Annotationen:Radical Constructivism and Teaching
This seems to me a good reason to take a brief look at the history of epistemology. There were single thinkers who already at the beginning of Western philosophy suspected that realism and its claim of objective knowledge were untenable. The sceptics have persistently denied such a possibility for more than two millennia. Most philosophers acknowledged the incontrovertibility of the sceptics’ arguments, but nevertheless went on hoping to find a path that would lead towards unquestionable truth about a real world. The paths they chose always led into the realms of metaphysics, that is to say, they tacitly implied some form of mystical belief.
Plato’s famous metaphor of the cave is a good example. In this fairy tale, human beings are chained in a cave the entrance to which they cannot see. In front of them, on the wall of the cave, they see shadows, and from these shadows they must guess what things there are in the world outside and what goes on there. But plato added that God, had instilled latent truths into the souls of humans, and if they learned to use their intuition, they would come to acquire truths about the real world. This metaphor is powerful because it presents a poetically plausible situation, without making it clear that this situation could be described only by a god, for only a god could know what lies beyond the domain of human experience. The Italian philosopher Giambattista Vico said this very nicely at the beginning of the 18th century: “God knows the world, because He created it, human beings can know only what they themselves have made.” The treatise from which this statement is taken, is the first constructivist manifesto. Immanuel Kant, some seventy years later, wrote in the Introduction to his famous Critique of pure reason: “Human reason can grasp only what she herself has produced according to her own design” (Kant, 1787). Neither Vico nor Kant, however, was able to shake the general belief that somehow we must be able to discover how the real world really is. In my view, the persistence of this belief springs from the fact that we all have a lot of knowledge that we consider reliable, knowledge that we trust when we make decisions about how to act. When we run down the stairs, we feel confident that the next steps will be there where we need them. And we have no less faith in much larger contexts. When I stepped into the plane to come here, I had no serious doubts that it would bring me to Geneva and that the old city would still be the one I know from many previous visits. Such faith in the permanence of objects and circumstances is essential in our everyday living – in spite of the fact that things do not always turn out quite as we expected.
We simply have to believe that, by and large, the world we experience is a stable world. But this belief should not lead us to assume that the world we experience must be like a reality that lies beyond.Because mental operations necessarily take place in someone’s head, they cannot be depicted. But I can reproduce a picture here, that will show that it is you yourselves who produce your specific perception.
To most viewers, this picture will seem meaningless at first. But the moment it is turned round, you recognize something familiar. You will probably say: “It’s a dog!”
In fact, it is nothing but a collection of irregular black splotches. Thus the question is: Where is the dog? In case someone still thinks that the dog must be somehow on the sheet of paper, I will show you another example. I suppose you are familiar with the constellation that is called Cassiopeia. It’s a big, capital double-U or, if you turn it round, an M. You can see it near the Polar Star, opposite the Big Bear. The Greeks called it the Crown of Cassiopeia, and it has survived the three thousand years since then without noticeable change. It’s as permanent and durable as one might wish. Yet, again, I would like to ask you: Where is it? – “It’s in the sky, of course”, you might say. Like President Clinton, speaking about his unsavory amorous exploits, I want to question the meaning of “is”. The constellation consists of five stars that astronomers designate by Greek letters.
Alpha and Delta are about 40 light years from the Earth. Gamma is twice as far, Beta, three times as far – and the distance to Epsilon is 520 light years, which means it is more than ten times as far from Earth as the first two stars. Imagine now that you are traveling in a space ship in the direction towards Epsilon. What happens? After a few light years, the double-U that you saw from Earth has spread so much that you have difficulty in connecting the five stars. After a tenth of the journey, Alpha and Delta lie behind you. The constellation, whose existence you confidently relied on when you sailed your boat at night, has disintegrated. Seeing the double-U depends on two things: 1) a specific point of observation; 2) carrying out specific perceptual operations. Piaget always maintained that perception was a form of action. Silvio Ceccato suggested that it is the movements of attention that generate the form and shapes we perceive (Ceccato, 1974, p.231). Attention, he said, is not like a spotlight that illuminates objects, but it is a pulse that focuses on sensory differences; and by moving from one point to another it produces outlines. Thus, once you have picked stars out of the darkness of the sky, your focus of attention connects them by moving from one to the other. With the five stars of the Cassiopeia, there are several possibilities. Here are two:
How does a child come to use the plural form of words correctly?
Imagine a two-year-old girl who a little while ago learned to say the word “apple” when it encounters a round, more or less red object that it can bite into and that sometimes tastes quite good. She now comes into the kitchen, and there are several apples lying on the table. With a certain amount of pride she points at one, and says “apple”. Then she points at the second one, and says “apple”. Maybe she repeats this with all of them. “Yes, my dear,” says the mother, “they are apples.” Perhaps the little girl notices the difference in the word the first time. In any case, she will hear the plural form of the word in other settings – and lo and behold, it does not take long before she uses singular and plural just as the linguistic convention demands it. How does the child learn it? All the apples she sees correspond to a kind of ‘recognition matrix’. This matrix is what Piaget called an empirical abstraction – and it is the one with which she has associated the word ‘apple’. But none of those individual apples can tell her that it belongs to a plurality which the adults call ‘apples’. The difference literally has to be conceived. It is not a matter of visual perception; it can be made only by a reflection on one’s own operating. Apparently, this was deemed so obvious that, as far as I know, no writer on developmental psychology has mentioned it. Yet, this does not mean that it was properly understood. The concept of plurality requires at least the following operations. Having recognized an object as, for instance, an ‘apple’, attention has to be focused immediately on at least one other object that fits the same recognition matrix. The salient point is that one and the same recognition matrix can be applied successfully more than once within the context. This repetition does not reside in the objects. Each of the apples in the example lies on the table and gives no indication that there are others. The repetition can spring only from something the perceiver does. This is to say, in order to apply the plural correctly, the little girl must in some way become aware of her own operations of recognition.
Ceccato coined the expression ‘consapevolezza operativa’. It stands for ‘operational awareness’ and is, I think, in many ways similar to what Piaget, somewhat less transparently, called “thematization”.I have used the example of the plural many times, because it is the simplest and clearest that I know. Concepts such as ‘beginning’ and ‘end’, ‘duration’ and ‘change’, ‘space’ and ‘time’, and in my view all abstract concepts, can be explained in this fashion. True, they require different and sometimes highly complex mental operations, but it is always the experiencer’s attention on his or her own operating that brings them forth. If you find this analysis appropriate, you may agree with my claim that it has a variety of consequences for teaching. As soon as it is clear that students must build their concepts by their own reflections, the notion that concepts can be conveyed by means of language is no longer tenable. As I tried to show at the beginning of my talk, the words that someone utters are interpreted by others in terms of the concepts these others already have. Only if their first spontaneous interpretation does not seem to make sense, will they be likely to attempt a new conceptualization.
This brings me to a point which, I believe, is indispensable for didactics. There is no infallible method of teaching conceptual thinking. But one of the most successful consists in presenting students with situations in which their habitual thinking fails. I shall lay out an example of this method that was developed by Leonard and Gerace at our institute at the University of Massachusetts.
Here you have a schematic representation of an apparatus that reminds me of a game that we passionately played as children, if there was a big heap of sand or at the beach. We made a kind of bobsleigh track, and let our large glass marbles roll down to see which was the fastest.
What you see here, are two tracks on which steel balls can roll with almost no frictional loss of energy. The two tracks are not the same, but start and finish are on the same height for both. The question is, which of the two balls will reach the finish first, if they are started at the same time?
Many of the beginning physics students to whom the question is put, say that number 1 will arrive first, because number 2 has a longer path. Others predict that the balls will arrive simultaneously, because, although number 2 gains a lead on the downhill slope, it will lose it when it has to roll uphill. Very rarely one answers that number 2 will win the race. Hence there is considerable surprise when the balls are actually let roll, and number 2 arrives first every time. Some of the students laugh and say that we have somehow managed to build a trick into the display. We assure them that there is no trick, and ask them to describe, as accurately as they can, what happens on each of the sections of the track. At first it is often not easy to get them to talk. But when we assure them that this is not a test and that we merely want them to share with the others what they are thinking, one or two begin, and then others join in. It usually does not take long for them to agree on the following descriptions: At point A, both balls arrive at the same moment and with the same speed. The slope from A to B accelerates number 2 and it therefore reaches point B before number 1. – “Number 2 has a lead?” we ask. Yes. At point B, number 2 has a lead – but then it has to roll uphill and it loses its lead. – We ask: “And when number 2 reaches point C, does it roll faster or slower than number 1? Usually this triggers a longer discussion, but eventually the students agree that the negative acceleration on the uphill section equals the positive acceleration on the downhill and therefore the two balls should have the same speed at point C. This is the moment when some catch a glimpse of the insight that number 2 rolls faster than number 1 over the entire stretch from A to C. The lead it gains does more than cover the longer path, and therefore it arrives first at the finish.
Of course, not all the students are immediately convinced. But those who have seen the solution are usually indefatigable in explaining it to others. In the end, most of them understand how, as physicists, they have to conceptualize the situation.Some of you may be shocked, if I now say that you do not have to look at a foreign language to find such conceptual differences. They frequently impede mutual understanding among people who speak the same language. This should not surprise anyone who has taken to heart Ferdinand de Saussure’s fundamental insight that words do not refer to things of a real world, but to concepts in the heads of those who use language. And if you consider Piaget’s extensive analyses of how concepts are built up by means of empirical and reflective abstractions that each child has to make for itself, it becomes clear that it would be something of a miracle if the conceptual structures in different heads would be exactly the same.
This seems to me a good reason to take a brief look at the history of epistemology. There were single thinkers who already at the beginning of Western philosophy suspected that realism and its claim of objective knowledge were untenable. The sceptics have persistently denied such a possibility for more than two millennia. Most philosophers acknowledged the incontrovertibility of the sceptics’ arguments, but nevertheless went on hoping to find a path that would lead towards unquestionable truth about a real world. The paths they chose always led into the realms of metaphysics, that is to say, they tacitly implied some form of mystical belief.
Neither Vico nor Kant, however, was able to shake the general belief that somehow we must be able to discover how the real world really is. In my view, the persistence of this belief springs from the fact that we all have a lot of knowledge that we consider reliable, knowledge that we trust when we make decisions about how to act
I became aware of conceptual difficulties very early in my life, because I had the good fortune of growing up with more than one language. Here in Switzerland you have the same wonderful opportunity. Many of you will almost every day come into situations where you have to compensate for conceptual differences between French, German, and Italian. Whether you actually become aware of what the differences are, is another question.
I had been teaching Genetic Epistemology for quite some time at an American university where I had to use English texts, before it dawned on me that this translation was unsatisfactory.
Let me give you as example a conceptual difference between French and English. In the last section of La construction du réel chez l’enfant, Piaget wrote:
L’intelligence ne débute ni par la connaissance du moi ni par celle des choses comme telles, mais par celle de leur interaction, et c’est en s’orientant simultanément vers les deux pôles de cette interaction qu’elle organise le monde en s’organisant elle-même. (Piaget, 1937, p.311) In Margaret Cook’s English translation, the last line reads:
“intelligence organizes the world by organizing itself.” (Piaget, 1954, p.400)The Italian philosopher Giambattista Vico said this very nicely at the beginning of the 18th century: “God knows the world, because He created it, human beings can know only what they themselves have made.” The treatise from which this statement is taken, is the first constructivist manifesto. Immanuel Kant, some seventy years later, wrote in the Introduction to his famous Critique of pure reason: “Human reason can grasp only what she herself has produced according to her own design” (Kant, 1787).
This agnostic position can be justified by all sorts of epistemological considerations. But as I want to focus on concepts, I shall quote something Albert Einstein wrote exactly fifty years ago: Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world. (Einstein & Infeld, 1950)
