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On the popular weekly television series "Tool Time", almost every episode contained a segment in which Tim's knowledgeable neighbor, Wilson, explained the social practices of another culture or of another era in order to help Tim put his own personal problem of the day into perspective. And every week although Tim might get the ultimate point Wilson was trying to make from some other aspect of the conversation, he would totally misconstrue the specific information Wilson dispensed, so that if Tim were concerned about the clothes his sons wanted to wear, Wilson might try to point out that fashion is a very subjective, harmless matter of mere taste, by saying something like, "The Maori's covered their arms with bearskins to ward off bad luck," and Tim would invariably repeat it as something like "The mayor's bears got their arms on his skin; so I shouldn't care what the boys wear as long as they wear some covering to prevent that kind of bad luck from happening to them."

Periodically some teachers circulate collections of "student bloopers" which are compilations of student homework or exam answers that show the same kind of misconstructions and lack of understanding. While some object to such collections as teachers' mockingly disparaging students' ignorance, I believe, they exaggeratedly demonstrate two serious, actually universal, pedagogical problems of a totally different sort -- problems involved with trying to teach for understanding:

(1) Unless there is some meaningful "engagement"⁽¹⁾ of material by students, they will often misconstrue it or they will miss important points in it. Bloopers are only the comical manifestations of such misunderstandings, but I think misunderstanding, or lack of understanding, is the norm for most student learning, and that the only reason it does not show up more frequently is that situations do not arise that would allow it to be demonstrated.

More importantly, (2) interpreting or explaining anything (whether it is art, literature, science, experience, or any sort of phenomenon) is different from learning an interpretation or explanation by being taught or told that interpretation or explanation. To learn an interpretation of anything is not necessarily, and not usually, the same thing as interpreting it. This is, in part, because of the phenomenon described above in (1), but it is also because there is something fundamentally very different involved in working out one's own understanding versus learning someone else's understanding -- between explaining something to yourself and having something explained to you.

First, the more one reflectively thinks, writes, or talks about any mistaken information, the more likely s/he is to notice (or get someone else to notice and point out) that something is wrong. Some idea using that information is more likely to be juxtaposed with some other information in such a way that the student or teacher will realize there is a misunderstanding occurring. If students only study for tests and don't use or actively think about the material they are learning, they are less likely to make the kinds of mistakes before the exam that would show they have learned something incorrectly. (I met a third grader one time who had invented her own way of subtracting, and it had always given her the right answer. Her second grade teacher had encouraged such inventiveness in math, but had not realized the girl's method was flawed, though it gave the correct answer in most "normal" second-grade math circumstances, but only because of a coincidental feature. It did not work for all subtractions, however, and that would have shown up sooner if the girl had worked more problems in second grade and had someone check them.)

Second, if students are meaningfully engaged with, and actively thinking about, a topic, they will be more likely to notice when something does not make sense to them - not because it leads them necessarily into a noticed error, but because they will more likely have a feeling of lack of understanding -- that vague feeling of discomfort that something is not quite right, not quite complete, or not quite satisfying about an idea, principle, belief, perspective, or set of propositions. Instead of memorizing things that make no sense because one's information may be incorrect, incomplete, or incoherently organized, they will more likely question their own understanding until they can make sense out of a topic. That does not guaranty that they will understand it correctly, but it increases the chances.

There are two exceptions to their being a difference in understanding between figuring something out for yourself and having someone else explain to you what they have figured out about it:

(1) if you have tried to figure out a phenomenon yourself and have made great progress but have either just made some mistakes or have simply not come across a key discovery or a key perspective, you are then often primed for, or receptive to seeing, someone else's explanation just as if you had made your own discovery of it. Scientists, for example, working on the same problem independently of each other will often be able immediately to understand and appreciate the findings of the first to solve the problem. Or a student who has worked unsuccessfully on a homework problem in an involved way and has tried all kinds of different approaches and ideas will often immediately understand and appreciate, either someone else's solution, or an explanation of how that solution differs from his/her attempted solutions in a key way. Other students in the class may have no grasp of either the correct explanation or how it differs from the failed attempts.

(2) there are certain cases, perhaps even certain methods, of explanation that tend to help students see problems in a way that helps foster or guide their own understanding as the explanation is being given. These are methods that are various forms of guided discovery (such as the Socratic Method of teaching, as explained at www.Garlikov.com/Soc_Meth.html) and certain lectures that lay out the explanation in such a way that students can anticipate what comes next, while then internalizing or absorbing what came before, if they are paying careful attention. However, this latter is particularly difficult to do. A great many lectures or audio/visual/graphic explanations try to do this, but no matter how technologically sophisticated they may be, they tend to leave out key steps or explanatory steps for most listeners/viewers. Textbooks and math problem solutions done in classrooms often are extremely difficult to follow because one cannot see where some particular step comes from. Sometimes a teacher will combine two or more steps without realizing it to get the next line of proof, or sometimes a teacher will put down a step that comes from something previously known outside of this problem, and students will not realize s/he has done that, and not understand where that "step" comes from.

PBS showed a series of physics classes that had a really dynamic and interesting lecturer and that had animated televised graphical derivations of formulas and illustrations of phenomena that were really impressive, but they were virtually impossible to follow because they didn't make any intuitive sense as you went from step to step. They would show mathematical derivations and manipulations that ended up being useful somewhere, but it wasn't clear why you were doing any of the particular manipulations as you went along. When they showed animated sequential diagrams of the elements involved in a reaction, causal chain, or sequence of events, it wasn't clear why the elements in the animations proceeded in the sequence they did, rather than going in some other path, that, often in the diagrams looked like they would be even more likely. And there was often no explanation of the evidence for (or reasons to believe) the sequence progressed in the particular manner shown.

It is extremely difficult to make explanations you can simply communicate to someone else in a way they can understand it, even if they are paying attention.⁽³⁾ Sometimes too many steps will get someone confused if they cannot keep those steps in mind. A long set of navigational directions for someone who is lost, will often fly over their heads. Or they may contain a vague or ambiguous element, such as "turn right when you get to a large church" and it turns out there is a fairly big church before the much larger church you had in mind.

Most of us have been faced with a request for directions to a place we know how to get to, but don't know how to explain how to get to - especially in a way that we can be confident the other person will be able to follow; hence, the response often given "You can't get there from here," which, I believe, means that the way is quite complicated and it may be very difficult, or even impossible, to tell you in a way that will actually help you get there. There are various ways to try to overcome such difficulties: (1) pointing out the general directional path the person will be taking before giving them the specific streets, distances, and turns, (2) directing them part way and telling them to stop for the remainder of the directions at a place it will be easier to see what people mean as they give them, (3) trying to discover some landmark or area closer to the destination that the person may already know how to get to, and then trying to guide them from there, (4) telling them that you will send them an easier way to follow though it will be a longer way, etc.

Someone who writes down all your directions and follows them to the letter, arriving at the destination, may still not "understand" how s/he got there or know how to get back home or how to find this destination again without those directions in hand. However, after possibly just another trip or two to that destination, the person will often become quite familiar with how to get there and back without even needing the directions.

The same thing happens with putting things together from a kit by following a manufacture's instructions and diagrams. Often the first time takes far longer, and is more difficult to figure out, than any subsequent times. Once one "sees" what is intended and how the object is actually assembled, one understands the instructions in a way one just often cannot from simply reading them.

There is something more going on in understanding instructions or other people's explanations than just reading or hearing the words. The fairly simple cases of following straightforward, linear directions in making a trip or assembling a product show that it is not the directions alone that produce the understanding. I believe that one learns or figures out other things for oneself as one follows these sorts of instructions, and that it is these other things, and this part of the process, that produce understanding. It is not just having the instructions or even being able to recite them that produces the understanding. One picks up other cues in following directions to a destination so that when one is finally able to repeat the trip without the directions, one relies on what s/he has noticed for him/herself that are not in the directions, and that one may not even be aware one is noticing. In this way, instructions are only a guide or a heuristic to understanding, not the understanding itself, nor even the simple direct cause of it.

Similarly, in learning a skill that requires a certain touch, it is not enough just to watch someone perform the task in order to know or understand what they are doing in a way that is meaningful enough to be able to do it oneself. If one watches an expert spread window putty or ice a cake or cut wood with a jigsaw, there is more to it than meets the eye; and there is something to it that is different from just receiving instruction. One has to learn and gain one's own "feel" for the process. Golf and tennis swings are other cases where verbal instruction, demonstration, and even video demonstrations of one's own swing are, by themselves, normally insufficient to produce a proper swing; it takes an understanding and a "feel" for the stroke by the person learning.

In tennis, for example, one of the difficult concepts, even as a concept, for people to understand is the concept of "accelerating through the ball", as opposed to just swinging hard or hitting the ball in a way that seems to be hard. In numerous sports, the concept of accelerating is confused with the concept of velocity; and it is also frequently taught as "follow-through" instead of as acceleration. If, for example, a runner gets to a finish line and stops just on the other side of it, that has to mean that s/he was going slow or slowing down as s/he approached it because momentum cannot be stopped that quickly otherwise. Similarly in swinging a bat or a racquet, if there is no "follow-through", the bat or racquet had to be slowing down as it made contact with the ball, producing less acceleration into the ball than it might have. However, when you tell people to follow through their shots, they tend to simply keep moving their bat or racquet as an afterthought, at some relatively slower velocity. They tend to put the power into their swing at the beginning and then decelerate as they hit the ball, and then just simply keep the swing moving so they look like they are following through, whereas a real follow-through is a different sort of thing and indicates a different sort of swing. If one is speeding up as one hits a ball, s/he simply will not be able to stop the motion of the swing until there is nowhere else to go with it. Your arm and body will stop the swing when you run out of room to extend the swing any further. A true follow-through is caused by being unable to stop due to momentum - not by a merely voluntary continuation of motion. There are various ways to try to explain this to learners, such as getting them to imagine they are trying to drive the ball through a point out in front of it, or trying to get them to imagine the point of impact as out in front of the ball, or to tell them that it is like throwing a bucket of water at someone far away. But such instructions don't tend to get the point across. Something normally has to happen where the person does it somehow and feels what they are doing, and then they will usually say something like "Now I see what you are saying!" or "Now I see what you have been trying to tell me!" This latter expression seems to me to be particularly indicative of the fact that understanding is not embedded in the explanation itself in such cases. The student may even have learned long ago to repeat the instruction, and yet not actually have comprehended it even though they thought they did.  Knowing the words of the instruction/explanation, and knowing how to use them correctly in sentences, is not necessarily the same thing as "seeing" their meaning.

A possible analogy, though it is possibly a different process, may be the difference between being able to play the written musical notes of a musical score versus understanding how to make those notes really come alive and be able to play not just the notes but the music those notes represent, in an interesting, artistic --musical-- way. Notes in music are only a representation, and an approximate representation at that, of the sounds that are intended; they are a guide to the production of that sound. Similarly, an explanation is only a representation or an approximation, of what is usually a richer understanding than the words alone can convey. Explanations serve then as guides to understanding, not as substitutes for it.

This may be particularly true for more complex or abstract intellectual ideas. Physicist Richard Feynman often talked about the difference, for example, between working out something in physics mathematically on the one hand, and understanding it intuitively somehow on the other. Often he would solve the math of a problem before he understood it intuitively. After he worked out the mathematics of a phenomenon, he would then try to see why the math worked out the way it did.

I had one funny experience with that sort of thing in high school once, where there was a trick problem presented to us that went something like this: a train leaves New York for Los Angeles averaging 52 mph. Eight hours and 37 minutes later, a different train leaves LA for New York that will make fewer stops, so that it will average 89 mph. When the trains pass each other, which will be closer to New York? I worked long and hard on the problem, and finally had to resort to some approximations in order to deal with fractions involved (in the days before calculators), and I finally figured out how far they would each be from New York when they passed, and it turned out the same distance. I thought someone had constructed the problem so it would work out that way, so I changed one of the amounts to see how that would have changed the answer, in order to see how they may have gone about their construction to get it to come out equal to each other. And it turned out that, though that changed how far they both were from New York, it did not affect the fact that they were still the same distance from New York as each other. I thought this was really weird that the math would work out that way. Then I thought about it some more and looked at the problem again, and realized what a fool I had been. The question did not ask how far each was from New York, and it did not require that difficult calculation; nor did it ask whether they would be closer to New York or to LA when they passed. It was a trick problem, since, when they are passing each other, no matter where they were, of course, they would be the same distance from New York, since they are in the same place when they pass each other, and that place will only be some specific distance from New York for both of them. Even if the New York train got to LA before the other train left for New York, they would still both be the same distance from New York.

As the material one is trying to learn or follow becomes more complex or more abstract, the difference becomes even more pronounced between knowing an explanation or instructions in a verbal way and truly understanding what they mean. Some friends of mine sent me the following e-mail math phenomenon one day almost simultaneously, and after I followed the instructions, I looked at it to see whether I could tell in some short time why it worked. It turned out to be fairly easy to figure out how it worked, but the explanation turned out to be so laborious when put into words, that it made it look very complex. This is in part because I gave the explanation for each step in the order the steps are given; whereas the discovery of how it worked did not precede in that way. (See the footnote in my explanation of step 4 for the way I worked out the discovery.)

So, of course, to be a smart-aleck I sent the explanation of it back to them, because I knew they would not likely care to even try to follow it. In words, the explanation is very complicated, though the idea it attempts to describe and communicate is not difficult at all. I am going to describe this whole process because it will illustrate the point between having an explanation and having an understanding, and because I then want to go on to describe what tends to happen during instruction when explanations are given and learned, without understanding's occurring. Often even a person who knows and gives an explanation will not have the understanding.

Here, first, are the directions they sent me:
Do not scroll down. Read and work out as you go:

1. First of all, pick the number of days a week that you would like to eat out.

2. Multiply this number by 2.

3. Add 5.

4. Multiply it by 50.

5. If you have already had your birthday this year add 1749. If you haven't, add 1748.

6. Last step: Subtract the four digit year that you were born.

You should now have a three digit number:

The first digit of this was your original number (i.e. how many times you want to go out each week). The second two digits are your age!!!

1999 IS THE ONLY YEAR THIS WILL WORK SO PASS IT ON TO YOUR FRIENDS!

The following is what I sent back to them. The ">" symbol designates the lines I quoted from the above:

> 1. First of all, pick the number of days a week that you would like
> to eat out.

The reason for this is that the number you pick needs to be a single digit whole number (i.e., a number less than 10), so that it will end up the first of the three digits at the end.

> 2. Multiply this number by 2.

This is so that when you multiply again by 50 (in step 4 below), it will give you the number that is 100 times your original number, since you are multiplying it by 2 now and then by 50 later, which is the same thing as multiplying it by 100 (which will make it the first of the three digits you end up with later).

> 3. Add 5.

This gives you 250 more than 100 times your original number after you multiply it by 50 in step 4. Then when you add 1748 or 1749 from step 5 below, that will essentially bring this part of the number up to 1999 or 1998 (whichever year was your most recent birthday) because when you add
250 to 1748, you get 1998; and when you add 250 to 1749, you get 1999.  So, of course, when you then subtract your birth year you will get your age (as the last two digits -- which, of course, assumes you are less than 100), and you get that original number as the digit in the "hundreds place" because you have multiplied it by 100 and (by just looking at it as a digit, are then dividing it by 100 essentially -- e.g., the first digit in 300 is "3", which is the same as dividing 300 by 100).

> 4. Multiply it by 50.

So, if you started with X as your first digit, you now have 50(2X + 5), which is 100X + 250. ⁽⁴⁾

> 5. If you have already had your birthday this year add 1749. If you haven't, add 1748.

And this will then give you 100X + 1998 (or 1999)

> 6. Last step: Subtract the four digit year that you were born.

> You should now have a three digit number:

This gives you 100X + your current age, and they are assuming that you are less than 100 years old.

> The first digit of this was your original number (i.e. how many times you want to go out each > week). The second two digits are your age!!!

Your age usually is given by subtracting your birth year from the year of your most recent birthday! And multiplying any single digit by 100 and then looking only at the first digit in the answer, will normally give you the digit you started with....

> 1999 IS THE ONLY YEAR THIS WILL WORK SO PASS IT ON TO YOUR FRIENDS!

But you can do it for the following years by adding 1749 or 1750, etc.; OR we could simplify the whole thing to:
1. Pick a single digit number

2. Multiply it by 100

3. Add 1999 to it if you have had your birthday this year, and 1998 if you have not yet had your birthday this year.

4. Subtract the year you were born.

5. You will have your current age plus 100 times the original single digit you chose.

Or better yet, just skip the original number and the multiplying it by 100, and just tell people to subtract the year they were born from 1998 (i.e., last year) if they have not already had their birthday, or from 1999 (the current year) if they have; and, lo and behold, they will have their age. Works for everyone; truly amazing!

As I said before, the point here is that it is unlikely that this explanation will provide much understanding to anyone who simply reads it, without thinking about the issue in depth; and it may even be easier for a person who thinks about the original problem (i.e., why the recipe in the original e-mail works) to solve it by him/herself in some inventive way of his/her own than to come to an understanding of why the recipe works by trying to follow my explanatory steps.)

However, what tends to happen in school (and other places where explanations are given) is that students believe, or are led to believe, that knowing the explanation of a phenomenon is all that is involved. So they try to memorize explanations to whatever extent they feel they need to or care to. When they memorize it or recall it in some mistaken way, they then on occasion commit the sorts of bloopers common to "Tool Time". More frequently, however, they either learn the explanation well enough that they make no mistakes when they use or repeat it under normal, superficial testing situations (even if they do not understand it and do not realize they do not understand it). Or they use it in ways that show they don't understand what is going on, that show they probably don't even realize that, and that show that they don't even seem to realize what they say or write is supposed to make sense. It is entirely possible that a student might memorize all the explanatory steps above and still have no real understanding of how this mathematical recipe works, so that if one asked students to construct the same sort of recipe so that it would work in the year 2045 and/or end up with a four-digit number in which the first two digits were a number chosen at the beginning of the recipe, they would not have any idea where to begin.

The problem with many school tests is that they test for knowledge of the explanation without testing for understanding. Students can often "regurgitate" or repeat what they have read or been told in class without actually understanding it in any meaningful way, and without being able to use it. They can also repeat interpretations of other students or of teachers, or of secondary sources (such as text books, book reviews, or commentaries found in works such as "Cliffs Notes") without really knowing why those interpretations make sense or might be mistaken. They can state an interpretation without having interpreted the story themselves, and without "having an interpretation" in a meaningful sense of that phrase. Having someone else's explanation of something in your memory is no more to have an understanding of it than is having that explanation written down in front of you, like the one above about the age-determining algorithm.

Unfortunately, even when teachers realize this, it is difficult to construct tests that show whether a student understands a phenomenon or is merely applying some sort of memorized formula or technique to work with it. When students work math problems, or give literary interpretations, they may do it in a formulaic manner that makes it very difficult to tell whether they are merely adhering to a pattern they have learned or are writing from true understanding. Only when the formula or technique, as they may have learned it, does not fit the exam question, does it show they have not understood what they are doing, but are merely following a recipe or pattern. The difficulty is that you need to test them on the material you have taught them, so you cannot stray too far from what you have taught; yet this allows students to answer by mere recipe or pattern recognition. But more importantly, if you have in mind some application that really does test their understanding of the material, it seems to me somewhat unethical not to have raised it in class while teaching them. In other words, if you have taught them everything you believe relevant to the topic (for their current level of absorption and assimilation), it then is difficult to construct an exam that tests for their understanding of the topic rather than their knowledge of your explanations. That is a serious problem in a teach-then-test environment where the testing is important for grading, rather than being important for teaching. (See www.akat.com/Evaluating.html)

One reason many students have difficulty with unfamiliar or unrehearsed word problems in math is that they learn to follow recipes for calculating answers for math problems that are already set up for them, without having any idea why those recipes work or what they essentially mean. Then they don't have enough understanding about the workings of the calculations themselves to be able to apply them in a new situation or a situation where the specific format for the recipes is not already identified, set up, or rehearsed (the way it is in math book chapters). Also, I think students often have trouble in algebra because they have learned to manipulate numbers by particular techniques without having understood how numbers are constructed and without understanding various relationships within and among numbers. Many times adults themselves, including teachers, don't have any understanding of the principles involved, though they have learned how to do the manipulations. (See www.Garlikov.com/PlaceValue.html) I have an essay (www.Garlikov.com/Soc_Meth.html) that explains "place-value" in a way students can understand it. The essay explains binary math by looking at analogies with our normal (decimal) math system. Teachers who have studied the part of this essay dealing with the binary math often say it lets them understand the decimal system construction of numbers for the first time, and, moreover, lets them see that they had not understood it before although they always assumed they had simply because they knew column names, could write numbers, and could use numbers in all manners of calculations.

In social studies courses, even students who learn particular factual material will often not see that it is relevant to contemporary events it might parallel or mirror. In literature classes, students will talk about characters, settings, and plots in stories in the same way a robot who knows nothing about human behavior might describe these things. Yet if their friends ask them to describe a movie or television show, the students will bring it to life and describe all the elements necessary for a good literary analysis in a way it never occurs to them to do with literature itself. Yet it never occurs to students that a required classroom literary analysis should make the same kind of sense to them that their own similar reports of movies or tv shows do. They don't see school literary reports as needing to make any sense; such reports seem to students just to need to follow some format they have been taught, and if that satisfies a teacher or makes the report seem meaningful in some way to the teacher, that is sufficient.

The corollary to this is that while teachers may be trying to get students to interpret a literary work, or a science experiment, students don't see the enterprise as being about interpreting anything. They see it as trying to figure out, or guess, what answer the teacher has in mind - not because it makes any sort of sense, but because it is "the answer". When a teacher gives an interpretation of some phenomenon or evidence then, whether it is literary, mathematical, scientific, or in any other realm of human knowledge, students don't see what the teacher is doing as "interpreting"; they see it as being told what the answer is (supposed to be). They then don't look for the answer to make sense; they merely look for an answer they need to learn in case the question comes up on a test. Parents and teachers often unwittingly encourage this by not explaining material in the early grades, and sometimes by being arbitrarily authoritarian or inflexible in their explanations, even when they are mistaken, so that when students reach high school they have often even quit trying to understand things and just believe they are only supposed to learn explanations, not seek to make sense of anything related to school.

If you ask students to explain almost anything they have learned in school in any detail at all, particularly if you ask follow-up questions, you will see they rarely understand much, though they will think they do. Exams for college teachers are often disappointing because that is often the first time they learn their students did not understand material the teachers thought was made perfectly clear.

But very few things are easily made clear to someone else just by telling them or having them read something. Misunderstanding is extremely common. And this is true even for adults, about simple, ordinary matters. Just today a woman told me that she had told her husband she would be unable to take one of their children to a soccer practice. He said he would take the boy. He did. But he didn't pick him up, because he thought she only meant she couldn't take him to the practice, not that she couldn't pick him up once he was there. She, of course, had meant she couldn't work the soccer practice into her schedule at all. When they finally realized neither of them had picked up their son, they went to retrieve him where he had been anxiously waiting since practice had ended and everyone else had left.

If you bring into the mix more abstract matters, or more complex matters, understanding is particularly difficult to achieve, especially if the student is not even trying to make sense of material but is just trying to learn to be able to repeat at an appropriate time what you are telling him/her.

What is preferable is not doing away with explanations, but (1) finding explanations that foster understanding and (2) getting students to attend to explanations as a guide to fostering their understanding, not as a substitute for that understanding. Students always need to be in environments that enable them to see that repeating explanations is not an end in itself, to see that explanations only serve as a means to understanding, and to realize that they have to achieve that understanding for themselves, whether they use the explanation given to them or whether they use some other method of discovery. Teachers and students need to know that understanding is not something that normally can be passively given or received. Students need to learn to seek understanding of theoretical, conceptual, logical, derived, and evidentiary kinds of material. And students who already know to look for understanding should not be taught in a manner that discourages their ability or desire to continue doing so or that stifles their appreciation for discovery.
 
[For more about this topic from a different approach --why students don't seek understanding-- see my "Understanding, Shallow Thinking, and School" (www.Garlikov.com/teaching/Understanding.html) For other essays of mine about education, visit www.Garlikov.com.]

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1. By "engagement" I mean something like an intrinsic interest or fascination with material that is sufficient to foster thinking about it, and often communicating about it, because one wants to understand it, know it, be able to apply it, etc. This is different from learning something one does not really care about or think about other than just to complete an assignment or to learn for a test. One may study for a test or for an assignment in an engaged way, but many people do not, so the distinction is about one's active interest in the material, not about why one is studying it or whether one is studying it.  (Return to text.)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

2. On rare occasions students understand more than they think they do (an example and explanation of it is given in the paper on "Evaluating Students, Teachers, and Student Teachers" at http://www.akat.com/Evaluating.html), and on some occasions students "feel", in thinking about material, that they have an understanding of it that they actually do not. This is different from the cases where students merely believe they understand something because they have not thought about it very much and just assume they understand it. Normally, however, when one thinks carefully about material, one has a pretty accurate idea of whether one understands it or not.  (Return to text.)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

3. Also, in some cases, different people's minds follow different sorts of organizations of material better. When I took introductory level history of art, I had great difficulty following the details in our textbook. Many students were able to follow them just fine, but I could not. The material was organized so that you studied a particular, somewhat short, period of development of the paintings, then sculpture, then architecture of a region of Europe. Then the next chapter would do the same thing with a different region of Europe. Once all the regions were covered, they would repeat the process with the subsequent time period. I could not keep enough information in mind to see patterns. But I found an art history book that organized the material for each art form -painting, sculpture, architecture- from each region separately and "longitudinally" so that I could see the development over two or three hundred years, for each art form, of the art of each region. I was able to follow that better and be able to see patterns and remember them. Then it was easy for me to compare the art of the different regions with each other at a given time. I could do the temporal cross-sections after seeing the whole structures through time, but I was unable to construct the whole structures from the temporal cross-sections. (Return to text.)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

4. This line actually is closer to how I discovered the way this recipe works, rather than the way I am explaining it after making the discovery. What I did was to let "x" represent the number of days a week they asked about and then to follow the steps in the recipe algebraically from there. That let me see that eventually you got down to this line. At that point it was easy to see that the "250" and the "1748" or "1749" were just going to get you to the years 1998 or 1999, plus 100 times the original number. Then when you subtract your birth year from these numbers, you will end up with 100 times the original number, plus your current age. (Return to text.)