" (showing next card)

D: "Twenty."

R: "And this."

D: "Thirty-two."

R: "And this."

D: "Forty-five."

Then I would return to the hard one, six times nine, which I had unobtrusively set aside when Denton first missed it.

R: "And this?"

D: "Oh ya, . . . uh . . . that's . . .uh. . . . uh . . . I know it . . ."

Obviously Denton didn't know it very well, so I would keep coming back to it after every three or four cards. After about the third or fourth time of being faced with six times nine Denton's thinking would start to change. He had been assigning low priority to six times nine. He would let the answer, fifty-four, drop from his mind immediately after he had said it. After all there are a hundred cards in the deck, and six times nine is only one of them. You can't worry much about only one in a hundred.

This type of thinking - you can't worry about only one in a hundred - would not be a conscious part of Denton's thinking, but it would be the premise he was acting on, an unconscious and unverbalized premise. It is a mental habit, a pattern of thinking that one uses without thought. By bringing back the same card unexpectedly, and repeatedly, I could force a change in this thinking. Denton would start giving higher priority to that one card. By giving it higher priority he would learn it. Once his priority system was changed in this way it was only a matter of time until he learned all the cards with the fluency that he needed. I would estimate about half my students in the prison school would go through this process.

I think the situational nature of this correction warrants emphasis. I could have simply said, "When you find a card you don't know, drill on it. Don't forget it!" But this would probably not produce much result, any more than a smoking habit can be easily broken by a few words. By using a situation - bringing up the difficult card unexpectedly and repeatedly - I got the desired high priority that was needed.

This "situational correction" I just described is close to what I call the operational process of concept formation. It is in contrast to what I call the definitional process of concept formation. Learning the flash cards is not really a matter of concept formation. The concepts, presumably, are well formed. Rather it is a matter of developing fluency. However the operational versus definitional aspect is pronounced. A definitional approach would be to verbally tell Denton to drill on the hard ones. And that approach, I have argued, would not work as well.

Explanation and practice are short term methods to correct a specific and immediate learning difficulty. Other steps can be taken in response to more generalized problems. One of the more important of these steps is to have a high level of coherence built into the course, as I have already discussed at some length. Another step, which I will now discuss, is to adjust the texture of the course. Texture, as I am using the term here, refers to how closely or openly the feedback is spaced, to whether the material is presented in big chunks or little chunks, to how long a cycle of presentation, practice, and feedback takes. I will speak of a "close" or "fine" texture if knowledge is presented in small chunks and feedback comes often. I will speak of an "open" or "coarse" texture if knowledge is presented in larger chunks and feedback comes less often.

In my general math classes at the prison school the texture of the course was fairly close for most students. A student might go through a cycle of presentation, practice and feedback several times in one class period. For example, Smith begins a workbook lesson but doesn't think he understands the explanation in the book. He brings it up to me with a generalized "I don't get it" type of question. I explain and have him do a problem or two and then send him back to his desk. After fifteen minutes he's done a few problems and wants me to check them just to be sure he's on the right track. I find a number of wrong answers and try to get him back on the right track. Later in the class period I help him again and find this time that he's finally got it all together. His answers are coming out right and he seems to understand it. This type of close texture takes a considerable amount of effort on the teacher's part, but it does produce results.

Other students in the same class could take a much coarser texture. They would hand in several lessons a week with no more feedback than the corrected lessons. Still others, usually those studying algebra, would study a whole chapter by themselves and get no feedback at all, other than perhaps having an occasional question. Then they would take the chapter test and pass it.

Coarser texture yet is exemplified by most college courses. College students often have only three of four tests a semester for feedback. In extreme cases students may have only the final exam as feedback for an entire course. Such an extreme is sometimes necessary, but hardly desirable, even for college students.

I have been using the term "texture" to refer to the spacing of feedback as it actually occurs in a course. We may also use the term to refer to the spacing of feedback in a planned course of study or to the spacing of feedback in a textbook. One may design a close textured course by assigning daily homework and grading it conscientiously. Or one may design a more open textured course by having tests every week or so but no daily homework. Or one may make the texture even more open by having only a few tests a semester. Similarly a book may be written with a close texture by putting problems or questions on every page or so. Or one may write a book with a more open texture by putting problems or questions only at the end of chapters. Or, of course, one may write a book with no problems, questions, or exercises at all, in which case the idea of texture would not apply.

What came out in the sixties under the name of "programmed instruction", or "programmed texts", is really nothing new. A traditional textbook is programmed in the sense that it is a coherent and methodical presentation of subject matter and in that it presents information and then asks the learner to respond to a problem, question, or exercise. A "workbook" may be defined as a textbook with a close texture, and perhaps with a few other differences such as paper covers and space to respond to the exercises in the book. A "programmed" text may be defined as simply a workbook with extremely close texture.

It can be argued that a finer texture provides more feedback and hence is superior to a coarser texture. Feedback is certainly important - that is the basis behind the whole idea of the management perspective of teaching - but it does not necessarily follow that closer feedback is better than more open feedback, or even that more feedback is always better than less feedback. My viewpoint is that texture should be adjusted to the needs of the learner, and that is why I am discussing texture in this chapter on corrections.

There are advantages and disadvantages to having a fine texture, and, of course, there are advantages and disadvantages to having a coarse texture. To illustrate these advantages and disadvantages consider the following analogy. A man is to move a mountain of sand from one place to another. He is to use nothing but a shovel and his own muscle power, but he does have a choice of what size shovel to use. If he moves the sand with a large shovel he will make rapid progress for a short time, but will soon wear himself out and quit. If he moves the sand with a small shovel he will not wear himself out so quickly, but neither will he make much progress. If he chooses a medium sized shovel he will make the most progress with the least effort. The large shovel, of course, corresponds to a coarse texture and the small shovel corresponds to the fine texture. The best progress is made when the texture is neither too coarse nor too fine, when the chunks of information, like the shovelfulls of sand, are neither too large nor too small.

To extend this analogy a bit we can consider the extremes. On the one hand suppose the man chooses to use a garbage can for a shovel, reasoning that each "shovelfull" will move so much sand that he will soon be finished. Unfortunately a garbage can full of sand would weigh hundreds of pounds. If the man ever manages to move one such "shovelfull" he surely will not attempt another, and his efforts will produce little results. On the other hand he might choose to use a teaspoon as his shovel, reasoning that each such "shovelfull" would be so easy to move that he would never get tired. Unfortunately he would still get tired eventually, not because of the weight of the sand that he moves, but because of the effort he expends in moving his own body. Thus a teaspoon would be an exceedingly inefficient shovel to use in moving a mountain of sand.

As another example, coarse and fine texture may be compared to high and low gears in a car. When going uphill in a car you shift to a lower gear. When going "uphill" in learning you similarly must shift to a "lower gear". This is accomplished by shifting to a finer texture. When you have an open road to travel you shift to a higher gear. When you have an "open road" in learning you shift to a "higher gear" by shifting to a coarser texture.

The advantages of a coarse texture are speed and efficiency. The disadvantages are that more intense effort is required and one may "stall out" completely when the going gets rough. The advantage of a fine texture is leverage. Any obstacle will yield if one can only get enough leverage against it. The disadvantage of a fine texture is inefficiency. One must expend considerable effort just to keep the gears moving, so to speak. Learning proceeds best when the optimum middle ground between these two extremes is found.

This perspective gives another reason why "programmed" instruction never really caught on. The texture is much too fine. Learning by programmed instruction is like moving sand with a teaspoon. Each step is easy, but far too many steps are required to get anywhere. Or, as another example, trying to learn a large body of knowledge by programmed instruction is like driving a car down an open road in low gear.

Next one might ask how a teacher can recognize when the texture is too fine and when it is too coarse. I can give no clear cut and consistent signs to look for, but that doesn't mean there is no use in looking. With a little experience one can watch a man shoveling sand and form an opinion as to whether he would do better with a larger or smaller shovel. Similarly with a bit of experience one can watch a class of students for a few days and form an opinion as to whether the texture of the course should be made either coarser or finer.

There are some signs indicating a need for texture adjustment that I think are worth mentioning, though they are certainly not as specific or dependable as one might like. If the texture is too coarse students want to quit entirely if at all possible. The job requires too much intense effort. If they can't quit they may start fooling themselves about what they've accomplished. They may decide they're ready for a test when in fact they have no idea how or where to begin studying. Or they may extract a few catch phrases from the material and think that by throwing them out now and then they show that they understand the material.

If the texture is too fine the students may again be frustrated, but it is a different kind of frustration. They are frustrated by lack of accomplishment, not by the intensity of effort required. They might want to quit, but only because they have already gone through a stage of wanting to speed up and get somewhere. Sometimes the students will simply tell you, in one way or another, that they texture is too fine. They may say that the exercises are too easy, or that they would like to work more independently, or that they don't see the sense in so much repetition. Perhaps the most important sign that the texture may be too fine is that students lose respect for the subject matter. They don't feel that their accomplishments are very significant and therefore decide that the subject itself is not very significant.

Unfortunately all these signs of too fine or too coarse texture can also be signs of other problems - too much or too little practice, or incoherence, or too much or too little strictness in discipline, and so on. Therefore, I conclude, one must try to "feel" his way in the regard. With a little experience I don't think this is too hard to do, providing one has an open mind and is willing to experiment a little.