Es kann also nicht mehr darauf ankommen, irgendwelche Problemlösungen sicher zu begründen, sondern nur darum, sie kritisch zu überprüfen, also sie im Hinblick auf mögliche Verbesserungen zu beurteilen, sie in diesem Zusammenhange mit alternativen Lösungen zu vergleichen und nach Verbesserungen zu suchen. Eine solche Beurteilung setzt Maßstäbe – Bewertungsgesichtspunkte – voraus, die sich nach der Art der zu lösenden Probleme richten müssen. Die Problematik absoluter Rechfertigung hat sich damit in ein Problem komparativer Bewertung verwandelt. [11]
Tag: learning
What counts in the future
Perhaps the greatest of all pedagogical fallacies is the notion that a person learns only the particular thing he is studying at the time. Collateral learning in the way of formation of enduring attitudes, of likes and dislikes, may be and often is much more important than the spelling lesson or lesson in geography or history that is learned. For these attitudes are fundamentally what count in the future. The most important attitude that can be formed is that of desire to go on learning. If impetus in this direction is weakened instead of being intensified, something much more than mere lack of preparation takes place. The pupil is actually robbed of native capacities which otherwise would enable him to cope with the circumstances that he meets in the course of his life. We often see persons who have had little schooling and in whose case the absence of set schooling proves to be a positive asset. They have at least retained their native common sense and power of judgment, and its exercise in the actual conditions of living has given them the precious gift of the ability to learn from the experiences they have. What avail is it to win prescribed amounts of information about geography and history, to win the ability to read and write, if in the process the individual loses his own soul; loses his appreciation of things worthwhile, of the values to which these things are relative; if he loses the desire to apply what he has learned and above all, loses the ability to extract meaning from his future experiences as they occur? [48-9]
Theirs not to reason why
The traditional scheme is, in essence, one of imposition from above and from outside. It imposes adult standards, subject-matter, and methods upon those who are only growing slowly toward maturity. The gap is so great that the required subject-matter, the methods of learning and of behaving are foreign to the existing capacities of the young. They are beyond the reach of the experience the young learners already possess. Consequently, they must be imposed; even though good teachers will use devices of art to cover up the imposition so as to relieve it of obviously brutal features.
But the gulf between the mature or adult products and the experience and abilities of the young is so wide that the very situation forbids much active participation by pupils in the development of what is taught. Theirs is to do—and learn, as it was the part of the six hundred to do and die. Learning here means acquisition of what already is incorporated in books and in the heads of the elders. Moreover, that which is taught is thought of as essentially static. It is taught as a finished product, with little regard either to the ways in which it was originally built up or to changes that will surely occur in the future. It is to a large extent the cultural product of societies that assumed the future would be much like the past, and yet it is used as educational food in a society where change is the rule, not the exception. [18-9]
How to train your chimpanzee
The main problem with school mathematics is that there are no problems. Oh, I know what passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Here is how to solve it. Yes it will be on the test. Do exercises 1−35 odd for homework.” What a sad way to learn mathematics: to be a trained chimpanzee.
But a problem, a genuine honest-to-goodness natural human question—that’s another thing. How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a number? How many ways can I symmetrically tile a surface? The history of mathematics is the history of mankind’s engagement with questions like these, not the mindless regurgitation of formulas and algorithms (together with contrived exercises designed to make use of them).
A good problem is something you don’t know how to solve. That’s what makes it a good puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions. A triangle takes up half its box. What about a pyramid inside its three-dimensional box? Can we handle this problem in a similar way?
I can understand the idea of training students to master certain techniques—I do that too. But not as an end in itself. Technique in mathematics, as in any art, should be learned in context. The great problems, their history, the creative process—that is the proper setting. Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique. But not too much. [40-2]
The art of explanation
By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity—to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs—you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes. [29]
An educational utopia
If I thought of a future, I dreamt of one day founding a school in which young people could learn without boredom, and would be stimulated to pose problems and discuss them; a school in which no unwanted answers to unasked questions would have to be listened to; in which one did not study for the sake of passing examinations. [40]
We’re not in it for the ‘win’
Serious critical discussions are always difficult. Non-rational human elements such as personal problems always enter. Many participants in a rational, that is, a critical, discussion find it particularly difficult that they have to unlearn what their instincts seem to teach them (and what they are taught, incidentally, by every debating society): that is, to win. For what they have to learn is that victory in a debate is nothing, while even the slightest clarification of one’s problem – even the smallest contribution made towards a clearer understanding of one’s own position or that of one’s opponent – is a great success. A discussion which you win but which fails to help you to change or to clarify your mind at least a little should be regarded as a sheer loss. For this very reason no change in one’s position should be made surreptitiously, but it should always be stressed and its consequences explored.
Rational discussion in this sense is a rare thing. But it is an important ideal, and we may learn to enjoy it. It does not aim at conversion, and it is modest in its expectations: it is enough, more than enough, if we feel that we can see things in a new light or that we have got even a little nearer to the truth. [44]
Fruitful Discussion
I think that we may say of a discussion that it was the more fruitful the more its participants were able to learn from it. And this means: the more interesting questions and difficult questions they were asked, the more new answers they were induced to think of, the more they were shaken in their opinions, and the more they could see things differently after the discussion – in short, the more their intellectual horizons were extended. [35-6]
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