I am not sure what is meant by constructivism. I’m not sure everyone who uses the term are talking about the same thing. The explanation of constructivism that I have heard a number of times is that every student must construct his or her own knowledge in his or her own mind.

      This makes a lot of sense, I always thought. Of course every student must construct his own knowledge in his own mind. A teacher or a textbook cannot do it for you. They can provide the parts, and give guidance, even detailed instructions, of how to construct knowledge out of these parts, but the actual construction must be done by each learner in his or her own mind. One personal memory that is elicited by this explanation comes from when I was in the first grade. I don’t remember the details, but it goes something like this. The class was working on arithmetic, perhaps doing addition flash cards. The problem was 8 + 3. I didn’t know the answer. The teacher told us, “It’s eleven. Eight and three are eleven”. “Now I know the answer”, I thought. “Eight plus three is eleven”. Yet there was something wrong. The teacher said the answer was eleven, so obviously the answer was indeed eleven, but it didn’t quite click in my mind. Somehow I still didn’t know it. In a few seconds it did click in my mind. Probably I visualized eight and three, or maybe I counted on my fingers. Or perhaps I simply remembered what I had learned before. The only thing clear in this memory is my thinking that even though I know the answer was eleven, there was some way that I didn’t know the answer was eleven, at least until I figured it out in my own mind to my own satisfaction.

      Constructivism explains my feelings. It is not enough to be simply given the fact that eight plus three is eleven. I had to construct this knowledge for myself. It had to “click” in my mind. And it did click after a few seconds. Of course it would. I had all the parts. I knew the idea of addition. We had probably done a lot of addition by this time. I knew how the parts should go together, but I had to put the parts together in my own mind. I was not enough that the teacher simply said the answer was eleven. I had to figure it out myself. And if I couldn’t put it together in my own mind, then the knowledge was not there. It was as simple as that.

      As a math teacher I often try to help students individually in my office. It happens again and again that after a few minutes the student gets an “Aha!” expression on her face. That’s good, I think to myself. I have been successful! She understands! That means I have explained, doesn’t it? That means I have put together a string of words that conveys the meaning, the idea, the math, whatever it is that I had been trying to convey. But very often at this point I can replay the last few minutes in my mind, and conclude that putting a string of words together that succinctly explains the situation is not the essence of what just happened. The idea is the important thing, but the idea was not conveyed by my succinct verbalization of it. Indeed, very often I realize that I have been verbally stumbling around, grasping for the right words. I think of myself as pretty good with words and explaining things, but some ideas are very difficult to put into words. And putting the idea into words is not all that I am doing to help the student. I pick a problem, tell the student about one thing, ask the student about another thing, correct on another thing. The student follows my directions. Some of those directions she follows blindly. Other steps make sense to her. The directions are not necessarily linear. We may change, or even reverse directions. She may ask questions, and I may ask questions. With every step I do my best to put into words what we are doing, what principles we are applying, and how, and why. But often I am still struggling to find better words when the students says, “Oh, I get it!”

      Putting ideas into words is a very important part of teaching, but it is not all of teaching. Many times putting the student in the right situation is an important part of teaching. The right situation means that the student has the parts, the parts that when constructed into the whole, form a new idea. Constructivism simply means that this assembling of parts must be done by the individual student. The teacher may guide in this construction, and may provide the parts to be assembled together, but still the construction takes place internally in the student’s mind. So constructivism makes a lot of sense.

      But as I heard more about constructivism over time it became apparent that others have different ideas about what it means, because they cite constructivism while advocating practices that seem to me to have little to do with constructivism. So what, I wonder, are their ideas of constructivism?

      Collaborative group projects is one common idea I hear from others as being a “constructivist approach”. The usual pattern, as I understand it, is that students are formed into groups of perhaps three or four people and are given a project. The project, presumably, differs in some way from what could be called simply an assignment. I believe a project is supposed to be more open ended that a simple assignment, which could be interpreted as simply following directions and therefore not quite qualifying as critical thinking. The members of the group then meet to plan their cooperative effort. I am not sure if division of labor among members of the group is envisioned, or perhaps more a matter of brainstorming. The vision probably includes the idea that the strengths of one member may bolster the weakness of another, and vice versa. The vision surely includes the idea that “two heads are better than one”, though the thinking may be no deeper than that. The result of the project will be a written report of some sort, something a little more elaborate than simply solving the problem.

      I had a bit of experience with group projects in my last teaching job. The students were college freshmen, and the two courses involved were college algebra and finite math, which is a collection of mathematical topics intended for students who do not need calculus or any higher math for their major. We were required in these courses to require group projects of our students. For me, from the vantage point of a teacher who is not enthusiastic about the idea, the experience was not particularly positive, but it was not a bad experience by any means. It was just a requirement of the job. The students accepted the projects as just part of what you do have to do to get a grade. I believe that probably about 50% of the time it was a positive experience for the students. People are social animals, after all. They enjoy working together.

      But what does all this have to do with constructivism? .

      I think ideally group projects are meant to be a way to learn, not just a way to demonstrate or apply what has been learned. I think that was the ideal back in the 1920’s and 30’s, when the “activity method” or the “project method” was new and exciting. I suspect it is the ideal today, though I really don’t know. But the group projects I had to administer in my experience, really were not a way to learn. They were simply another assignment. The learning had to happen otherwise, and then the learning could be applied to the group project.

      Of course this could be said of any homework assignment. It can be argued that doing a homework problem is not a way to learn the topic in question. The actual learning must be done otherwise, and then it can be applied to the homework problem. This is true, I think, but I would also argue that there is a very intimate connection between the learning and its application on the homework problem, at least if the problem is well chosen. Doing a problem often takes a very tentative hypothesis and turns it into an idea that makes sense. The type of problem that we would choose for a group project, it seems to me, is a bit more removed from the learning. The project will be a bigger problem than a homework problem, and therefore less a matter of a way to learn the material and more a matter of being an application of it.

      There is certainly a case to be made for doing some “big” problems, as well as some “small” problems. A typical homework problem usually requires applying only the immediate topic to be learned. But isn’t there a place for problems that require the application of more than that, perhaps requiring the application of the whole chapter? So why not put such big problems on a group basis? Many students want to study together anyway, and they seem to profit from studying together. Why not put it all together and assign group projects.

      I admit to not being enthused about group projects. There are problems, both theoretical and practical. But when well done they can be valuable in the way that regular homework problems are valuable. But my question remains, what does all this have to do with constructivism?

      It seems to me that if anything, group projects would be the opposite of a “constructivist approach”. Once one member of the group figures out the project it is not at all certain that the others will figure it out, because they don’t have to. There could be division of labor. One person does the brain work, another types the report. I have no doubt that in many cases with the group projects I assigned, some students never did figure out the problem.

      There is an argument to be made that it would work the other way. Once one member of the group figures it out, that person can help others in the group figure it out. If a project is well designed, all members of the group would want to understand and are in a better position to do so because of the help they have available from the other members of the group. The nature of the group project gives them all the motivation to help each other. In this scenario the goal is not to have division of labor, but simply to facilitate cooperation among the students.

      Or perhaps advocates of group projects have something a little different in mind for constructivism. If so, I’m not sure just what it is.

      I still like the definition of constructivism as the idea that knowledge must be constructed individually in each student’s mind. So I will continue along this line.

      The “box of one escape” is a concept I think useful here. Envision this situation. A chimpanzee is in a cage. Outside the cage, hanging just out of reach, is a bunch of bananas. The chimp has in the cage with him a stick and a chair. If the chimp is smart enough he will eventually take the stick, climb on the chair, and use the stick to pull the bananas within reach. Does constructivism apply here?

      This type of situation, or something like it, was studied by Wolfgang Kohler early in the twentieth century and gave rise to gestalt psychology. I believe Kohler’s emphasis was on problem solving and intelligence. My emphasis is more on learning and knowledge. The chimp may at first be frustrated. He wants the bananas but cannot get them. He lacks knowledge of how to get them. But he somehow manages to construct this knowledge, through trial and error perhaps. The important thing is that once he has this knowledge he can act immediately next time to get the bananas. And another important thing is that the chimp had to figure it out for himself.

      I may have set up an undesired duality here. It may appear that in figuring out a concept or an idea the student either figures it out in a flash of insight, or the teacher explains it and the student doesn’t have to figure it out. That is not quite what I intend. Either way constructivism is called for. The student has to figure it out either way. An idea that is easily verbalized by the teacher, and easily understood by the student, still must be constructed in the student’s mind. I have given the example of when my explanation is stumbling but the student figures out a problem anyway as a good illustration of construction going on in the student’s mind. But construction goes on the student’s mind in any learning, as my example of adding eight and three in the first grade illustrates. As another example when a stranger asks directions to a post office and you tell him to go three blocks south and one block north, that knowledge must be constructed in the learner’s mind. Evidence of this construction would be a look of concentration on the stranger’s face for a moment as he repeats the directions, before he then brightens up, says “thanks” and takes off in the right direction. Constructivism is just as necessary with a good explanation as with no explanation at all.

      As another example, you can put a chimp, who understands no language, in a cage with the stick and chair and bananas out of reach, or you can put a math teacher in the cage with a stick and a chair and bananas out of reach. In either situation the mind must construct the idea of how to use the stick and the chair. If you have the math teacher in the cage and carefully explain every step of the way, the math teacher’s mind still must construct the knowledge. The language ability of the math teacher gives him a great advantage over the chimp, but the knowledge must still be constructed.

      The “box of one escape” idea is that if you want a person to learn something one way to accomplish this is to put him in a box in which the only escape is to figure out whatever it is you want him to learn. This doesn’t sound too appealing. Isn’t there a better way? Can’t we be civilized and use language? Just tell the person what you want him to know. Yes, we should be civilized and use language. Teaching includes a lot of telling. But remember my example earlier in this article. Quite often when helping a student I discover that while I am groping for just the right words there is something else going on. The student is like the ape in the cage with the stick and the chair. Simply by choosing a problem to work with, and bringing to mind various ideas to work with, I have provided the student parts from which knowledge may be constructed. While I continue groping for words, the student continues to try to put those parts together in a meaningful way. When she manages to do that she gets out of the “box” she is in. Hopefully she puts those parts together quickly and easily because my succinct verbalizations tell her exactly how to do that. But sometimes, it seems, success comes more because I put the right props in the box. Some things are hard to verbalize. Perhaps it is especially true in math that a verbalization that is technically and mathematically complete and accurate may sometimes be little help in learning.

      A well-posed problem is like the box of one escape. To get out of the box, to solve the problem that is, you have to have the right parts, and you have to put the parts together in the right way. Hopefully this is not a trial and error process. The teacher and the textbook not only put the learner in the box with the right parts to be assembled, but also gives detailed assembly instructions. The text and the teacher, in other words, provide explanation.

      So my idea of constructivism fits right in with the concept of direct instruction. Good teaching requires not only the parts be made available, but good instructions also. Then the learner can most efficiently construct the ideas that are desired.

      I had not heard the term “direct instruction” until recently. Apparently it arose in opposition to some education trends and ideas, that, though exciting to some educators, have led to dissatisfaction and frustration among parents, students, and some other educators. So having some idea of practices they felt were inappropriate or counterproductive, they use the term “direct instruction” to mean practices that do work. And not surprisingly practices that do work have been around for many, many years. This leads some critics of direct instruction to think that it is very traditional, and traditional can be taken to mean stuck in the past. I would argue that traditional is neither good nor bad, simply descriptive. To me saying that direct instruction is very traditional is like saying that the route from Fargo to Omaha is very traditional if it goes through South Dakota and Iowa. It is indeed traditional, but tradition is not the reason we do it. It is pure practicality. It works!

      Another thing I have heard related to constructivism is the “discovery method”. This was an important part of the “new math” of the 60’s and 70’s. The idea here was that students should discover math on their own. I don’t know the history of this idea. I suspect it goes back early in the twentieth century in one form or another. Many progressive educational ideas in the twentieth century have tried to reduce the role of the teacher from a taskmaster to a “facilitator”, or something along that line. This would fit with the idea of the student discovering math on his or her own.

      Could it not be argued that the “discovery method” is the same as what I have described as the “box of one escape” idea? In both situations the student is put in a situation with some “props”, and must assemble those props, or parts, or facts, or data, or concepts, or whatever, into an idea. All of this fits in with the idea that the learner constructs his own knowledge. But I just said that direct instruction also fits with this “box of one escape” idea.

      I think the difference is in the relative emphasis. The discovery method emphasizes a lack of structure. Students are not necessarily expected to discover one right answer. They are expected to discover whatever they might discover. Direct instruction emphasizes guidance by simple direct language, and students are expected to discover what the teacher plans for them to discover. In the discovery method the conclusion is withheld until the student makes the discovery. In direct instruction the conclusion is plainly and directly stated from the beginning. The two methods do have something in common. No matter how direct, or indirect, instruction is, students still have to put parts together in their minds. Constructivism applies either way. If students are to learn that eight plus three is eleven, a teacher using the discovery method would be slow to confirm that eleven is the sum, while a teacher using direct instruction would be quick to confirm that answer. But either way the student must do some mental work. Even in something as simple as this, a succinct verbalization on the part of the teacher is not sufficient in itself to effect learning. The idea of direct instruction simply says that succinct verbalization can be a valuable aid.

      Returning to the chimp in the cage idea, the discovery method would be giving the chimp lots of props in the cage to work with, but no help whatsoever, while direct instruction would be giving only the props that will be needed and as much verbal help and guidance as possible.

      In the “Standards” published by the National Council Of Teachers Of Mathematics there is the ideal that students will discover by themselves (probably in groups) solutions to various problems. It is emphasized that alternate methods of solving the problems are to be welcomed. The students are to “construct” knowledge, but what knowledge they come up with is variable. Here, again, we can use the comparison to the chimp in a cage. The chimp in the cage has one goal, to get the bananas. Correspondingly a student in a math class may be said to have one goal - to solve the problem. But this one goal may not be satisfactory to the teacher, or to society at large. Suppose the chimp in the cage obtains the bananas by an alternate solution. He ignores the stick and the chair, picks the lock on the cage, and runs out of the cage and grabs the bananas. Is this satisfactory?

      At this point we have to ask about goals. Do we want the chimp to learn to use the chair and the stick? If so, then picking the lock is cheating. To teach him to use the stick and the chair to get the bananas we’ll have to put him back in the cage with a better lock. The NCTM ideal of discovering alternate solutions would accept the lock picking solution as success. The direct instruction ideal would not.

      Here is an example of “picking the lock”. In teaching college algebra it is desirable that students be able to do written problems. That means take a stated problem (“Train A leaves the station at 60 miles per hour . . .”), translate it into an equation, and then solve the equation. I discovered in putting such problems on tests that if I didn’t carefully construct the written problem a fair number of students would solve it by trial and error and arithmetic, not by algebraic means. Sometimes their paper trail of checking different answers would make this pathetically clear. It would mean, of course, that they understand the problem, which is good. But my goal for them is that they understand the algebra that systematically leads them to a solution. Solving the problem by trial and error is not what I had in mind. It corresponds to the chimp picking the lock, when I want him to learn to use the stick and the chair.

      In grading such problems on tests I would be faced with the question of giving full credit, or little credit. It’s hard to withhold full credit when the student is arguing that the answer is absolutely correct. But it also makes some sense to give no credit because the student demonstrated no understanding of the algebra that I are trying to teach. One solution to this problem is to give the answer along with the problem and state that full credit will be given only for translating the problem into an equation and solving the equation. This is like saying to the chimp that picking the lock is not allowed. He’s still got to learn to use the stick and the chair.

      So does constuctivism enter in here? Certainly. I want the students to construct algebraic ideas. By stating the problem as “translate the problem to an equation . . .” I am putting the student in a box in which only a knowledge of algebra will get him out. Trial and error will not get him out of the box. My argument is that constructivism always enters in, at least my idea of constructivism. But constructivism is best used by using language as much as possible to bring the parts to be assembled to mind, and to guide the assembly process. The “discovery method” can easily degenerate into undirected and unproductive (and sometimes frustrating) efforts. Direct instruction still depends on the efforts of the learner to take parts and assemble them into meaningful wholes, but the process is to be done with careful guidance, and therefore a minimum of confusion and frustration.

      The “Socratic method” might be considered a constructivist approach. Presumably Socrates taught by leading questions, questions that when carefully considered inevitably led to the conclusion Socrates was looking for. This seems very similar to the “discovery method”. In both cases the teacher does not want to provide the conclusion for the students. But in both cases the proper parts to be constructed must be somehow brought to mind. The student must use these parts to construct the whole.

      So by my idea of constructivism, every method requires it. Direct instruction, the discovery method, the Socratic method, the lecture method, the activity method, the project method, and any other method we can come up with requires constructivism.

      I understand there is a psychological meaning of constructivism, though I don’t know what it is. It seems to me the meaning of constructivism that I started out with is a good concept, and “constructivism” is a good name for it. However it does not lead, in my opinion, to group projects and discovery learning. I don’t think it leads to any particular teaching method. Rather it simply gives a bit of explanation to learning. It describes learning, but there is no particular magic in that description. To say “I am going to use a constructivist approach to teach world history this year”, is like saying “I am going to use a mechanical approach to auto repair this year.” It’s true enough, but hardly needs to be stated. If anything, I would argue that constructivism leads to many teaching practices that have been used for many years, because they are effective. And I think “direct instruction” is a good name for those practices.

      My best hypothesis is that constructivism, to educators, really does not have a well defined meaning. Rather it is a term primarily useful by its imprecision. It can be used to justify what the teachers want to do on other grounds.