In many classrooms, student culture is such that students merely memorize, or haphazardly memorize, material without really thinking about it or trying to make sense to themselves out of it. By high school, or the end of middle or junior high school, they are accustomed to learning things that don't necessarily make sense to them, getting a grade (which usually is in some sense acceptable enough to them) on it, and moving on. At some point relatively early in their student careers, many students give up even thinking the material is supposed to make sense to them.

Or perhaps even worse, they think learning a few facts or a few statements about something, is to make sense out of it. A friend of my older daughter, told me in one sentence with just a few phrases, what the French Revolution was about, and she and my daughter thought that was all there was to it. Both are honors students and get good grades. When I pointed out that they couldn't even characterize their 16 year old friends in that simplistic a way, they said they were telling me what their teachers and texts had said, as if that should end the matter.

That same daughter went to a renowned university "debate camp" this summer, for high school students, and when she told me some of what she had "learned", and showed me her notes, and the outlines they had passed out, about ethical and social philosophy, it was all simplistic in the same way -- the highlight or headline approach to ethics and (social) philosophy; Mill and Kant in 50 words or less. She could talk about the material in vague and general terms, and could repeat the examples and statements they had given her, but she had no real understanding of the problems or how the works of the philosophers whose names she knew addressed or solved them; and she could not answer (or think to ask) reasonable questions involving applications of the material. She was not able to make the material relevant to everyday ethics or social dealings with people, even though she has a wealth of social experiences from school, to Girl Scouts, to her church youth group, all of which she is actively involved with. But it does not occur to her that what she learns in a school type setting about social philosophy is really supposed to be relevant to her personal social and group experiences.

Similarly, when students study science in school, or study literature or math, they learn the material so they can answer test questions about it, then move on without worrying whether it really made any sense to them or not. My younger daughter explained that potential energy was the energy you added to something when you raised it to a height. She thought you were actually putting something into it. So if you asked how something was any different if you dug a hole under it, from how it was when it didn't have a hole under it, without ever touching the thing, she couldn't answer how "the potential energy got added", or what the potential energy even was. But she was sure it was something "because there are two kinds of energy 'potential and kinetic' ".

Or listen to students answer questions in literature, grammar, or math classes. They can parrot what they have heard or read, even adapt some things they have previously heard or read to their current material, but it is all, or primarily, surface adaptations. They aren't saying anything of any real meaning or interest to themselves or to anyone. And this is not an affliction limited only to students. Adults, too, often have rather simplistic or naive ideas about how things work in the world or how they ought to be, while thinking they understand them better than they actually do. In a seminar I attended one time, one of the men came in all excited because he had just come across a quotation he thought very insightful -- that it was not hate that was the opposite of love, but that indifference was the opposite of love, because hate was at least still an emotion. I chuckled, and when he asked why I was laughing, I pointed out to him that both hate and indifference were opposites of love, just in different ways, that whether someone hated you or was indifferent toward you, in neither case did they love you.

In another ongoing seminar one time, taught by a supposed child development expert, the class was listed as a course in "The Moral Development of Children", but it did not take long to see that what the teacher meant by "moral development" was simply what is often called "socialization" or "enculturation". She was only interested in teaching about how parents could get children to behave in the ways they wanted or that society considered appropriate at the time. She was adamant that that was what "moral development" was. It was not about encouraging, fostering, and honing things such as moral reasoning or moral sensitivity, but about how to train children in the mores of society or the family in the most efficient ways.

Similarly, elementary arithmetic is usually taught as a series of methods or recipes for writing and manipulating numbers rather than as encouraging and enabling students to see mathematical concepts or understand (the logic of) numerical relationships. For example, (1) most adults have no real understanding of place-value or why it works the way it does, though they know how to use it to write numbers, and they can name the various columns (e.g., hundreds, tens, ones, tenths, hundredths, etc.). (2) Most textbooks and arithmetic classes summarily introduce and casually treat fractions as alternatively 1) parts of a whole, 2) proportions (e.g., 9 out of 10 dentists recommend....), and (3) division (e.g., 3/5 = .6 or 60%); yet these are all quite different things, and it takes fairly complex reasoning to show that they are equivalent to each other. Casually treating them as though they were obviously equivalent makes it more difficult for students to understand fractions than if they were introduced as separate things which "will turn out later to be equivalent, though that is difficult to explain at this point."  This is because those students, who don't see how talk about fractions in all these different forms as though they were all the same thing -- "fractions," will generally just feel either there is something wrong with their ability to understand or they will think there must not be anything that needs to be understood. So if they learn to deal with fractions at all, it will tend to be by virtue of recipes they have to remember, and know when to apply, rather than because any of it makes any sense to them.

Many students get "lost" in math by the time they get into algebra because they haven't learned about those numerical relationships which they could have understood from early on, and because they were not nurtured to even seek relationships or see that there might be relationships among numbers; they only learned numbers and numerical manipulations as "facts" or mechanical recipes. It is not that children need to be taught all possible number relationships; it is that they need to understand there are such relationships, by having been taught, or introduced in some meaningful way, to those they can learn, and by learning that there are many more to be discovered or learned.

What Socrates found out nearly 2400 years ago in Athens is just as true of adults today in the United States; it doesn't take very deep questioning to expose flaws and holes in people's understanding about a great many things, even sometimes in their areas of expertise.  The above examples about "potential energy" in science, fractions and place-value in math, the French Revolution in social studies, etc. are just a few examples of the myriad of things that students learn in school that they don't really understand.

However, since most people aren't going to become mathematicians or literature majors or historians or scientists, most schools considered to be "successful" do teach enough for people to become contributing members of society because they learn sufficient math to balance their checkbooks; they learn to read well enough to read the newspaper and business correspondence and to write well enough to send e-mail or business memos. But such success is at a relatively low level for 12 - 16 years of schooling. What I worry about is what is lost, because it seems to me that considerable is lost by not having students learn a great deal more. And I think students would learn far more if they were taught for understanding, and if, instead of memorizing material they readily forget, they were given the inspiration, opportunity, and nurturing to think about material they are taught and to put it into meaningful frameworks and perspectives.

School, as it is often taught, tends to promote such shallow thinking in a number of ways:
(1) Grading by assigning a letter or numerical grade, instead of discussing the content of mistakes and encouraging or compelling students to deal in some significant way with their errors or lack of understanding.

Students tend to see receiving the grade as the end of the study of the unit, not as a mile marker on the road to learning or knowing the material. In their view, when their chapter grade or six weeks grade or semester grade is in, they are essentially done with that material; they have covered it. If there was something they didn't learn, it is "too late" to "bother" about learning it now, and no need or point to doing so, since their grade has already been determined.

(2) Teachers' moving on even though a student may not have learned something, even something that might be crucial to know later.

Generally (except for rare teachers or rare circumstances) teachers tend to see giving grades as the end of the teaching of the unit, not as information about what additionally needs to be done to help the students more fully learn the material. The exceptions to this are those relatively rare teachers who use exams to try to find out what they might have done wrong or what students might have misunderstood that needs to be cleared up or revisited in a different way. Also, on occasion grades may be so low on average that a teacher, or a department head, may feel something was wrong with the teaching instead of the students, and may look for ways to have the material re-taught in order to try to remedy the students' deficiencies.

But even in various forms of "mastery learning" situations, where students have to reach a level of competence in a unit before they move on to the next unit, the level of competency is generally a grade or numerical score of some sort, such as "80%". But it is my contention that a student can sometimes get a score of 80% on material and not know the material as well as they should, either because some crucial material or questions were not covered on the test or because some of the crucial material fell under the 20% the student missed. On many tests it is quite possible for a student to know enough facts or to know enough formulas by rote to be able to work, say, 80% of the problems correctly, but miss those questions which require understanding general principals and deriving formulas from them that were not previously taught. If the teacher is only looking at numerical scores, s/he will not likely see that the questions the student couldn't answer correctly are far more important, and indicative of later success with the material, than the ones the student did answer.

(1) and (2) together tend to make school something like a contest where once your score is judged, the contest is over and you either came in first or last or somewhere in between, but it is done; and you are done with a particular chapter or a particular grading period. There is often little carry-over or continuity either in reality or in students' perspective about the material, so it doesn't "really matter"; it is "just school stuff". "I got a 'C' on it; I passed; I'm glad we're through with it." Or in response to what they got out of certain material, students might say "I got an 'A' on the exam." The grade is all that matters to many students about their school work, if that much.

(3) Teachers often save applications of theoretical material to use as exam questions instead of dealing with them in class in a way that might help make the material more "relevant" or interesting to students. In this way, teachers ask those questions that might have been most helpful in fostering student learning at the time that the students are not going to be studying the material or thinking much about it any more. And it is a rare student who will be disturbed enough by a question he could not answer to his own satisfaction on an exam to continue to think about it after the exam is over until he can answer it.

(4) In "traditional" math classes, often recipes or algorithms are taught just to give students methods for arriving at the right answer, without showing them how or why those methods work.

(5) In some "constructivist" math classes, children's supposed discoveries about math concepts are not challenged or deepened, but are accepted at face value if they help the student get the right answer. Sometimes students' understanding is not guided, or if it is, it is only guided until the students get to where the teachers want them to be without directly telling them. I have even seen cases where students developed methods of calculation that only worked with certain sorts of problems but which they thought were universal. Previous teachers had accepted these flawed methods as universal because they didn't try them out with a variety of problems and because they didn't try to understand the mechanisms the student had come up with.

(6) In history and literature courses, the kinds of questions teachers ask, and the kinds of answers they accept and even sometimes praise, are often devoid of any relationship to real life. Historical and literary figures are made, or allowed, to seem unreal or ungenuine to students and that is considered acceptable.

Even take the analysis of a movie such as Spielberg's Saving Private Ryan.  Critics ignore that the actions of the soldiers just don't make much sense. And the impression is given that the soldiers who died, while on the mission to bring Private Ryan back out of harm's way, died because of that mission. Yet, it is fairly clear if you analyze what happened that most of them did not die because of that aspect of the mission. One of them was shot by a sniper when he was not being careful, and it could have happened at any time he was in Europe during the war. The rest were killed attacking positions, or defending positions, that were well-fortified or heavily armed, and that had nothing to do with their order to find Private Ryan and bring him out alive.

Further, some questions were raised in the movie that were never answered. A letter from Lincoln was read that supposedly answered why Ryan's life was worth risking the lives of others to get him out, but the reading of that letter in that scene was just a very long non sequitur; it didn't justify the mission. The beach landing scene raised questions as to why they landed in daylight, without protective bombing, smoke protection, or any sort of cover, against machine guns and other weaponry pointed right at them, with their landing craft doors opened facing those guns, instead of turned some direction that would have afforded them a better chance to get off the ships dispersed, instead of being shot like fish in barrels.

Much about the movie, and perhaps about that aspect of World War II just did not make sense. But none of that seemed to matter in the critiques of the movie. Schools treat book characters and plots the same way. And book characters and plots -- particularly of certain books -- are often already one or two steps removed from sense or reality to most students in the first place, so studying them is like studying some other species in a zoo lesson, rather than studying how people just like them behave in ways that are not all that dissimilar to how the students themselves behave, or would behave, under relevantly similar circumstances.

One of my daughter's teachers assigned the Oedipus trilogy for summer reading, with no introduction to it, and the students, of course, predominantly thought it was boring and stupid because none of them would kill their same sex parent or want to marry their opposite sex parent. They bogged down in the specifics of the plot rather than seeing the universal patterns of behavior and the self-deception or denial about such behavior. They did not read the plays as being about how choices made because of your nature or character -- even rational and intelligent choices-- can lead you to the most unacceptable circumstances that you would never have knowingly chosen.

(7) Much of the adult behavior in schools seems arbitrary and unreasonable to students, but they are not often allowed to question it or seek explanations or justifications. Even less, to argue about it. Life in school is about adapting to what teachers and administrators "want" (you to do or say), not about understanding of what is really true or right, reasonable, or expressing what you really think and why you think it. If you try to show many kids in school a better way to work through problems or a better way to think about material, or if you disagree with some idea they have, they tend to say things like "but that is not what the teacher wants", "that is not the way we are supposed to do it", or "that is not what the teacher/book says".

The mentality goes beyond students; one university philosophy professor one time asked me to read and critique a draft of an article he was working on. I did, pointing out what I thought were errors and how he could amend them. His comment was that I was probably right but that he thought a particular journal he has his eye on would accept the article the way it was written. Having "his answer" accepted was what mattered to him - getting a good grade -- not publishing something that he could fully justify as correct.

(8) Even when teachers think they are trying to help students understand material, many of them are really just trying to get students to know and believe their explanation, rather than trying to get kids to understand the phenomenon in the way they do. If a teacher says the French Revolution was fought over A, B, and C, and students just memorize that and can repeat it when asked the point of the French Revolution, that doesn't mean the students really understand why the revolution was fought, not on a meaningful or human level. If a teacher says that raising a mass imparts more energy to it because one has then increased its potential energy, and kids can say that when asked why elevating an object increases its energy, that doesn't show kids have any understanding at all about the concept of potential energy.

Similarly in literature, if an author or a teacher works out and states his/her own interpretation of a literary work, that is a very different kind of experience than students stating those same interpretations simply because they have been told them and have accepted them without thinking, or have memorized them. Experiencing an interpretation is not the same thing as interpreting an experience. Stating an understanding of someone else's is not necessarily the same thing as having that understanding for yourself.

What happens when students just learn spoon-fed explanations is something like when a husband only hears what his wife is saying, but doesn't think about it enough to understand for himself its implications. So if a wife says she will be out of town for two days, the husband may be able to say in answer to someone's asking for his wife, that "she is out of town for a few days", without its occurring to him that means he is now responsible for doing certain things she normally does, such as feeding the dog or taking the kids to a piano lesson, or making sure they have breakfast and are reasonably dressed for school in the morning. In cases like this, the husband knows the material that was told him, in some sense of "knows", but he doesn't fully understand the significance or the ramifications of that material. And those ramifications or that significance might be so elementary or so important that we tend to say that in the most important sense of "knows", he does not know the material.

In that very same vein, students can write "E=mc²" without realizing that because "c", the speed of light, is such a large number, that "c²" then is such huge number that a tiny bit of matter will convert to a huge amount of energy. The famous stereotypical story of this sort of learning without understanding is the one where engineering graduates are shown through a plant that makes refrigerators, and they are brought into the (durability) testing room, where they see rows and rows of refrigerators all running with their doors removed, so that their compressors will run continuously hour after hour, day after day. In the ceiling of such rooms there are huge air conditioners; and the engineers invariably ask why, if all these refrigerators are cranking cold air into the room, air conditioners are needed. "Won't the cold that the refrigerators are turning out" make the room cold? What are the air conditioners for? They cannot apply what they have "learned" or studied to the situation. Refrigerators and air conditioners do not create cold; they merely transfer heat from one place (e.g., the inside of the refrigerator) to another (the outside of the refrigerator), and since motors are not 100% efficient, they even add additional heat to the room. So, without the air conditioners' transferring heat from inside the room itself to outside the room, the temperature in the room from the operation of the refrigerators, would get quite high.

Or, in teaching photography, if you teach students about telephoto or zoom lenses, they sometimes forget that they can make an object appear closer in a photograph by actually walking up to it with a normal lens. They take pictures where the main subject is very far away, and when you ask them why they didn't shoot it closer, they say they didn't have a telephoto lens with them. When you then ask why they didn't just go up closer to the object, they look at you stunned and say "I didn't think of that."

I visited a friend of mine's college English literature class once and the students sat passively and non-responsively while an attractive, young female teacher asked them questions about interpretations of the works they had read. Sometimes, one student might venture an answer, but even one response was rare; most of the time the students just sat there saying nothing. When someone did respond, the other students took no notes and made no comments, but just sat with no expression, waiting for the teacher to give her answer. Eventually the teacher would answer her own question and the students would write down what she said. Then the process would be repeated with the same result. The day I was there, the teacher happened to become exasperated by this and asked why virtually no one responded. No one responded. I later told her I thought that as long as she graded them on what she said about the material, and as long as she gave her views when they were quiet, since they were interested in the grade and not the material, it was only reasonable for them to be quiet so she would tell them her "interpretations" of the material that they needed to know for the tests. If she wanted responses from them she was going to have to quit giving them the answers when they were quiet, and/or she was going to have to quit grading them on their ability to repeat on tests what she had said in class; and she would have to convince them that their grade would not be based on their ability to repeat things she said..

Relatively few students above the primary grade level find most school material sufficiently interesting or meaningful enough when it is presented in a textbook or a straight narrative style lecture to think about in ways that will help them see its point or the point of its structure and presentation, and extract their own meanings, ideas, consequences, ramifications, conclusions, corollaries, or extrapolations from it. And these are some, many, or perhaps all of the sorts of things that comprise "understanding" material. Some of my children's school textbooks recently have had some really interesting "sidebars", containing material that helps show the point of what is being studied and that draws interesting conclusions from it. But those things are interesting to me, not my children, because those things (1) address issues I, myself, have already thought about or puzzled about, and so I appreciate the author's pointing them out, or they (2) raise interesting points about the material for people who already have a basic understanding of that material, and are ready to "see" ramifications or corollaries that others point out. But to my children, all those sidebars are just "more information" or more material, not derivations or ramifications of the material. They are more material, not helpful insights into the main material.

I have written elsewhere (http://www.Garlikov.com/Soc_Meth.html) about using one way, the Socratic Method, to help foster reflection and understanding, so I will not repeat that here, but there are teaching techniques to help get students in the mode to think successfully about material in order to be able to understand it, not just passively learn to repeat it under school conditions. There are also many psychological barriers to overcome in order to convince most students that you really do want them to think and that you care about their ideas, especially students who have previously had teachers that did not care what they really thought or who graded them lower for having ideas the teacher did not accept or seem to understand.

(1) they may understand the material without necessarily seeing on their own at a particular moment some particular ramification or consequence that you see or have in mind, especially if the way you present a problem or question throws them off-track. For examples of questions that can throw almost anyone off-track about things they know, see footnote #5 in my paper on "The Concept and Teaching of Place-Value" in math: http://www.garlikov.com/PlaceValue.html#N_5.  Although the questions there are purposely formulated to make anyone think along the wrong lines, questions that accidentally have the same effect on some students are common on exams.

(2) they may know the idea you are thinking about, but only because they have been told it, not because they derived it or "see" it.

So whether a student answers a particular question correctly or not, in neither case does it show whether s/he understands or does not understand the material it relates to or not.

What is necessary for a teacher to do to evaluate student's understanding is to have as much discussion with that student as possible so that if the student does not truly understand the material, significant gaps in their knowledge will show up. A few questions or a few unsolicited comments by the student may show such a gap, or it may take more than a few questions to see whether a student has the idea or not.

When my older daughter was in second grade, she missed all the questions on a test that asked her how much change there should be when one gave certain amounts of money for purchases having various prices. The teacher said that my daughter did not know how to make change. That struck me as odd, since for more than a year I had been letting my daughter pay for some things after asking her how much money she should get back for, say, a five dollar bill if the item she was buying was $2.87. She could do those in her head, and almost always was right. I asked my daughter why she missed all those questions on the arithmetic test and she said "I don't know what 'change' means." I had never referred to it as change; instead I had always said "How much money should you get back if...?" When I told her "change" was just another word that meant "how much money you got back" in such situations, she could answer them all easily. But the teacher had not had enough discussion (none, actually) to see that there was no problem with my daughter's being able to compute change, and that her only understanding problem was in knowing the meaning of that particular word. The teacher mistakenly believed her test showed that my daughter did not know how to calculate change. But her test question was not sufficient to demonstrate lack of understanding of calculations. Follow-up discussion might have shown her that, but she did not engage in it.

Similarly an acquaintance told me her daughter in first grade was supposed to write the first letter for each object in a picture the teacher handed out. There were apples, bananas, etc. in the picture, and the child said each one of them started with an "f". The teacher called the mother in for a conference and told her the child did not know any of her letter sounds. The mother knew otherwise and asked the child why she had marked an "f" for each of the objects. The kid's reply was that it was a boring assignment and since all those things were fruit, she just marked "f" for each of them as fruit to get done with it. Another acquaintance had a daughter do almost the same thing, except it was in pre-school and the child was supposed to color a picture with the normal color of the objects. The child colored everything purple. The teacher called in the mother to explain the child did not know colors or understand the concept of coloring. The mother knew differently and asked the child later why she had colored everything purple. The child explained that she thought the directions were stupid and she didn't see any reason to have to show the teacher she knew which colors to use. While these two children may find future student/teacher relationships problematic from a conformity perspective, the point remains that these teachers, because they did not pursue the matter with the children involved, made mistaken inferences about what the students' had demonstrated.

In a more pedagogically important case, that is typical of student misunderstanding, the same second-grade teacher who had said my daughter couldn't calculate change had given a test that year that asked children to be able to sort out "sentences" from groups of words that were "not sentences". My daughter missed one that some kids got right, and she missed one that all the other kids also missed. She had said "Rico bats" was not a sentence, and she (and everyone else) had said "Tom is sleeping" is not a sentence. I happened to have a parent-teacher conference just after the teacher had graded this test and she remarked that she didn't understand why my daughter missed the first and why everyone missed the second. I suggested she ask the students why they answered as they did. I don't think she did that, since my daughter had to think about those things when I asked her later. I had a suspicion about the first one, and she confirmed it when I asked her why she had said "Rico bats" is not a sentence. She said "Bats are flying animals, sort of like birds; and 'Rico flying animals' doesn't make any sense." I asked "What if it means 'bats, as in batting in baseball', meaning that Rico is up to bat?" and she just looked at me and said "I didn't think of that; that would be a sentence."

The second one, "Tom is sleeping", was even more interesting. Her explanation was "The teacher told us that a sentence has to have two things, a 'naming word', and an 'action word'. 'Tom' is the naming word in that group, but there is no action word because sleeping is not an action. When somebody is sleeping, they aren't doing anything."

The "incorrect" answer to first question did not demonstrate lack of understanding of what a sentence is, but the incorrect answer to the second one did. However, if the teacher only marked the grades and did not have any discussion with the students to find out how they misunderstood what she said, she would not likely address their misunderstanding.

On the other hand, students can accidentally give correct answers to questions without understanding why they are correct. Multiple choice ("multiple guess") tests are obvious examples of this, but so are tests where a student can answer using enough material they have memorized or remembered, or sometimes even fabricated, to be able to give an answer that fools a teacher into thinking they are demonstrating understanding. I have come across students who gave good answers on exams, but who, on follow-up questions, demonstrated they had only been lucky.

Or one might even appear extremely gifted though that might not be the case. When my younger daughter was in third grade, I had already done lots of math with her and we were continually doing more, often in the car as I chauffeured her about. I would make up problems that I thought would be fun and that might illustrate some point or other. Because she'd had considerable experience with math, that accompanied any natural logical thinking ability she had, she was pretty good at it in school. Also, she was enthusiastic about participating in school and she loved her third grade teacher. One day, however, the teacher got a little exasperated (though mostly in fun) that my daughter always had her hand up and always knew the math answers, and she said she was going to come up with something the kid could not possibly answer. So she created a quite complex "progression" or "pattern" problem - one of those problems where there is a string of numbers that progress in a certain pattern and you are asked what number comes next, which usually requires you to see the pattern. She chose a double-progression series -- one where numbers are related to the number on the other side of the adjacent number; i.e., every other number is related to each other.  My daughter answered correctly immediately, and that really tickled and frustrated the teacher, who finally asked "How did you know that so fast!" And my daughter replied very matter of fact "My dad and I did one just like that in the car yesterday on the way to school." The teacher gave up. But had she not asked how the kid did the problem so fast, she might have mistakenly thought she had a math prodigy on her hands. Now, my daughter had solved the problem herself in the car, but it took her a bit longer, and it was after doing some other progressions I gave her that in some sense led up to the really difficult one to make it easier.

It also turns out that students may understand something perfectly well, and not be able to demonstrate that during a particular exam simply because the ramifications or answers you had in mind as the teacher are not the ones they happened to think of until ten minutes after they turned in the exam, if then. There is often a difference between understanding something and being able to perform in a particular way at a particular time by seeing the helpful relationship -- as the questions on the "Place-Value" web page demonstrate.  I replaced the motor in my clothes dryer one time, and a friend came over to watch and to assist where an extra pair of hands might be needed.  I had the motor replaced and was trying to put the large, rotating drum back into place, but was having great difficulty even though this is something I had done previously fairly easily when I had replaced a belt.  But this time I was unable to hold it up in the air in its place against the back panel while closing the front panel on the dryer to keep it in place. The longer I tried, the weaker I got, and I was not able to suspend the drum with one hand (which is all there was room for) while trying to manipulate the front panel.  My friend finally said "This may be a stupid question, but what if we laid the dryer on its back?  Would that make it easier?"  Easier!  It made it totally easy!  And I felt like a total idiot for not thinking of it.  It was an obvious solution -- at least once it was thought of; but I would have never thought of it at the time, if ever.  When you laid the dryer on its back, the drum just sat in place against the back panel, and all you had to do was simply close the front panel to hold it in place when the dryer was turned back upright.  Laying the dryer on its back made the back temporarily be the bottom, and the drum then stayed there just by gravity.  This let gravity work for me, not against me.  Even the easiest and most obvious answers don't always come to someone, especially if they have thought up, and become enamored of, some difficult or wrong way to attack the problem.

Moreover in an academic setting, because of the nature of the grading system, there are often other forces at work that may impede students from showing their understanding. Students may not feel their answer is the one the teacher wants, and they may try to give the teacher an answer they think s/he will like, and get it all messed up because they are not really saying anything they believe or understand. Students have so many experiences in their careers with inflexible teachers who cannot appreciate ideas that diverge from theirs -- no matter how reasonable or well-presented, and who grade such ideas as wrong, that only the most independent and/or perhaps naive students will persist in telling teachers in papers or, especially, on exams what they really believe when they think the teacher believes something different.

I once had a student come to me as an academic advisor seeking permission, late in the term, to drop my introductory philosophy course, because, she said, "I don't understand anything in the course". She had seemed to be pretty good in class and I was surprised to find out that she did not understand anything. Since she had some time to talk, I asked her for an example of something she didn't understand.

She picked a topic and told me what she did not understand about it. Then, because of the way the conversation about that topic went, I asked about another topic, and we discussed it, and another, until we had covered all the major topics in the course. What it turned out was that she understood everything perfectly, but she disagreed with me about every topic because she had objections to points I had made about each topic. She sat in my office for two hours, without preparation or warning, and without notes, telling me what we had discussed in class, what the book said, and what I had said about each of these topics and why she thought I was wrong.

What she didn't understand was, essentially, why I held the views I did about these topics because she thought I was clearly mistaken. And, yet, because she also knew that I was well-educated in the subject matter and not totally unintelligent, she felt she therefore must not understand the topics and that there must be something wrong with her that she just couldn't comprehend philosophy. By the end of the session I was practically laughing out loud; this student who "didn't understand anything" had demonstrated she understood everything and had learned as much from the course as I could have ever hoped for a student to learn. I tried to show and explain to her that what was going on was that she had really insightful and excellent objections to my views, but that there were responses that could handle all her objections, but they were complex in ways that would take additional (advanced) courses to be able to adequately pursue.

When I tried to answer her questions/objections as we went along, it was hard for her to grasp the details of my responses because there were too many unfamiliar concepts required in too short a time to reflect on and be able to absorb. All I could do was try to show her that not only were her objections good ones, but that they were part of the history of these topics, and that philosophers had recognized them as so significant that they needed to address them, and had. I am not certain I succeeded in showing her that, though I did say it. I told her not to drop the course, and not to take the exam.

She probably still does not understand why she received an "A" for the term, but my view was that in the course of our intense discussion she had demonstrated complete understanding of what we had covered in that introductory course. Yet, I feel that because she was confused, not about the material, but about what she thought I wanted her to believe, she probably would have done quite poorly on an exam because I think she would not have been "true to her own convictions" and would have instead tried to write something that she couldn't believe in a way that sounded like she did, and because the typical written exam does not allow for interactive, follow-up questions that might elicit what a student really knows, really means, or would really like to say.

Understanding something is often an open-ended or perhaps relative characteristic.  One may understand something well enough to apply it in some circumstance but not another.  One may have full understanding of a concept or phenomenon in regard to using it in a particular circumstance, but not think to use it in another circumstance, or not realize it has even greater application or broader ramifications.  Or one may have more understanding of some phenomenon than anyone else, and yet find as one grows that one's understanding also grows, and that what seemed to be complete understanding at one time, turned out to be only very partial and limited.  It may even be impossible to tell in some cases whether one's understanding of a phenomenon is complete or not.

Evaluating someone's (or one's own) understanding is often therefore going to be difficult, and will generally require sufficient testing to be done to show with some degree of probability that (1) if someone "fails" the test it will be most likely because they simply don't have sufficient understanding of the issue, and (2) that if someone "passes" the test it will be most likely because they were relying on their understanding and not just a lucky guess or the repetition of someone else's answer they had heard to this problem.  This usually takes more questions, and/or a greater variety of questions, than is given on the average quiz or exam.  And it is a particular problem for standardized tests that attempt to distinguish understanding and abilities from particular knowledge, in those cases where students have access to past tests that are not changed much or where students are given preparatory training by people who have "psyched out" the previous tests.

The arguments you need to be able to make for demonstrating understanding or its lack are the following:

For demonstrating understanding:
A) Jones gave correct answers to the questions he was asked about phenomenon P.
B) The questions he was asked covered sufficient aspects of P in sufficient ways to test for understanding of it.
C) It is highly improbable (or impossible) Jones could have given such answers unless he understood phenomenon P.
Therefore,
D) It is highly probable (or certain) Jones understands phenomenon P.

For demonstrating lack of understanding:
A') Jones did not correctly answer the questions he was asked about phenomenon P.
B') The questions he was asked covered sufficient aspects of P in sufficient ways to test for understanding of it.
C') It is highly improbable (or impossible) for Jones not to have been able to answer correctly if he understood P.
Therefore,
D') It is highly probable (or certain) Jones does not understand P.

For B or B' to be true, sometimes a substantial number of questions need to be asked or a substantial number of circumstances need to be asked about.  And for C or C' to be true, all the kinds of things (and perhaps more) of the sort described above, need to be accounted for.  It is often very difficult to do either of these in non-interactive testing situations where students are simply given an exam and their answers graded, but not followed-up in meaningful ways.

This topic is pursued from a different approach in the discussion "Evaluating Students, Ed Students, and Teachers" at www.akat.com/Evaluating.html and to see more about the difference between understanding and merely learning an explanation see www.Garlikov.com/Interpretation.html.  For other essays about education by Rick Garlikov, visit www.Garlikov.com/writings.htm.

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