Pedagogy and Content:
A Dialogue About the Relative Importance Between
Subject Matter Knowledge and Instructional Technique

The following is an edited version of a private e-mail dialogue between Bill Hunter and Rick Garlikov. It was begun after the following public response that Rick Garlikov made on the Internet educational group EDPOLYAN to comments by Samuel Lubell:

Lubell had said:
"There's a reason why teachers need courses in teaching that are separate from their courses in English or mathematics. And when people are brought into the classroom without this training, their subject matter knowledge doesn't help them."

To which Garlikov responded: "You are assuming nothing can be learned, and no skills developed, without a course that is taught by someone else. That is often not true. My point is not that all people with knowledge can teach; but that those who can, and who can demonstrate they can, should not have to take courses in order to begin a teaching career."

But, moreover, Lubell had said:
"Of course, ideally a teacher should have extensive subject matter knowledge, the ability to emphasize [empathize] with others, and training in teaching methods. However, a very skilled teacher could teach a course in something he or she knows little about, by requiring students to go out and do their own research and teach the teacher."

And Garlikov had said: "If this is teaching, we don't need schools; just libraries and assignments. I thoroughly disagree with this. It is NOT teaching."
  At this point, Bill Hunter began the discussion with Garlikov.

Bill Hunter: I don't want to engage this onlist, but you said,
"If this is teaching, we don't need schools; just libraries and assignments. I thoroughly disagree with this. It is NOT teaching."

We are far apart on this one. It actually seems that it is you who is now saying that learning involves the transmission of known stuff from one person to another. A teacher is a person who fosters learning in others. This can sometimes be done with relatively limited subject matter expertise. It involves far more than giving assignments. It may include listening very carefully and restating what has been heard, raising questions about what seems not to be clearly understood (either by the learner or by the teacher), finding sources (including other teachers and subject matter experts) that will help the learner to advance, responding to assignments as an intelligent reader (and perhaps finding experts to respond to content), encouraging effort in times of depression, restraining undue optimism, demanding critical analyses, and a good many other activities.

I would wager that you have had the experience of having a student come to you and say "Can you help me with this? I don't understand..." and then either 1) solving the problem her/himself as they explained it, 2) finding a solution as a consequence of your trying to understand the question (and intuitively asking good questions in the process) or 3) getting the solution as a result of suggestions you made about general problem solving strategies. None of these require that you have subject matter expertise. Such students will thank you for teaching them and they are right. The very fact that they approached you (and not someone else) is evidence of your capacity to teach, something they count on to help them learn. Your questions and general suggestions constitute teaching the learning process, something with generic utility. I see this often on graduate defenses. Even though the student, who is maybe two hours away from being a doctoral level expert on the subject, may completely fail to understand a question, it is entirely possible (and quite common) for an external professor from a completely different field to comprehend the question and restate it in a way that leads the student to a new understanding. How is this NOT teaching?

  Rick Garlikov: You are right that those things you so eloquently, passionately, and accurately described are instances of teaching, or at least of helping someone learn. And though I wish there were far more of that in all our everyday lives and in school, I don't think that sort of thing would generally serve very well to teach a class a subject, such as math. There are times in a math class it would, but it would not serve to teach the subject overall in some sort of "efficient" way. Making everyone have to discover everything for themselves would take each of us more time to learn than each of us has. And while I don't think that telling is necessarily teaching or that knowledge is necessarily "transmitted" or fostered by telling, I do think that in many courses the point IS to facilitate the learning of a particular body of material (or at least introduce it in a way that students can begin to assimilate it, reflect on it, etc.)

And if one is not versed fairly well in that material, one cannot likely teach it very well. For example, my wife's expertise is in teaching reading; and she is very good at that. But although she uses many appropriate techniques to get kids to think about math or science, she just simply does not know enough math or science to be able to help kids even "look" in the right places for understanding or for even factual knowledge. And, as I wrote in the Place-Value paper I sent you last year, I think that one of the reasons it is so difficult for children to learn place-value is that most adults (including teachers) don't really UNDERSTAND it and therefore don't do things that facilitate children's understanding it, and probably do things that actually hamper children's learning it.

At the age level I was thinking about (roughly K-undergrad), I think students need to be guided by someone with some subject matter understanding. That is less the case in a grad student seminar or in a dissertation preparation; or with colleagues, where as you said, mature, reflective questioning may spur someone on to understanding you yourself don't have about the particular subject.

Suppose you suddenly awoke one day with the desire to learn to play the violin or to learn how to rebuild an engine; and lo and behold a flyer arrived at your house that day describing the new term's courses for the local college's adult studies program with introductory courses offered in violin and in auto mechanics. $75 each. You dig into your savings and fork over the $150, and eagerly go to your first classes. They are both taught by a person with a Ph.D. in Education -- a person who knows how to read and knows how to inquire and knows how to reason. But s/he has never studied music nor mechanics; but is there to help you learn these things. Are you going to be happy about having spent that $150 this way? I don't think so.

That is the sort of thing I was talking about with regard to "teaching"; I meant teaching a course, not just helping you figure out some sort of problem by being a good listener or a good reasoner, or having general knowledge that is helpful. Does this make sense?

Bill H: I hope I did not seem to be saying that subject matter knowledge is not desirable (your reply seems aimed at refuting that assertion) for a teacher. What I want to say is that it is not the sine qua non (which your earlier post seemed to suggest). To take your examples:

  Math: fascinating that you chose this as your first example of an area where knowledge is necessary. You recall I have some interest in the area, but little actual education (I read perhaps 1/3 of a calculus text on my own at a point when it became clear that absence of calculus was limiting my understanding of statistics). Indeed, this is an area in which I have found I can be helpful in areas that I don't understand myself. This has been mainly with my youngest son, now a math major in his final year (coming). He will say to me " Dad, what do you know about "X"?" and I'll answer "Almost nothing. Tell me about it." or"Never heard of it, what do you want to know?" The ensuing conversation involves my helping him through his own thinking. Fairly often (maybe 30%) of the time, he gets the answer he wants. Often, I still don't understand the question. Now, perhaps he is like a grad student in many ways, and he has the benefit of a math teacher he can go to (but note, he finds math profs singularly ineffective in communicating math). More importantly, some mathematician has POSED the problems he is facing and that is a most critical part of the teaching.

  Music and Mechanics: Excellent--two areas in which I am abysmally ignorant. I grant you that on day one I would be worried about my $103.00 (converting to Canadian). But if I did not withdraw, I can see the distinct possibility of getting my money's worth. Do you know Los Indios Tabaharas? Two brothers, I think, from a South American (I forget where, Peru maybe) rain forest, never exposed to white folk, who found a guitar in the jungle. Not only did they teach themselves (one another?) to play well enough to make it to international concert halls, they also built their own stringed instruments and, upon getting exposure, learned to play classical music. It would be taking things a bit far to say that the person who left the guitar was a teacher, but certainly this kind of independent learning suggests the possibility that powerful learning can sometimes take place with only a little teaching. So, if my music teacher knew how to inspire and support his kind of independent learning, I could end up happier than if I were exposed to someone who wanted to force me to perform in public when I was five (and who was possibly a better musician).

  (Art and music are areas where special skills and knowledge do seem to be an extremely important part of teaching, though, since teaching one to see or hear differently almost requires that the teacher have those perceptual skills themselves. Still, I have supervised student teachers in both art and music and have felt that I contributed to their growth as teachers in their subject area--again, though, they had a classroom teacher knowledgeable in the subject to work with.)

  Mechanics: as a student, I worked one summer on diesel busses. The chief mechanic was a guy who was one with engines, any engines. He had started on Model T's and knew every development from there forward. Marvelous guy. His method of teaching was a bit like Myogi-sahn (Karate Kid)--he gave me a starter and said rebuild it. He checked back from time to time to show me how to use a tool or to indicate a part that needing replacing (say brushes) and when I had finished he said, OK, now there are four more starters, I can't take the time to help you with those. Later, it was brakes, fuel lines etc. My initial clumsiness and ignorance appalled him (how could you reach 21 and know so little about mechanics?), but by the end of summer, he wanted me to quit university and apprentice with him (he said he had never had any body working with him who only had to be told or shown something once). Working with him would have been fun and my curiosity is pretty much endless, but despite how much I learned and how much I admired this old guy, he had not managed to spur in me any desire to learn more. This may be more like what you are calling teaching--he had knowledge to give that I was reluctant to take. I learned what I needed and no more. I was happy to get out of there even though I was paid pretty well and I liked all the people--I still hated engines. He was a good mentor, but lacked something as a teacher (or I was a hopeless case--a very distinct possibility in this realm).

  I think it is clear that you see that knowing the content enables one to have a sense of what is to come and how that ought to be organized for most people. What I am saying is that there are some generic skills in learning and that a knowledgeable teacher knows how to get others to use them. One of these is finding sources. Another is seeking help. Another is organizing content for oneself. Another is testing the applicability of what is learned. And so on. These things can be done with groups as well as individuals. I am still persuaded that the best teaching I ever did was when I first taught educational computing. I was only a few steps ahead of the class and I was often able to say, in all honesty, "I don't know--how could we find out?"

  I have often thought that what I would really like to do is to teach a high school class like we teach elementary--all the subjects with me. Of course, I'd want to do this in a building where there were people who knew a lot more chemistry and physics than I do, but I think I could get some dramatic results. And I am sure you could too (given what you have said about your approach to teaching philosophy). The real difficulty would be in assuring that kids got a balanced education, since the tendency(for me) would be to really work on the poetry with those who were passionate about it and ignore the chemistry and vice-versa. Of course, British secondary education does precisely that --much earlier specialization. And they survived Margaret Thatcher.

  Well, I've wandered greatly. If I may try to state your case for you, I think that what you actually lament is teachers that have little or no subject expertise in _anything_ (this is why elementary schools seem to trouble you more). People who would claim to know how to teach but who know nothing TO teach. Here, I am in full agreement with you. A teacher should be a skilled learner and that learning should be evident in knowledge of something. In Canada, many of our elementary teachers have a B.A. or B.Sc. in something else before they become teachers. They can think about questions in history as they relate to what they know about science or math (or vice-versa). This is not a guarantee that they will be able to help students in other areas, so they do get teaching methods in all of the subject areas (at least the academic areas) which includes attention to the curriculum content at the various grade levels.

  Sorry, I am rambling. My point is meant to be simple: I acknowledge that subject matter expertise is part of the "baggage" of a good teacher (and I would not hire a teacher who did not have it). However, I think some powerful teaching takes place even in the absence of that expertise. Part of the reason that I don't want to go on list with this is that I am as likely to be arguing the other side of this case as you are--especially with respect to our Keegstra case--the high school shop teacher turned social studies teacher who taught that the holocaust never happened. He is, at least in part, a victim of a system that did not require him to have specific subject matter expertise and so he read right wing propaganda with all the naive faith of a lay person unable to judge the quality of evidence in the discipline. And then he taught it with a veneer of "critical thinking"because he was challenging the status quo and getting kids to "think for themselves."

  I am not looking to persuade you here, Rick. More like thinking out loud in response to your own provocative thinking.

  Rick G.: Fascinating! You are still right. But there must be something we are still leaving out. You have some of my part of it, but not quite all; but I am not sure what is missing. I agree with you about much of what you say, and I think I teach the way you do in regard to the kinds of things I feel I am able to teach. But, given my knowledge and my "general" inquiry skills, etc., there are many subjects I believe I could not teach/(help-anyone-learn) ADEQUATELY at all. Whatever is missing in those cases (and I am sure there are such cases) is what is missing in what I am trying to say. I think.

Perhaps it is knowing WHAT to inquire into, or knowing a reasonable way to structure the inquiry. Your mechanic mentor gave you something that COULD be repaired, for example. And your son's math professor knows which problems are important or significant and meaningful to assign. If they did not know the subject very well, they could not do that.

When I write isolated pieces I forget that they can be read as isolated pieces. I may never have written where you might have read it, that I think there are two main aspects to what good teaching is: (1) inspiring students to want to learn more, and (2) helping them acquire the skills/knowledge to do so; especially on their own. So I did not really intend to imply I thought subject knowledge was the sine qua non of good teaching. I think it is important in most subjects to do both of the above, however. Except in the kinds of cases you point out. I wonder whether there are not two "mitigating" factors in those kinds of cases.
(1) Students who are perhaps a bit more readily motivated to seek answers on their own, or in companionship of the sorts you describe,
(2) teachers knowing enough about the subject or something so closely related to the subject (i.e., logic for math; or logic plus some math; or one kind of math for another).

I might be able to teach calculus to some kids, but not to kids who already know calculus fairly well; they would be ahead of me. And though I might motivate them or sometimes ask some challenging, prodding, helpful questions, I don't think they would ultimately appreciate having taken calc from me instead of someone as motivating and prodding as me/you who ALSO knew calculus and could steer them toward relationships I would not even know existed.

I took a broken camera apart once. When I undid one something-or-other, billions of little pieces shot out everywhere, like some sort of chain reaction run amuck. That camera was good and broken then. I learned from that, that camera repair was much more complicated than met the eye. I don't think I could teach camera repair in any way --though I MIGHT be able to help some really desperate people learn it if they had lots of time and if we had sufficient money to go through enough cameras till we got it right.

Most other things I have been able to repair even without knowing how they worked at the beginning of the endeavor. The "man of my dreams I have met" is a guy who operates a parts supply place for almost anything in your home. What is really cool is that not only does his store have all this stuff, but he helps you get what you need, and tells you what to look out for when you install it. Some of it I would figure out on my own, but not all of it. Some of it is tricky and you have to have screwed up once or broke it once or simply seen someone do it once in order to know how to do it. I consider him "teaching me" the things he explains -- and more efficiently than I would have learned it, though I also understand what he is teaching better when I get into the operation myself.

Bill: Your insertion of ADEQUATELY makes a big difference. Adequate for what purpose and for whom? See, a lot of what we (educators all) do is decide what is adequate for others--a useful function given our expertise and one that actually does require expertise. But many times, the learners have other objectives (as I did as a mechanic) and our definition of adequate may be either insufficient or far too demanding. I think a lot of what I am saying applies to the circumstances in which the learner is free to determine adequacy (that is, all learning not taking place in school or as a job requirement, I think. No, see, that's the kind of stupid thing I could say and have it appear onlist when I don't really mean it. Clearly in the arts, one learns for one's own enjoyment and yet trust's to someone else to determine what is adequate because the learner know that he/she lacks the criteria to judge adequacy, but the learner ought voluntarily to yield this right--maybe that would apply to all learning, but I suspect that society (school boards, parents, etc) might take a different view.

  Rick: By "adequately" teach a course, I mean teach the course in a way that helps the students learn it in some relatively efficient time frame -- relative to, say, how a good teacher would be able to help students learn it. For example, when I teach the photography, I take two lessons, of about two hours and one hour. UAB Special Studies takes one or two hours each week for six to ten weeks, and doesn't cover much of the stuff I do. I would say they don't teach it adequately. My friends who took intro photography at Michigan were not taught it adequately because it generally killed their interests in it and made them do a lot of unnecessary, and unhelpful work. And they didn't learn much about taking good pictures. And the people that taught the material were knowledgeable in some ways about photography, but not in how to teach it.

On the other hand, if I tried to teach molecular biology, though I might have some more interesting questions for students than a molecular biologist, I am starting from scratch and don't have any knowledge at all about what the "highlights" might be to get "into" the subject in some efficient or reasonable way. By knowing enough to "adequately" teach photography, I mean being able to teach it in such a way that the student realizes s/he is getting his/her money's worth -- not because they learn a lot of material (actually, they don't), but because I help them UNDERSTAND (how to do) photography. My big mission in teaching anything is making/keeping it interesting, and making the subject UNDERSTANDABLE. I would change what I said before to what I seek to do is to organize the material or the "instruction" (including the tasks and questions I give to students) in such a way that the students will understand the subject, or have a framework of the subject, in such a way that they can fill in their own knowledge gaps.

Some subjects are difficult to learn on one's own without a good resource --whether teacher(s) or book(s). In the place-value paper I think I told about my history 101 course where MY lecturer didn't do anything to frame the 80 pages a week we had to read. I read all the stuff twice each week, and it just made no sense to me at all; it was just an endless succession of popes, and kings, and wars. I couldn't keep them or their causes straight, and couldn't see any threads or anything on my own to organize them with. My friends had a lecturer who provided frameworks and patterns, etc. And they were able to use that to organize the 80 pages of stuff into something meaningful to them. I couldn't see my European history course as somehow a way of progressing from early times to the present; it wasn't progress for me, but merely change. Art history I saw differently; I could see patterns. But like my learning BASIC programming, it took two differently organized books for me to be able to begin to see the framework, etc. (I learned BASIC from books on my own, but it took me a year to do it. Had I wanted to learn more quickly, and probably more, I would have been better off taking a course from someone who knew it and who could teach well.)

I definitely don't teach so that students remember stuff; I can't do that. They have to do that. I may try to make stuff interesting, and that may make it somewhat memorable; but I see my role as trying to help students understand stuff as they think about it; and as trying to make thinking about it interesting, stimulating, and fruitful. Are we getting any closer? I have the suspicion we teach very similarly, but that we are each having a devil of a time describing in words what we actually intend to be doing --in a way that the other sees what we mean. This is really challenging to me.

Bill: I think I was avoiding the use of the word "evaluate." When you said you could not teach something adequately, it implied to me that YOU would make a judgement about how well something you had tried to teach had in fact been taught (learned?). What I was trying to get at here is that this judgement (of adequacy) is often part of the professional service we expect of a teacher--I have sometimes said it is a major portion of what students "buy" from us. So, yes, making that judgement (as a teacher) depends on subject matter knowledge (and in some areas -- the arts in particular since subjectivity is a large element of the judgement--an overwhelming part), but it is also POSSIBLE to conceive of this judgement as rightfully belonging to the student--"Ah, good question, now I see where I have to go with this--thanks--I won't need any more help." This latter case MAY be more typical of the situation in which subject matter expertise is less crucial--if I don't have to make judgements about how well you learned or how well I taught, I may be instrumental to your learning without knowing a lot about what it is that you need to learn. My math son is a case in point--I could ask "metacognitive prompt" type questions and HE could make the judgement about whether or not he knew more, understood better, etc.

Yes, we are getting closer. And your closing lines related to my last question. The difficulty we have in communicating with one another about good teaching is a function (I think) of the difficulty of teaching. We probably do some things similarly, but I also know from previous notes that you do some things very differently from me. nevertheless, I would be delighted to take a course from you and would be fully confident that I would learn a great deal (understand a lot) and enjoy it. I would, however, be hard pressed to choose between photography and molecular biology. You "know" how to teach photography. You would be learning both molecular biology and the teaching of molecular biology relatively concurrently with me and I would find that invigorating. I would feel that my questions might be useful to you and ask more of them. I would delight in seeing new insights emerge on your face as you taught. I would want to be responsible for contributing to your understanding so I would work hard. I grant you I am an unusual student (I also wouldn't care if you gave me a B- and I would likely protest little over a C+--though I imagine I would get neither of those), but I like to think I am one model of a kind of successful student, a kind of student we ought to foster. I would wager that what I didn't learn about molecular biology wouldn't really matter too much and that what I did learn would matter a great deal. But that is because YOU were the teacher--I would not trust every teacher or every prof to pick up molecular biology and teach it to me. There are probably a good many molecular biologists from whom I would as soon not learn. But you would be a studious teacher of that which you were also learning, no doubt putting in many more hours than I would be as a student and sharing your understandings as they emerge would challenge mine as they emerged. I can't help but think this would be great fun. But remember, I would not hire you to teach molecular biology even though I would register for it if you did.

  What strikes me is that when you want to make your case strongly, you return to the classroom. This is where we make those judgments about adequacy and empower others to assess the sufficiency of our learning. If "teaching" is confined to these quarters, then content knowledge is very important indeed.

  How about a contrary example. I reached a point in the teaching of statistics at which I came to understand that I no longer understood what students needed to know. My own curiosity and interest had driven me to the point that I sought to include all the neat stuff I was understanding and I lost sight of what students actually needed of me. They still were learning a lot and I still got good ratings from them, but I had to take a break from the course to insure that I could renew my sense of where they "came from." Do you see that too much knowledge can sometimes impede teaching? I suspect that anyone who has ever been to university can understand that.

  Rick: Good example. Yes, some teachers are too specialized to teach more general courses. Presumably they have the knowledge they need, but don't/can't use it. A professor I knew, call him by the alias of Foreman for purposes of this discussion, was one of, I think, the worst teachers possible, especially for intro philosophy, which he taught too regularly, and for many "lower level" grad courses. Recognized as one of the world's greatest historians of philosophy, he taught stuff NObody cared about. I was one of his Teaching Fellow's one time. Without going way outside of his material, it would have been almost impossible to foster any sort of interest in philosophy among the students in the course. In a grad school course one time he said that he was probably the only person alive who knew what David Hume thought about something or other (I don't remember what he was lecturing about at the time I awoke) "and I'm not going to tell anyone!" We were all SO disappointed.

It is not that I keep returning to the classroom for validation of some sort or because that is where judgments get deferred in some peculiar fashion about what is important. It could be one-on-one stuff too. I teach photography in TWO lessons and charge $150 that they pay at the end of the first lesson if they feel I have taught them their money's worth. They get additional sessions for no additional cost, as they shoot pictures and come in to discuss them with me. That is when what I had organized for them in the first two lessons begins to sink in. I have the stuff organized in much more meaningful ways than the local college adult studies people do who charge much less than I do. But students leave them to come to me sometimes. I "tell" a lot and demonstrate a lot, and ask them questions about stuff I show them in those first two sessions. If I did not, I don't think I would be teaching them photography in the way they want to learn it. After they start coming in with their pictures, it gets more into the sort of mode you discuss, but that is when they are to a certain point that they can self-learn, reflect, etc. about photography. I, by the way, am self-taught as a photographer; but the people who come to me don't want to, or can't, "self-learn". They want to be taught; however that is done.

  Bill: "We were all SO disappointed."

  I can't tell if this is sarcasm. Few of my profs had that kind of distinguished reputation. When one of them could talk first person about someone like Hume (in my case, Karl Pearson or Spearman or Cattell), I really perked up.

  "...stuff I show them in those first two sessions. If I did not, I don't think I would be teaching them photography in the way they want to learn it. After they start coming in, it gets more into the sort of mode you discuss, but that is when they are to a certain point that they can self-learn, reflect, etc. about photography."

  Yes, of course, all that structured stuff you do is valuable--that is what they perceive themselves to have paid for (I'd say they paid for the changes in themselves).

  "I, by the way, am self-taught as a photographer; but the people who come to me don't want to, or can't "self-learn". They want to be taught; however that is done."

  A fascinating point. I can't take pictures that are any good at all. Mainly, I think, because I have never bothered to keep records of what I have done in order to see the effects. I've had some good tips from people like yourself, but most often, I forget to take a camera with me anywhere and then months go by and I forget whatever I had learned. Conclusion: I don't care enough to really learn how to do this. A sad comment on me. Taking a course would help. I have taken adult courses in Japanese, Finnish, and ballroom dancing, all stuff I would have trouble learning on my own (and none of the teachers were good at teaching, but I didn't really care, though I did offer some advice to the ballroom dance instructors). But I do teach myself all sorts of other things. My wife, on the other hand, wants a course any time she wants to learn something (she talked me into all of those courses). It is a running joke around our house: Mom thinks the only way you learn anything is in a course. None of my sons buy that (they were coursed to death with swimming and self defense and cartooning etc.). All are good at teaching themselves (guitar, role-playing games). I wonder if this preference for learning environment is related to thoughts on teaching. Kay would agree with you and, while she is an outstanding teacher, her ways are radically different from mine. Much of the difference centers on how much the teacher organizes stuff in advance--doing that makes students HAPPY, but I am persuaded that putting the burden for organizing material on the students encourages them to learn (I think that if I actually kept records of my settings and results, I would learn to take better pictures in a way that would stick with me--if I took your course, I'd learn stuff that I could use and then forget by six months later when I took the camera out again.) This does not relieve the instructor of planning responsibilities, but the plans become "how can I get them to organize this information?"Rather than "How can I organize this info so they will remember it?"

  Rick: But what if I say I think my role is to "organize my part of it so they will know what to organize for themselves and have a leg up on being able to do it"?

  Even when I teach myself stuff -- such as BASIC programming, or art history, or boomerang throwing -- there are books, etc. that are organized in better or worse ways. Some are better or worse just for my peculiarities; i.e., what is better for me may be worse for you, and vice versa. But I THINK that some are organized so badly it would make it very difficult for anyone to learn, and some are organized in ways that make it easier for most interested people to learn.

  *I* could teach you photography so that it would not require notes, record-keeping, etc. And you would remember the ideas --though probably not stay in practice enough with the details to be able to do it real well, but that would be because of lack of interest. Lack of interest makes learning real hard, no?

  I love David Hume's writings. I consider him one of the finest thinkers and one of the greatest writers of the English language. And to show you then what prowess Foreman had as a teacher, HE made hearing about Hume from him boring to tears. Probably no other mortal could do that. The man gave new meaning to teaching! One could not help falling asleep. Pedantic and mono-tonal! Incredible.

  Bill: Regrettably, I know all too well the kind of teaching you attribute to Foreman --have been guilty of it myself on (rare, I hope) occasion.

  Yes, of course it helps to see how others organize stuff, especially how someone who REALLY knows the stuff organizes it, BUT there are two problems: 1) all too often the task is "learn MY organization of this info" rather than "learn TO organize this info" 2) too, often, the info overwhelms the organization.

  I learned this when one of my grad students gave a brilliantly organized lecture on Jerome Bruner's work. She put it all on overheads and the board as she went with headings and subheadings abundantly evident. I walked around looking at the notes of the 100 or so undergrads listening to her and I found no one (I didn't see them all) who paid any attention to the organizing concepts and principals. The task, as these students understood it, was to "get the facts, ma'am" and nothing but the facts.

  I am trying to remember the name of the woman who studies university teaching by having faculty and grad students from one discipline attend lectures in a beginning class in a very different discipline. She finds that they can't see the organization either. For example, an engineering prof sitting in a history class will say "What's important here? I don't get it. The lecturer used nothing but white chalk--how are you supposed to know what is important?" Or the history prof in a turnabout, will say: "The board was a confusing jumble of distracting colors. Why doesn't he underline the important concepts or put them into an outline form?" Probably we should learn that it is not enough to organize but to talk explicitly about how to find the organization and how to go about creating one's own.

  It is entirely possible that you do this and that it contributes to your success.

  Writing this, I am remembering the first note I saw from you. A lament about teachers and teacher education. I think we had an exchange in which I said I thought maybe you thought it was a lot simpler than it really is (because so much seems obvious to you). I no longer think that. It is clear that you understand the complexity of teaching. I wonder how much of that understanding is emerging from the discussions here and how much has been there all along.

  Rick: Interesting about the complexity of teaching. I don't know how I view that. I still see teaching as, in some ways, much simpler than people make it, though difficult only in the sense of trying to find out what "will work" to help a student understand, whatever. It just seems there are all these obstacles that people impose on that process. It may be that teaching is simple for those who understand it and difficult for those who don't. That may be true of many things. I think photography is really simple, but I realize that is true only for people who "have an eye" and an interest. It is ungodly difficult for people with no interest or "feel" for it (mixing metaphors now, sorry -- at least all in the sense organ mode, though that is probably the WRONG mode...).

The guy who repaired my Achilles tendon said that he found 3-D anatomy knowledge really easy. I never did, but I have trouble with spatial relations in general, especially 3-D ones -- which may account for why I am good at photography; it is easy for me to "see" in two dimensions, which is what photos are, so I can tell ahead of time what will make a good photo -- which is most of the trick; that is, figuring out what makes a good subject. Anatomy, taught at least from a book, or from my own team decimation of a cadaver (the way I studied it in a medical school course), is not so much"complex" to me as it is simply extremely difficult to "see" or to remember.

I think that some of the things we try to teach ed students to do is that same way for many of them -- either because we are teaching some unnecessary things or because some of them will never have a knack for teaching; and the steps are too difficult/complex for THEM. I see the specifics of teaching something to someone as difficult, but not the understanding of what you are trying to do as difficult or complex. Like, it is hard to hit a golf ball long and straight, but the principle is not hard -- unless someone tries to teach you merely by having you memorize a bunch of unnecessary mechanics. I suspect poetry came easily to Shakespeare; but trying to write poetry like he did from studying thousands of analyses of his writing would probably be difficult/impossible, and would SEEM complex. In short, there may be a difference between complexity and difficulty. And though teaching may be impossible for those with no knack, and though it may be difficult with a given student at a given time, I don't know that I would say it is therefore complex. A Chinese puzzle is complex, though it may be easy to do after some practice. I am thinking out loud here, so.... I think we are doing our "molecular biology" course together right now, though about teaching instead of about molecular biology.

I still think what you are calling teaching in the sense you mean it is what I would call collaborative learning, but.... I do understand that someone like you or me teaching a course we came in knowing very little about is preferable to someone like Foreman's teaching it or like the person you mentioned who had reams of notes, overheads, etc. By teaching, I certainly do NOT mean "inundating with information", no matter how well-organized that information might be. UNDERSTANDING stuff is different from knowing facts. I'll repeat an example I gave once (I think); during one of the NFL seasons, the last week still had most teams with a chance to make the playoffs. There were 6 zillion permutations. ESPN felt obligated to run through ALL the possibilities of what had to happen for each team to make the playoffs. (E.g., for A to be in, they had to beat or tie C, but D had to beat E, unless F tied G OR H lost to I....) It took them about 10 minutes, using visuals, and RACING through them to get them all out. Only the ones involving the team you might be rooting for had ANY meaning to you at all, But the announcer, when he finished and took a deep breath, said: "There will be a quiz on this tomorrow." Cracked me up. He realized it was a semi-pointless exercise. But many "teachers" don't see that.

What I try to organize is a "way" of looking at the material, not the details. I deal with details as they arise in some context. I think good teachers tend to do that. The guy who taught music appreciation told us on the last day of class to forget all the details he had pointed out; they weren't what was important about listening to music; they were merely illustrations of the structures he had talked about; structure in general, and how it contributed to enjoyment, was what was important or interesting. He said a parrot could be taught to repeat the details. I found from watching students that freshmen tend to write down everything a lecturer says, but that seniors tend to sit back and listen, and just jot down something as a reminder periodically. I think lots of students tend to distill information as they get further and further through college (except perhaps in courses where the lecturer is presenting a billion facts that WILL be tested and that you can't find in the book). Attend a history of art lecture where there are a mixture of grade levels, and you will see the freshmen/sophomores writing down every detail the lecturer points out about a work of art; seniors just look and think about it. (Generalization, of course.)

  Bill: Good stuff. I like the difference between complexity and difficulty. The former may have to do with understanding and the latter with performance. In developing expert systems (or studying expertise) it is often noted that "expert" performers are rarely able to give accurate explanations of how they do what they do, be it tennis or medicine or mathematical reasoning. So, I hear you saying that for some, the performance of teaching may seem to come quite easily even though discussions of what makes it easy for them may prove perplexing. Is that close? This is what I think I tried to say to you some time ago, but we did not have the right language to put it succinctly.

  I also like the idea that we are taking our course now (first note), but this is more like collaborative learning since neither of us is ignorant of this field--we just take turns teaching.

  Ah--you deal in photography with that which I have always thought most important and least attended to (composition). What you call "having an eye" I have learned from my son (an artist) may be the product of elaborate training in how and what to perceive. I am getting better (that is, attending more to the visual nature of my world) as a result of discussions with him. Still, I rarely take a very satisfying photo because my photos are usually intended more as historical documents (we were here and here and here) than as art. When I treat them as the latter, I think I get an occasional good shot. And judging from the number of shots professional photographers take, the odds on good shots must be fairly low for even them.

  Rick: I am not certain that I want to break complexity/difficulty down along understanding/ability lines. It may be that is what it will come down to, but I can't quite see it. For those with ability to do 'X', 'X' is not necessarily complex; it is only complex for those who have to analyze X into some sort of components, components that the person with natural talent (or easily acquired talent) may or may not recognize.

The problem is that doing this backwards --going from the components TO the ability-- is not always easy or even possible. It is like learning to read phonetically, where kids say each sound with a breath in between them, and the breath sort of masks what the words are, as in saying "wuh-or-duhs" when trying to "sound out" "w-o-r-d-s". It is difficult to say whether we make the transition or whether we finally find a way to read without going through the phonetic rules. But is seems to me that as long as one has to go through the rules at all, one is not quite reading; or at least not reading very well. Reading may not really be about "sounding out" words --sounding out words may simply be a difficult and complex way of looking at reading; but not something that readers do.

Greg Camilli wrote a post one time listing many of the things they teach in ed school in order to turn students into teachers. It seemed to me there were too many things for anyone to be able to keep in mind, and that if those things were what teaching was about, nobody would be able to do it. Plus his list seemed to me to miss the main features. (I don't recall any of the particulars now, however.) It was like teaching by algorithm --without any understanding of what it was all supposed to be about. So a student teacher might know all the "complex" steps and be able to do them, but would still not be teaching --just like the kid who says "wuh-or-duhs" is not reading. But this would not be because teaching is difficult, but because assembling components in certain ways is not the same thing as teaching. The difficulty in reading Kant is not a phonetic difficulty. The difficulty in teaching a twelve year old algebra is not a technique difficulty.

Now it may be that teaching components and having students drill or practice, will get them past that stage into being able to be more automatic and accomplished. I think there are some things where you can start out slow and then merely build speed in order to "put it all together", as in learning to play difficult piano passages. But I am not certain that this works in cases where the complexities are "constructs" rather than components. I think phonetics is a construct. Similarly, Greg's list of ed subjects. Or when golf pro's teach someone golf by trying to explain all the components of a swing. Instead of doing this latter now, one golf teaching technique is to have a student take a bucket of water and, using two hands, throw the water out of the bucket to a certain place without spilling any water in the "backswing" or beginning of the "delivery". The idea is that the same motion ---WHATEVER that is--- is the same in hitting a golf ball. So instead of putting on some "overlay" of constructs or individual components and teaching from that, one tries to go more "directly" into the whole thing, but in a way that seems more easily learned or more natural for people to do.

If you watch one of these robotics contests they have -- where a self-guided machine has to perform some sort of task (e.g., traverse some sort of a course with turns and obstacles, find and pick up some particular object, and retrieve it into some other part of the course), that is a monumentally difficult and complex task to get a machine to do. But you can train a dog to do it fairly quickly. You have to break it down into components for the machine, but not for the dog. If you tried to teach the dog by using the same components you used to "teach" the machine, it would be hopeless, I think. It may even be that something is going on in the dog's brain like in the machine's computer program, but a teacher cannot put it in there like that. The difficulty in teaching the dog to fetch and retrieve and store an object is not the same kind of difficulty in getting a machine to do it. Both may be complex and both may be somewhat difficult to figure out how to do, but the tasks are very different.

Playing chess well is not the same thing as planning individual moves --even if one can break the strategy down into individual moves once one has seen the grand design. Teaching the dog to fetch is not the same thing as teaching the dog every nuance of motion, even if the dog has to go through every nuance of motion. Nobel physicist, and human extraordinaire, Richard Feynman said that he watched his (less mathematically bright) cousin being taught algebra by someone. And the tutor asked the kid something like "if 3x + 2 = 11, what is x?" and his cousin would say "3", and the tutor would say, "Well that is right, but you didn't get it 'by algebra' -- you intuited it; you have to learn to go through the algebra in order to get it algebraically. You have to subtract 2 from both sides and then divide both sides by 3." That drove Feynman crazy. He maintained there was no such thing as solving something "by algebra" --there was just solving stuff. He said the rules of algebra were just some imposed technique to help kids who couldn't solve stuff be able to get answers to problems of certain sorts.

Someone asked one of Feynman's colleagues one time what method Feynman used to do physics, and (I think it was) Gellman said: Physics or any problem -- you write out on paper or on the chalk board as clearly as you can what you think the problem is; then you think very hard about it for as long as you need to; then you write down the answer if you get one.

I think teaching is "simply" about getting people from one point in their understanding/ability to a "further" point. Now that requires having some idea how to do that. Generally it requires knowing how to figure out what the person already can do, figuring out what sorts of things can build from that with that person; and usually it requires (except in the cases you mention, which are more a kind of collaboration) knowing some ways to get from A to B, or knowing what B is, or in the cases you mention, knowing how to point someone toward B even if you don't know what B is yourself.

So, for example, the golf pro can see that a student cannot swing the club very well at all and cannot hit the ball. The pro may therefore bring out the bucket of water. But maybe the student cannot even do a bucket of water. Now the pro has to figure out something else, maybe half a bucket of water; maybe some sort of arm splints. One old joke is that to keep golfers' heads down, one pro used to have them tie a handkerchief to their belts, put a knot in the other end, then bite down over the knot as they swung. One guy, in his effort to look up as he usually did, ended up pulling his dentures out. But what tends not to work is to teach a student all the individual component motions of a good swing. The golfer who addresses the ball with a list of things he has to do in order to hit the ball well, won't hit the ball well. And these are not things you can practice one at a time and then just work up speed and automatic-ness. They all come into play at the same time.

When you teach bicycle riding, there is not some thing that the kid does that makes him/her be able to ride; not some thing s/he knows when s/he can ride but does not know when s/he cannot ride. You have to teach "the feel", not some knowledge, not some component. There is information you can give (like, "balance with your butt, not your shoulders --when you start to fall left, move your butt to the right, not your shoulders; your balance is in your butt, not your shoulders; when you move your shoulders to the right, your butt goes to the left, so you will fall over faster by trying to balance with your shoulders"). But you cannot say how far, or when, the kid needs to move his/her butt. The kid has to learn to feel that; and though you can help him/her learn to feel that, you cannot do it by explaining the physiological or physics steps to the kid, even if you knew what they were.

I think much of teacher ed is explaining artificially constructed steps of what teachers do, not teaching teaching. That makes teaching complex in a way it is not. And it makes learning to teach difficult in a way it does not have to be. Perhaps. What I said a moment ago also has to do with "having an eye". I don't think having an eye has to do with knowing "what and how to perceive", though one MIGHT (doubtfully) acquire an eye that way. I remember three instances of being told some "eye" rules -- I mentioned to a museum curator one time that the "balance" seemed off in a certain painting, and she said it was because the artist intentionally had used something other than "the golden triangle" in order to cause a certain kind of tension, etc. "I know you know all about the golden triangle." Nope. Didn't have the foggiest notion of what she was talking about. Apparently there are some proportions that seem more pleasing to most people, even though no one is taking measurements. To me, having an "eye" means recognizing something pleasant or interesting or whatever, not recognizing the (perhaps accidental) components of what might be normally associated with its looking interesting or pleasant or whatever.

An older photographer visited my studio one day and, looking around at the more than 200 photographs on the walls, the vast majority of which are in black and white, said, "You need to get some brighter white areas in your pictures. A good black and white should have the full range of shades from bright white to deep black." I don't think so; and the judges who have given my pictures awards apparently didn't think so either; nor the people who buy them; nor the subjects in them who have won photogenic contests with them. I have tried to imagine in many of the pictures where one might put white, bright or not, to good use, and I just cannot see it at all. The rule he was pointing out seems to me to be false. And even if it were true, having it in mind is probably not the way one would go about sizing up a scene or subject to photograph.

In history of art, they pointed out an interesting thing one time. Someone took slides of famous portraits and then asked people to view the slides and describe what personality characteristics they thought the subject in the portraits might have. What was interesting is that when they showed the slides reversed (i.e., backwards in the projector), people saw VERY different characteristics in the subjects. Apparently, for example, people facing to your left (as you look at them) in a picture seem warmer and friendlier than those facing to your right. But as a photographer, when I am looking for a person's best angle, I am not looking for such rules to follow --and that one does not often actually work anyway. Most people tend to look better from the side on which they part their hair (or from which their hair 'sweeps' if they don't part it). But that is not always true. Most people tend to naturally sit down in the direction that when they look at you, is from the side they part their hair --though they have no idea they do that, and sometimes cannot point to which side their hair is parted on without looking in the mirror or raising their arm as though they were about to comb their hair, to see which way their arm goes. Some times people tell you their wrong side is best, because they are used to seeing themselves in a mirror, and they get confused about what they remember seeing.

Anyway, when I try to find someone's most pleasant or interesting look, I don't go through a set of rules, though I may use some "rules" to look for pleasant looks. What I am after is what pleases me, not what fits some rules. (Translating this to two still dimensions is sometimes difficult, so what looks good in real life because of motion or depth, may not work out in a photo, but that is a matter of practice, not following rules --you practice trying to see things "flat" and as they are at the moment.) Sometimes what pleases me is what is counter to the normal practice. There is one client of mine that I never could get to look as good as I thought he could look, until one day while I was talking to him as I sat to change film, I noticed that looking up at him made a much better angle at him than being level or looking slightly down. That is almost never true of anyone. Now when he comes in to update pictures, I always begin by looking at him at an upwards angle.

I think that the people who make the best photography students are people who look at pictures --their own, other people's, magazines, etc.-- and who have likes and dislikes about them, and who want to make pictures of their own that they really like. They don't have to know what it is about the picture that makes them like it or dislike it; they just have to "feel" some sort of difference. Those are people who have an eye. They just need to be taught how to translate what they see (or think they see) to film, so that their pictures show up with the "feeling" or "image" they saw, or thought they saw. Drawing/painting is different from doing photography. I cannot draw or paint. Can't even trace well. I don't know how to see as a painter, just as a critic (i.e., analyst, not a complainer). I see photography as merely capturing the "things" in life that from an art critic's point of view look good naturally (or with a little help). I don't seem them as lines and shades or colors, but as complete subjects, finished works. I just copy them into the camera. (The fun for me is finding out which angles, lighting, expressions, etc. make them look their best -- not the details of what composes those things. So I don't like doing "straight" copy work, usually.) I like finding the landscapes everyone else drives by without noticing; or showing others the beauty of some woman that no one else would even have glanced at twice, previously. I don't like doing children as much because they are just almost always cute, and it takes no special "help" from me to make them look better. The trick in doing kids' pictures is not messing them up. The trick in doing adults is helping them look their best; so adults are more fun for me to do because it involves more of "me" in getting the picture.

What I like to do in my photography is to show people how I "see" things when I think it is not things they would have seen for themselves, and that they would like to see. But whatever one's interests, seeing what looks interesting is not a question of finding what fits certain rules -- even if what is interesting generally ALSO could be analyzed and FOUND to conform to those rules (when they do.) So learning the rules is not learning to see what looks interesting to you. Back to the previous post then. Good teachers may do certain things, but those things are not likely to be following certain procedures, even WHEN what they are doing could be analyzed and FOUND to conform to those procedures (when they do). So learning the procedures is not learning to teach. Just like learning grammar and vocabulary is not learning to speak a language.

  Bill: I was tempted to think you were saying that having an eye could not be taught, but I think you clearly think otherwise. You just don't see it as a matter of learning some set of rules. I agree. I did not mean to suggest the contrary, but I do think that one can learn how an what to perceive and develop "an eye"in the process. some of this is like what I just said about teaching people that there IS a "b" toward which to work. I have always had artist friends and for a long time assumed that the difference between their perceptions and mine had to do with talent--something inborn. Then I ran into a guy who works mainly with language, but who used to paint and take pictures. He and I had an artist friend in common, and that enriched our interactions. I discovered that he saw a page of text something like a painting--he notices the white spaces, the differences in fonts, whether it has been laser printed at 300 dpi or 600 dpi. Somehow, that told me that there was a "B". I began to realize that I also could see differences and can now respond to photographs the way you describe, have very strong reactions to paintings, even if I do not know why, notice the light in old black and white movies etc. I have been getting a visual education over the last ten years. I have also developed more of an ear during this time span. In my youth, I was uni-dimensional (or bi-dimensional if language and math are different things) and lived an aesthetically impoverished life. I cannot now produce in any of those other forms, but I am learning to appreciate them.

  (Maybe this is just aging--I could say the same of my palate, e.g., I used to have to really strain to taste any difference between wines of enormously different quality, but now, I have strong preferences, sometimes between different brand or years of the same wine. Likewise beer or cheese. But I guess I know this is not a result of aging because my sons have discriminating tastes that they have acquired, in part, by paying attention to why I would prefer X over Y. Somehow in my own youth, I missed such experiences and I was firmly convinced that everyone else faked these discriminations--possibly the conclusion of an over-endowed ego.)

  Rick: Bill, I couldn't remember whether you subscribe to AERA-C or not; apologies for repeat of this message I am forwarding if you do. I thought this was like something we had talked about last year or so. W. Gary Martin wrote, as a small part of a longer post:

  "Now, I am ready for a firestorm on this one, but it is my experience that teachers who were quite average in school often make excellent teachers because they can understand the difficulties their students are having. While high-achievers in some cases find their students' thinking baffling. If we went solely by academic achievement, we would not necessarily be getting the right persons into the teaching profession. There are other important qualities."

  Bill: Yes, that is like a talk we had. In fact, I have had some outstanding student teachers who were good precisely for the reason that they had difficulty as students and could therefore understand student problems. However, I would not advocate this as a procedure in selecting teachers unless it were possible to do some very rigorous testing--like a year's internship. Despite my position in our current talk, I think there are great risks in having teachers who are not knowledgeable about their subject matter. And the subject matters--Kay (my wife) has a grad degree in Human Nutrition (but teaches English, for which she has an undergrad degree) and went through several roofs when a local health teacher told her class that teenage girls should not drink a lot of milk because they did not "need" all that fat in their systems. (Lack of calcium in adolescents is a major contributor to osteoporosis in old age.)

  Rick: SKIM milk is, of course, the solution; at least that is what I am told. All the calcium and protein and other goodies without the fat. Yes?

  And, of course, if we decide that there was some substance in the comment from the AERA-C posts, we would have to be VERY careful how we proposed a program, since I could just see some reporter saying we were saying that poor students make the best teachers -- "Ed Schools Recruiting Worst Students; Say They Teach Best". I suspect that, of course, it is not that these students had trouble learning; it is that they had trouble learning and OVERCAME the trouble in order to learn in spite of it. That is the sort of thing I was talking about however many months ago we had that discussion where I said learning things was always difficult for me.

Bill: BINGO!. Interesting that I have never thought of the importance of that kind-of-obvious now-that-you-mention-it wrinkle. I forgot your saying you had difficulty learning. I find that difficult to believe, mainly because you often sound quite a bit like me and learning came easily to me. Or, I avoided learning anything I found hard to learn (like an instrument or painting or photography). Things that more or less require persistence and practice, I shunned--weak character, I guess.

  "SKIM milk is, of course, the solution; at least that is what I am told. All the calcium and protein and other goodies without the fat. Yes?"

  Exactly. Except for the lactose intolerant.

  Rick: Ahhh, your last comment about shunning things that took persistence and practice gets me into my "video game theory of learning". I too shunned what I thought took persistence and practice and felt lazy for doing so. BUT I worked hard (though it was fun, and didn't seem like hard work) at working out the things I thought I could learn, wanted to learn, and felt like I was making progress at --or thought I soon would. There were some things that seemed too difficult, whether they were or not; but there were some things that though they seemed difficult, also seemed reachable; and the quest, and the progress, were fun.

Video games are like that. The first time you play one, you get clobbered quickly. But the second time you do a little better, and the third, better yet. There is progress; and kids think they can master this and they keep trying. Like golf, it does not matter if mastery is impossible; it is the lure of success that keeps one going, and the progress or seeming progress. Golfers will go out and hit some great iron shots one day, and come back saying "I know I can hit those irons now; and when I play tomorrow, if I get the woods working with them, and just putt a little better, I should break 80." Well, in golf it ain't gonna happen; but in video games, it does --so kids will spend hours doing it. The things I quit on were generally things I was frustrated about seeming to get nowhere; the things I stayed with seemed to yield progress to the effort; and the effort and the progress were fun, not work.

I remember how much fun it was working on the Rubik's cube, though I never got the very last little bit to go; still I felt I mastered it as well as I could. Maybe some day I will solve the very last little twist. My latest one was some math extra credit problem my 10 year old brought home. Took me till 1 a.m. to solve it, but....I learned ping pong by playing against a neighbor who beat me 21 - 0 quite regularly at the beginning, but I felt even then that I COULD figure how to beat him. Gradually, I got three points, then five, then 8 or 10, then 17 or so, then some very close games. Then I finally got to where I could beat him most of the time, but always in close games. I just always felt the incremental progress was not like practice, and did not require patience; I was having fun. There is probably more to this, but so far that is what I have. In regard to it, I think part of the art of teaching is structuring material (or the introduction of it, or the learning environment, etc.) for students in such a way that they can experience the confidence and excitement of incremental progress so that it either does not seem like work, or seems like very worthwhile work.

By the way, when I took the WISC or whatever the test is called -- the part that has the cubes you have to make pictures with: the last one was harder than hell, and I took forever to solve it, but I solved it in my head before I made any moves, because I didn't know how else to go about it. I figured if I just started to manipulate cubes, I would get really screwed up. Some really old psychiatrist was giving me the test, and he said "I have a four year old grandson who could have done four of those in the time it took you to do that one; but in all the years I have been giving this test, you are the first person to solve it in his head." I didn't ask how else one would solve it; and I still don't know. That is the kind of thing I meant about things being hard for me; I have to do everything by analysis and logic. Well, most everything.

  Bill: Much of what you talk about in the complexity message (golf, cycling, etc.) is what Gilbert Ryle (I assume you know his _Concept of Mind_) called "knowing how" as opposed to "knowing that". And maybe that is where complexity resides. "Knowing that" is perhaps reducible to memorizing whereas "knowing how" involves levels of abstraction and application.

  I am not taking issue, just responding. For me, in the kinds of cases I want to say are teaching and you want to say are collaboration, the question of getting from A to B, as you described it, is useful. Sometimes what I may do is point out that there is a B, that they must know where B is (or find out if they don't know) and that getting FROM A means taking steps, even random steps, and trying to decide if they are getting you closer to B. Sometimes you might take forty steps before you realize that B is now further away, so you backtrack. Sounds like I am talking about teaching people how to engage in independent learning. So, I am applying a kind of expertise that is different from the content expertise.

  Your Feynman algebra anecdote is good. There was one guy in my class better at math than me. In grade five he went to the board to "do" problem 5 and he wrote something like "247" then sat down. I meanwhile had "done" problem 9, writing the givens, what I needed to know, indicating a formula that contained both knowns and unknowns, substituting and solving. The nun asked Tom to show his work. he was quite upset since he didn't "know the work" he only "knew the answer." The nun knew he was capable of whatever this was, but she said "No, Tom, just show us how you found the answer--how did you solve the problem?" His answer was "I read the question and I knew the answer." I could do that too for alot of the questions, but I could also work backward and fit the model. I spent two years learning how to know an answer WITHOUT being able to specify the steps and even then, I could only not know the steps for as long as I refused to try to figure them out. I think Tom must have learned during that time how to "do the work" because in high school, we both had developed the strategy of answering math multiple choice questions by picking the right answer intuitively and then working through the problem backwards rather than carrying out the whole process from the start. This was much faster.

  As I typed the above, I wondered: "Is it possible that we are so good at teaching math in the way you describe accurately as analytical that we lose fluency in math? That is, if math teaching were as mysterious as language arts teaching, would we end up with a lot more people like Feynman's nephew? Do our imposed algorithms for math and for the teaching thereof stifle mathematics communication?

  This seems to be a good time to mention a theory of my own-- that our game choices say a lot about our "learning personalities. For example, it did not take long for me to tire of the Rubik's cube --- I think two sides was all I ever did. And I hate those story puzzles where John only eats eggs and the drummer sings out of key and Fred dates a musician and the question is who ate the strawberry--you know the type?

  And yet, I will spend nearly any amount of time on a Chinese puzzle box or an intriguing math questions.

  Rick: I am not certain what you mean about the "mysterious"-ness of language arts teaching; perhaps that it is less rule-oriented. Of course, however, remember the teaching of grammar. There may be a parallel here, in that though grammar may help us learn some things about language, I doubt many people, if any, can learn a language by means of studying its grammar. Grammar is more readily understood, if at all, once one can speak a language; it does not tend to be a heuristic device for acquiring a language facility.

  Yes, I, too, hate those chain puzzles too (who ate the strawberry ....)

  My younger one, Lydia, when she was about 8 or so, put the Rubik cube into perspective. Sorry if I have told you this one before. I gave her the cube to play with. She fooled around with it a while and then brought me one face done. I showed her the sides of the "edge" cubes and explained that they were all different. She looked at me and the cube a minute and said "You mean you have to get all the sides the right color at the same time?" I said "Yes." She thought about it a second and then said "That could be real hard." Then she walked away and, I think, never picked up the cube again. Yet, she too will work a long time on a math problem. Or on doing origami. She taught herself origami from a book and loves making the paper objects. *I* can't get into THAT; too tedious for me. Interesting theory you have there. Now you just need to give more details and explain how which sorts of interests relate to what sorts of learning, etc. Should be easy.....

  Bill: I assume that your "...should be easy..." was tongue in cheek.

  No, you had not told me about your daughter and the cube. My mathematician buddy in Nova Scotia is a group theorist, as is Rubik. He looked at the cube and said "Yeah, interesting. Comes from my field of mathematics, you know" "Does that mean you can solve it?" "Probably. Well, maybe. No. I don't know, but I can prove to you that it CAN be solved." And therein, his interest ended.

  Rick: Yes, the "should be easy ....." to go from game choices to learning needs was meant tongue in cheek. Very.

  And I loved your friend's Rubik cube comment -- that he could prove it could be done, and that that was all the interest he had in it.

  Suppose you teach me how to engage well in independent learning. A good thing to do. However, suppose now I want to use ALL my skills, including the new ones you have helped me learn, to learn "electronics" -- not just how to build a satellite dish for which I have a schematic diagram that I could follow --but I want to know how to design components to do things of my own desire, etc. I want to know how all this works. I suspect I could rummage around in books or even some courses, but it seems to me that the best/fastest/easiest way to learn is to find someone who knows what I want to know, appreciates what I want to know, and is able somehow to answer my questions or point me to specific things to learn in perhaps a specific order, and serves in general as what Michael Wylie seemed to refer to as a "tour guide" in the subject. I would think it is often better to have someone like that, no matter how good one might be at independent study. And that difference in speed/ease/quality, etc. is what I guess I am calling "teaching", as opposed to collaborating, etc.

  I don't think the distinction I am making about complexity/difficulty is about "knowing how" versus "knowing that", which I think normally IS the distinction between ability and understanding. I wish I could get a handle on what I am trying to explain. I keep thinking I have it, until you ask another question or make another comment. Then I see I need to do more work.

Today I did a simple algebra type problem with my 10 year old. She got it right away by "intuition", or by trial and error. Once she got the answer it was easy for her to see it was right, by essentially just plugging back in. (I have a bag with two more things than you have in your bag, and together we have six; how many do each of us have? 2 and 4) Now she is not at the stage where she understood that "you have x, and I have x + 2, and therefore x+(x+2)=6, which means 2x+2=6; 2x=4 (by subtracting two from each side, because....), and x=2 by dividing both sides by 2, etc. She knew the answer, and she could explain how she could tell it worked, but she could not do, explain, or understand "the algebraic" way. BECAUSE that was NOT the way she got it. She had a way of getting it; just not that way. And THAT way was harder than her way. "Greg's way" of teaching teachers to teach seems to me harder than is necessary to teach teachers to teach, even if his way could sometimes be made to work. It may have something to do with the old saying about forests and trees.

When we had the long discussion a year or two ago about "testing" in order to see what students know, it seemed to me that a number of people were relying on test results to mean something they didn't necessarily mean, because they didn't have any feel for what they were doing, and were just relying on tradition and on formulas that may or may not fit. The man that inadvertently got me to give up formal tests was a man from Michigan's Resource Center for Learning and Teaching (I think it is called --something like that anyway). He merely said that when you design a test you want to design it in such a way that you can see whether the students know or can do, what you want them to know or be able to do. And the more I thought about it in light of tests I had given, I decided I could not design such a test. The material, and the skills I wanted students to be developing were just too open-ended.

Plus, I knew that kids took philosophy tests in screwy ways that tended to not be their best or most natural effort. But what this man said made more sense to me than any algorithms for making tests. One of the problems as I saw it was that if I asked questions that just called for repeating what we had covered in the class or the book, that would not show me kids were thinking. But if I tried to ask something that required a "leap" of intuition or logic, the fact that some kids might not be able to make that leap during those two hours of exam conditions just showed me they couldn't make that leap at that time under those conditions. I didn't think a philosophy grade should merely be a reflection of such conditions or of such luck. Then you would have kids who knew enough to say some "okay" things using the right terminology, but they didn't quite say as much as they might have, and you couldn't tell if they knew more or didn't or if they were just using some terms they thought relevant, and were playing it cozy. Top that off with the fact that I covered everything I thought it important to cover by using as many ramifications as I could during the term and that didn't leave me a whole lot of new ways to approach material that they hadn't already somehow been exposed to it.

The teachers and students who participated in the discussion on testing at the beginning were talking words like "validity", etc. and I had the feeling they didn't really understand the ramifications of them. It may be that the average assessment or evaluations courses make all this clear, but I suspect they make it harder. When I read the constructivists' stuff about teaching math and then see how they go about it, I see many of them following some sort of plan they think kids should be able to respond to, but *I* don't see their plans as what kids necessarily DO respond to, any more than kids can necessarily FOLLOW a lecture or some pre-worked out proof. Lydia brought home the following from an NCTM publication: Make a square with rows and columns. Put numbers at the top of the columns and the beginning of the rows. Where the rows and columns intersect, fill in the space with the PRODUCT of the row and column numbers. Then select one and only one such product from each row and from each column (e.g. one of the diagonal four intersections) and find the PRODUCT of those, call it the grand product (though they had some other term) and it will turn out (though you are supposed to see this yourself first) that no matter which squares you pick, you will always get the same grand product. They are taking kids through all these steps, and assuming kids can follow the directions (not easy) and that they make no mistakes in multiplying all this stuff (not easy), and that they are struck by this phenomenon.

Well, I am able to see with a bit of thought that the reason this works is because by using one and only one row/column intersection from each row/column that you are essentially multiplying all the row numbers and column numbers together using each once and only once. They try to hint at this by saying "Using the generic margin numbers, a,b,c,d,e,f,g, and h, what will your rows and columns be, and what will the grand product be?" Well, since I have already seen this, I can do it; but my kid is going "what does generic margin numbers mean?" and "they didn't give any numbers, just these letters." So I explain what that means (which I only knew because I figured the relationship I had seen was the one they were going for). So we work out some squares, etc., then we get to a problem that says "using a three by three square, and using only positive integers, construct one with a grand product of 257."

Well, in having done a bunch of squares they gave us earlier, a few numbers gets to a pretty high product really quickly. So this one has to have pretty low numbers. None of them can be even because 257 is not even. Six 3's are too high; 5 3's and a 1 is too low. I figure there is a misprint. I can't even find a factor of 257 as I trial-and-error it, hoping for some sort of breakdown. Nothing. They also have the next problem which asks for nine totally different numbers which will give a grand product of 257. Gotta be some sort of screw-up. But I keep thinking about it. In the shower at 12:30 a.m., it dawns on me that 257, and five 1's will do it. Damn! Simple, but they had psychologically mislead us with the other squares we had been working with.

Now, earlier on they had two different squares that were related in such a way that the column numbers of the first square were twice the column numbers of the second, while the row numbers were half: and these had the same grand products. So that meant (or showed) that you could always make "related" squares by multiplying and dividing. I did that to get a square with fractions in the rows and multiples in the columns in order to get the second square with 257. Feynman would have seen it all right away. Many would never see it at all. I only saw it because it bugged the heck out of me and I couldn't let it go. This seemed to me to be one hell of a hard exercise to show fairly simple principles, and it would have been much more interesting showing it "forwards" than making kids work it "backwards" the way they did. You almost had to know the principles in order work this out.

My point is that they took something that is fairly simple, and they constructed a way of demonstrating it that is very difficult and very complex. But that does not make the principles complex. Doing something a hard way, even if it can work, does not mean the thing you did was, or had to be, difficult. Now if we are trying to teach some math point; or we are trying to teach teachers how to teach a math point, there may be harder and easier ways to go about doing this. If we pick the hard way, that does not mean we were teaching something difficult; perhaps we just made it difficult. Perhaps teaching about assessment does not need to have statistics in it. Perhaps teaching about teaching math does not have to have constructivism in it. Perhaps getting enough vitamin C does not mean we need big pharmaceutical factories to make vitamin C pills. Perhaps we only need to grow more oranges -- something which is not too difficult unless you are trying to do it in a lab using only basic chemicals, and no oranges to start with.

Or suppose we figure out how brains learn and remember things; and suppose this is a terribly complicated process, though easy for the brain to do. Do we need to know all this complicated stuff in order to teach a kid a second language (or even a first language)? Of course not. Ignorant people teach their kids language; and they don't know either "how" or "that" to do it. They just do it by talking in front of, and to, the kids. So my question is this: if you were going to try to teach someone how to teach something, shouldn't you start by letting them try to teach it and see what happens that they need to correct if they haven't taught it (or if they try to teach in some way that there is reason to believe is damaging)? Maybe kids who come into an ed program CAN teach some things already. If so, how; if not, why not? What needs fixing? Maybe it won't take much. Maybe we have analyzed and theorized teaching into something more difficult than it needs to be or than it is.

  (And, in a subsequent message:)I don't know about the "developing an eye" part. I am not sure what we are actually developing when we learn to discriminate one picture from another. A friend of mine asks me how I arrange my wedding pictures so that the groups look so balanced. I do it quickly, and she watches me, but doesn't see how I do it. To me, I just see what is out of balance and then try to shift it. E.g., if some guy is too high above everyone else, I ask if he can step down a step. If he can, he does, and then I see if that makes him more in line and looking okay. If it does, I shoot. If there are two girls on one side and three on the other, I try to do something that will make the space look even, though the numbers are different, maybe put two bigger girls on one side and three smaller ones on the other; or put three in a second row, and have one on each side of the bride so that there is no imbalance.

But I don't know how to teach someone that cannot see imbalance to begin with. If I show them samples that I CALL imbalance, do they just learn to imitate this, or do they actually feel the kind of imbalance and cognitive dissonance I do when I see imbalance that is not right to me somehow? If all that is happening is the former, I have taught them to mimic my eye, but not to have my eye, no? The palate issue you brought up is interesting. Perhaps you could distinguish wine and cheese tastes before but couldn't remember what went with which. If you couldn't discriminate at all, perhaps you hadn't eaten sufficient different cheeses to be able to notice there were some categories. Maybe you didn't notice what you noticed; like some kid who goes with his family every year to the grand canyon because they live near it, and at the end of the summer says he didn't do anything interesting or special all summer. I have a palate infirmity of sorts. All alcohol tastes like rubbing alcohol to me --bitter. Now, you can give me beer, which is sort of malt and bitter, or daiquiris, which are sort of fruity and bitter; or rum, which tastes really bad, and bitter; etc. I can distinguish one drink from another, but I doubt it is the way drinkers distinguish. I am missing something. I don't think I can be taught what I am missing, though I might be able to make the same kinds of verbal judgments alcohol connoisseurs do. When they first advertised the non-alcoholic beers with all the taste of beer, but none of the alcohol, I thought that seemed really stupid, since beer tastes so bad to me I can't drink more than half a can when I am totally parched and it is 100 degrees out. I have never been drunk because I cannot stand the taste of alcohol; but I assumed people drank for the buzz, not the taste. Some day I may try a non-alcohol beer to see what the "taste" of beer is. But I won't know if I have it, will I?

  Bill: Trying to respond to two in one and to keep it brief.

  First, alcohol. What a blessing you have. As one of my sons says, NOBODY likes the taste of alcoholic drinks at first--we force ourselves until we get to like it. For many, you are right-- it is the buzz. While I have occasionally gotten carried away (very rarely), it has nearly always been because I drank a fair amount very quickly at the outset and that resulted in reduced inhibitions and more drinking. I do not like the buzz, except in the mildest of forms. But I do like good beer (that is, heavy, British-style ale or stout). So I buy expensive beer and usually have only one or two at a sitting. But I recall not liking the taste and I recall vividly thinking it all tasted the same. Perhaps you would do better to try non-alcoholic wine, but I'll bet you won't like that either. All of that stuff is first made with alcohol which is then removed. You probably dislike the effect the fermentation has on the taste of the beverage as much as you dislike the alcohol. In general, I think the ability to perceive and appreciate distinctions adds zest to life, but I also know that it is perfectly possible to live a very contented and happy life enjoying a few things a great deal (as the Amish do). While you would not want to forget what you "know" about seeing a picture, if you never had it, you wouldn't be likely to miss it much.

  On teaching teachers: if you have not already read Schon's The Reflective Practitioner, you may want to take a quick look at it. The process you describe is precisely the process he claims is at the heart of all professional practices--find someone who knows, watch, try, make mistakes, try again. Given the passion with which you describe the failures of some of your daughter's teachers, though, are you prepared to set prospective teachers loose on her with NO attempt to prepare them in advance? Actually, this is very much what I would advocate--a one-on-one experience with a good teacher in which they watched a lot, helped a little, slowly took on more responsibility and took courses at a slower pace over a longer period of time. I understand that lots of professionals were trained like this in revolutionary Cuba because it was cheap and made maximum use of human resources.

  Rick: No. Not the approach I favor; though in a sense, by having had many teachers who were good and many who were bad -- if we had been paying attention, we would have learned a lot that way. But it is really hard to learn simply by watching, because some times we see the "wrong" things. I was supposed to have learned biochemistry from watching a biochemist; he was anal retentive, and I could not tell "what" he was doing by watching just the outward things he did. There were reasons for his particular actions; and if I wanted to do something similar to what he was doing in a different area, I needed to know what he was thinking, not just what he was doing. Otherwise I could only imitate him, not be like him. When I photograph adults, I generally have them say "sex". It makes them laugh at first, and then smile. But it is the element of surprise under the circumstances, not the word "sex" by itself that does it. One university president's wife would always tell the people she wanted me to photograph "Okay get lined up so Rick can take this picture; he's going to get us all to smile great by having us say 'sex'." Well, that messed it up for me, since she took out the surprise, and just saying "sex" is not all that funny without the surprise; usually. People who imitate me often do like she did -- they have them say the word, but don't say it under the right conditions, or take the picture during the laugh, not the ensuing smile. Watching a good practitioner of anything may be helpful, but not as helpful as watching and discussion, etc.

Mentoring, in this sense may be useful; agreed; but not just watching. I am FOR teacher education; and what you call preparation prior to entering the classroom. However, I think there is a difference between unnecessarily complex teacher ed, and good teacher ed. Louis's [Louis Schmier] latest post, with its 43 questions he wants to ask himself everyday, and think about is too analytic. If he really thinks about all 43 of those every day, he won't get ANY other work done that day. That is too detailed. Louis probably has a "feel" for what those 43 questions actually articulate; but I doubt he can go through the list of them in the way they are actually spelled out.

  Wait; maybe that is the word I have been looking for "DETAILED". Overly detailed things tend to be overly complicated in a way that ruins what the details were intended to enhance. Like when mothers and daughters and wedding directors go into such detail that they ruin the emotional aspect of the wedding for each other. Everyone becomes so concerned about the details that they become anxious about and focused merely on the trivial -- which side to hold the flowers on, where to stand, when to move or say something, etc. They then don't get to have and notice the emotional joy of the moment and the sharing it with friends. And the actions look contrived instead of flowing naturally and spontaneously. Overly detailed golf instruction ruins a swing. Overly detailed pedagogical instruction can "make" teaching be too complex to succeed -- can make it more difficult than it is. I think that is what I have been wanting to say.

  Bill: OK, overly detailed helps. Especially if we also understand that the level of detail that is "over" may be a function of the learner's interest and readiness. Maybe what _I_ have been trying to say is that I fear you are trying too hard to understand the role of the teacher without sufficient attention to what the learner brings (which always changes everything as your thoughts on photography, math and philosophy consistently show).

  Louis is an interesting case in point. He seems to be so focused on the learner that he doesn't care what the subject matter says. I am sure that history is so much a part of him that he works with it naturally and effortlessly and only THINKS that it doesn't matter. But, see, this is kind of what I have been trying to get at too. Those who teach well and who are reflective and analytical about what they do are not necessarily right in their analysis--so we have Louis seeing the process one way, you seeing it in a very different way and others are on other tangents. Maybe that is a good metaphor--there is this thing called "good teaching" and each of us is sitting on a tangent.

  Rick: "OK, overly detailed helps. Especially if we also understand that the level of detail that is 'over' may be a function of the learner's interest and readiness."
Yes; AND also a function of the need -- that is, if you are looking for directions to someone's house, and you know how to get to a landmark he mentions that is a block away, he only needs to give you sufficient directions from that landmark. He does not need to give you minute directions or compass headings from where you are, or from the landmark. I think in too much teaching, teachers give more detail than is necessary or than students are ready for. And that detail impedes the learning.

  "Maybe what _I_ have been trying to say is that I fear you are trying too hard to understand the role of the teacher without sufficient attention to what the learner brings (which always changes everything as your thoughts on photography, math and philosophy consistently show)."

  I don't mean to do that, though my focus may have implied I was leaving that out. Sorry.

  "Louis is an interesting case in point. He seems to be so focused on the learner that he doesn't care what the subject matter says. I am sure that history is so much a part of him that he works with it naturally and effortlessly and only THINKS that it doesn't matter. But, see, this is kind of what I have been trying to get at too. Those who teach well and who are reflective and analytical about what they do are not necessarily right in their analysis--so we have Louis seeing the process one way, you seeing it in a very different way and others are on other tangents. Maybe that is a good metaphor--there is this thing called "good teaching" and each of us is sitting on a tangent."

  Hopefully a better description would be that we are writing about different tangents or focusing on different tangents in particular writings. Hopefully we know many of the same things even if we don't write about them in these relatively short compositions. I wrote once before that I presume Louis actually does teach history and that he does it in such a way that kids learn some things about history, even though he NEVER (as far as I know) mentions any aspect of his subject matter, other than the themes the students give some sort of presentation about. I too expect kids to learn in some sense on their own, even though I WRITE about what the teacher needs to do to present the material to facilitate that (learning). And to foster interest in doing so.

  Bill: Yes, perhaps we all need to become pan-tangential, if we will not then be accused of circular thinking.

  Rick: Or of pretending to be descendants of England's King Henry II.

Bill Hunter

Rick Garlikov