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|There are some cases where one is teaching something that does not
require the student to have special prior knowledge (besides vocabulary),
and that prior knowledge does not really help. For example, teaching
a passage for memorization and recitation, such as a nursery rhyme to a
small child. One might need to explain it or answer questions, such
as what a "tuffet" is or what "curds and whey" are but one can basically
just recite the nursery rhyme a few times to a child and the child will
learn it in no time.
But many, perhaps even most things, one teaches, are more easily learned if the student has some sort of previous understanding that helps make sense of the new material. This can be something relatively simple. When I was in first grade, my father would have me read to him out of The Weekly Reader we received on Fridays. One night, I got to the word "doughnut" and could not recognize or pronounce it. He had me sound it out by asking what "D-O" spelled. I said "do" (as in due). Then he asked about "nut", and I said that word (as in walnut or peanut). He told me to put the two words together. "Do nut." "What does that sound like?" I said "Do not." But that didn't fit the context. Then he said "No, it is like a bagel." "What is like a bagel?" Then I was totally confused because I had never heard the word "doughnut", nor had I ever seen or eaten one before. My father found it hard to believe; but I never had had doughnuts at home or anywhere else I had been where someone handed me one and said "Here's a doughnut." or "Here, have a doughnut." or "Those doughnuts in the bakery case look delicious."
But generally it is more involved. Much of arithmetic, for example, is somewhat cumulative, and it is easier to learn certain concepts or ways to calculate answers if you already have understanding of some previous concepts or calculations. Converting decimals to fractions, for example, requires that students can read decimals and state them in fraction form, such as .03 as being 3 one hundredths, which then makes it easy to write it as 3/100. To then get the simplest fraction something like ".5" represents, students need to be able to turn 5/10 into one half in some way, say by dividing both numerator and denominator by the same number -- in this case, 5. If they cannot do that, or do not understand why multiplying or dividing numerators and denominators by the same number results in no change of the value of the fraction, you have to explain and demonstrate that in some way. Or they have to see that 5/10 can mean "5 out of 10" and perhaps see that as half, because they have worked with quantities of five and ten, as in changing a $10 bill for two $5 bills. This is all about the teachers' understanding the logic, or a logical learning/teaching progression, of the material and the concepts involved.
Included in understanding what I call the "logic" of teaching the material, is knowing the order of what one would likely have to learn (or what would be helpful, even if not necessary, to know) in order to learn the next step. For example, typically to teach students to divide, you would teach them to multiply first because it is very difficult to divide in your head without knowing multiplication "tables" or what I guess today they call "multiplication facts". But you do not need to teach multiplication tables to show how division works -- you can do that by having kids divide things among themselves, or ask kids how they would divide, say, a bunch of M&M's among themselves evenly. (Typically they would let everyone take, or be given, one, and then everyone gets another, etc. until they are all gone -- like dealing out a deck of cards.) Then at some point you can show them there is another way to do it, by knowing how to do "division" in their heads or on paper. You then will have primed them to want to learn about multiplication so they can do division without having to divide up thousands or millions of things one or two at a time.
Too often, teaching is not conceptually understood in the above way, and that often causes teachers to be ineffective with students, even though they are doing what they are trained to do or what research supposedly shows to be "best practice." One has to keep in mind the point of teaching, and therefore of any teaching practice or method, I think, in order to do it well, because generally just giving students more information that really does not mean anything to them (information they cannot absorb or assimilate) does not help them learn what you want them to know. The point is to lead or take students from the knowledge and skills they have to the greater or higher levels of knowledge and skill that you want them to have -- and to do it in a way that inspires and enables them to continue to learn, for it is not good to poison their interest just to achieve modest gains in their knowledge or skill when doing so curtails their long term growth in something for which they might otherwise have talent and enthusiasm. Nor does it do any good to give them short term increases at the cost of the foundation or supports on which to build much better later. In order to build upon what students already know, one must know how to find out what they already know, and one must know how to use what they know in order to get them to where you want them to be. This means also one must know the subject matter well enough oneself to know what knowledge is foundational, helpful, or necessary for understanding more.
It seems to me that by attempting to make it easier in some cases by giving teachers "methods", teaching is made, and then conceived of as being, much more difficult than it needs to be, particularly with regard to mechanical techniques or forms of teaching or with regard to the current trend of what is referred to as "learning styles" which I have criticized in a different way previously. I do not think that for teaching most things learning "style" or any rigid, prescribed, mechanical or technical approach is as important as is teaching in a way that allows students to assimilate, absorb, and/or remember the material -- and that requires teaching them by relating material, in ways they can understand and find meaningful, to things they already know or to experiences they have already had (or heard about in meaningful ways or seen dramatized in movies or tv, etc.). That means teachers need to be adept at knowing or figuring out what students already know that can be useful in teaching them the material at hand. In many cases the teacher will also need to supply the information or experiences upon which to build. For example, in teaching elementary reading, a teacher might first have to enrich students vocabularies so that when they come to a written word they have not seen before in print, they will be able to make sense of it once they have, say, sounded it out. It does no good from a reading (comprehension) standpoint for a student to be able to pronounce a written word, if it is a word he does not recognize even when he says it correctly. Sometimes a teacher might have to give a demonstration of a phenomena so that students can see what the topic is about. Sometimes the teacher might need to cause cognitive dissonance in order to bring to life a problem that students may then be more interested in solving or resolving.
I often get questions about how to use the Socratic method to teach students material based on their being able to think logically about things they already know, because of a well-received essay I have on the web about it, but the Socratic Method is only one form of teaching that utilizes what students already know and can reason out in order to help them attain more knowledge and understanding. There are plenty of other ways to teach starting from what people already know. For example, if one is trying to teach someone to play bridge, it helps, but is not necessary, if the learner already know how to play the card game "hearts", which is similar in some ways to bridge.
Now there is no point in teaching someone hearts in order to teach them bridge (just as there is no point in teaching someone how to get to a landmark from which you might direct them, had they already known it, if you can just as easily direct them to your place from somewhere else they know), but if they already know how to play hearts, it will make teaching them bridge easier (just as if they already knew how to get to a nearby landmark, it would easier to give them directions from it). If they do not know how to play hearts, it would be faster just to begin teaching them bridge. Similarly if someone is looking for directions to your home or place of business, if you are near a landmark they might know, it will probably speed it up giving them directions and make it easier for them to find you, by saying I am across 21st Street from, say, the belltower. Normally in giving someone directions, you need to find out where they will be coming from, and what things in the area they already know how to find, if any. Then you direct them from the place closest to you that they already know how to find. If they don't know any of the places anywhere near you or on the way, you might have to start them just from their home the way Mapquest does. But typically you can start people from somewhere in between you and them. And normally it is somewhat more efficient to spend at least a little time trying to find such a starting point than to start them from their home. You can even give driving directions to a group of people if they all know some common (starting) place between you and them; and typically you can find such a place.
Teaching is essentially the same way. The trick is to think of things related to your topic that the person you are trying to teach may either already know or be able to figure out for themselves in part (perhaps with some error even), or that you can readily demonstrate. So you have to know the foundational elements of your subject matter, but you also have to know, or be able to find out, which of them your students already know or could easily grasp. You just have to keep thinking and trying out some questions, etc. until you come up with something that tends to work; and generally if you can get it to work with some kids, it will work with all or most of them or at least be a place to start. Then you have to respond to their answers by thinking about what their answers may show about their understanding, and how you might use it to advance them. I don't think there can be any pre-scripted recipe for this. There is an art to this.
I don't know that there is "training" in the Socratic or any other method of the sort I am describing anywhere, but I do not think it is necessary either. The way it works is that you have to understand the facts and logic of the material you are trying to teach and just use that as the basis for your questions. There is a certain amount of psychological and logical creativity involved, and I doubt that can be taught, but it also may be within anyone's grasp already, and may only need to be practiced, after one understands and appreciates the concept of teaching in this way. Then one can be looking for things that students already might know (or that can be taught to them easily) which might help them absorb and assimilate the new material. Luckily, in any given cultural setting with a group of students from relatively similar backgrounds, what works for one students will work for many. In many cases you can even teach people with different backgrounds or of different ages (such as children and adults) at the same time, because adults may have no greater background in the particular topic than children do. In teaching philosophy, for example, children will often do just as well or better than adults. In teaching "place-value" in the essay linked above, children often can follow the logic there far better than many adults.
In some cases, you have to "backtrack" fairly "far" to get students to have the sorts of experiences or information they need to be able to absorb and assimilate the material you want them to learn. What you might have to teach in order to help students learn depends on what the students already know. And that means you have to find out what students already know that is either relevant to what you want to teach them or that is necessary for them to know first. While that often takes some time, particularly with students who are new to the teacher, that time is well-spent, because it saves having to re-teach the material many times, and it also lets the children learn the new material more quickly and with far less misunderstanding.
Many people say that teachers, particularly elementary teachers, need to know "how children learn." I think they learn the same way adults do, but just do not generally have as much background knowledge or experience. I believe adults and children learn many kinds of things faster when they already have knowledge or experiences that relate in a fundamental way to the new material they are trying to learn, and when teachers can show them that. In some cases, however, adults will let previous background knowledge get in their way, if they either have a wrong perspective on it, or if they did not learn it correctly, or if related, but different, things confuse them. In those cases, children without all the confusing clutter in their backgrounds, will be able to learn the new material faster. In some cases, children will have background experiences that may be helpful which adults do not have, such as in learning technology based on conventions and tools first used in children's games or entertainment. Or when children come from bilingual homes and have an advantage over monolingual adults in learning a new language because of that.
Even in presenting what might be considered to be strictly factual material, such as anatomy, it helps to organize the material in a way that people can make sense out of it or retain it. Some of these are extrinsic to the material, as with acronyms or other mnemonic devices to help people at least recall the first letters of the list of things they need to know, as in "On old Olympus' towering top, a Finn and German viewed some hops" where the first letter of each word is also the first letter of the each of the twelve cranial nerves in order of position. But sometimes intrinsic explanations are even better, where one shows how and why something works. An example is the "Demonstration of Why or How (a+b)(c+d) = ac+bc+ad+bd."
Sometimes demonstrations or games or just very careful or dramatic explanations can help students learn factual material more easily than just their trying to memorize it. For example, games like blackjack help students learn to add with greater facility many numbers that sum to twenty-one. Games with multiple dice faces added together can help children learn to add single digit numbers from one to six. There are variations of many common children's card or dice games that make practicing adding and subtracting or multiplying be fun -- as in having children turn over two cards at a time, and the person with the highest sum gets all four cards (a variation of the cardgame usually called "War", in which opponents each turn over one card, and the one with the higher card keeps both; and this is repeated till someone gets all the cards, or you both get tired of playing). Or the winning pair of cards could be the pair that multiplied together forms the highest product.
Even in coaching athletic skill, or other sorts of physical skills such as violin playing, it often helps to explain techniques in terms of things already, or more easily understood. One interesting approach to describing a correct golf swing I heard one time from a pro was that the motion should be something like tossing out to the side a full bucket of water so that you got the water away from you as far as possible without spilling any first. One of my children had trouble holding her bow arm correctly when playing the violin, and her teacher at the time could not get it corrected by telling her repeatedly for a year what to do, and having her try exercises that never solved the problem. A different teacher fixed the problem in two minutes by telling her to imagine that she held a soccer ball or basketball between her arm and her body while she played.
Typically, material that we try to teach contains not only "strictly factual" elements but has a conceptual or logical component. Some material is even primarily conceptual or logical, or the difficult part is primarily conceptual or logical. Place-value in arithmetic is primarily conceptual, though it can be taught by rote in a strictly factual way. Teaching it by rote (e.g., telling students what the column names are, and teaching them rules or recipes for adding and subtracting, regrouping where necessary, etc.) makes it more difficult for students to understand or use it though and it takes them much longer to learn it than does teaching them for understanding in a way that they can assimilate and absorb.
In my paper on "The Concept and Teaching of Place-Value", I describe a method for teaching place-value that is much easier and far faster for more students to learn it. It involves giving them experiences with color value, using either poker chips or color tiles, and then showing them how place-value relates to color value. Place-value is a very difficult concept for reasons I explain in that paper, but for some reason children have no problem with using poker chips (or color tiles for those squeamish about using something possibly related to gambling) whose values depend on their colors, when you simply tell them what the exchange values of white, blue, and red chips are. (E.g., a blue chip is worth, or the same as, ten white ones; a red one is worth ten blue ones.) They can quickly learn to add and subtract amounts represented by poker chips (or by an abacus, though an abacus represents a higher degree of difficulty). After you have taught poker chip addition and subtraction in the way I describe, you can point out that column values in the way we write numbers are the same as the different color chips. And each number in a column represents how many poker chips there would be of the color corresponding to the column. That is, the ones column tells how many white chips you have; the tens column is how many blue chips you have; the hundreds column is how many red chips you have, etc. Children can make that transition very quickly then with a little practice doing simultaneous column additions on paper and poker chips additions with the same numbers.
The reason I challenge that we need to know "how" children learn is that I have no idea why it is or what the mechanism is that lets children be able to add poker chips together and make exchanges between 10 of a lower value with one of the next higher value. But I know they can do it easily. And if you, as the teacher, have understood for yourself just how place-value works, and if you know kids can work pretty easily with poker chips and their values, you can see how to devise ways to teach columnar place-value by using the poker chips first. It is not about how children learn, but about what knowledge they have or can more easily have that will let you elevate their understanding to include place-value. If they already understand about poker chips, you can show them how columnar place-value is like using poker chips, but if they don't already know about color or poker chip values, than you have to introduce that to them first. That makes the transition to their understanding and being able to use place-value correctly possible, easy, and rapid.
It seems to me what is required for good teaching is to know how to find the connecting points between the subject matter and what kids know, or could more easily know. That means being able to know the sorts of things students tend to know in general and being able to ferret out which kids do not know that or what more things some kids might know already. And it also means knowing your subject matter so well that you can link it in some way to whatever you can find kids already know. And it means being able to figure out things you could easily teach them that would allow them to make the transition.
Sometimes opportunities to do this arise naturally. For example, when students in a middle school held a student council election one year, the eighth graders won all the officer positions because eighth graders in the school seriously outnumbered the seventh graders, and tended to vote for members of their own class that were running for general offices -- president, vice president, treasurer, etc. The seventh graders felt the election was "unfair", but teachers dismissed their complaints with indifference, pointing out the election was fair and that in democracy the majority wins. But democracy does not require simple majority rule, and often goes to great lengths to avoid it. The U.S. Constitution devised an elaborate set of procedures for making sure that very little is done by simple majority rule. If the social studies teachers in the school understood the nature of the U.S. government and the inherent problems with democracy that are, for example, discussed in The Federalist, they could have turned that election into an extremely teachable moment for a whole lot of kids, but they did not do that, either from lack of knowledge or from lack of consideration that teaching should be about linking more sophisticated concepts and knowledge to student experiences and understanding. Had this situation been handled properly, the students could have devised a much fairer student council election process, and would have learned valuable lessons about government that might have helped them make serious political contributions to society, in issues of fairness, when they become adults.
Teachable moments arise at all kinds of times, sometimes induced by a teacher. One common trick among those philosophy teachers who really want students to learn, is to give an "F" to students who turn in papers which argue that relativism is right or that everything is subjective, or mere opinion, etc. The students invariably come in to argue the unreasonableness and unfairness of the grade and show why they deserve not only to pass but to get an "A". The teacher will let them give all the arguments they want, and then will say something like "You realize, don't you, that you are totally contradicting the point you make in your paper, because you are offering me all kinds of objective and factual evidence for what you really believe is an objective truth -- that your paper deserves better than an "F". If you really believed the point of your paper, you would have to say that my giving you an "F" is as valid as your thinking it deserves and "A", and you would not be here arguing for the truth of your claim about the grade. Now I actually gave you a much higher grade than what I put on your paper because it is well-written, and well-argued, but it is simply wrong for reasons you did not consider, but which I hope you consider now, since you obviously do not really believe the conclusion you tried to establish. I only pretended to give you the "F" so that you would come in and argue exactly as you have."
These last two examples also relate to the issue of knowing what is interesting or significant or meaningful to students. My view of teaching is that it should not only be informative but, whenever possible, also inpirational so that students want to learn even more. When you have a "live" moment as in the above two cases, you already have the students' interest and attention and do not need to fish for something interesting to them to relate your material. In most cases, though, you need to spark some interest first, perhaps by stoking a debate among students over some issue they cannot resolve themselves or to present material in a way that students will likely find interesting, challenging, puzzling or somehow otherwise stimulating, perhaps even humorous. This again requires you to know both your material and students' likely interests.
And by "students' likely interests", I do not mean something from popular culture (though that might work), but something of likely universal interest to people. For example, there is a very strange phenomenon that few people know about, and if you mention it to students, they will test it themselves because they will not believe you, or if you demonstrate it to them, they will be amazed and want to pursue it further. That is the fact that if you drop a raw egg (a regular egg from a carton of eggs you buy at any grocery store), from a second or third story window (or sometimes even far higher), or if you throw the egg way up in the air as far as you can, the egg will not break when it lands, as long as it lands in grass, and does not hit a bare spot of ground or a stone or something. There is something about the physics of eggs and the way grass distributes the forces or something that makes eggs seldom break when you do this. It is extremely difficult to believe, even if you have seen it or done it several times yourself. It does not generally work, though, for "jumbo" or sometimes "extra large" eggs which often have very thin shells -- the kind of eggs that sometimes break even as you are getting them out of the carton. And it does not work for eggs that are cracked, of course.
Another surprising thing for students, which can help them get interested in areas of circles is the answer to this question:
Suppose that the earth was perfectly smooth all the way around, with no hills or valleys or bumps of any sort, and suppose you tie a ribbon around the equator after pulling it tight so that it fits snugly to the surface all the way around. Now imagine splicing in an extra piece of ribbon --exactly one yard (that is, 36 inches) long-- and then smoothing out the entire ribbon all the way around the globe, pulling out all the slack so that the ribbon is now everywhere the same height above the earth's surface, if it comes above the earth's surface at all. True or false: the ribbon will still be so close to the ground all the way around the earth that the extra thirty six inches you have added to its length will be virtually unnoticeable?It turns out that if you do, or look at, the math, you will see the ribbon will be nearly 6 inches off the ground all the way around the earth. That is so surprising to most people that they want to understand the math and the phenomenon better.
Also, for example, one of the ways I begin an ethics course, or a section on ethics in an introductory philosophy course, is to ask "When is it right to break a date (or an appointment, or promise), and why?" It is amazing sometimes the answers students will give, many of which other students will disagree with." It leads to some interesting issues and yields some insights into students' views of ethics in its own right, but it also gives you, as the teacher, a way to get into the more general and useful topic of formulating general principles of ethics. In some cases one does that by seeing what more general justifications students will give for their specific reasons that justify breaking a date. In other cases a teacher might show how their specific justifications fall into, perhaps, two broad categories, or perhaps a larger number of slightly less general categories. The point is to relate what you want to teach to something in which they are either already interested or in which you can arouse their interest.
Another way to stimulate or reach universal interest is to tell a story or situation first that illustrates a problem, then state the problem in question form, rather than just beginning with the question. For example, I used to hate when teachers began a history or psychology course or some such by coming in the first day and asking "What is history?" or "What is psychology?" That usually puts all but the most avid "teacher-pleasers" into a stupor, usually saying something to themselves like "It is the stuff in this book" or "Who cares?" or "I don't know; why don't you just tell us, since you have the degree in it?" Instead if a history teacher were to come in to class and say instead something like, "Is it part of history whether Shakespeare owned chickens or not, or whether he ate eggs on the day he began to write Romeo and Juliet?" or "Suppose all the history books ever written and all the diaries and journals we have were totally lost, would there still be history? And if so, what would it be?" or "Are the things you have forgotten about your own life still part of your biography or history?" or "What if some of the things in history books are false? Are they still part of history?" or "Commentators and politicians often say things like 'History will judge whether this is the right policy or not.' Well, when does history begin, or get to make that judgment? How long do we have to wait for 'history' to kick in? Or who in the future gets to say?" Questions like those might generate a much more lively and a much more productive discussion than "What is history?"
One can also use the concept of probing students' knowledge in a way they find interesting and stimulating, to discover what they have learned from what you have taught or assigned -- to see what they have "got out of" an assignment or your instruction. And to see what they might have missed or misunderstood. I have argued elsewhere (in "A Common, but Terrible, Mistake in Teaching Math and Science" and in "The Immorality of Giving Tests for Grades in Teaching") this should be part of your teaching them, not of your testing them for a grade. It should be part of your evaluating how effectively they have learned the material in order to see whether you need to do some more or follow-up teaching or whether it is okay to move on to different material, particularly material that might require knowledge and understanding of what you have just tried to teach.
For example, I teach that the "Golden Rule" is not a feasible rule for determining what is right because what you want may not be what others want, and because even if you all want the same thing, it may still not be the right thing. I try to show through various, usually memorable, examples (e.g., if you want a girl you just happened to see on the street for the first time to kiss you, does that mean you should run across the street and kiss her?) that the Golden Rule fails and why it fails, pointing out that it only seems to work because it coincides with the right results when what you want for yourself is right, not because you want it, but because of other reasons that make it right. I do point out that one way of taking the Golden Rule is that one should remember other people are human beings with feelings, just as you are, and so they should be treated decently and rightly, just as you would want yourself to be treated and just as you and everyone else deserves to be treated. But that understanding of the Golden Rule does not really help you determine what is right, or what is the right way to treat others. It only lets you know that once you know what is the right way to treat others, that you ought to do it, just as you would want them to treat you in ways that they know to be right.
Anyway, I go through a whole litany of examples, and students themselves often have similar examples of their own -- such as the boy whose girlfriend gave him two bookends in the shape of frogs because she is into "frog" items, frog knick-knacks, frog decor, frog designs, frog accessories, anything to do with frogs. His comment to the class was "Why is that a good birthday present for me? I don't care about frog things; she does! That would make a good present from me to her for her birthday, but not from her for me on mine." So we go through all this in class, with my explanations and examples and the students' examples and comments or questions, etc. and at some point, I think I have "taught" the point to them. How can I test whether I have got the point across to them?
Well, if I really didn't care to find out the truth about what they have learned, I could ask them "What do you think of the Golden Rule?" or I could ask them to tell me what is wrong with the Golden Rule, or I could ask whether the Golden Rule is a good principle or not, or whether they should use the Golden Rule to determine what is right or wrong. The odds are that they will give me the right answer on that because they will simply parrot what I have told them and what we have discussed in class. What I need to know, though, as a teacher is whether or not they really "get it." I have found a question that tends to show whether they have or not, or whether I need to re-inforce the material in some other way. So I might ask them to write about the following (I have them write first so that they each have to give an answer, not just the first person or two who might speak up in class): "Okay, we have spent a lot of time covering the flaws in the Golden Rule and explaining and seeing that it doesn't work in those cases where what you want is not what the other person wants or where what both of you might want is still wrong for some other reason. But apart from those cases, is the Golden Rule okay? Which kinds of cases can you use it in in order to determine what is the right way to treat other people? In short, when does the Golden Rule work to let us know what is right and what is wrong?"
The answer is supposed to be "Never!" because it is a flawed rule from the beginning, for determining what is the right thing to do in a particular situation. But students who have not really seen the light of that will give all kinds of cases where they think the rule works, and one has to then take that as another teaching opportunity to make the point more clearly, convincingly, and effectively. After one has done that, one can go on to ask additional questions to see how well the students can apply the material.
Thus, there are a number of sets of questions or uses of questions for teaching. The initial set of questions for beginning to teach new material is to find out what the students' state of knowledge is with regard to the topic at hand. The second set of questions, after, or while, teaching the material, is to find out how the students are assimilating what you say or what the students have learned and can do with the material.
It is often difficult to know where to start or to figure out there is a misunderstanding or lack of understanding on the part of students, just as it sometimes is difficult to know when there is a mistake or misunderstanding in real life in general. I had a strange occurrence in the supermarket recently. I saw a man I had not seen in a long time and I re-introduced myself by saying "Hi, Doug, I'm Rick Garlikov." He said he recognized me and commented that he still had the photos I did of his daughter when she was young, and we talked for a while before I asked him how his wife's sister was that had cancer. He said none of her sister's had cancer. I said "Then maybe it was her aunt -- her mother's sister, and I might have been confused." Again the answer was "No, none of her relatives." I pointed out the last time I had seen them was with the aunt or sister, when I had taken a family picture of her mother's whole family, and he said "No, that wasn't us." Turned out that I had mistaken him for another man I knew who, by sheer coincidence, was also named Doug, and who looked a lot like him, had a daughter I had photographed, etc. After we got that confusion on my part straightened out we had a better conversation. But the point of this story is that it took us both a little while to see I was mistaken in thinking who he was, and if the right topics had not come up to point that out to him, I would have gone home and told my wife I saw Doug A. when really it was Doug L. that I had seen. He would have gone home and told his wife he ran into me and that I had remembered and recognized him after all these years even though I only had met him once or twice before.
It is hard to devise questions that will let you know what children already really know and understand about material --or questions that will let you find mistakes in students' thinking-- until you learn in what ways kids or adults tend to misunderstand stuff in the material you are teaching. Fortunately with any given topic, most people tend not to understand, or to misunderstand, it (when they do) in the same or similar ways. Once in a while you will come across a student or even a group or generation (young or old) with a fairly unique misunderstanding. But whether you are starting out as a teacher or whether you have a somewhat unique student or group of students, my point is that, no matter how difficult or challenging it might be to find out what students already know that is relevant to the material you want to teach, it seems to me that it is much easier to become a good teacher by keeping this model of the art of teaching in mind than it is just to follow scripted techniques or methods that really just are general mechanical ways of trying to do this in cases where one has nothing more useful s/he can discover or devise. In the best cases of scripted techniques or of teaching "methods," they are simply shorthand ways, not just of finding out what students do not know, but ways of also finding out the kind of knowledge students already have that might help you teach them what they do not yet know. In such cases what I have explained above will not conflict with the method or technique you are using, and it might help you understand it better and use it more wisely and efficiently, perhaps varying somewhat from it in a useful way when that might likely yield better results, or might at least be worth a try.
In an abacus typically there are (at most) only
10 beads of each color on a row, whereas with poker chips, you can have
more than ten. So a child can add 5 white chips to 8 others and get
13 white chips, before exchanging ten of them for a blue chip. But
with an abacus, you can only add either two of the five to the eight, or
five of the eight to the first five, and have to hold the other three in
reserve (in your mind) till you have exchanged the 10 beads worth one each
first row for a bead worth ten on the next row. Then you have to
add the three you had not yet added. That is much more difficult
to learn. (Return to text.)
Two things somewhat similar to the "learning styles" fad, is the "brain chemistry/physiology" field of education and the concept "how children learn." We have had better and worse teaching methods long before the brain was even understood to be the organ of intellect or knowledge, and knowing brain physiology does not necessarily help you figure out a good way, or a better than already known way, to teach some particular skill or information. It is extremely difficult to translate studies in brain physiology or neural networks into effective teaching techniques.
Moreover, it is one thing to know how to use what children know in order
to teach them something else or more complex; it is quite another thing
to know "how children learn" or what the (particularly neurological) or
self-learning process is that they utilize in learning the first thing
or in using that first thing to understand the more complex thing.
Trying to use neurological or other sorts of processes to teach, seems
to me, not totally dissimilar from trying to learn how to drive a car by
studying the mechanics and thermodynamics of the internal combusion engine.
It is neither necessary nor sufficient. (Return to
The ribbon will be between 5 and 6 inches off the ground all around the whole earth. As you will see below, anytime you add X amount to the circumference of any circle, you add X/2pi to the radius of that circle, which, since pi is a little over 3, is roughly X/6.25. Since we are adding 36 inches to the 24,000 mile circumference, we will be adding 36/6.25 inches to the radius, which is approximately 5.75 inches.
The circumference, C, of any circle is 2piR, where R is the radius of the circle.
Dividing both sides of C=2piR by 2pi, we get the formula for the radius in terms of the circumference: R=C/2pi
So take any circle, and designate its circumference as C1 and its radius as R1
Now, if we add 36" to the circumference of that circle we get a new circle with a new circumference, call it C2
The new radius, call it R2 will be C2/2pi
and since the new circumference is 36 inches longer than the first
circumference, we can substitute (C1+36") for C2 to get: R2 = (C1+36")/2pi
Since a/x + b/x = (a+b)/x and vice versa, (a+b)/x = a/x + b/x, we can rewrite our result to get: R2 = (C1/2pi)+(36"/2pi)
Since (C1/2pi) is the old radius, this means R2 = R1+(36"/2pi)
And since 2pi is 6 when rounded off to a whole number,
this means that the new radius is the old radius plus approximately 6 inches.
And, in general, the new radius of any circle will be the old radius plus
one-sixth of the amount added to the circumference. So whether you add
36 inches to the circumference of a dime or to the circumference of the
earth, you increase the radius of each of them by the same amount, roughly
six inches. Hard to believe, but true. Hence the radius of the ribbon with
the 36 inches added to it will be nearly 6 inches more than the radius
of the earth, meaning that it will be that much higher off the ground all
the way around the earth. (Return to text.)
In the early 1970's, one semester I had a group of students who answered "When it is right to break a date?" with "Anytime you want to, because it is better to be honest than not to be. If you do not want to go out with someone, you should tell them, rather than leading them on." I, of course, asked what about 30 minutes before the senior prom, when the girl has bought a dress (or the guy has rented a tux and bought her flowers, made dinner reservations, etc.) and is looking forward to going to the prom with someone. I thought that would show them they needed to refine their first answer. It didn't work. They were a group, and the only group I ever had, who were wedded to this principle of total honesty no matter what the consequences (because of government lying and cover-ups in the news at the time, etc.), and no matter what outrageous illustration of their principle I came up with to show them the error of their ways ("Is it okay to murder someone if you want them dead?" "Yes, as long as you are willing to take the punishment if you get caught.") they held to their principle. Nothing I could do that class period got me anywhere with them.
So the next class period, two days later, I went in and told them, contrary to what I had said at the beginning of the term, 12 weeks ago, they would have a comprehensive exam Monday and Tuesday (this was homecoming weekend) that would count 75% of their grade. I gave reasons for why I was doing this, etc. I expected them to object with demonstrable outrage, but instead they became docile and only asked whether they needed to bring blue books (test booklets) and which topics would be covered on Monday and which on Tuesday. I told them to bring blue books and that I would not tell them which parts would be covered which days, because they were supposed to have learned it all as we were going along. I was pretending to be as cruel as I could be, since I was supposedly disappointed, frustrated, and angry with them because, as I told them, "you have brought this on yourselves by not studying the way you were supposed to," etc. -- all of which they believed about the other members of the class. It took everything I could do to keep from laughing, because this was all a hoax on them to make a point.
When they had no further questions and were sitting there feeling sorry for themselves, and angry with me, I asked them how many thought this was wrong for me to do, and that I was a real jerk for doing doing it to them. They all raised their hands. I then told them that it was all a lie, and that there would be no test. Then they got angry! And they wanted to know why I had tortured them for thirty minutes in class. I pointed out that according to their main ethical principle of two days ago, it is right for me (and everyone) to do anything I want to do whenever I want to, and that if I want to break a promise, at any time, that is okay. If I want to give a surprise exam covering twelve weeks of material after saying from the beginning there would be no exams, according to their principle, that would be what is right for me to do. "How many of you still believe that?" All but two of them gave it up on the spot. The two diehards said "You're the teacher so you can do whatever you want?" And when I answered "But does that make it right?" they responded with "That's just the way it is." So I had more work to try to do with them when I could figure out what that might be.
The point here is that I was not prepared for the lengths they would go to defend what they thought was the central principle of ethics. One always has some students who believe at first that people should or must do what they really want to do (ethical egoism, often based on the false theory of psychological egoism that deep down inside whatever we do is what we really want to do), but I never had a whole group believe that or be so unperturbed by the logical extension of their principle -- until it put them on the "receiving" end of it. Fortunately for teachers, it is unusual in teaching to have one group of students be so different from all others in covering a topic. So even though teaching is an art and has to be tailored to each student or group, normally the variations of student backgrounds and ideas are not so unique that the teacher has to have totally separate lesson ideas for each student, even if the teacher wants to move different students at different paces or have some additional stimulating material for some that others might possibly not yet be ready to do. (Return to text.)