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Everyday Classroom Practice

      In this chapter I will go a little deeper into just what a daily classroom practice could or should include. I will attempt to relate almost everything I have talked about in all the previous chapters. I have discussed a wide range of topics. Obviously not every one of these topics will be relevant in every class period. A given hour in the classroom will consist of a synthesis. There are many things that a teacher might do in a given class period, and there are many possible combinations of these things. And perhaps more importantly, there are many things a teacher could do, but will not do, in any given class period.

      I have tried at various times previously to indicate what actually happens in the classroom. I will not attempt to do that again in this chapter, but I will digress for a moment to describe some superficial, though popular, conceptions, or misconceptions, of everyday activity in the classroom.

      One commonly repeated version of what goes on in the classroom is what I call the “encyclopedia salesman’s perspective of teaching.” In this scenario teaching consists of assigning “reports” or projects on one sort or another, which sends the students looking for sources of knowledge. Encyclopedia salesmen would understandably find this perspective of teaching attractive, because encyclopedias are traditionally a prime source of information. However it is not encyclopedia salesmen from whom I hear this perspective described. It comes from a wide variety of sources. Whenever a local TV news team wants to do a story on education this perspective of teaching and learning is likely to be assumed. They will talk about the students looking for information in order to make “reports”. This scenario has some validity, of course, but it is only a small part of teaching and learning in everyday life. The essential feature of this perspective is the idea that finding a source of information is most important, as if a source of information is the limiting factor in learning. But that doesn’t fit reality very well. Available information is seldom the limiting factor in learning any particular topic. In most subjects the students do not go searching for sources of information. Rather the information they need is very neatly packaged in one book, the text book. The problem is to learn the information, not to find it, to take that source of information and somehow use it to build a structure of knowledge.

      Unfortunately this perspective is sometimes adopted in educational theory. Thus a teacher who sticks to one text book is considered unimaginative, and the teacher who will bring in alternate “sources” is considered progressive. And to bring in a guest speaker is considered the ultimate. But, as I discussed in Chapter Six, coherence may be decreased by using many sources of information. A single source, if it is complete and coherent, is much better. Multiple sources automatically lack coherence.

      We hear much today about the wonders of the internet. It is seen to be a prime source of information, and those touting it can sound very much like encyclopedia salesmen. We can find information on just about anything on the internet, of course, but that does not mean that it will revolutionize education. Every algebra students lugs around all the information about algebra he or she can possibly use in a school year in one book. Finding information is not the problem. Assimilating it is. The encyclopedia salesman’s perspective of education is not very applicable to everyday teaching and learning.

      Another common perspective of teaching is what I call the “comedy writers’ perspective”. In this picture teaching mainly consists of asking the students questions. Comedy writers use this as a set up for jokes, usually rather unsophisticated jokes. I have seen skits on tv in which the teacher goes wildly from one subject to another, each question being nothing more than the set up for a punch line. After about three or four of these jokes the skit is done. Obviously asking questions is an important part of teaching, but it is only a part. Asking questions is a matter of prompting, which is, of course, an important part of teaching. It is part of practice, a part of evaluation, and often even a part of explanation. But it is not the sum total of teaching.

      Yet another popular perspective of teaching is what, for want of a better term, I will call the “philosophers perspective.” By this perspective teaching and learning consists of discussing big ideas. Therefore what you do at school is to discuss. The teacher facilitates the discussion, but does not direct or control it. If there is a tragedy affecting children, TV reporters covering the story are likely to adopt this perspective. They will come into a classroom and report on how the teacher and students are “dealing” with it. Again this perspective has some validity, but many limitations. It may be a common activity in a course such as “Democratic Living”, but have very little to do with a course such as trigonometry.

      The “Socratic” method can be seen as a variation of this philosopher’s perspective, though the discussion is directed toward a specific conclusion in this method. The essential idea here is that the teacher asks a series of very directed questions, as presumably Socrates did, and the answers lead the students to some conclusion. This method rests on the idea that learning is a matter of extending logic, and of course there is at least some truth in this, but it is no more the sum total of teaching than is the encyclopedia salesman’s perspective that learning is a matter of finding information.

      Once in a while we hear about teaching by parables, which is a little like the philosophers perspective. Presumably it worked for Jesus. But that does not mean that it has much to do with everyday teaching and learning in the classroom.

      So what does go on in a classroom everyday, if these popular misconceptions miss the mark? Again, everyday classroom practice is a synthesis of many possible activities. Next there are two terms which I think are worth a bit of investigation. They are “recitation” and “method”.

      I have previously mentioned the term "recitation" and spoke of it as synonymous with "going over the lesson”. I have found it a very convenient term at times. However I have rejected the term as being very useful, on the idea that to many people it is synonymous with “catechism”, which means rote memorization of given answers to given questions.

      I gather from readings in the history of education that at times in the past students would “recite” in a formal way. The teacher would ask a question and call on a student to answer. The student called upon would stand in a proper manner and give an answer in a prescribed form. Any deviation from prescribed form in either demeanor or response would call for correction by the teacher. I get the impression that such a recitation was deemed proper for formal situations. If a visiting official were to be properly impressed then the students must be able to properly demonstrate learning. However I do not think it follows that such formal behavior would be the everyday rule in the classroom. Such a formal performance is a way of demonstrating knowledge, not a way of acquiring it. Of course we may also conjecture that in an age in which there were few textbooks and materials that we take for granted today, it may well be that much practice took a form such as this. Whereas today a teacher might use the questions at the end of a section of a history book both as an assignment and as a way to structure a half hour’s time of “going over the lesson”, in earlier times perhaps these questions at the end of the section simply did not exist. Perhaps the question that is to be answered formally when a visiting official comes is the same question that is used to introduce a topic to the class.

      Educators are often quick to argue that in the past teaching never went beyond the requiring of rote memorization. This does not seem realistic to me. It does seem reasonable to think that the ratio of rote memorization to understanding might be somewhat different in the past than in the present, for various reasons. But it does not seem realistic to me to imagine that good teachers in the past did not also teach for understanding. A competent teacher helping a group of fifth graders learn to add fractions would explain what they are doing, whether they are in a one room school house in 1850 or in a modern school today. Such a class and teacher in 1850 might take it for granted that their learning about fractions would be turned into a formal “recitation” at some point. I can only guess at that. But it does not seem reasonable that such “recitation” would form the bulk of their everyday activities in the classroom.

      Catechism, that is a formal list of questions and answers to be memorized, has little place in most education. Recitation, meaning a much more informal activity of questions and answers, may have a definite part in many situations and topics in modern education. It is a form of practice, and practice is important. The next level in this progression does not have a name, so far as I know, but it is very important. In this final level there is a lot of questions and answers, but is directed by the teacher both to elicit feedback and to give practice. It is this third level in which all I have said about prompts and responses is very important. And it is this third level which forms a lot of everyday activity in the classroom, and which usually goes under the name of “going over the lesson.”

      Teachers often fall into patterns that are repeated everyday, simply because they are patterns that work. Then when asked “what method do you use?” don’t have a ready answer. And it can work the other way. Teachers sometimes read about some “method” and decide that is going to be their method, when in fact they modify it so much from the ideal that it seems questionable if they really are using that given method.

      "Method" can have several meanings in education. One meaning of "method" is a recipe that tells the teacher just what to do. It can also mean a more general strategy of teaching. I would argue that one should be wary of adopting any particular "method" of teaching. Rather, teaching is like a stew made up of a number of ingredients. The basic ingredients are:

      some form of presentation of new material,
      some form of practice or drill,
      some form of low level diagnosis and correction, and
      some form of evaluation.

      These things, I would argue, are the basic things that occur in everyday classroom practice. Normally when one speaks of a "method" one is simply emphasizing one particular practice or aspect of the teaching, not defining anything definite. To say "I use the Socratic method of teaching", is about like saying, "I make my stew with pepper." The phrase may be descriptive, even informative, but it is hardly definitive. Socratic prompts are one ingredient that can be used in teaching, just as pepper is one ingredient that can be used in stew. Pepper may be important enough in a recipe to determine a name - "pepper stew". But "pepper stew" will not really be stew if it contains only pepper, and if the other ingredients are poorly chosen pepper will not be a magic ingredient that will make it a good stew.

      "Lecture" is sometimes mentioned as a method of teaching, but it is not really a technique of teaching so much as a method of presentation. By itself it does not elicit or evaluate feedback from the learners. It does not provide for practice or review. It does not manage the time and efforts of the learners. By itself it has more the nature of a textbook than a technique of teaching. One might call it a "verbal textbook". It may be an important ingredient in the process of teaching. In many college course it may be by far the most important ingredient, but it is not the whole thing.

      "Discussion" is much idealized as a method of teaching, but like many other things that are overidealized it falls short. By discussion I envision a teacher eliciting thoughts and opinions from the class. Each thought or opinion is carefully listened to, but the teacher does not generally want to label it right or wrong. It assumes the “philosophers perspective of teaching”, as I discussed above, but that is incomplete as a form of teaching. It does include the essential ingredient of feedback, which lecture does not, but it fails to include the directedness that good teaching involves. Discussion, as we generally think of it, does not include a careful, direct, intensive, and extensive explanation of a topic by a person who understands it to students who don’t. And discussion is not drill, but in many subjects drill is sometimes needed. Thus we don’t teach algebra by discussion. Discussion may be an ingredient in various methods, and at times it may be a valuable ingredient, but it is not the only ingredient in any method.

      The "project method" says something about the organization of the subject matter, but says little about presentation, feedback, or evaluation. The same is true for the “spiral method”, as I discussed in a previous chapter.

      None of these methods, in actual practice, are pure. A teacher who insists that he lectures, will answer questions at least some of the time, and he must give homework and tests at times. A teacher who insists that she uses "discussion" will exert some effort to stay on the subject, and will lecture, give Socratic prompts, give tests, and so on. A teacher who thinks of the project method as the central strategy he uses will surely be forced to break up the project method at times. And none of these methods are definitive. They are simply adjectives that are informative to a greater or lesser degree, just as a term such as "beef stew" is informative, but not definitive. "Beef stew", "turkey stew", "pepper stew", "Irish stew", "Uncle Fred's stew", and many other terms may be meaningful terms, but they are not definitive. Every stew must have some basic ingredients like meat and potatoes. And every teaching method must have basic ingredients, as mentioned above.

      So I do not use the terms "method", or “recitation” as definitive. Classroom practice needs a more general term, a "generic stew" of teaching, something that means "going over the material". Daily classroom practice can include a lot of things and it can omit a lot of things. It is a combination of ingredients. And, just like in a stew, one combination of ingredients may be sublime, and another combination of ingredients may be mediocre.

      The point of this chapter is to discuss the important considerations that are involved when one contemplates "going over the lesson". And apparently “going over the lesson” is the best term we have.

      When a teacher begins a class she may be thinking, "Fifty minutes to fill. What in the world will I do?" Or she may be thinking, "So much to do and only fifty minutes available. We'll have to work fast." I have been in both of these extremes. It's definitely much better to be in the latter situation than the former. Most teachers on most days, I think, are somewhere in the middle on this continuum, but usually closer to the “how can I squeeze it all in?” end of the continuum. For the moment I am going to speak from the former perspective. How can you fill 50 minutes?

      You fill the fifty minutes with the basic ingredients of presentation, practice, diagnosis, correction, and evaluation, and sometimes you add some not so basic ingredients for a special flavor.

      Going over the homework assignment is a very common way of beginning a class. If the homework was carefully assigned so that it elicited a good effort from the students, there will be some questions, or points to be clarified. If questions are not spontaneously forthcoming from the students the teacher may call on individuals to give answers to problems or questions. The teacher presents prompts, either high level or low level, or perhaps some of both. The answers not only provide feedback by which problems may be identified and dealt with, but activity itself provides practice. This takes time.

      Next there is usually some new material to be presented. This presentation typically is by lecture. This takes time. Usually there is something in the new material that needs some directed practice. This can be pure drill, such as when a teacher drills the class with the addition flash cards. Or it can be by going over a few basic ideas by questions from the teacher and answers by students. Directed practice shades into pure discussion, as when the teacher throws out a very general question "to get the ball rolling" and then a genuine wide ranging discussion ensues. In most situations the "discussion", or the "recitation" is somewhere between the two extremes of formal drill and free discussion. I think it is a very important point at that both of these extremes and every point between them can have a valid and beneficial place in the classroom. There is a continuum between the highly structured drill of flash cards and the completely unstructured open discussion. Any point on the continuum may have value, but only as one ingredient in the stew. No point on the continuum is optimal for all situations.

      So it is not at all hard to fill fifty minutes. With a little experience a teacher comes to know the things that need to be done for effective learning. Then good planning is needed to fit all these things into the available time. When things are going well the time goes fast, and the teacher must stay focused or time will slip away leaving important things undone.

      In conducting a lesson (or recitation, or activity, or whatever one might wish to call it) one uses all the principles I enumerated in previous chapters. I will briefly recount some of them here. In Chapter Two and Five I discussed structures of knowledge and how this concept affects how information is presented and practiced. In chapter three I discussed diagnosis. An experienced teacher comes into the class each day with some idea of how the students' thinking might be wrong. The problems shown by last year’s students will probably be about the same as the problems students experience this year. Perhaps most importantly there will be fragile structures. The teacher should not assume too much fluency. Perhaps last night when doing homework the students learned the basic facts about the battle of Trenton. But this morning those facts have deteriorated. Some have simply disappeared from the students' mind. Some have lost their grounding. Going over the lesson will once again establish those facts. Some knowledge will disappear again over the summer vacation, perhaps, but perhaps a core of knowledge will remain for a while. Facts are an important first step in history, but only that. Do the students know the importance of the Battle of Trenton? Why is it covered in the chapter? What are the antecedents, and consequences? That will have to be checked out by questions, answers and discussion.

      In Chapter Four I discussed correction of learning problems. Much correction will take place in going over the homework. More correction will take place when the teacher decides what must be done on a class basis. Will the class move on to the next lesson in the book? Or should they spend another day on the same lesson? Is more practice needed? Where will more practice come from? Must the teacher write out and copy an extra homework assignment? Is more explanation needed? Should it be a repeat of previous explanation or should a different approach be used?

      Chunk theory should be considered. Was yesterday’s new material just the right amount? Was it too short, or too long? Was today's homework the right amount, or too much or too little? Were the problems too easy or too hard? Was the intensity of mental effort too much or too little?

      So far I have been discussing classroom practice from a one-day perspective. Of course a much larger perspective is needed. How does today's topic fit into a coherent course of study? How does it relate to what came before, and how will it relate to what is to follow? I discussed this in Chapters Seven and Eight.

      As time permits the teacher may give very careful thought to the subject matter in terms of the concepts I discussed in the chapters on types of learning, skills, habits, and reasoning. She may consider whether an operational or definitional approach would be more appropriate for a given topic. She may consider the response levels that are most beneficial for different topics. She may consider the prompt levels that are best.

      All of the above discussion, of course, is from the management perspective of teaching, not from the performance perspective. But the performance perspective is also important. A good recitation is like a good church service. It is not primarily entertainment. It’s primary purpose is to deliver a message. It may be entertaining, but its entertainment component should never supplant or diminish its primary purpose. When done right it is more than entertaining. It is educational in the best sense of the term.

      With this basic perspective of daily classroom practice, I now want to introduce a few more concepts. First I will introduce the idea of "slide" versus "catch" and explain how these concepts fit into daily work in the classroom. To introduce these two ideas I will present two scenarios:

      Scenario #1 A kindergarten or preschool class of about ten children is gathered around a table with the teacher. On the table the teacher has a pan of water, some tin foil, a metal pie plate, and perhaps a few other items. She talks to the children about floating and sinking. She shows that the metal pie plate will easily float if laid carefully on the surface of the water, but will sink if slipped under the surface of the water. She gives a similar demonstration with the tin foil, showing that it will float if made into a boat-like shape, but will sink if not. She explains, as best she can, the principles involved. She asks the students to think, observe, and analyze along with her. After about fifteen minutes of this they move onto to some other activity on their daily schedule.

      Scenario #2 A high school physics teacher is talking about floating and sinking. On the blackboard he has diagrams of a cube under water. He leads the class through calculations about water pressure as a function of depth, about pressure on the sides of the cube balancing out to zero, about pressure on the top and bottom of the cube not balancing out, and about floating and sinking as a resultant of total force on top, total force on bottom, and weight of the cube. This takes about twenty-five minutes. During this time the students ask some questions, and he answers them carefully.

      In scenario #1, the learning can slide. In scenario #2 the learning must catch. To "slide" in this context means that whether the students understand or not, the class will continue. Those who do not understand will not be subject to any censure or indication of failure. It is hoped that many of the students will understand and learn at least something about the immediate topic, but it is not required, or really expected. The goals in the kindergarten are very generalized. An actual understanding about floating and sinking is not an end in itself, but is seen as simply one means to these more generalized ends. In the high school physics class the goals of the day are important. If Joe doesn't understand the calculations and principles he is expected to persevere until he does. The material will appear on homework, quizzes, and tests. Some sliding may occur and be tolerated, but not a great deal. Students who slide by substantial amounts of the learning every class period will make poor grades in the course and certainly will not continue in any field that requires physics. Someone who slides too much will probably not even continue on in the course. The learning must catch if the learner is to continue. In the kindergarten scenario, in contrast, a great deal of sliding is easily tolerated.

      Sliding and catching refers primarily to discrete, measurable learning. In a music class, for example, appreciation of music may be an important goal, but it is very hard to measure. We cannot demand that music appreciation "catch", at least not in specific instances. In playing an instrument, on the other hand, progress is easy to measure. If the teacher wants the clarinet players to be able to execute a particular series of notes in a band piece, he can easily determine whether they have or have not learned it. He can then decide whether that particular passage is important or not, whether to keep at the clarinet players until they "catch" it, or whether to give up and let the learning "slide". Decisions such as these may occur many times during one class period.

      There is a continuum between lower and higher education in this regard. In kindergarten a student may do a great deal of sliding. In second grade this has changed considerably. The addition facts much catch. It is not acceptable for a student to slide through the whole school year without learning them. In high school it is even more important that knowledge catch.

      The “Slide versus catch” idea is similar, but not quite the same, as the “learn-and-remember versus learn-and-forget” idea. Many small bits of learning have an immediate purpose, and thereafter may be forgotten. Looking up a phone number can be an example of this. One must learn the number in order to dial it, but once the number is dialed that bit of knowledge may be dropped from mind completely. Such bits of learning are not framework structure. They are not even supporting structure. They serve only as a temporary supporting structure, totally dispensible once their function is served. But many, many details in any subject are of this nature. For example one might look up the battle of Trenton in the index of a history book. The bit of knowledge that it is on page 342 need be remembered only seconds before that page is found, and that number can be forgotten. But it is not only extraneous learnings such as a page number or a phone number that may be quickly forgotten. In a given context many details about the battle of Trenton might be momentarily learned, and then forgotten because they are unessential. One may be interested in the Battle of Trenton only as it bears on some other historical even that is of interest at the moment.

      “Slide versus catch”, as least as I am using the term, refers more to understanding, or at least fluency. An idea about why things float or sink, for example, is not immediately understandable. A difficult clarinet fingering, as another example, is not immediately accomplished. These things cannnot immediately be learned. Other things to be learned, such as a page number, are very easily learned, but also very easily forgotten. I do not use the “slide versus catch” terminology for these learnings. “Slide versus catch” concerns structure building. “learn-and-remember versus learn- and-forget” concerns brain packing. To say that an idea or concept doesn’t catch means that it is not understood. To say that an idea or concept is forgotten usually means that it was understood, it is just no longer needed.

      Separating essential learnings from non-essential learnings is an important part of teaching. This is the reason that the concepts of “slide versus catch” or “learn and remember versus learn and forget” are relevent to everyday classroom practice. Essential learnings must be given emphasis, or practice, or orientation. Refer again to Figure 2 in Chapter Two, the diagram about going to the movies. Every element in that diagram must be present in the learner’s mind if the learner is to understand it. That is to say that every essential element must be learned and retained, not learned and forgotten. Only when each essential element is present in the learner’s mind can the idea make sense, the learning catch. Interspersed among these essential bits of knowledge in the diagram would be a zillion other details about the people involved and their lives, many of which enter and leave the mind on a learn-and-forget basis. Imagine the mother in this diagram, whose is intimately familiar with the lives and happenings of the people involved, explaining the situation to a friend who is not so intimately familiar. Understanding by the friend is not possible until the essential facts are retained in mind. In a discussion many more facts may come up, but may be forgotten. The mother trying to explain the situation is a teacher, and to be an efficient teacher must have clearly in mind which of many details that might be discussed can be forgotten, and which details are necessary for understanding. A teacher explaining a math problem must have clearly in mind which facts or ideas must be retained in the minds of the learners if understanding is to occur, and must help the students learn and retain these facts or ideas.

      It may appear that any bit of knowledge may be put in the “learn and forget” category depending on the context and the goals of the learner. Indeed after a final examination a whole course, or at least what seems like a whole course, may enter the “learn-and-forget” category. But that misses the point of the structure of knowledge idea. After a course is done and the final grade recorded most people are no longer trying to build a structure of knowledge. It is when the structure of knowledge is to be built that the distinction between essential and nonessential parts of the structure is important.

      A word about direct or indirect teaching seems in order here. Most of the teaching we do, we do directly. That means we can readily identify the learning we want to occur and we directly teach that knowledge. If it is conceptual information it can be put into words. Putting it into words is certainly not the sum total of teaching - I have dwelled at length on that - but it is usually an early and essential part of teaching. If the learning is perceptual or motor, words are invaluable in guiding the learner, and teaching is still very direct. But there are some learnings that can only be approached indirectly. Appreciation is a prime example. How does one teach music appreciation? A personal example that comes to mind is a course my wife took in college and felt was very much worthwhile. It covered, I believe, the American musical theater in the first half of this century. I think it is fair to say that the teacher's goal was music appreciation, or more generally, artistic appreciation. In my wife's case at least, this goal was accomplished. But I think it is also accurate to say that it was accomplished primarily indirectly. One cannot teach appreciation directly. One can teach information directly. And this is what the teacher did. Of course it was not a catechism. A structure of knowledge was built, a structure of accretion primarily. The course consisted of learning information, of organizing, understanding, and assimilating information. A structure of knowledge was built, and it was built very directly. But appreciation, which is the real value of the course, cannot be taught directly. It has to be a by-product of the structure of knowledge.

      Learning activities can be put on a continuum of directness. Drill, such as multiplication drills with flash cards, is very direct. Doing a page of problems, or a set of questions in history, is very direct. A classroom game or contest in which the students respond to prompts to win points is a less direct. Doing reports or presentations to the class is less than direct, though perhaps not yet deserving of the term “indirect”. A project of some sort may be a little less direct yet. In fact it may be so indirect so as to be ineffectual. A field trip to a factory may be very indirect, having the goal only of gaining a few observations. Teaching appreciation of a subject may be so indirect that it is impossible to measure.

      It would seem sensible to always have the rule to teach as directly as possibly, as directly as the nature of the subject will allow, but that does not mean that only direct goals are worthwhile. Indirect goals, such as appreciation, may be very important in the long run, even decisive. Good citizenship is a very important goal of social studies, indeed of all education, but I am not sure it can ever be approached directly.

      Next I want to expand on the "concentration principle" that I introduced in Chapter Seven, and to join it to a complementary principle. In that chapter I simply pointed out that effort must be concentrated if learning is to be efficient. Generally in building a structure of knowledge one works with a tiny little bit of the structure at any one time. If the learning is to catch, as discussed above, then one must concentrate on that tiny little bit of structure. I discussed this in chapter Seven in relation to the "activity method" of teaching. I argued that an activity such as baking cookies might be a context to provide some practice in adding fractions, but that the activity would not provide the concentration needed to attain an adequate degree of either understanding or fluency in adding fractions. Only the old fashioned activities of lecture, homework, and drill can provide that.

      Failure to isolate and concentrate is an invitation for the learning to slide. In seventh grade math the learning should not slide. Every year in math is a foundation for the math to follow. If this foundation is faulty the more advanced math cannot be developed. The learning must catch.

      But learning does not all occur in isolation. If we are to build a coherent body of knowledge the little isolates of learning must be connected. That is the whole idea of having a structure of knowledge, rather than just a collection of facts. If the seventh grade math students can fluently do the problems of dividing fractions on one page of their book, but have no idea why they are doing them, or how those problems fit in with other topics of math, then we have failed. We have to be able to relate this one technique to the larger body of knowledge. For example, a student might be able to take the prompt:

      8 3/4 ÷ 1 3/8 =

      and go through a process ending up with the right answer. This ability is very important, but it is incomplete in itself. Suppose he is faced with a prompt like this:

      John has 8 3/4 gallons of gasoline in the tank of his car. It takes 1 3/8 gallons to drive to work. How many times can he drive to work before he must get more gas?

      Will he immediately translate to the division problem stated above? If so he has connected the process of division to a greater structure of knowledge. If the teacher is not careful, however, it is quite possible for the students to be able to do the division problem, but not to be able to do the written problem. By isolating and concentrating one can build a small structure of knowledge. But next one must spread and relate. The "concentration principle" is only one half of a duality. The other half of the duality is just as important. One must tie the small structure of knowledge to the larger structure of knowledge. It is not enough to "isolate and concentrate". One must also "spread and relate".

      Sometimes the relatedness of an element of a structure to the rest of that structure is easy and obvious. For example, suppose a mailman learning a new route must find "Pine Circle". Once he finds it, as coming off Smith Street, for example, then the relation to the rest of the map in his head is obvious. No special effort is needed to “spread and relate”. But in other cases the problem is not so simple. The connection of one particular bit of knowledge to the rest of the structure can be very obscure, yet important. In such instances considerable thought of how to spread and relate may be needed.

      Going back to the dividing fractions example, there may be one page of drill, simply problems of dividing mixed numbers and fractions. On the next page of the book, after a bit of explanation, there may be a set of ten written problems that are solved by dividing fractions. This relates the technique of division of fractions to at least part of the rest of the structure of knowledge. But this is not the final step. The student knows that on every problem on this page he must divide one number by the other. The only question is which number goes into the other. What will he do with a page of problems that are not all division? Suppose he gets all the problems right on the page of written problems of division, and then misses most of the problems on the next page because he doesn't know whether to divide, or to multiply, add, or subtract. The successful page of division problems, in this scenario, may be the result of the isolate-and-concentrate principle. It does not necessarily follow that he can relate this bit of knowledge to the larger structure of arithmetic. The failure to be able to do and understand the next page of broader problems in arithmetic would be evidence of this. Isolate and concentrate is always important, but never enough. Relating should never be taken for granted. Any topic must be learned with some degree of isolation and concentration, but that topic should not be considered done until it is related to the rest of the structure of knowledge in the student's mind.

      It is important that the spread-and-relate goal is accomplished primarily by the teacher, not by the textbook. The three pages of division of fractions problems that I described may be a valuable tool for the teacher, but only that. In a class of bright students well prepared in arithmetic those three pages may be more than enough. In less ideal conditions the teacher may need to do a lot of explanation and make up more problems for practice.

      Relating is often more than just making a connection. It is often differentiating, integrating, or contrasting and comparing. When learning about the Civil war there ought to be some comparisons to other wars, and cause and effect connections to other things of that time. When learning the fingerings on a trumpet one ought to be able to relate the fingers in the lower octave to the fingerings in the upper octave. When learning the quadratic formula, one ought to understand how it derives from “completing the square” and how that relates to factoring.

      The process of teaching typically oscillates between the "isolate and concentrate" mode and the "spread and relate" mode. Usually this is easy and automatic. But it should be kept in mind in every daily lesson, for there are plenty of times when it is neglected.

      Practice prompts are an important part of the "isolate and concentrate" strategy. Differential prompts are an important part of the "spread and relate" strategy. Practice prompts are a form of “brain packing”, as discussed in Chapter Five, very necessary in many topics. Differential prompts are an important part of “structure building”, also discussed in Chapter Five. They form the connections that are necessary for a structure of knowledge rather than a collection of unrelated facts. Many prompts have some element of both.

      A final pair of complementary ideas I would like to discuss are the “box of one escape”, versus the “teachable moment”. The idea of the teachable moment, I believe, is well established in educational thinking. It is the idea that there are times when events bring about a situation which is very conducive to teaching some important idea. For example a blizzard may provide the opportunity for a sixth grade class to learn something about weather. Or, on a more personal level, a student’s misbehavior that causes injury to a classmate provides an opportunity to learn some important lessons about safety and responsibility. I do not wish to denigrate this concept. Teachable moments do arise from time to time, often completely unexpectedly, and no opportunity to teach important lessons should be passed over. However I will argue that such teachable moments have very little to do with everyday classroom practice. Normal everyday life will never provide the impetus for learning all that we want children to learn. Rather we must contrive the situations. In math such contrived situations go under the name of “story problems”.

      Ideas are normally presented verbally or in text. In most subjects we expect the students to be able to apply those ideas. Indeed applying ideas is a very important way that we spread outward from the idea and relate it to the rest of the structure of knowledge. So we contrive situations. And being contrived, these situations can be tailor-made to do the job right..

      The box of one escape is the idea that a contrived situation, a problem, a question, or a prompt of some sort, puts the learner in a box, and the only escape from that box is to apply the idea that we are trying to teach. A math problem is a box, and the escape from that box is to get the right answer in the right way - by applying the idea that we are trying to teach. Any prompt, high level or low level, puts the learner in a box. To escape the box, if it is a well designed box, the learner must either recall information, or apply an idea, or in some other way demonstrate learning. In an hour of class time a teacher may devise many boxes, and may modify or refine many of these boxes for better learning. This is just to say that prompts are very important in hard core teaching and learning, and should be given careful thought..

      The teachable moment idea fits into the project method, or the activity method. Presumably an activity like baking cookies (to again use a well worn example) provides a teachable moment for learning many things. My argument, that I have made many times, is that the net result is not at all adequate for the amount and depth of learning that we should expect. The box of one escape idea fits into the normal classroom practices that teachers use, lecture, practice, homework, prompts and responses, quizzes, tests, and so on, that are sometimes called “direct instruction”.

      I would like to end this session with a review of the many progressions I have mentioned along the way, with some mention of how these progressions are often overdone in the educational literature.

      Learning progresses from a very fine texture at the lower level to a very coarse texture at higher levels. In the first grade perhaps no more than a minute of information may be given before the students are asked to do something with it, either respond to a question, or start some activity. In seventh grade students may be expected to deal with much larger chunks of knowledge. At the college level students may have to sit through a whole semester of lecture before being asked to demonstrate evidence of learning.

      Learning progresses from the concrete to the abstract. In math, for example, students in the first grade must be very concrete, dealing with objects as a way to eventually get to number as an abstraction, at least in an operational way. In algebra numbers may seem very concrete, but equations, variables, operations, are abstractions to be dealt with. In college these concepts seem concrete and enable us to deal with even more abstract concepts. This is not to say that there are no abstractions at the lower levels of math, however. Getting the idea of number as a result of learning to count is to make an abstraction, and that is a very important abstraction. Getting the idea of addition is to make a further abstraction, again a very important one.

      As part of this concrete to abstract progression, we progress from descriptive to analytic. In kindergarten social studies would be primarily descriptive. History would be descriptive and narrative. Science would be primarily descriptive. But by seventh grade history should not be only descriptive and narrative, perhaps not even primarily descriptive and narrative. It can be much more analytic, and good teachers will make it so.

      Integration of subjects to differentiation is a progression that takes place with advancing education. In the fourth grade students study “science”, no further differentiation required. At the high school level this is differentiated into biology, chemistry, and physics. At the college level these subjects are subdivided many times more. Under buzzwords such as “interdisciplinary” or “project based” we sometimes try to go backwards in this progression. Perhaps there are times when this is justified by the topic being considered, perhaps required in order to spread and relate, as discussed above. But in general we differentiate more as we progress. That is just a part of the isolate and concentrate idea.

      Learning must spiral a great deal at the lower levels of education. We don’t expect students to be exposed to nouns, verbs, and adjectives once in the fourth grade and then never again. Rather we expect a considerable amount of spiraling over a number of years. In a college course that is not the case. We expect to present a topic only once.

      Student control of their own learning is an idea that is often proposed. It often seems the “progressive” thing to do, as if it were a brand new idea. Of course it is never a brand new idea. It is a part of any education, but the degree varies with age and experience. In first grade students can take little responsibility for their own learning. In college students must take a great deal of responsibility for their own learning. An educational fad that advocates letting fifth graders direct their own education is just advocating a readjustment of a natural progression. If the fad calls for too much control by the student, a degree of control that is not realistic, then it is an abdication of the teacher’s responsibility. If the fad calls for the teacher to minutely control everything the learner does, then it just frustrates everyone, for fifth graders are capable of taking some control of their own education.

      The whole idea of the management perspective of teaching, indeed of the idea of teaching itself, is that having one person direct the learning of the other can result in efficient learning. This is most true at the lower levels of education, where the student bears very little responsibility for their own education. But it is true to one extent or another at any level of education. Even at the college level there is much a teacher can do, by making well thought out assignments, to assist learning.

      In Chapter Ten I discussed the idea of “cortical lead” in reference to learning skills. This term also fits in with academic learning and the control of that learning. In the lower levels of education there is very little “cortical lead” on the part of students. They primarily do what the teacher tells them to do, and learning follows. But by middle school, for most students at least, that changes considerably. Students understand that they can decide to learn something, and they can act on that decision. They understand that they can make certain choices of what to do with the end of learning in mind. This can certainly be called “cortical lead”. In the fourth grade it may not extend to a learning task beyond learning a list of spelling words. In high school it may involve deciding to keep working on a chapter in a history book until it makes sense. In advanced education it may involve choosing a problem to investigate for a thesis or dissertation.

      I think this progression of responsibility for learning from the teacher to the student is social as well as academic. It is entirely proper to scold a fifth grader who doesn’t do an assignment. College students don’t want to be treated that way, and college teachers don’t want to treat them that way. Good teachers at any level do more than just present information and then test on it, but the degree to which they can impose control of learning depends on age and cultural norms, as well as the nature of the learner and the nature of the learning task of the moment.

      All these progressions enter into daily classroom activities and decisions. Everyday, indeed every hour, the teacher makes choices. There are ways to be a little more abstract, or a little more concrete, in presenting a particular topic. There are ways to be a little more or a little less controlling of the students activities. There are ways to make the texture a little more coarse or a little more fine. There are ways to isolate and concentrate a little more, or to spread and relate a little more.