For we do not understand math
One great concern to issues Edsger Dijkstra was the teaching of mathematics (and computer as part of this). In this brief article, gives us what could be the "tip of the iceberg" in search of why higher mathematics (and sometimes not-so-superior) are all too difficult to understand and master.
Why Johnny can not understand
few years ago I heard a presentation on the structure of the tests. Without hesitation, the speaker became very graphic and evidence became directed graphs with arrows the antecedents to consequents. [Mathematics Inc. has marketed the product in those days as Understanding By Computer Assisted Animation Arguments.] After fifteen minutes he directed our attention to the fact that some tests were flat, while others were not. Then he showed how simple changes in other tests could logically equivalent to change its "flatness", but instead of concluding that, therefore, the flatness of the evidence was probably not a relevant concept, embarked on a study of arguments inherently non-flat, etc.
dissertation was the most absurd I've heard in years. (For still remember that.) The poor guy was a serious victim of his education: confused the directed graph as a subset of ordered pairs, with graphical representation of arrows between points. [If it had been instructed on the incidence matrices could have lectured on the eigenvalues \u200b\u200bof the evidence.]
This is what happens to us again and again. When we introduce a new concept we are given several examples of a hopefully familiar context, or give us one or two models in which the new formalism, its objects and operations can be "understood." And it really encourages us to make those interpretations to convince ourselves that the new formalism "is sense. "They fail, however, warn that such interpretations tend to be misleading because the models are sobreespecíficos, in understanding that such habits are completely baffling when accompanying visualizations mislead the imagination and the mental burden of moving to and from the formula and its interpretation, should best be avoided. In fact, one can only hope that increasing familiarity with the formalism, the model is quietly vanish from our consciousness.
This had already begun when we learned the numbers natural. We did not learn that 2 + 3 = 5, first learned - graphically! - that two apples and three apples are five apples and then for pear, pen, litter, trees and elephants. The apple model is woefully inadequate, since, to give rise to the product, the apple has to be squared, and thus-and fortunately-fades, but not before he had created an obstacle to the negative integers. One can argue that we are still paying the price, that is, if we consider the invisibility of zero in the model of the apple as being responsible for all mathematical complications due to consider the 1 as the least natural. (In comparison with the Greeks have been lucky: with its line segments could multiply very little, unfortunately not enough to pull your model. Greek mathematics and eventually died because of poverty and complex conceptual graph: a lesson for us all.)
I seriously doubt that the detour through the block model is essential to teach young children the whole, but even in that case I see no reason why a learning process that may be suitable for young children should be also for the adult mind. And this seems to be the operating assumption on which most writers and many adult readers. Mi-sad conclusion is that the most widespread patterns of understanding have not been selected carefully for their effectiveness and can be best described as habits addictive, many of whom deserve a warning in general surgery.
My observation common to see people who feel more comfortable with the specifics unnecessary. When confronted with a partially ordered set, they think mentally "for example, integers. While I was trained to avoid reading a text examples, as they may be superfluous and in any case, distracted, "I see people who feel more uncomfortable fuando face a text examples. People who have difficulty understanding a building that contains a natural parameter k has assured me that this parameter presents an additional obstacle that may initially replacing remove k by a small value, say 3. I have no reason to doubt his word, the strange phenomenon probably was connected to the fact that k did not occur in a very simple, but as the string length or number of arcs that meet at a vertex (contexts in which they are used to manage in terms of graphics.) In the same way a permutation "arbitrary" created similar problems would have preferred a specific, possibly followed by a comment at the end to indicate that the choice of the permutation does not really matter. It's very strange, even disturbing, to see people upset when left open questions whose answers are irrelevant.
A final observation suggests to me that, in fact, blame the education system. I well remember the introduction of the idea that the teacher's task to motivate students. (I remember it well because I thought the idea was too absurd.) Now I find young scientists trained under the motivational system, which has a major disadvantage: its ability to absorb information is motivated not limited to 10 lines. The object and purpose are different things, but they did not learn to distinguish and are now unable to separate these issues. It is a frightening example of how education can instill psychological needs become a major disadvantage.
Xxx dissertation was the most absurd I've heard in years. (For still remember that.) The poor guy was a serious victim of his education: confused the directed graph as a subset of ordered pairs, with graphical representation of arrows between points. [If it had been instructed on the incidence matrices could have lectured on the eigenvalues \u200b\u200bof the evidence.]
This is what happens to us again and again. When we introduce a new concept we are given several examples of a hopefully familiar context, or give us one or two models in which the new formalism, its objects and operations can be "understood." And it really encourages us to make those interpretations to convince ourselves that the new formalism "is sense. "They fail, however, warn that such interpretations tend to be misleading because the models are sobreespecíficos, in understanding that such habits are completely baffling when accompanying visualizations mislead the imagination and the mental burden of moving to and from the formula and its interpretation, should best be avoided. In fact, one can only hope that increasing familiarity with the formalism, the model is quietly vanish from our consciousness.
This had already begun when we learned the numbers natural. We did not learn that 2 + 3 = 5, first learned - graphically! - that two apples and three apples are five apples and then for pear, pen, litter, trees and elephants. The apple model is woefully inadequate, since, to give rise to the product, the apple has to be squared, and thus-and fortunately-fades, but not before he had created an obstacle to the negative integers. One can argue that we are still paying the price, that is, if we consider the invisibility of zero in the model of the apple as being responsible for all mathematical complications due to consider the 1 as the least natural. (In comparison with the Greeks have been lucky: with its line segments could multiply very little, unfortunately not enough to pull your model. Greek mathematics and eventually died because of poverty and complex conceptual graph: a lesson for us all.)
I seriously doubt that the detour through the block model is essential to teach young children the whole, but even in that case I see no reason why a learning process that may be suitable for young children should be also for the adult mind. And this seems to be the operating assumption on which most writers and many adult readers. Mi-sad conclusion is that the most widespread patterns of understanding have not been selected carefully for their effectiveness and can be best described as habits addictive, many of whom deserve a warning in general surgery.
My observation common to see people who feel more comfortable with the specifics unnecessary. When confronted with a partially ordered set, they think mentally "for example, integers. While I was trained to avoid reading a text examples, as they may be superfluous and in any case, distracted, "I see people who feel more uncomfortable fuando face a text examples. People who have difficulty understanding a building that contains a natural parameter k has assured me that this parameter presents an additional obstacle that may initially replacing remove k by a small value, say 3. I have no reason to doubt his word, the strange phenomenon probably was connected to the fact that k did not occur in a very simple, but as the string length or number of arcs that meet at a vertex (contexts in which they are used to manage in terms of graphics.) In the same way a permutation "arbitrary" created similar problems would have preferred a specific, possibly followed by a comment at the end to indicate that the choice of the permutation does not really matter. It's very strange, even disturbing, to see people upset when left open questions whose answers are irrelevant.
A final observation suggests to me that, in fact, blame the education system. I well remember the introduction of the idea that the teacher's task to motivate students. (I remember it well because I thought the idea was too absurd.) Now I find young scientists trained under the motivational system, which has a major disadvantage: its ability to absorb information is motivated not limited to 10 lines. The object and purpose are different things, but they did not learn to distinguish and are now unable to separate these issues. It is a frightening example of how education can instill psychological needs become a major disadvantage.
