"Contrary to the empiricist-associationist assumptions on which traditional mathematics instruction is based, relationships cannot be put into children's heads from sources external to them. Relationships must be created by children through their own mental activity. Therefore, demonstrating figure 3 to children will not contribute to their learning about the system of tens."

While many bad teachers equate telling students something with teaching them, it has been known long before "constructivism" that does not always work.  Many math students have, through the years, complained about teachers who "can't explain" things in ways that makes sense. Of course, some students do not work to understand any explanation, but there are plenty of students who really do want to understand material but who cannot understand the bad or unhelpful explanations they are given.  Normally when one is given a bad explanation, one seeks out a better one, perhaps from someone else.  At some point one may, of course, give up and feel the matter is too difficult for him/her, but typically one makes the distinction first between bad explanations and good ones.  It has long been obvious that bad explanations do not generally teach well or give understanding to students, but that does not mean there can be no good explanations which do impart understanding.  When Kamii and Joseph say "Therefore, demonstrating figure 3 to children will not contribute to their learning about the system of tens," the connection implied or stated by the word "therefore" does not follow from the claim that children must create relationships or understanding by their own mental activity. The word "therefore" is superfluous in that sentence, since it should be fairly obvious that particular diagram is not really likely to be a useful teaching device to children (or to adults) under any theory of learning; but that does not mean there are no teaching devices and explanations which would be useful and would give, or at least foster or promote, understanding.

To contradict the above assertion, there are clearly two ways that "relationships [can] be put into children's heads from sources external to them." And constructivism, at least as many people express it, seems to fail to see that.  That is what I want to discuss here.

First of all, most people have experienced asking a question and having it answered in a satisfactory way; even in math.  And most people have experienced being told something in a way they learn from it and understand it. If I say "that stack of cookies looks like it is about to fall over, so you might want to straighten them or divide the stack into shorter stacks" that is generally pretty easy for another person, even a child with the past experience of such things, to grasp.  If I am giving a mathematical proof or demonstration, there will be times some students will be able to follow it, even if not all students follow it. There is no reason to make the strong assertion that relationships cannot (ever or generally) be learned from external sources by being told.  And I take the phrase "put into children's heads" to mean something like "told to children so that they see, or understand" what you are telling them.  Good explanations do expand knowledge, which I take to be "putting knowledge into" people's heads, though of course the "putting into heads" is not meant literally in some physical manner, as in surgeons' putting a metal plate in someone's head or a criminal putting a bullet in someone's head.  It is simply not true that giving an explanation never gives understanding.

Of course, the person receiving the explanation has to have sufficient background to be able to understand what is being told, has to be attentive, receptive, and may even have to think a bit to make some connections.  But sometimes it takes very little thought, if any, because the explanation given is so clear.

Now the constructivist view is somewhat correct in that children, as anyone, have to "develop" understanding with regard to many concepts or phenomena. Many things take some thought and exploration to see, even if one is being given an explanation at the time.  If I am showing you, for example, how a mechanical clock mechanism works, I might direct your attention to the spring and to the different size of the cogs that connect to each other, and it may take you a minute to "see" the relationships I am pointing out.  You may even want to take some measurements and make some calculations. In cases of some phenomena or relationships, particularly perhaps conceptual or abstract ones, one might need to examine an idea or an explanation or a phenomenon or relationship for a long time even with an explanation.  It is not uncommon for someone to say "I hear what you are saying, but I need to think about it on my own a while, because I am not quite seeing it." Then one may come back with a question or one might see it on one's own after some reflection, with the help of the information one has already been given. 

Even in something as straightforward as one person's asking driving directions to somewhere from another person, one might take down all the directions, but then in order to understand them better, say something like "It looks like you are telling me to go near the Old Mill. Is the place I am looking for somewhere near it? Because if it is, I know how to get there, so just direct me from it."  Even if one is in a totally unfamiliar city, one might look at a set of directions and say something like "You have me going two blocks west and then a block north and then another two blocks west. Can't I just take this street four blocks west and turn right for a block?" The answer might be that none of the east west streets go all the way to the desired street.  But this is a case where the recipient of the instructions is thinking about them, not just following them by rote.  And the thinking helps give that recipient a kind of understanding.  On the other hand, one could just follow the directions by rote and get where one is going, if the directions are correct and easy to follow.  In either case, the directions themselves are still crucial, and they give the recipient valuable information.  In the case of the "no through east-west streets", the original directions might have even made that question unnecessary, by being something of the form: "You are within five blocks of where you want to go. It is four blocks west and a block north, but there are no streets that you can go all four blocks west on from here, so the easiest way is to go two blocks west on this street, then go a block north, and then take the remaining two blocks west on that street."  Even in giving driving directions to someone, there are better and worse, clearer and unclear ways of doing that.

And while many people who cannot explain things well or who cannot teach much perhaps do not appreciate the difference between good explanations and bad ones, and generally do not know how to ferret out what their students already know or do not know that is relevant and necessary for an explanation to be helpful -- in order to know where to "start" or what to say -- many people do know how to give useful and helpful explanations that impart knowledge in a fairly direct way. Direct explanations with demonstrations, drawings, models, etc., where necessary, can certainly work. It may even take more than one kind of explanation. And different explanations may work for different people. Constructivists are certainly correct when they point out that there may be more than one way to understand something and that what makes sense to one person may not make sense to someone else.  They are correct to say that one should not necessarily expect just any given explanation, even if accurate and precise, to impart understanding.  But if they mean to say that no explanation or way of expressing someone's understanding can make sense to another person by telling it to them, that seems clearly false.

But there is something far more important to point out about successful teaching with regard to helping students understand.

First, it is certainly not that students must discover everything about math for themselves or on their own.  As I have stated many times, it would be foolish to expect every student in K-12 (or K- college) to discover on his own what it took the finest minds collectively 2500 years to discover.  The only question in teaching is not whether to (try to) foster discovery, insight, and understanding, but which techniques and methods and explanations work, or work best, or get through to the greatest number.  New methods and techniques continue to be invented and evaluated.

What I want to point out here is that there are many methods, such as the Socratic Method, which foster insight without simply telling students information directly.  Good teachers know how to direct students' attention to those factors and in those ways that let the "leap" of insight most likely occur. So even if learning involves self-discovery, there are teaching methods that are better or worse for likely fostering such discoveries.  Good teaching is often not about telling someone relationships or connections, but in setting up a situation, or describing information, that gives a higher probability to their making the connection or seeing the relationship themselves.  What one says or does is crucial for teaching and for providing understanding, even if the manner of presentation is not in some sense direct or merely descriptive.

Now it is not clear what it is that gives anyone insight, in the sense of the mechanism. One can be working at full concentration for days on a problem that one cannot solve or grasp, and then while not attending to it at all, or while doing something else -- perhaps driving,  mowing the lawn, taking a shower, talking about something totally different on the phone, or even sleeping -- suddenly come to see the connection or answer one has been seeking.  It seems to just "pop into one's mind" out of the blue.

But often once one has the insight, one can see what the elements were that needed to be juxtaposed in one's attention, in order for it to occur. And when trying to teach others what one has figured out, it is more effective sometimes to try to bring those elements to the attention of a student and let the student have the same insight, than it is to try to directly or descriptively spell out the insight one has had.  Sometimes it is difficult to know why a particular method might work to help students make discoveries or gain insight, but the important thing is that it does work. Understanding is not just a function of the students' mind working by itself, even if you give it something to work on (such as manipulatives). Understanding can be fostered or prompted by what a teacher says to a student, even if the teacher is not giving a direct explanation.  So if the above passage by Kamii and Joseph is meant to say that nothing the teacher says is helpful, crucial, or essential to giving the students understanding, that is simply false.

But clearly, teaching is not always about explaining something. It is often about getting students to experience something you have experienced in a way that helps them learn what it is you want them to know.  Teaching bicycle riding to children who find it difficult to learn to ride, is a good example. Some children apparently can get on a bicycle the first time and just ride it, but I am not talking about them. I am talking about teaching a child to ride who has great difficulty learning to do it -- the child who tries to ride but cannot do it. You can explain all you want to about "balance" to a child, but it generally will not help, because "balancing" is really just another word for "riding" since it means "staying up while pedaling the bike (or coasting on it)", which is what the child does not know how to do. A precocious child could very well say "I know I need to balance the bike, but I don't know how. When I asked you to teach me how to ride the bike, I want to know how to keep my balance. Don't just tell me to balance; teach me how." 

I find there are two things for teaching bike riding that are much more effective than giving verbal directions, though one includes some verbal explanation. 

What I have found works well to teach bicycle riding is doing both of two things. First, you have to help children "feel" the balancing point. I do that by walking or running beside them holding up the bicycle while they pedal. I hold it at the seat, rather than the handlebars (though sometimes I have to have one hand on the handlebars part of the time to help hold them up at the very beginning), for a number of reasons. The main reason is that if you are holding them up by the bicycle seat, you can feel how well they are balancing, and how much you need to support them.  When you hold by the handlebars, it is fairly easy for you to keep the bike upright even if they are not helping much.  But if you hold it by the seat, and they are not balancing very well, it takes a lot of force on your part to keep the bike upright.  When it takes very little force on your part, they are balancing pretty well. So you can tell by the amount of force you need to apply how ready they are to ride on their own. 

The other reason to hold the bike up by the seat rather than the handlebars is that they cannot see you let go, when you can tell it is time for them to solo.  Many times fear will cause students to fall if they know you have let go.  It takes a child, normally, much longer to notice you have let go of the seat from behind him than to notice you let go of the handlebars.  And most children do fall a while after you let go of their seat, and they get upset with you for letting go after they have fallen, but if you let them ride on grass or some other relatively soft surface, and if you point out ot them how far they got on their own and show them they were riding the bike on their own, they get over their anger and fear and are then ready to ride on their own. (Then you only have to teach them a bit later how to "start" the bike going on their own.)

But the other thing I have found that helps children learn to ride, or learn to balance, is to both tell them and show them till they can feel it, that their "balance is in their butts, not in their shoulders" -- a phrase children tend to find both funny and memorable in a way that helps them.  What I mean is that the normal tendency for children who find riding difficult is to try to balance their bikes by leaning their shoulders in the opposite direction the bike is starting to fall or tip over.  That is, if the bike starts to fall over to their right, they tend to lean their heads and shoulders left; and vice versa.  That normally accelerates their fall rather than preventing it. The reason is that when they lean their shoulders to the left, they almost always simultaneously put their hips and hindquarters to the right.  Since their weight is centered closer to their hips than to their shoulders, the direction their hips go tends to be the direction the bike will lean.  So what is helpful is to tell them the bike leans the way their "butt" leans, and then to get them to see and feel that, by holding up the bike while it is not moving, and letting them wiggle their hips right and left so that they can feel how the bike goes with their hips, and that it takes very little movement one way or the other to tip the bike in either direction.  That helps accelerate their learning to balance with their hips while riding because they can concentrate on moving their hips instead of leaning their shoulders.

So there is nothing about the way children learn, and the fact one needs to develop one's own understanding, to mean that a teacher cannot do anything to accelerate that understanding by what they say or by the kinds of materials they present and the way they present them.  Good teaching helps provide insight either by direct explanation that is effective, or by helping the child focus on elements in ways that help the child's understanding "snap" into place.  There are a potential raft of ways to do these things.  And they do not necessarily impose the teacher's views or understanding on the student.

But there are times one is trying to teach a particular concept or is trying to teach how the concept is understood by others.  And it is important then to teach in a way that the student understands how the concept is used.  It is not enough, sometimes, for a student to simply understand material in his/her own way; s/he also needs to know the conventional understanding of the concept, if s/he is going to discuss it with others.  Kamii and Joseph discuss ways for students to learn to add two column (two digit) numbers together, but they really do not discuss in their article the teaching of what makes "place" important in "place value".  Place value is actually a strange and complex subject that is separate from simply knowing the "names" of columns and being able to add or subtract multi-digit numbers.  So if one is going to explain place value, which is a convention of a certain sort, not just a logical relationship, one has to tell and show students certain kinds of things. But that does not necessarily mean spelling everything out in some boring, linear purely descriptive manner.

Finally, constructivism, "learning style" research, and "hands on" learning have pretty much given "lecturing" an undeserved bad name in education. That seems extremely short-sighted to me because, first of all, as above, some lectures are quite effective teaching methods because they are put together in such a way and delivered in such a manner that students can and do follow them.  Second, lecturing does not have to mean droning on in a bunch of boring declarative sentences that precludes thinking, discovery, and insight, instead of stimulating it.  For example, one can present material that causes cognitive dissonance, so that the listeners have to scramble to figure out what is wrong. Presenting challenging paradoxes is one way of doing that. A lecturer can even ask the audience some questions. Lecturing does not have to be totally one-sided from a speaker to a listener.  And even when it is totally one-sided, there are better and worse ways to present information.  Movies and books, for example, are totally one-sided in that material is presented to a viewing or reading audience. Yet clearly some movies and books help audiences and readers understand, feel, and experience emotions and ideas they had never thought about before. It is not the one-sideness of transition that prevents anything from being an effective teaching tool. 

In short, while telling other people information does not necessarily get them to understand what you are telling them, there are ways of presenting material and structuring a presentation so that it is educational and does foster insight and understanding, even excitement and the joy of discovery.  And it is often necessary to teach with such presentations to make learning be efficient and productive or to occur at all.