This is a petty use of questions that tends to discourage student enthusiasm and learning because it shows students when teachers merely grade them and then move on to other things, that whether they have learned something or not was not important to the teacher other than for its grading and ranking value. It was mocked perfectly one time by sports news anchor Chris Berman on ESPN when on the final week of a tight NFL season, he listed all the possible playoff permutations, depending on which teams did what in the final week of play. There were dozens of possibilities. He went through them at lightning speed, because obviously people were only interested in their own particular teams. At the end, when he finally took a breath, he then said "There will be a test on this tomorrow." There are billions of facts in the world. It is always possible to give tests to people on any of them, but often such tests are mere entertainment (as in playing Trivia) or they are tests of mere memory, often short term memory at that, and their results have very little practical meaning though vast importance is imputed to them.

This article is to introduce, first, additional purposes for asking questions, questions which are more imaginative in most cases, and, second, a different perspective on grading and evaluating students in asking them questions.

The first use I only want to touch on here: using questions to develop interest in a topic. The Socratic Method paper, just mentioned, demonstrates some questions that do that, the initial or main question for students in that paper being something like what kind of numbering system could aliens with only two fingers (one on each of two hands) be able to have, and how many numbers would they be able to have in it? There were then leading questions asked to steer students toward the answer. The students who participated in that lesson became very enthusiastic about trying to answer those questions. 

Sometimes it is not so much the initial question that will make students interested in the material, but showing problems with their initial answers that gets them involved. This can happen when different students come up with diametrically opposed answers to a question and are stunned to see that could happen, and they begin to argue with each other. Or it can happen when the teacher shows flaws in their answer by showing how their reasoning leads to consequences they would not themselves accept. One of the questions at my site www.garlikov.com/Philosophy.html, intended to introduce philosophical thinking to students by e-mail through the use of questions they might find interesting, is whether a man is going around a squirrel or not in the situation where the man is about five feet away from the trunk of a tree on which a squirrel is climbing on the opposite side. As the man goes around the tree, keeping his distance from it, so does the squirrel, so that the trunk is always between them. Both the man and the squirrel are obviously going around the tree, but the question is whether the man is also going around the squirrel or not. At first, this question, in a classroom setting, does not generate any particular excitement. However, kids will disagree and begin to show what is wrong with each others' positions, and worse, when they turn to me to adjudicate, I usually pretty easily show that their explanations are both wrong. Once either or both of those things happen, the students become animatedly involved because they are sure they are right but that they just aren't saying it quite right. They become frustrated everyone else cannot see they are right and they want to convince everyone. 

It also bothers them that both answers seem incorrect and that that must be impossible. Paradoxes serve the same sort of purpose because paradoxes are obviously impeccable chains of reasoning from obviously true premisses to obviously false conclusions, and that cannot happen, so paradoxes tend to really bug or eat at students until they can figure out how to resolve them.

The same sort of thing happens with the question on that site, when used in a classroom, as to which is more important in songs that have words, the lyrics or the music. Students are astonished to see other kids disagree with them, and that usually no answer they give will hold up.

Sometimes, such as in the "song" question, you do not have to lay any groundwork; their own experiences will allow them to think about the question initially. In other cases, however, you have to set the stage for them in order to be able even to ask the question, such as in the squirrel question, or in a question I use to get them to think about an issue in economics: Suppose (even if this is pure fantasy) that an automobile company was able to discover a way to inexpensively produce a safe, non-polluting automobile that was also inexpensive to maintain and operate -- virtually trouble-free, ageless, indestructible, and that efficiently used cheap, readily available fuel. This would seem to be a wonderful invention, but given that so many people's livelihoods depend on automobile manufacture, sale, maintenance, repair, insurance, and fuel production and distribution, etc., such an automobile would cast large numbers of these people out of work, would ruin the economy, and would probably cause social chaos. Something that seems like it ought to be a perfectly wonderful boon would turn out to be a tremendous burden. Why should that be? Or how could that be?

In the article "Fighting for the Higher Self", I discuss some of the differences between the kinds of questions that tend to interest students and the kinds of questions that do not.
 

The second use of questions is to give students a map for self-recognition of reaching milestones of understanding as they study a unit, by asking, when a unit is begun, some difficult things that require understanding and insight into the material to be able to answer. Let the students know these are not easy questions to answer, and that the answer will not be directly given as the material is presented, but will be logical consequences of the real significance of the material. Students will then be able to see whether and when they reach these milestones of understanding and insight⁽¹⁾ as a self-measure of their own learning. I will give two examples, the first for a physics class involving the study of heat transfer; the second for a geometry or algebra class:

(1) I do not know whether refrigerator models are still quality-tested this way or not, but at one time, manufacturers would test refrigerator designs and/or production quality by having a large room where refrigerators were kept operating with the doors removed so that they ran continuously, in order to see how long they lasted. There would be dozens of refrigerators running. But in these room, there were also huge air conditioners, not for testing purposes, but in order to keep the room cool while the refrigerators ran with their doors off. The obvious question, indeed the one often asked by engineering graduates when they toured such facilities, is "Why do you need air conditioners if the refrigerators are running with their doors open? Wouldn't that alone make the room cold?" Why do you need the air conditioners?

(2) Suppose the earth were perfectly smooth instead of having mountains and valleys or rough terrain. Imagine that you tie a ribbon around it at the equator (say, 24,000 miles long) and that you pull the ribbon tight when you tie it, so it is snugly against the ground all along its path. Now suppose that you splice in, at some place, one additional yard (or meter) of ribbon - 36 inches (or 100 centimeters) - of ribbon, so you have this little loop at that place on the earth. It is not a big loop, but you want to get rid of it by smoothing the ribbon out all along the earth so there is no place where there is more slack than any other place. The question is "How high will the ribbon then be everywhere above the earth?" Or put another way: "Would it be high enough to trip over it?" And "Finally, what if I told you that the ribbon would be nearly six inches off the ground around the whole globe? Could that possibly be true, and if so, how could that be?"

It is surprising sometimes how many teachers do not think to ask students direct questions about what the students might be thinking. A college teacher one time asked a group of us on the Internet whether any of us might know of a format by which he could ascertain what students thought of his teaching, without having to give them a form that would have to go through his department. I wrote back and suggested he simply ask them, and that if they wanted to give an answer anonymously to have them mail it to him. He couldn't imagine that approach, but he tried it, and he wrote back with astonished glee that they had opened up to him and given him really helpful critiques and analyses.

Most teachers will understand the concept I am referring to by the designation "differentiating questions", though they usually use such questions, not to teach, but to evaluate (here meaning "grade -- assign a letter grade to") student learning. Such questions may be part of a test or they may appear as a question for "bonus points" because it is harder. For example in arithmetic, after a teacher has taught students to subtract one two-digit number from another, s/he might give a problem on a test that has a three-digit number subtracted from another to see which students can extend the concept of what they have learned, or at least to see which students might think to apply the procedure they have been taught, whether they understand it or not.

Or a teacher who has taught subtracting that requires "borrowing" or "regrouping" using non-zero digits, might throw in a bonus question on a test that has some zeroes in the number that is being subtracted from. An even more difficult question than that would be one in which the number that is being subtracted from has consecutive zeros in it.

In a literature class, a teacher might ask for any potential significance of a seemingly minor character, or a particular plot development, or perhaps a particular passage, in a book they have been studying -- a minor character, plot development, or passage that was never discussed during the "teaching" of the book.

In any given subject matter or topic, one can usually come up with a series of questions that it is possible to rank roughly in order of general degree of difficulty. I say "general" degree of difficulty because there is always the possibility that a question which is difficult for most people turns out to be fairly simple for one or two students, not because they are brighter or more diligent, but because their circumstances are such that a minor, obscure point made a great impression on them. E.g., they may remember the unusual name of a dog that was only mentioned one time in a 400 page book, because that is their own dog's name. But normally, we can rank questions as being generally more, less, or equally difficult.

When questions are more difficult because they require greater understanding or understanding of something more complex, rather than requiring recall about something less likely to be noticed or remembered, the order of difficulty of the questions allows for distinguishing differences in student knowledge and understanding. It allows for differentiating between what students know and understand about a topic and what they do not know or understand. The wise use of differentiating questions can help teachers diagnosis student needs, and, as in the Socratic Methods article, help teach students by guiding them into having insights for themselves. 

More importantly, using differentiating questions diagnostically lets you know where to start an explanation with any given student or group of students with the same needs. You can see how far they have gone into the subject matter and/or where they had misunderstandings or where they got confused or lost. It has been my experience that most people get "hung up" or confused, or have misunderstandings at the same "places" in almost any topic. That means that "individualized" instruction can really become small group instruction for a number of students at one time, rather than having to be actually different for each and every student. This should significantly reduce the work needed by teachers who wish to individualize their instruction. Moreover, diagnosing students' precise needs by using differentiating questions makes instruction far more efficient because it allows the teacher to address only what needs to be addressed, and it addresses what the student is likely to be ready and prepared to learn and thus be able to absorb more readily.

Normally, in using questions to teach, as in the Socratic Method, or in any method where you use questions to get students to notice or focus on something that should be helpful to them, so they develop insight, you progress from questions about concepts that are less complex or difficult to those which are more complex or difficult. You go from what the students know to what you want them to learn, through a series of steps that they can easily traverse with their own thinking. But in diagnosing student needs, and what they know or don't know, in a face-to-face encounter, it is normally helpful to proceed in the opposite direction, from the more difficult and complex part of the particular topic at hand to the less. In part, this is so that students do not try to guess what they should say, and thus throw off your diagnosis when they get an answer correct by accident, and, in part, so you can see a genuine flash of recognition on their faces when you finally get to something familiar to them. This order also lets students see what some of the things are that they do not know, and that they will learn. It helps psychologically groom them to learn sometimes - if there are not so many things that they will be overwhelmed. 

There may be some diagnostic cases where you want to go in both directions in some sort of alternating fashion in order to try to close in more quickly on where you need to be, particularly if you do not know at all where they are starting. Usually I start with the more complex, but if after a couple of steps I draw total blanks without even a glimmer of recognition of the concepts, let alone any understanding of them, I then usually skip to a much simpler aspect of the topic to see whether they have any understanding of any of it.

So suppose you have a group of third or fourth grade students whose education about fractions you are to continue from their instruction the previous year. You need to know what they learned and remember. It is not likely they learned to divide by fractions. But you might start out by making sure of that, asking perhaps "Does anyone know how much 1/2 divided by 3/4 is?" or perhaps slightly easier "Does anyone know how much 12 divided by 3/4 is?" Probably no one will know. So then you might 'drop back' to finding out whether they can multiply by fractions: "Does anyone know how much 1/3 times 5/6 is?" Or, again, slightly easier, "Does anyone know how much 8 times 3/4 is?" Probably not that either. You might want to see whether they have a more intuitive grasp of that 'same thing', so you might ask whether anyone knows what "3/4 of 8" is. Suppose they do, but they don't get it from multiplying; or suppose they do not know at all. You might want to give a subtraction problem and/or an addition problem involving simple fractions, such as ½ and ½, or ½ and ¼. If it turns out they cannot do any of those things, you may need to find out whether they know what a fraction even is. You may want to ask whether they know how to determine what the top or bottom number of a fraction should be. To do that, rather than asking for some sort of arcane definition, you might line all the kids up along the walls and have them tell you one at a time what fraction of their family at home that they are, and how they figured that out. E.g., "I am one fourth of my family because there is me and my sister, and my mom and my dad; so I am one out of four people." If they can do that, then you can give them "harder" similar fractions to figure out: e.g., what fraction of all the males or females they are, or what fraction of all the children, or of all the living animals and humans together in their family they are. They would each have to give an answer and then explain how they got it. E.g., "I am 1/6 of the living things in my house because I have a mom, a dad, a sister, a dog, and a goldfish. So there are six living things in our house and I am one of them; that makes me 1/6." If students cannot do these things, then you probably need to teach them fractions from the beginning again.

Or suppose your students are about to study rate/time/distance phenomena. You might want to see whether they understand any of the concepts intuitively at all. In this case you might want to start "simple" and work your way up, not in order to lead them but in order to see what they know or think about the concepts of rate and time. You might ask how long it takes to do a trip if you drive half as fast. Or you might ask how much more water you put on a garden with your hose if you water it twice as long at the same water pressure. If they can answer those questions, you might ask how long you have to drive 60 miles an hour in order to drive 240 miles. 250 miles? You might ask how fast the earth turns in miles per hour at the surface at the equator, letting them know, only if you have to, that the earth is roughly 24,000 miles in circumference at the equator. (Since the earth makes one complete turn in 24 hours, it turns at the rate of 1000 mph. Doesn't feel like you are going that fast though, does it?) You might ask a really hard one: how fast do you have to drive the second of two miles in order to average 60 miles an hour over both miles, if you drove the first mile at the rate of 30 miles an hour. (The answer to that one is that you can't do it no matter how fast you drive, because in order to drive two miles averaging 60 miles an hour, you need to do it in a total of two minutes, but since you used up all two minutes in the first mile driving 30 miles an hour, you can't recoup.)

Or you might want students to paraphrase (parts of) a famous speech or document, but you might need to ask first whether they understand what paraphrasing means. Or you may give them some very short passages or samples and ask them to paraphrase those for you, so you can see how well they do it. Perhaps you may need to explain and to give some examples yourself, and then give them some examples to do to see whether they now grasp the idea. You might want to make it more challenging - asking them to do both a plain, boring paraphrase, and also one that tries to capture some of the beauty, excitement, pithiness, or general character of the original in tone as well as meaning.

Or, if you are trying to see how they think about books in general and how they tend to read them, you might ask students how they have done class book reports in the past. If they say something like they had to discuss characters, plot, and setting, you might ask them whether that seemed a satisfying way to talk to someone about what the book meant to you. If so, why; if not, why not. When they told friends about some book they liked, did they talk about character, plots, and settings? What about when they told friends, if they ever did, about movies or about television programs they had watched; did they talk about character, plot, and setting, or something else altogether?

There are zillions of questions you can ask about subjects to find out what kids know or think about them, and what ideas they have, what progress they may have made, etc. The trick is to devise questions that show you just what and how much or how little students may actually know and understand about something. Those are what I am calling "differentiating" questions. 

Devising differentiating questions requires two things: (1) Understanding and insight on the part of the teacher either into the specific material; or at least the propensity to seek it and ability to find it when problems seem to arise in student comprehension. This is crucial because generally teachers who do not really understand or have underlying insight into the material they are teaching (or at least do not have the propensity to look for them) will not be able to know what to look for in trying to diagnose student difficulties in learning. 

For example, one second grade teacher who was having difficulty getting her students to understand mathematical place value had a fairly expensive prop she had purchased, that had columns of sliding balls or tiles on it, each ball or tile with a number on it. So you could make the number 21, say, by lining up the 2-ball/tile in the second column from the right, and the 1-ball/tile in the right-hand column. She did not understand that columnar place value just made no sense to her kids, no matter how it was physically presented, and that it was not because it was written on the board or on paper that made it difficult for kids. She knew what place value was and how to calculate using it, but she did not have any understanding of the underlying concept of place value, which is a complex and unusual concept (see "The Concept and Teaching of Place Value"), so she could not do any more with her students than to say the same thing over again or to show it over again though perhaps in different physical ways. I had a college teacher one time who, when students asked questions about material he had just presented, repeated what he said louder, or  wrote it on the board again using a different color chalk, as though the reason students did not understand in the first place was because they were deaf or color blind. He couldn't see what "steps" in logic or psychological intuition they were not grasping because he did not know what steps there were to be grasped.

(2) It is helpful for teachers to know what the typical student misunderstandings are likely to be and what the typical difficult places are for students to understand material. That way they can anticipate misunderstandings or lack of understanding at the appropriate time in the instruction and look for it. They can then use differentiating questions as they are proceeding in order, first to diagnose its occurrence in order to address it in their teaching, but  also to help students see they need to think just a bit more about the immediate topic at hand, or in order to get students to see there are some distinctions they have not yet grasped about this topic which they need to know in order not to make mistakes or be confused. This is a morally fairer and more pedagogically rational way to use such questions than to "save" them for a test just in order to be able to give curved or differential grades to students. It is an all too typical practice among those teachers who do understand the concept of differentiating questions to use them to grade students differentially when it is too late to teach them what they need to learn, rather than to diagnose their needs at the time they could fulfill them through additional teaching. It seems to me to be heinous at worst, and uncompassionate at least, for a teacher to know that there are likely pitfalls in student understanding and to use those pitfalls only for grading purposes instead of for teaching purposes.

The question is: Should students' initial (or intermediate) lack of understanding or lack of knowledge be held against them, gradewise, even if they have remedied it by the end of the term and have learned the material by then?  Should early quizzes or tests be averaged in with late ones? Or should a student receive an A if s/he learns something by the end of the term even if s/he missed it previously in the course on a quiz or test?

First, there is the issue of fairness involved in deciding this. If a student has learned material before the term ends, is there any good reason or justification to give him/her a lower grade on it because s/he did not learn it when it was first tested? My answer is that it is unfair to incorporate the original test answer in the student's final grade because most people read final grades as a measure of whether students have learned or not, not whether they learned quickly or early or not. They interpret grades as a measure of knowledge and ability (gained during a course), not as a measure of how quickly that knowledge or ability was gained. If diagnostic questions are used in part to assign letter grades to students, late bloomers will be at a disadvantage even though they learn the same things. And those students who come into a course knowing the material will be given a designation that shows they learned more than kids who had to struggle to learn the material during the course, even though the reverse may actually be true.⁽²⁾ If one wants to use grades to distinguish not only knowledge, but speed or age of learning, there should be multiple sets of grades or assessments given on student report cards, not just one. My own view is that if students are capable of learning faster or learning sooner (because of past experience, not because of intelligence), they should be taught more, and their additional knowledge should be reflected in their transcripts as such, not as a mere grade. If one wants to note that a student is also a bright or fast learner, that too should be noted somehow in writing, with the evidence given for it, not as mere letter grades in individual courses or on individual tests.  It is my view that if a student demonstrates knowing or understanding something at any time during a course, they should receive credit for knowing it, and that the times s/he did not know it should be gradewise ignored and those grades discarded.

And not only are grades normally taken to signify one's knowledge, they ought to mean that when they can. There is, in some sense, only an arbitrary difference in most cases between having learned something correctly in the second week of class as opposed to having learned it the third week or the sixth week. (The only time that is not arbitrary is when learning faster allows one to learn more, but there are many courses where learning faster does not necessarily mean one would learn more. That is too complex to go into here.) Moreover it seems to me that it is better to learn things well than to merely learn them quickly. Many famous and extremely competent people were late bloomers or did poorly at first. Churchill was held back in English many times because he was, or appeared to be, poor at it, and he later attributed his oratorical and writing prowess to having labored so long and hard over the simple English sentence instead of moving on early to study Latin and Greek with his classmates. Schools should be places of teaching, as they were for Churchill, not of weeding out slow learners from fast one, or those with poorer backgrounds from those with better ones. Questions should be used to help students be taught what they need, not to  discard students because they have not yet learned what they need.⁽³⁾

Second, and equally important, there is a whole psychology involving grades that is detrimental to learning, and to teaching. Kids tend to study for grades rather than for knowledge and understanding; teachers tend to give assignments and tests to be able to give grades, not for any useful teaching purpose. One of the results of this is that material is taught at a certain pace whether kids need it or not, and whether kids are ready for it or not, and whether kids need some other material or not. If you get a C you move on with the class, instead of going back to remedy why you got that C. If you top out a test with an A, you feel you have learned all you needed to, even if you did not really study very much or tap into the potential to have learned far more. When either of these things happen - and they happen often - grades are counterproductive to teaching and to learning. If teachers do not use differentiating questions as diagnostics instead of only for summative grading, they are doing a disservice to those students who could have learned what they did not, and to those who could have learned much more than they did.

Insight and understanding come when one suddenly "sees" relationships or implications of material one did not see before, even though one had all the same information or knowledge in some sense that one had before. It is one thing to know something and quite another to see how to use it, or to be able to make use of it. I one time wrestled with getting the drum back in place in a clothes dryer that I had replaced the motor in for myself. I could not hold the drum up in the air in place and at the same time close the front panel that would secure it. The drum was too heavy to hold with one hand while closing the panel with the other. There was not sufficient room for two people to work on it together, even though a friend was there watching me repair this dryer. Finally my friend asked whether it would help to lay the dryer on its back. The minute I heard that question I knew it was the solution, and I felt like a total idiot for not having seen it myself. I was struggling against gravity when I could have simply used gravity. It was not that I needed to be taught any more about gravity or about dryers or weight; it was that I needed a boost to have insight into using what I already knew.

In a case even more trivial, I was trying to develop photographs in my dormitory bathroom one night when I found out the sink trap was leaking badly. I was able to remove the trap in order to try to re-install it better, but when I finally got it removed, I lost track of my purpose and when I finally stood up and needed a place to discard the water that was still in the trap in my hands, I, of course, did the dumbest thing and poured it down the sink drain from which I had just removed it. That gave me the opportunity to mop the floor again. But having done that once, I have never made that mistake again. It is one thing to know one has removed a pipe from a sink drain; it is something else altogether to realize that means one should not be pouring water down it. It seems obvious, of course, but only if one has thought of it or been made to think of it. Just being told it, will often not do the job. You can tell someone to be sure not to pour water down a sink they have removed a drain pipe from, and they may even think you must think them stupid for telling them that, but the odds are good they will still do it because they will not really have "learned" or "understood" it even though in some sense they "know" it.

Obviously insights into academic subjects such as math, science, and communication skills are far less trivial, but the principle is the same.  Archimedes is said to have discovered his principle of buoyancy during a bath.  It is not uncommon for insights and epiphanies to arise in the shower, or while driving or walking, or even upon awakening in the middle of the night. They come while one is thinking, even if unconsciously, about what one knows; but they are different from, and more powerful than, the knowledge itself that prompts them.

Like the engineering example about the air conditioning in the refrigerator-testing room, most subjects are such that students gain knowledge without gaining insight into what that knowledge means. Questions can help determine whether students have insight, and they can even often help students gain insight, as in Socratic teaching. But insight can never be automatically taught nor can it be automatically diagnosed, so even though I have been writing here about using questions for these purposes, it has to be understood there is at least as much art as technique in these endeavors, and that one will not always be successful at either diagnosing or teaching for understanding. Still it is important to try; and it will often be successful.  When it is successful, those are momentously rewarding occasions for teachers and for students.

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1. With occasional exceptions, most teaching is done in order to impart specific knowledge to students, not understanding. Kids, for example, are taught what place value in mathematics is, and how to use it, but they are not taught the underlying basis or significance of it or why it (or some substitute for it) is necessary. Most adults, including teachers, have no real understanding of place value, though they can use it perfectly well.

While teaching to impart specific knowledge seems to work, it does not work as well as teaching the same material for understanding. Many students have trouble in algebra because they learned merely to manipulate numbers for calculations, rather than being helped to develop insights into numerical relationships as they learned about numbers. While learning for understanding does not guarantee one will have any particular insight when it is needed, it makes it more likely.

Minds are interesting things in that sometimes they will develop insights on their own from working with, and perhaps thinking about, particular information and applications, but this would probably happen far more if students were groomed to look for understanding in general as they go through school, as well as having teachers try to foster in them the understanding and insights into particular subject matter. Too many students leave school without even thinking that information and material ought to make sense to them. They then end up accepting some things that do not even make sense and that they should not accept. But they also then do not get as much potential use out of any actually accurate information as they could.

Moreover, while teaching for understanding sometimes takes longer initially, it usually ends up being much more efficient because material does not have to be repeatedly re-taught and because it generally allows students to move faster through ensuing material because they can grasp it much more readily. Often the difference between people who seem much brighter than others is that they have more understanding and can thus draw more inferences or draw them more quickly from the same original or initial information.
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2. One of the societal abuses of grades therefore is using student standardized test scores as a measure of school or district teaching performance, when often it is easy to see that high scoring students, particularly in primary grades, came into school with much of the knowledge, and yet the school is getting credit for having taught it to them. (Return to text.)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

3. Unfortunately what schools teach and what students need are not always the same thing; that is addressed in some of the articles at www.garlikov.com/schools.htm also. So just teaching more effectively may not mean one is teaching better. The effective teaching of inappropriate material is not an ideal to be sought.  Schools need to be able to distinguish better between what sorts of things students really ought to learn and what sorts of things are not essential.  (Return to text.)