左右互搏――老顽童

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左右互搏――老顽童 关于人生、社会、世界的思考。尤其是组合数学、数学教育、思考问题的方法的讨论。 首页 Why The Professor Can (16) 方法 (28) 妙文拾趣 (33) 网站日志 (5) 影视音乐 (8) 数学 (3) 经典欣赏 (11) 小说连载 (34) 评论 (8) 2005 年 1 月 Sun Mon Tue Wen Thu Fri Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 最后更新 沙僧日记――秀逗前年9月30日 此间的少年――乔峰（Ｉ）：一次篮球赛 沙僧日记――秀逗前年9月28日 此间的少年――乔峰（Ｉ）：关系铁 沙僧日记――秀逗前年9月27日 此间的少年――郭靖（Ｉ）：图书馆 沙僧日记――秀逗前年9月14日 此间的少年――郭靖（Ｉ）：朋友 窃喜即偷欢 风雨再访金文明 最新评论 dwsd : aerqwerqw eqr. 冷峻散势 : 左右互搏就是自己. 金庸 : 多少风情， . 涓涓流水 : 这篇文章不长，但. darkevan : 搞个链接不就行了. ivy : that is a good o. mojaves : 用了我最喜欢的海. isgaryzhu : 你是纪晓岚剧组的. 存档 内人 E-Mail to 老顽童 我的链接 分页: 第一页 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] Why The Professor Can't Teach――第九章　基础教育的曙光 - 2003-05-31 23:34 CHAPTER 9: Some Light at the Beginning of the Tunnel: Elementary Education. I had been to school.. . and could say the multiplication table up to 6 x 7 = 35, and I don't reckon I could ever get any further than that if I was to live forever. I don't take no stock in mathematics, anyway. Mark Twain (Huckleberry Finn) Of all the levels of mathematics education, from elementary to graduate, the elementary is the most difficult to teach. The primary reason is that we do not know enough about how young children learn. We know little more about how older children learn, but they are no longer as much dependent on the teachers. In view of the lack of any sound knowledge of pedagogy it is not surprising that from the very founding of primary schools in the United States until about 1950, sheer ignorance determined the method of teaching arithmetic- drill in skills. Certainly, in the nineteenth century and prior to that there were no knowledgeable professors to train elementary school teachers and, in fact, most were trained in normal schools, which could hardly be considered college-level institutions. However, even when the training of elementary school teachers was taken over by the liberal arts colleges and the schools of education, no improvement was made. The liberal arts professors were really not concerned about the mathematics that should be taught to prospective elementary school teachers, and the professors in the schools of education knew no more about how to teach than did their predecessors in the normal schools. Educational psychology was introduced early in this century as a formal discipline in the schools of education, and prospective teachers were required to take a course in this subject. But the professors had nothing to offer. They endorsed and even applauded rote learning. The leader of this school of thought, Professor Edward L. Thorndike, maintained, as we noted elsewhere (Chapter 2), that students should be taught to respond automatically to any given question or problem. Repetition of the stimulus-response would develop skills. In fact, Thorndike maintained, in The Psychology of Arithmetic (1924): Reasoning is not a radically different sort of force operating against habit but the organization and cooperation of many habits, thinking facts together. Almost everything in arithmetic should be taught as a habit that has connections with habits already acquired and will work in an organization with other habits to come. The use of this organized hierarchy of habits to solve novel problems is reasoning. The value of this methodology can readily be tested. One has only to ask an educated adult, How much is 3/4 divided by 2/3? Inexplicably, teachers continued to demand rote learning in accordance with Thorndike's psychology while at the same time advocating the values of mental discipline and transfer of training in reasoning to other fields. A new era was inaugurated in the 1950s when mathematics professors decided that the elementary- and secondaryschool curricula needed reform. Arithmetic became an abstract, logically developed subject. To present this new approach teachers had to be trained or retrained. And so new courses in content were offered by mathematics departments. Because set theory had been introduced as a topic to be taught to elementary school students, prospective teachers had to be taught set theory. The ludicrousness of what was and is, in the main, still taught may be illustrated by the attention given to the empty set, usually denoted by 4). Thus, if no apples are present, the emptyset of apples is denoted by 4). Some texts for teachers ask the students to name several different empty sets. Others ask them to prove that the empty set is unique. Further, there is strictly a logical distinction between an object and set of objects. A set or collection of books is not a book. Likewise, there is a distinction between the empty set and the set containing the empty set. See here More generally, these newer courses stressed the deductive structure of mathematics, which means proof made by deductive reasoning from axioms. Presumably, without such proof nothing can be asserted. But in the elementary school deductive proof should play little, if any, role. Inductive arguments, physical evidence, measurement, and pictures are the basis for learning and understanding. However, the prospective elementary schoolteacher, sitting at the feet of a distinguished professor, is in no position to reject or refute his teachings. Hence when the student becomes a teacher he feels like a traitor if he induces young children to accept a process or technique on the basis of, say, pictorial evidence rather than deductive proof. If the material just described is the best that mathematics departments can offer prospective elementary school teachers, then they should offer no courses in academic mathematics to these students. One defense of such college courses is that teachers must prepare children for the uses that will be made of mathematics twenty and thirty years from now. The professors also argue that understanding of fundamental principles will narrow the gap between elementary and advanced knowledge. Just what these future uses and advanced mathematics have to do with the education of children is not treated. The content, then, of the courses usually addressed to prospective elementary school teachers is decidedly poor. To make matters worse, the teaching of such courses is often entrusted to graduate students or young instructors on the ground that the subject matter is simple and does not require the mastery of mathematics that the older professors have. The real reason is that the professors don?t wish to bother with courses for nonmathematicians. The psychologists hastened to defend the new approach. Jerome Bruner's statement, in The Process of Learning, "We begin with the hypothesis that every subject can be taught effectively and in some intellectually honest form to any child at any stage of development," has been quoted endlessly and used - in particular by the proponents of the New Mathematics - to defend teaching abstractions. The saving feature of this statement is its vagueness. What is an intellectually honest form of teaching Immanuel Kant's Critique of Pure Reason to six-year-olds' It seems fair to say that most parents can discern as much about how children learn as psychologists have taught us. That all people, children in particular, must be amply grounded in concrete experiences before they can appreciate an abstraction is readily observed. That children differ in the interest they take in learning, in the rate of learning concrete material, and in the ages at which they can grasp even simple abstractions, whether these differences are genetic or conditioned, is obvious. With or without help from psychologists we can readily appreciate that the hardest part of any subject is the beginning, because it is strange. It is also the most important part, because the patterns of thought and the attitudes acquired at this time become established in young minds and then are hard to change. The obstacles to the learning of arithmetic are especially formidable. The first is that it deals almost immediately with abstractions. The number 5 is a meaningless symbol to most five-year-olds. More difficult is the question, Is 7 greater than 5? One might as well ask a child to calculate the distance to the star Sirius. Moreover, the abstractions become more remote from experience in the higher grade levels. Negative numbers are far more difficult to grasp and eluded the best mathematicians for over one thousand years after they were introduced. Another obstacle to the learning of arithmetic stems from our ingenious but unnatural way of expressing quantity. Our number system uses positional notation; the two "l"s' in 151 have entirely different meanings. The sophistication of positional notation is evident from the fact that of the many civilizations that developed some system of arithmetic, only one, the Babylonian, conceived that scheme. The Europeans learned the idea from the Babylonians, and improved the scheme somewhat by shifting to base ten from base sixty. Though positional notation facilitates the operations of arithmetic, it makes comprehension more difficult. The rationale of long division is not readily understood and the usual square root process is even more difficult to comprehend. The elementary school teacher is obliged, if he is to avoid mindless drill, to apportion his efforts and the available time between teaching skills and imparting understanding. Neither task is easy. Still another difficulty is the precision of the language that must be employed. In ordinary discourse many ambiguities are tolerated and the meaning is usually deciphered from the context. Thus, a person who addresses a letter "Dear John and Mary" surely means that John and Mary are both dear to him, but grammatically the "Dear" modifies only John, or at least it can be so interpreted. Likewise, the banks that advertise that they are happy to extend small car loans are not specific as to whether they mean that they will make loans only on small cars or that loans on any cars would be small. Such ambiguities cannot be tolerated in mathematics; thus 10 - 7 + 3 is not the same as 10 - (7 + 3); if the latter is intended, the parentheses are essential. Likewise, 3 x 4 + 5 is intended to mean (3 x 4) + 5 and not 3(4 + 5). (Parentheses would, in fact, help; instead, elementary school students are taught that the strong operations - multiplication and division - must precede the weak operations of addition and subtraction.) Mathematical language must be precise even though it necessitates symbolism and stiltedness. The difficulties we have described cannot be evaded. In contrast to the situation at the secondary-school level, there can be little choice of contents in elementary school mathematics. Every adult must be able to perform arithmetic and use common geometrical formulas for area, perimeter, and volume; and in our society some knowledge of statistics and probability is almost as imperative. Whether these last two topics should be taught in the elementary school or reserved for the secondary school is debatable. In view of the content and the few principles we do possess about how children learn, what should prospective elementary school teachers be taught? A truly cultural liberal arts course, such as was described earlier (Chapter 6), would indeed be more helpful in the education of all people and certainly to teachers of elementary mathematics. To the latter it would give some perspective on the place of mathematics in human affairs. Some of the knowledge gained could even be utilized in their teaching. But the primary need of prospective elementary school teachers is re-education in arithmetic. They know no more than they learned in elementary school and it is commonly conceded that these teachers, like most adults, are insecure in that knowledge. Many even dread the period that they must devote to the subject. These failings seriously hinder effective teaching. Re-education would certainly help, particularly because as college students these prospective teachers are more motivated than they were as elementary school students. Beyond re-education, the eoursç could give insight into arithmetic and geometry, which would fulfill the essential requirement that a teacher must know more than he teaches. Thus, whereas positional notation in bases other than ten should not be taught to elementary school students, it should be taught to prospective teachers. The usual square root process cannot be made entirely clear to children, but it can be to teachers. Terminology such as inverse, commutative, associative, and the like can be taught to prospective teachers, though it should not be imposed on youngsters. Since the elements of statistics and probability are entering the elementary school curriculum, certainly future teachers should know more about these topics than they will teach. No doubt mathematics professors will object that such a course is not college level and will resent having to use their precious time and energy to teach such simple material. But the farce of teaching the kind of course they do give and their ignorance of the needs of elementary school teachers do not seem to trouble them. Apparently, bewildering students with endless questions on the empty set is college level. Despite the at present poor preparation of elementary school teachers, some progress in elementary education has taken place in the last few years, especially in the lower grades, whereas none is noticeable at the other levels. One reason may be reaction to the New Math. Of course, the New Math was a disaster at both the elementary and secondary levels, but the extent of the disaster was more evident at the primary school level. When students could not add 9 and 8, almost everyone was shocked. That a high school student could not solve x + 3 = 7 should have been equally shocking, but since solution of this problem is not required in daily life and is beyond most adults anyway, the failure to master simple algebra was not attacked as severely. In any case, teachers concerned with the elementary school seem at last to have learned the obvious. Abstractions, however simple, acquire meaning only in terms of a host of concrete experiences. Thus, texts of the last few years have pictures of people, animals, houses, candy, and other objects familiar to children, and the whole numbers and operations with whole numbers are discussed for a long time only in connection with these pictures. That is no longer based on the commutative law or taught by pure rote. See here Children are asked to look at a rectangular array of discs (Figure 1) and they see that two rows of three discs and three columns of two discs still total six. They soon recognize that the order in which any two numbers are multiplied is immaterial and thus learn the commutative property of multiplication. Fractions are parts of pies and it is visually obvious that 2/6 of a pie is the same as 1/3 of a pie. The student then readily understands the principle that one can multiply the numerator and denominator of a fraction by the same number and not change the value of the fraction itself. Building up arithmetic on the basis of real situations accomplishes more than the teaching of skills. If a child learns to perform the four operations but has not constantly associated them with real problems, he will be in the position of not knowing which operation to use in a given instance. If asked to give three children equal portions of 1 1/2 pies, should he divide? If so, should he divide 3 by 1 1/2 or 1 1/2 by 3? As the teaching of arithmetic progresses to the higher grades, wherein operations with large numbers, more complicated fractions, decimals, and percentage are the topics, teachers, poorly prepared in their college education, often ask, What real phenomena or situations can we employ to make the arithmetic meaningful and purposeful? Some suitable themes are the same as those suggested for the first two years of high school, but the questions raised would be simpler. For example, given the speed of sound in air, students could be asked how far sound travels in a stated number of seconds or how long it takes a sound signal to travel a given distance. These questions are not intrinFically exciting, but one can readily introduce some that are. Sound travels at about 5,500 feet per second through the earth but at only 1,100 feet per second through air. Hence, a distant sound reaches the auditor more quickly through the ground. This is why Indians put their ears to the ground to hear a rider on horseback sooner. One can ask for the difference in time if the distance is one mile. Another application of the speed of sound uses the interval of time required for a sound to reach a mountain and for the echo to return to calculate the distance to the mountain. The same principle is used to determine the depth of the ocean and of harbors, and to determine the distance of one submarine from another. Work with the speed of light, which is 186,000 miles per second, can also lead to some novel facts. Given the distance from the earth to the sun how long does it take light from the sun to reach us? A calculation would show that it takes light from the sun eight minutes to reach the earth. Having learned this, students might be asked, Do we see the sun where it is now? Of course not. We see where it was eight minutes ago. It may have exploded in the meantime. Speed, time, and distance can be applied to problems of sports. For example, a line drive - that is, a ball batted to follow practically a straight-line path - has a speed of, say, 100 feet per second and is fielded by an outfielder 350 feet from the batter. How many seconds does the outfielder have to get into position to catch the ball? A man on third base can run the 90 feet to home plate at 40 feet per second. A pitcher 65 feet from home can throw the ball to the catcher at a speed of 20 feet per second. Will the runner beat the ball to home plate? How many feet per second does a runner travel if he can cover a mile in 4 minutes? The distinction between mass and weight can be made and utilized in elementary school work. In daily life we confuse mass and weight and the confusion does no harm. But in these days of space exploration the distinction is important, and it is easily taught. Mass, roughly speaking, is quantity of matter. Weight is the pull that the earth or some other body exerts on mass. Thus, an astronaut on the moon has the same mass as on earth but weighs far less, about onesixth as much as he does on earth. Do we have to take children to the moon to demonstrate the difference between mass and weight? Film strips of astronauts walking on the moon would serve. There is also recourse to the child's common experience. An object placed in water is buoyed up by a force equal to the weight of the displaced water. This fact is known as Archimedes' principle. Thus, a swimmer is buoyed up by the weight of the water he displaces, which is just about his weight, and so he is weightless in the water. If a spring whose upper end is held fixed is placed so that the bottom just reaches and is attached to the swimmer, his mass will not extend the spring. If an object is heavier (in air) than the weight of the water it displaces, it will sink. Water weighs 62.5 lbs. per cubic foot. Given the weight in air and the volume of a piece of lead or other substances, one can readily frame arithmetic problems that ask whether the objects will float or sink in water. Archimedes' principle applies also to objects in the air and accounts for objects rising instead of sinking. Helium is lighter than air. If a balloon is filled with helium, its weight is less than the weight of the air it displaces. Hence, it is forced upward by the weight of this displaced volume of air, and since its own weight, which pulls it downward, is less, the balloon rises. It is easy here, too, to raise simple arithmetical problems on the motion of the balloon. A good case can be made for including in elementary school the elements of statistics and probability. The citizen of the United States is faced with statistics in practically every broadcast and in the daily newspapers. The most important decisions of life, as Laplace pointed out almost two hundred years ago, are made on the basis of probability. Even trivial ones, such as the decision to cross the street, are made on the same basis. The interpretation of statistics is by no means simple and the significance of a probability of, say, 0.75 depends very much on whether it applies to a horse race or a medical treatment. Of course, political leaders and others with axes to grind use statistics and probability to mislead people. Hence, some knowledge of these subjects and their applications to daily life is important to all citizens. Commercial problems are relevant. Certainly, compound interest should be understood by every citizen, as should the subject of discount. Young children may not be excited by such topics, but perhaps challenging questions would help. If a storekeeper reduces the price of an item by 10 percent and then increases the price by 10 percent, has he restored the original price? Which is the better buy: an item subject first to a discount of 20 percent and then to a discount of 10 percent of the reduced price or the same item subject to a discount of 10 percent and then to a discount of 20 percent of the reduced price? The elements of geometry are now introduced in elementary school. This is useful knowledge. But as currently taught it seems to be innovation for the sake of innovation. Children are asked to learn any number of terms - acute angle, right angle, obtuse angle, straight angle, and dozens of others - and to know what they mean pictorially. But nothing is done with this knowledge to give it purpose in the eyes of the children. Fortunately, there are dozens of interesting and simple applications. See here It would be helpful to see some uses of angles beyond their being parts of triangles. One of the commonest applications is to mirrors. A mirror reflects light in accordance with a very simple principle, called the law of reflection. See here This law can be readily demonstrated in a classroom by darkening the room, laying a small mirror flat on the desk, and using a pencil flashlight to show the incident and reflected rays. Since every child uses a mirror, the law should be of interest. See here To further impress the student with the importance of angles one can describe or even demonstrate the use of the law of reflection in a periscope. A ray emanating from A (Figure 3) is reflected twice and reaches B. The law of reflection also applies to billiard balls. If a ball at A (Figure 4)is to hit the ball at B by being bounced off the side of the table, the ball must be directed so that See here Elementary geometry can be applied to astronomy, preferably with the aid of a model of the solar system, which is commercially available. In this model the sun, the planets, and their moons revolve in precisely the orbits they follow in space and with the correct relative speeds. Such a model is worth far more than any verbal description, diagrams on the blackboard, or pictures in books. A number of mathematical themes - the velocities of the planets, the varying length of the day throughout the year, and eclipses - can be treated readily with the aid of this model. One might object that the subject belongs to astronomy. It belongs to knowledge, and mathematics is essentially involved. Still another area to which the elements of geometry can be applied is geography. Latitude and longitude are angles formed at the center of the earth. Students can compute distances traveled along a meridian under a change of, say, 5 degrees of latitude or distances traveled along the equator or a circle of latitude under a change of 5 degrees of longitude. (In some instances information such as the radius of a circle of latitude would have to be supplied.) Some applications are mundane but practical. Students might be asked to consider the areas of various rectangles with the same perimeter. Thus, if the perimeter were 100 feet, the dimensions could be 1 by 49, 2 by 48,4 by 46, 10 by 40, etc. By using arithmetic and the formula for the area of a rectangle, students would soon discover that the maximum area is given by a square with dimensions 25 by 25. This result is useful. If a farmer wishes to use only 100 feet of fencing, he can have more area for planting if he chooses a square. Or consider a fifty-story office building. The cost is determined mainly by the walls, but the income is determined mainly by the number of square feet of space that can be rented. Thus, the square shape pays off on each floor. Of course other factors, such as the shape of the land, may not allow the use of the square shape, but the closer one comes to it the more floor space will be obtained. Later, on the high school level, the proof that the square furnishes maximum area can be taken up. See here Thus, the roadway is only 27 feet longer than the circumference of the earth -exactly as much longer as the circumference of the fence is longer than the circumference of the garden. The introduction of scientific applications of mathematics is a lesser problem to the elementary school teacher than to the secondary school teacher. The elementary schools now teach some science, and since the same teacher teaches all the subjects, he or she knows the science involved. Why not teach what is relevant to mathematics in connection with mathematics? The rigid segregation of subjects, which calls for teaching arithmetic at ten o'clock and geography at eleven, is artificial and works against the major goal of education: the integration of all knowledge. The above examples of real situations - which provide motivation and application as well as the context in which elementary mathematics can be taught - may convince children that the subject, properly approached, is fascinating. As long as the child can think in physical or sensory terms he is at home. Symbols and words acquire meaning only in terms of sense perceptions. Fortunately, teachers and texts at the level of elementary education are becoming committed to the principle that understanding is intuitive rather than logical and are employing intuitive aids such as children's experiences out of school, activities in the classroom, pictures, measurement, and geometrical schemes to answer the question of why we do things the way we do. Conviction derived from seeing that the arithmetical processes do yield what is physically true, rather than proof in the mathematical sense, should be and is becoming the basis for children's acceptance of the processes. Laboratory materials are now used extensively as aids to pedagogy. The teacher no longer needs to bring apples to school. In the first two or three grades a teacher now has available any number of devices, including Cuisenaire rods, cards, discs, balances, scales, fraction bars, geoboards, clocks, tapes, various types of rulers to perform measurements, geometric models, spinners (to teach probability), and blocks of various sizes, colors, and shapes to represent units, tens, and so on. Games and puzzles motivate at the lowest elementary school level and it is pleasant to note that these, too, are currently being introduced. For very young children Plato gave this advice over two thousand years ago in The Republic: "Do not, then, my friend, keep children to their studies by compulsion but by play." Many commercial organizations are now manufacturing games, and school systems are buying them. Sadly needed are laboratory materials for the higher grades. One of these, the hand calculator, is coming to the fore and will undoubtedly aid in the learning of arithmetic. (Its precise value will be discussed later.) Laboratory materials, introduced only in very recent years, were advocated by great educators centuries ago. Montaigne and Rousseau advised us to teach the very young with the aid of real things - let children learn by doing. Through building up elementary mathematics on the basis of physical problems, real phenomena, and laboratory materials, we seek, of course, to teach understanding of the skills in the hope that the understanding will help in remembering the skills and aid in determining where to use them. However, full understanding of our present-day arithmetic is not easily attained, even with the best of pedagogy, because our system of arithmetic is sophisticated - at least for youngsters. Yet the skills should be acquired early in life. The resolution of this difficulty is not to abandon the teaching of understanding but to emphasize the acquisition of the skills. Though the child's understanding may not be complete, or even though he may forget the justifications of the operations, much has been gained by the presentation. Receiving it at least once has a psychological value. It sets the mind at ease, whereas in the absence of understanding the mind remains perplexed and balks at performing. The situation is somewhat the same as when an adult is called upon to donate money to charity. He wants to know why this charity and not some other. Having satisfied himself, he gives freely rather than grudgingly; and in future years he gives without questioning because he remembers not why he chose this particular charity but merely that he had satisfied himself that it was worthwhile. So it is with skills. Having understood why and having learned how to perform them, in the future one performs them without hesitation. One need not and, in fact, should not rethink through the why. As Alfred North Whitehead points out: It is a profoundly erroneous truism, repeated by all copybooks, and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle - they are strictly limited in number, they require fresh horses and must only be made at decisive moments. There are professors who still believe that we can give priority to the concepts and thereby impart understanding. Then, supposedly without much drill, the acquisition of technique will be achieved. Today some go further and argue that the teaching of arithmetic skills can be dispensed with entirely. Students and adults will use hand calculators. As is often the case with new devices, their value is exaggerated beyond all reason. They become the new fad and the excuse for countless worthless papers and talks. The prospect of hand calculators replacing arithmetic skill conjures up rather ridiculous scenes. Apparently men and women will carry calculators to shop in stores, to decide how much of a tip to leave to waiters and to cab drivers, and to decide how many nickels, dimes, and quarters one needs to pay a subway fare. Perhaps if we were training people only to operate modern cash registers in our grocery stores, the reliance upon calculators would do. Peculiarly, many of the very same people who favor a curriculum that will prepare students for college mathematics are the ones who would dispense with teaching skills and would substitute the calculator. But students who will go to college must go to high school. And there they will be asked to perform calculations such as 1/a + 1/b, a5.a3, and (a + b)2. The best rationale one can give to students of algebra is that letters stand for numbers, and if one knows how to perform the corresponding operations for numbers, the operations with letters are readily justified and performed. The addition 1/a + 1/b is carried out in precisely the same manner as 1/2 + 1/3. Hence, one must learn the skills of arithmetic to do algebra. Moreover, even after pushing buttons to perform an arithmetic operation, one should check to see that the answer is reasonable. One might have pushed the wrong button. But estimation as to the reasonableness of an answer calls for knowing a good many arithmetic skills. The hand calculator can be compared with the typewriter. One can use the latter and dispense with handwriting, but the typewriter does not tell us what to say and how to say it. Similarly, the calculator will not tell us what operations to perform or whether the answer is reasonable. Its value is limited. To one who knows arithmetic and how to use it, the calculator can serve to speed up long series of calculations or calculations with large numbers; it is useful as a check on calculations performed with pencil and paper and thereby serves as a teaching aid; and it does provide a gadget that youngsters can finger and so introduces some novelty. But it cannot replace the teaching of arithmetic skills. The training of prospective elementary school teachers should, of course, include an understanding of the physical phenomena that may be used to motivate students and to apply arithmetic and geometry, and it should include the use of resources such as laboratory materials. College courses for prospective teachers should also foster many more pedagogical aids, some of which can merely be mentioned here. Terminology should be and some of it, fortunately, is being improved. In earlier times, when students were taught how to perform the subtraction problem 24-19 they were told to borrow 1 from the 2 and to place it alongside the 4 to make 14. They could then proceed. The term borrow was a poor one. One might as well have chosen beg or steal. And borrowing, as Polonius said to Laertes, "dulls the edge of husbandry." Moreover, if one borrows, then one should repay. But there is no repayment. The better term is exchange. (Regrouping is also used.) One takes one of the two tens, converts or exchanges it for ten ones and adds these ten ones to the four. One has exchanged one ten for ten ones, just as one exchanges a dime for ten pennies. False application of techniques can be a challenging call for critical thinking. Students are taught, rightly, that division of one whole number by another is repeated subtraction. Hence, 7 divided into 28 might be carried out as follows: One tries 1 as a trial quotient and subtracts. This gives: See here But now 3 can be the next trial quotient and, in fact, 7 divides evenly three times into 21. The false procedure then is to write: See here and the answer is presumably 13. Though the flaw may seem trivially obvious to an adult, it is not so to the child. Of course, there are many subtler false arguments and processes that can be employed to advantage. At some stage in the primary school curriculum students should become fully aware that numbers are abstractions. Simple situations can be used to make the point. A man goes into a shoe store and buys three pairs of shoes at $20 a pair. The salesman says that 3 pairs of shoes at $20 a pair cost $60, and he expects the customer to hand him $60. But the customer instead replies that 3 pairs of shoes at $20 a pair is not $60, but 60 pairs of shoes, and he asks the salesman for the 60 pairs. Is the customer right? As right as the salesman. If pairs of shoes times dollars can yield dollars, then why cannot the same product yield pairs of shoes? (The physicist would give a dimensional argument, but this is a crutch.) The answer is, of course, that we do not multiply shoes by dollars. We abstract the numbers 3 and 20 from the physical situation, multiply to obtain 60, and then interpret the result to suit the physical situation. In this example the clerk's physical interpretation is the correct one, at least in our economy. The injunction to teach discovery has by now become hackneyed. The fact that it is difficult to do undoubtedly explains why it is rarely done. One must prepare a series of leading questions for each topic and, often without seeming to do so, reformulate a student's reply so that it leads to a useful suggestion. The process is also time-consuming, and many teachers complain that they can't cover all the ground of the course if they teach discovery. But covering ground that succeeds only in causing dislike and even failure certainly accomplishes little. It is not possible here to present the discovery approach to the many topics that have to be taught in the elementary school. But a simple numerical example may serve to illustrate the idea. Students can be presented with the following facts: See here They can then be asked to generalize. They need not write the generalization in symbolic form, but they might be able to state in effect that the sum of the first n odd numbers is the square of n. This conclusion can be supported by interesting diagrams that, in fact, were first used by the Pythagoreans of the sixth century B.C. (Figure 5). See here One sees in these diagrams how the addition of an odd number of dots, those above and to the right of the solid lines, produces the next square number. Further, granted the conclusion, the student can be asked to state, without adding, what the sum of 1 + 3 + 5 +...+ 97 + 99 is. There are many such examples, involving only simple properties of whole numbers, that can reasonably call for discovery. The application described earlier of finding the rectangle of maximum area with given perimeter can be posed as a problem calling for discovery of the fact that of all rectangles with the same perimeter, the square has the most area. Though education in arithmetic has made some progress since the 1970s began, there are still dragons roaming the land, and these must be slain if further progress is to be made. College professors attach very little importance to the courses addressed to prospective elementary school teachers. But these courses are as important as any that professors teach. How these elementary school teachers perform later is crucial. Children start to learn mathematics from the moment they enter school. Their success or lack of success will affect their attitude toward the subject, their confidence in themselves, and, very likely, their attitude toward all learning. Poor mathematics education may be the rotten apple in a barrel of sound ones and by contamination spoil the rest. It is in the course of their elementary school work that far too many people decide that they do not have mathematical minds. Actually, they are victims of teachers who have themselves been victimized by college professors. People "do not have mathematical minds" because they do not receive decent mathematical teaching. Meaningless mathematics will not penetrate any minds. And one who is frustrated by arithmetic will retain an inferiority complex toward all of mathematics, because the subject is cumulative. Even where arithmetic is not a prerequisite, as in large areas of geometry, the mere fact that these are also part of mathematics causes students who have been burned by arithmetic to refuse to let their minds be kindled by new themes. Mathematicians do like to foster the impression that there is such a thing as a mathematical mind that eo ipso is necessarily superior. Though the existence of such minds is highly unlikely, there is the probability that creativity of the Einsteinian caliber does require exceptional abilities. But it is certain that one does not have to have a mathematical mind to understand mathematics. The courses that colleges offer to prospective elementary school teachers are usually a waste of time. Typically, they teach the very same topics that the current so-called liberal arts course offers - the logical development of the real number system; set theory; transfinite numbers; Boolean algebra; truth tables; abstract mathematical structures such as groups, rings, and fields; finite geometries; and a heavy emphasis on axiomatics and proof. In fact, the very same texts that are written for the freshman liberal arts course are advertised and used for the courses addressed to prospective elementary school teachers. Even in the heyday of the New Math these topics were inappropriate. Elementary school teachers do not need to know this material and even get wrong ideas from it concerning what mathematics is about and how mathematics should be taught. Despite the shattering effects of past and current education of elementary school teachers, there is reason to be optimistic. Dragons, especially the mythical ones, have been slain in the past. The current educational dragons, even though devoutly believed in by professors, are also myths; these, too, can be slain. C&O 发表于 23:34 | 阅读全文 | 评论(0) | 引用trackback(0) | 编辑 合久必分 - 2003-05-31 15:13 好无奈！ 刚想将自己的两个内容相同的BLOG合并，或者说想放弃 BLOGCN ，因为他们的服务器不太稳定。不想 BLOGBUS 也连不上，使得好几天没更新了。希望这只是偶尔的。总不能让我“包二奶”后，又“狡兔三窟”吧 。 现在我打算将这两个BLOG的内容分工一下： BLOGBUS 将继续原来的内容，比较大众化，也经常更新。 BLOGCN 则将改版为更专业的内容，不定期更新。 希望大家对这两个姐妹不要有大小之分，有先后之分；她们是我的左右手，手心手背肉，我的生活不同面的记录。在她们之间我很快乐、充实 。 谢谢！ 有空给我发E-Mail： 老顽童 老顽童 C&O 发表于 15:13 | 阅读全文 | 评论(0) | 引用trackback(0) | 编辑 面朝大海，春暖花开 - 2003-05-28 11:12 面朝大海，春暖花开　海子 从明天起,，做一个幸福的人 喂马，劈柴，周游世界 从明天起，关心粮食和蔬菜 我有一所房子，面朝大海，春暖花开 从明天起，和每一个亲人通信 告诉他们我的幸福 那幸福的闪电告诉我的 我将告诉每一个人 给每一条河每一座山取一个温暖的名字 陌生人，我也为你祝福 愿你有一个灿烂的前程 愿你有情人终成眷属 愿你在尘世获的幸福 我也愿面朝大海，春暖花开 C&O 发表于 11:12 | 阅读全文 | 评论(1) | 引用trackback(0) | 编辑 成语和学习――水清无鱼 - 2003-05-28 11:02 26．水清无鱼 成语 “水清无鱼”源于东方朔《答客难一首》，“水至清则无鱼，人至察则无徒。”水过于清澈，鱼就难以生存。人过于明鉴，就找不到同伴。因此， “水清无鱼”比喻事物不可能绝对地纯。也比喻对人要求不可过高。 人们总是喜欢把自己认为好的一面显示在其他人面前。文学家、艺术家总是把他们认为好的作品献给人们，科学家也是把他们认为成熟的工作发表出来。另一方面，人们的作品、工作只有在有人(一般都是专家)认为是好的时候，才能被接受或者发表。那些被退回的稿子就是因为审查人认为不好。 高斯是近代数学伟大的奠基者之一。他在历史上影响之大，可以和阿基米德、牛顿、欧拉并列。高斯总是等到工作十分成熟的时候才公布出来。呈现在人们面前的只是完美无暇的结果，而省略了分析和思考的过程。 李白是我国最伟大的诗人之一，他的诗经历了久远的时间，不知给一代代人带来多少启迪和精神享受。 听一位好的老师讲课，学生们会觉得是一种享受。在这种感觉下能学到知识是每个学生所希望的，也是负责任的老师所追求的。 在那些完美的作品或工作面前，人们往往会望洋兴叹。人们不知道怎样才能创作出如此完美的东西。从李白的诗中，你很难知道他为什么会那样写诗。从高斯的文章中，一般的学者很难掌握他的思想方法。有人曾说过：高斯就象一只狡猾的狐狸，把它在雪地上走过的痕迹都用尾巴扫掉了。 人们都知道失败是成功之母，完美的作品是由不完美的东西逐步形成的。人们知道鼓励学生们在学习中、在生活中跌倒了要爬起来。写作文要一遍一遍的修改完善。但是这种失败走向成功，由不完美走向完美的过程，中小学生甚至大学生很少能够从他们的老师身上看到。老师在课堂上的讲课是经过准备的，学生的问题老师是明白的，老师讲授的内容是无懈可击的。 很多人认为，老师是不能被学生问倒的。不少老师不能回答学生提出的问题时会觉得很尴尬。作为老师当然要尽可能地熟悉自己讲授课程的内容和相关的东西，但是谁也不能保证自己不会被聪明的学生问住。出现这种情况正是老师表演自己如何另外，以倒行逆施的思维方式考虑问题，老师不可以对讲授的内容只是临阵磨枪吗?老师对讲授内容的理解一定要比学生高吗?当老师和学生一起学习师生都不会的东西时，老师不是正好可以向学生演示如何学习新东西的过程吗? 你偶尔会问我一些数学题目。当我会做的时候，你知道我会说：再读读题目、翻翻哪本书。当然题目太难是会给些提示的。当我也不会做的时候，就会明确告诉你，我一下子也做不出来。我们一起想一起做时，我很高兴你能发现我的错误。对于你的嘲笑：还是大学数学老师哪，连小学的题目都不会做，我面不改色，心不跳。 （全文完） C&O 发表于 11:02 | 阅读全文 | 评论(0) | 引用trackback(0) | 编辑 成语和学习――水能载舟，亦能覆舟 - 2003-05-28 11:00 25．水能载舟，亦能覆舟 成语“水能载舟，亦能覆舟”讲的是这样一个简单的道理：事物用之得当则有利，反之必有弊害。过去常把统治者比喻成舟，老百姓比喻成水，如果统治者不能处理好百姓的事情，就不会有稳固的政权。 在学习方面，也有一些事情属于水能载舟，亦能覆舟的情况。人们常说的“聪明反被聪明误”就是最明显的一例。除此之外，还有许多。 对于学生来讲，能够上一所好学校，遇到一位好的老师是值得庆幸的。但是如果完全依赖学校和老师，课上听老师讲课，课下做老师留的作业，一切都听老师的，完全没有了自我，那就要出问题了。 一个人知道得很多本是好事，但如果不善于把知识用于需要，那就没有什么用处。如果他注重的是知识的数量，而不是知识的质量，知道得很多很多，但却不知道最有用的东西，那就糟糕了。更糟的是被知道的知识束缚了思维，还不如不知道。 计算机是个好东西，它的应用范围越来越广，所以从小学习使用计算机是非常必要的。但是中小学生过多使用计算机有可能妨碍他们的智力发展和动手能力。另一方面，对大多数人来说，计算机只是工具，和电视机、汽车等的作用一样都是为人服务的，在中小学阶段没有必要学习太多的计算机知识。 我现在除了上课一般很少写字，写点什么东西只要可能就用计算机。有一段时间没有给学生讲课，结果我染上了提笔忘字的毛病。有一次居然想了一上午也没有想出“梦想”的“梦”字怎么写。现在有些学生很注意使用计算机，但却不愿意练习写字。由于有了计算机，一个人的字似乎不能再是他的门面了。但是练习写字尤其是用毛笔写字不仅能够使一个人的字变得好看，更重要的是练字的过程也是修身养性的过程。 现在有许多适合中小学生参加的竞赛，数学、作文、英语等等。应该说这些竞赛对于激发兴趣、开拓视野、训练能力等方面有着积极的作用。另一方面，在竞赛中获奖对于上好的初中、高中甚至大学都有帮助。成绩极突出者还能为个人，家长、老师，学校或国家争得荣誉。 因此，不少学生和家长对参加竞赛感兴趣，为此付出很多的时间和精力。 但是，如果为在某种学科的竞赛中取得好成绩而偏科畸型发展就可能是坏事了。曾有在数学竞赛中出类拔萃的学生上了大学后感到自卑，因为除了数学他好象什么都不懂。另一方面，竞赛的题目是超出学校教育的知识或能力要求的。对于一般的学生，甚至是优秀的学生，都不容易得到好的成绩。那种屡战屡败的感受很可能会使学生对某个学科丧失兴趣和信心，而将来他很可能在这一学科方面会有所作为。当然我不排除一些不服输的学生会屡战屡败的可能。 同学们想上大学是好事，把上大学当成阶段奋斗目标也是可以的。但是学习的目的仅仅就是为了上大学，就有点鼠目寸光了。那种把上大学作为一生的奋斗目标的想法更是要不得。大学对大学生的最大作用是提供更高层次的教育，使得学生在今后能有更大的作为。要达到这一目标，大学生在学校读书期间要更加努力地自觉地学习，尤其在大学一二年级。 也许是因为奋斗目标已经达到，也许是在中学阶段太苦了，也许是家长和学校以前逼得紧了，现在有不少学生上了大学以后，或者放松了学习，或者不适应大学的学习方式。待到醒悟过来时，最应努力的时间已经过去了。 C&O 发表于 11:00 | 阅读全文 | 评论(0) | 引用trackback(0) | 编辑 成语和学习――揠苗助长 - 2003-05-28 10:57 24．揠苗助长 揠：拔。成语“揠苗助长”意思是将苗拔起，助其生长；比喻不顾事物发展规律，强求速成，结果反将事情弄糟。 相比于欧美国家，我国的中小学教育的特点是讲授的知识深一些、广一些。而学生的家长对孩子的期望和要求有一个值得注意的倾向，那就是超前教育，希望孩子提早成才。 正常教育是根据孩子不同年龄阶段的特点和能力制定的，具有一定的科学性。尽管有极少数的孩子天资确实非常非常好，但是并不能代替所有的孩子。然而仍有众多的家长认为或者希望自己的孩子是神童，所以进行超前教育。另外，有些家长则认为自己的孩子智力一般，为了让孩子在正常的教育中能有好的成绩以便出人头地，也进行超前教育。 超前教育的特点是要孩子去做他的年龄特点及其能力所不能或不应该要求的事。譬如，学龄前要认识好多好多的字，学习小学一二年级的数学，背他不懂的唐诗。更干脆的就是早上学。超前教育的另一个特点是教育以记知识为主。 在我看来，超前教育就是“揠苗助长”，没有好处。而超前教育的最大害处是:扼杀兴趣，削弱自信。 让人去做他的能力所不能及的事情，最终的结果不是做不好就是费了九牛二虎之力才勉强做好。如果无人看重结果倒也无所谓，否则常这样做事情的人的自信心一定很差，其他人即便不会轻看他也不会重看他。谁能相信在这样的情况下他会对他所做的事情感兴趣?反过来，如果一个人很容易或者不是很困难地做好了事情，那他会觉得我做这事还行，这是我的强项，这时他的自信心一定很强。 有的家长认为，自己的孩子很聪明，早一点上学不会有什么问题。早上学表明他能力强，还能增强他的自信心。我完全相信有些早上学的孩子能够胜任学习任务的。例如，四、五岁上小学的孩子学习没有什么问题，少年班的大学生也完成了学业。但是有两个问题值得考虑。 首先，这些孩子有完整的童年、少年吗?其次，孩子的能力特别强是与同龄人相比的。如果他和同龄人一起走过学习生涯，那他的能力是出类拔萃的。但是与比他大的学生比较，他的能力就要打折扣了。 一家报纸的家教热线栏目中曾刊登过一则内容。有家长问：“我的孩子搭积木很难成功，总是搭到一半的时候全面推倒，做其它项目时也有类似的情况，请教专家怎么办?”专家认为是孩子的信心问题，建议家长先搭，表演失败，失败，最后成功的过程。专家的回答应该说没有问题，但是他没有考虑那个孩子究竟多大。他是否有能力搭好积木。 在正常教育的过程中也会存在超前教育的问题。俗话说：“十个手指有长短”，由于学生们能力的差别、基础的不同、用功的程度不同，他们的学习有好与不好之分，学习成绩也有高有低。对于学习成绩好的学生，尤其是班上前几名的，家长和老师自然没有太多可说。而对于学习成绩属于中下游的学生，有的家长就会不满意，想尽各种办法要提高孩子的成绩，这时候的家长就有可能在进行超前教育了。应该注意的是，造成成绩不理想的众多可能的原因中还有一种是学生没有觉悟。 这里所说的觉悟有两种情况。一种情况是孩子没有真正认识到学习的重要性，不能自觉地投入到学习中。另一种情况是孩子的知识和能力还没有达到所要求成绩的那个水平，还在积累，待到积累到一定程度就会产生一个飞跃，这个飞跃就是所谓的觉悟。在促进和等待学生觉悟的过程中，家长和老师切莫伤害了学生的自信心和学习兴趣。另一方面，学生自己则不能丧失自信。 我不赞成进行超前教育。超前是相对的，超前教育是一种急功近利的心态。就算是科学研究证实了儿童完全可以再早些开始接受现在的小学教育，也会有人对自己的孩子进行超前教育。 我们的家长、幼儿园、学校或多或少地关心神童的问题。尤其是有些家长特别关心神童的问题，生怕由于自己的原因耽误可能成为神童的孩子。 我不喜欢“神童”这两个字，当然也反对培养神童。我一直不明白培养神童对于神童本人、神童父母以及社会有什么好处?难道不明白，十年树木，百年树人。一个人对社会、对人类贡献大小不会看他儿时神不神吧?就算是光宗耀祖吧，年纪轻轻就大学毕业又怎么样?还不知道什么年纪才能成家立业呢! C&O 发表于 10:57 | 阅读全文 | 评论(0) | 引用trackback(0) | 编辑 Why The Professor Can't Teach――第八章　高中教育的误区 - 2003-05-28 10:48 CHAPTER 8: The Misdirection of High School Education Great God! I'd rather be A pagan suckled in a creed outworn; So might I,... Have glimpses that would make me less forlorn; William Wordsworth In a world in which changes take place with bewildering and discomfiting rapidity, mathematics education has provided one fixed landmark: the high school mathematics curriculum. This program is not only fixed; it is, thanks to the university ideology and practices that determine it, rigid and seemingly immovable. But the high school curriculum is certainly outmoded. Let us examine it. The ninth grade is devoted to algebra and the subject matter has been almost entirely a series of disconnected, atomized processes, a mélange of topics: factoring, operations with polynomials such as x2+5x+b, operations with fractions such as (2x+5)/(3x+7), laws of exponents, and the solution of equations of various degrees and with one or more unknowns. This approach to mathematics proper is most unfortunate for, on the whole, algebra is a means rather than an end. It is the spelling, grammar, and rhetoric of most mathematics, but it is not literature. If students of English were asked to spend years on preparation for reading - which is what the mathematics student does in eight years of arithmetic and the one year of algebra - without ever being introduced to the pleasures of reading, how would they react? Beyond the fact that algebra per se is a means rather than an end, it involves a specific difficulty that bothers almost everyone - the use of letters. Just as number symbols are a hurdle for elementary school children, so literal symbols are a hurdle for high school students. To speak of 3x = 5 is not quite so terrifying as to speak of ax = b, wherein a and b are any numbers. In principle the idea is simple. Instead of talking about John and Mary, algebra talks about men and women. But generalizations about men and women are meaningful only to those who have had lots of experience with individual men and women. The analogue for algebra is lots of experience with various manipulations of numbers. Unfortunately, few students have more than a nodding acquaintance with arithmetic at the time they are propelled into algebra. The use of literal coefficients did not occur to mathematicians until about 1600 A.D., roughly two thousand years after first-class mathematics had been produced by the Greeks. This fact in itself should warn teachers that students will balk at letters and that special devices are needed to make the transition from numbers to letters acceptable. Whereas algebra proper is both meaningless and pointless, geometry, which students learn in the tenth grade, does have intuitive meaning. But the pointlessness is still glaring. Here, many dozens of theorems are proved in a logical sequence and, to the students, the goal seems to be to prove as many as possible. They are confined to learning inflexible and obtuse trains of thought that permit no derailment. Just why anyone wants these theorems and how these theorems and their proofs were ever conceived is not treated. The eleventh grade curriculum repeats the ninth grade subject matter because students did not learn that material, adds more algebraic processes, and introduces trigonometry - in which, among other topics, the students learn many identities at the risk of losing their own. The twelfth grade contents, taken by relatively few students, has not been stable. But the material, whether solid geometry or the beginnings of calculus, is taught dogmatically and has been no more enlightening or enthralling than the preceding subjects. How did this curriculum come about? It was fashioned in the nineteenth century by professors. In fact, as we have already seen (Chapter 2), the subjects that are now taught in the high schools were, in the United States, first taught only in the colleges. They were gradually moved down into the high school curriculum, but retained essentially the content that had been taught at the college level. During the years of transition, and even long afterward, committee after committee re-examined the curriculum. In detail the recommendations made by these committees did differ; some would teach factoring before exponents, and others the reverse. But in essence all agreed upon this traditional curriculum. The arguments they accepted in favor of it are hackneyed. The value of mental discipline was never questioned, and this purported value - that mathematics teaches thinking - freed the committees from any real concern whether the content was right. In addition, utility in daily life, preparation for college (after the subjects were transferred to the high school), and the learning of the higher truths in the noblest branch of our culture were also offered to justify the status quo. There seemed to be no doubt that a2 - b2 = (a + b) (a - b) would uplift any soul. There has been one change in recent years. The ineffectiveness of the traditional education, made more evident by the needs of World War II, stimulated some university professors to devote themselves to the reform of the curriculum. But these professors were even less informed about high school education than their predecessors of the last century or the early part of this century. Hence the reformers, too, thought in terms of the values usually claimed for mathematics education, such as preparation for the mathematics to be taken in college. Their contribution was to impose rigor, generalization, abstraction, and an emphasis on structure onto the traditional content. All the arguments marshaled against such features in preceding chapters apply with greater force to the high school curriculum, because the students have yet to find out what mathematics is all about. Consider, for example, structure. Structure of what? The answer seems to be structure of abstract mathematical systems. It is true that the collection of positive and negative whole numbers and fractions - that is, the rational numbers - do have some properties that the whole numbers alone do not have. And the real numbers have the same structure as the rational numbers. Since these structures are all that high school students know, and not too clearly at that, what can the study of structure mean? One might as well ask students who have yet to see a dog to learn the structures of various species of animals. Moreover, what does structure mean in Euclidean geometry and trigonometry? The word is fashionable, but certainly not applicable to high school mathematics. The new curriculum did not significantly alter the basic high school material. It secured the rigidity of the curriculum by adding rigor to the proofs and by burrowing deeper into the foundations through set theory and symbolic logic. (See Chapter 6.) These "innovations" may have ensured the stability of the several mathematical edifices, but they also pulled the students so far down into the dark earth that they could no longer see the surface. Where light and air were needed, the new mathematics added steel beams. This reform, far from being an improvement, is more accurately described as a disaster. Fortunately, the new math is a passing aberration and there is no point to beating dead bones. To the mathematician all of this standard material, whether in traditional or new math guise, is self-justifying; to the student it is self-condemning. The contents of all four years, certainly as presented, is abstract, dull, boring, and intrinsically meaningless. It consists at best, to use Alfred North Whitehead's phrase, of inert ideas. That students should have difficulty with it is inevitable. Psychologists found years ago that human beings could not remember more than six or so nonsense syllables one hour after they were memorized. If students haven't already acquired a dislike for mathematics in elementary school, the high school mathematics, particularly algebra, will surely engender it. The aftermath of traditional mathematics is revulsion. It is incredible that mathematicians and pedagogues could ever have believed and still do believe that this curriculum has any value. To be sure, one or two students in some classes do take to mathematics, either because they like it or because they wish to guarantee admission to college, but mathematicians would be the first to caution us that one swallow does not make a summer. Let us review the arguments presented in defense of the traditional curriculum. One argument has been that this material is useful in later life. Does the average educated person use this knowledge in daily life? Do even mathematics teachers, who know the subject, ever use the quadratic formula, the Pythagorean theorem, or the trigonometric identities outside of the classroom? The honest answer is no. Of course, the students learn that a straight line is the shortest distance between two points. That's useful. But even a donkey knows that: Put some food at a distance from him and watch the path he takes. The subject matter of high school mathematics - that is, the subject matter per se - is worthless knowledge. One or two topics, such as the calculation of compound interest, may prove useful. But the exceptions do not alter the general assessment. And the high school courses have not taught thinking, pedagogues? assertions to the contrary. The traditional algebra teaches memorization of processes. Geometry purportedly emphasizes proof and therefore thinking; however, because the proofs are arranged in a logical sequence and this sequence is not natural, the rationale eludes the students and they are forced to memorize here, too. The students' appreciation of proof in geometry is epitomized in their oft-repeated remark that geometry is where you make proofs in two columns. There is another frequently advanced argument for teaching mathematics: The subject is beautiful. Whereas in the college liberal arts course topics can be chosen for beauty, what is taught on the high school level is not notable for its aesthetic value. The choice of topics was based on what is needed for further education in mathematics. All the preaching and rhapsodizing will not make such ugly ducklings as factoring, adding fractions, and the quadratic formula attractive. Even the fact that the sum of the angles of a triangle is 180° is hardly attractive; a sum of 200° would at least be a round number. Moreover, beauty is a matter of taste, and in the case of mathematics the appreciation of beauty calls for a certain sophistication. It is in fact fortunate for society that not too many people are attracted by the esoteric beauty in mathematics. Our world would be sadly affected if even 10 percent of American high school students should decide that they wished to pursue mathematics for mathematics' sake. One good medical doctor is worth a thousand mathematicians who pursue mathematics for its beauty. Omitted thus far is another value that champions of traditional mathematics cite: intellectual challenge. But should we sacrifice the 99 percent to the 1 percent who respond to this particular intellectual challenge? The clinching argument that advocates of the traditional curriculum have used endlessly is that students need high school mathematics to go to college. It is inexplicable how anyone could take such a position in 1900, or even in 1930 when no more than 25 percent of the high school graduates went to college. Though the percentage has increased steadily, it is still true that no more than 50 percent of those who graduate from high school enter college. Moreover, at least 25 percent of those who start high school do not graduate. Of those who go to college, at most 10 percent will need the techniques and theorems now taught in the high schools. The others either take no mathematics in college - many colleges no longer require a mathematics course for the bachelor's degree - or take a liberal arts course that does not use most of the currently taught high school mathematics. Do the colleges require the conventional high school mathematics for admission? They have not demanded it for many years. Some do not ask for any high school mathematics. Almost all of the others will accept any respectable two-year curriculum. Hence, one cannot justify the present high school curriculum on the ground that it prepares students for college. The argument concerning preparation for college has also been self-defeating. Faced with the dull traditional curriculum, fewer students have been taking academic mathematics. In 1928, when about three million students attended secondary school 27 percent took ninth-grade algebra and 18 percent took plane geometry. In 1934, the corresponding percentages for five and one-half million students had dropped to 19 and 12. Since 1934 the number of students attending secondary school has increased sharply, but the percentage taking academic mathematics has declined still more. Those who still take it do so only because most colleges, as already noted, require some courses, though the precise content is not restricted to the traditional subject matter. Moreover, how appealing is the motivation - preparation for college - to the students? What normal fourteen-year-old really knows whether he is college bound? His parents may have definite plans, but these may go awry for many reasons. One certainly will be the child's reaction to high school work, and the traditional mathematics courses are not likely to induce a favorable reaction. Even at the time of graduation from high school, many youngsters are still uncertain about whether they will go to college. They often decide to do so only because of external pressures or circumstances. They may imitate what friends do or follow the path of least resistance. To all of the arguments for the traditional high school mathematics curriculum the advocates of the new math added that our country's need for mathematically sophisticated manpower required the secondary school curriculum to bring students more quickly to the frontiers of research in pure and applied mathematics. In view of the sequential nature of mathematics, such an objective is preposterous. Whether or not the values claimed for mathematics are indeed present, certainly student readiness and capacity to appreciate these values should have entered into the considerations of the framers of the curriculum. The results of their failure to consider effectiveness have been notoriously poor. If we may judge by the results on the Scholastic Aptitude Tests, the performance of the best group - the college-bound - has been getting poorer year by year. More students are going on to college, but the high schools should be succeeding with all of these, at least. Additional evidence of ineffectiveness is the fact that remedial mathematics is now the biggest problem that the colleges face. In the two-year colleges 40 percent of the students taking mathematics are enrolled in remedial courses. One cannot dismiss the traditional material without taking into account that some high school students will eventually use mathematics professionally. Since students at the high school stage do not know whether or not they will need mathematics later, everyone, so the argument goes, should be required to take it. However, only about 5 percent of the high school students will ever use mathematics and their needs should not dominate. They can be met in a manner that will be suggested later. Is it, then, a mistake to require mathematics of all academic high school students? Not at all. The arguments against the teaching of mathematics are against the kind of mathematics that has been taught and against the justifications traditionally and currently given. At least in the United States, which is the only society that has sought to teach mathematics to all students, the vital material and the reasons for teaching it have been ignored entirely. The framers and reformers of the curriculum have approached the problem from the wrong direction. The values and objectives have been those that professors enjoy or respect. Before considering an alternative approach, let us note a critical distinction between elementary and high school mathematics education. Everyone has to know some arithmetic and a few elementary geometrical facts merely to get along in life. Arithmetic has basic practical value, much as reading and writing have. However, as we have already pointed out, this is not true of algebra, geometry, and the other high school subjects. Whereas we are obliged to present the elementary school material, we are freer to choose the material to be taught on the high school level. What are the objectives of an academic high school education? The courses in literature, history, science, economics, and foreign languages are intended to enable future adults to live more insightfully, wisely, and enjoyably. In short, they are an introduction to our culture. The mathematics courses should serve the same objectives. Hence, the mathematics we teach should be worth knowing for the rest of the lives of all students. It should contribute to a truly liberal arts education wherein students get to know not only what the subject is about but also what role it plays in our culture and our society. We must teach not just what mathematics is but what it does. Teachers will undoubtedly ask, "Where and what is the material that constitutes a liberal arts curriculum?" To an extent we answered this question when we considered the contents of a freshman liberal arts course (Chapter 6). Admittedly, the intellectual level of the high school program and the background that it presumes must be taken into account. But there is a mass of suitable material, much of which can be culled from diverse sources. A few examples may convince doubting Thomases. Elementary algebra deals with simple functions such as y = 5x, y = 3x2+6x, and the like. As purely mathematical expressions they are dull, devoid of interest; but they can be and are used to represent an enormous variety of motions, the motions of balls, projectiles, rockets, and spaceships. Motion on the moon, which exhibits striking contrasts with motion on the earth, provides an exciting theme. The motion of objects dropped into water is another readily understandable and interesting phenomenon. Further exploration of simple functions leads to the law of gravitation, and to remarkable calculations such as the mass of the earth and the mass of the sun. Functions are not merely symbolic expressions; they are laws of the universe, and they encompass the behavior of grains of sand and the most distant stars. Algebra can also be applied to the study of elementary statistics and probability. These techniques are used to obtain vital knowledge about the distribution of height, weight, intelligence, mortality, income, and other facts of interest and concern to every would-be educated person. The efficacy of medical treatments; the control of quality in production; and the prediction of future prices of commodities, population growth, and genetic traits such as susceptibility to diseases are achieved with the same tools. The reliability of conclusions reached on the basis of statistics and probability should also be taught. The well-known quip, "There are lies, damned lies, and statistics," should certainly be taken seriously in the study of statistics, because so many of the conclusions that are hurled at us are not supported by the data. The uses of high school geometry are manifold. The determination of the size of the earth, the distances of the planets from the sun, and the periods of the planets; the explanation of eclipses of the sun and moon; and the calculation of inaccessible lengths, such as the height of a building, the width of a canyon, or the distance across a lake - all, though utilizing only the simplest geometry, are remarkable feats. Equally accessible through geometry is the behavior of light. If a light ray travels from A to B (Figure 1) via reflection in a mirror, the path it takes, namely, the one for which angle 1 equals angle 2, is the shortest one it could possibly take. Thus the actual path APB is shorter than, for example, AQB. Moreover, since light travels in this situation with constant velocity, the shortest path requires the least time. In fact, in practically all situations light takes the path requiring the least time. Now light is surely inanimate. How, then, does it know to choose that path? Why does it seek "efficiency"? Here one can touch upon one of the grandest doctrines that man has proposed and in some centuries unquestionably affirmed. There is an order in nature, a design behind each phenomenon, and mathematical laws reveal and express that design. Who instituted that design? Perhaps best left unanswered, this question can be raised at the high school level. A beginning can be made toward teaching the power of mathematical abstraction. What the pure geometry of the reflection of light says is that if one wants to go from point A to line QP and then to B, the shortest path is the one for which angle 1 equals angle 2. Suppose now that QP is a railroad track and a station is to be built to serve the people of towns A and B. Where should the station be situated so that the total distance the people of A and B must travel to reach the station is a minimum? We have the answer. It is P. Suppose instead that a school or a telephone central is to be located along QP to serve towns A and B. Again, minimum distance or minimum telephone lines require that the school or central be placed at P. Thus, one geometrical theorem answers many different practical questions. Other uses of elementary geometry can deal with a description of the functioning of our solar system, geography, perspective in painting, the structure of atoms and molecules, architecture, elementary engineering, surveying, navigation, and even clothes design. If this cultural approach is extended to the third year of high school mathematics, wherein trigonometry is usually taught, many more values of mathematics can be demonstrated. When light travels from one medium to another, as from air to water, it changes direction; that is, it is refracted. The law of refraction can be simply stated in trigonometric terms. The importance of knowing this law is readily demonstrated because it is used in designing eyeglasses, and in understanding the functioning of the human eye and treating its diseases. The trigonometric functions enable man to analyze the pitch and quality of musical sounds, and this analysis is used in designing the telephone, radio, phonograph, and television. Of course, none of the above-mentioned applications calls for a knowledge of set theory. But teachers should be willing to make some sacrifices in behalf of 100 percent of the students. That mathematicians of the past were inspired by real problems and found the meaning of mathematics through them is beyond dispute. Equally beyond dispute is that applications to real problems are a pedagogical necessity. However, there is some question about which applications will be interesting and meaningful to students. Only experience will enable teachers to determine the best choices. Fortunately, the choices are so numerous and so varied, and interest in the real world is so much more widespread than interest in abstract mathematics, that attractive applications can surely be found. At the very least, it is easier to arouse curiosity about real problems than about mathematics. Though authority can be found to defend almost any position, the words of some of the most famous educators of the past warrant reading. Friedrich Wilhelm Froebel (1782-1852), the founder of the kindergarten, stated: Mathematical demonstrations came like halting messengers. . . . On the other hand, my attention was riveted by the study of gravitation, of force, of weight, which were living things to me, because of their evident relation to actual facts. Aristotle asserted that there is nothing in the intellect that was not first in the senses. The use of real and, especially, physical problems serves not only to answer the question of what value mathematics has but also gives meaning to it. Negative numbers are not just inverses to positive integers under addition; they are the number of degrees below zero on a thermometer. The ellipse is not just a peculiar curve; it is the path of planets and comets. Functions are not sets of ordered pairs; they are relationships between real variables such as the height and time of flight of a ball thrown into the air, the distance of a planet from the sun at various times of the year, and the population of a country over some period of years. To rob the concepts of their meaning is to keep the bra nches of a fruit tree and throw away the fruit. There is another value to be derived from developing mathematics from real situations. One of the greatest difficulties that students encounter in mathematics is solving verbal problems. They do not know how to translate the verbal information into mathematical form. Under the usual sterile presentations in the traditional and new mathematics curricula, this difficulty is to be expected. On the other hand, if mathematics is not just applied to the real world but is drawn from the real world, as happened historically, its applicability is no longer a mysterious process. The difficulty students now experience in applying mathematics to real problems is very much the difficulty a French youth, who knows only French, would have in translating his thoughts into English. If, however, he is brought up in a bilingual region as are, for example, the French Canadians, he can certainly express himself in both languages. The inclusion of applications offers still another advantage. As we have noted, most of mathematics arose in response to the desire or need to solve real problems. As the teacher treats such problems, he can include an account of the historical background in which the problems were tackled and even include some biography of the men involved. If skillfully interwoven, rather than added as disconnected appendages, the history and biography will not only enliven the courses but also will teach an equally important lesson: Mathematics is produced by human beings responding to a variety of problems. It is a living body of ideas that developed over the centuries and grows continually. The defenders of the traditional material often rejoin that they do teach applications. Let us look at just a few of them. There are work problems such as the ditch digger's dilemma: "One man can dig a ditch in two days and another in three days. How much time will be required if both men dig it together?" Such problems create pointless work. Then there are tank-filling problems for students who have no swimming pools to fill. And there are the mixture problems: "How many quarts of milk with 10 percent cream and how many quarts of milk with 5 percent cream must be mixed to make one hundred quarts of milk with 50 percent cream?" Such problems are perhaps most useful to farmers who wish to fake the cream content of their milk. And we shouldn't neglect to mention the time, rate, and distance problems, such as up - and downriver travel, for students whose desire to go anywhere except out of the classroom will not be aroused. Some authors of algebra texts point to "truly physical" problems. For example, Ohm's law states that the voltage E equals the current I times the resistance R. In symbols E = IR. Calculate E if 1 = 20 and R = 30. But if the concepts and the use of the law are not explained to the students, the current doesn't drive any mental motors. The proposal that the first two or three years of secondary mathematics contain all sorts of genuine applications may seem radical, but a little perspective may correct this impression. In the seventeenth century mathematics courses comprised astronomy, music, surveying, measurement, perspective drawing, the design of optical instruments, architecture, and the design of fortifications and machines. In the intervening centuries some of these topics lost importance and were dropped from the mathematics curriculum. The expansion of mathematics itself and of knowledge generally has compelled educators to drop other topics. But the isolation of mathematics from all applications relevant to our times cannot be tolerated, even if the inclusion of applications necessitates covering less mathematical topography. Beyond the demands of pedagogy, we must recognize that mathematics did not develop independently of other human activities and interests. If we are compelled for practical reasons to separate learning into mathematics, science, history, and other subjects, let us recognize that this separation is artificial. Each subject is an approach to knowledge, and any mixing or overlap, where convenient and pedagogically useful, is to be welcomed. The need to relate mathematics to our culture has been stressed by Alfred North Whitehead, one of the most profound philosophers of our age, and a man capable of the most exacting abstract thought. In his essay The Aims of Education, written in 1912, Whitehead says: In scientific training, the first thing to do with an idea is to prove it. But allow me for one moment to extend the meaning of "prove"; I mean - to prove its worth.. . . The solution which I am urging, is to eradicate the fatal disconnection of subjects which kills the vitality of our modern curriculum. There is only one subject-matter for education, and that is Life in all its manifestations. Instead of this single unity, we offer children Algebra, from which nothing follows; Geometry, from which nothing follows. . . . Our course of instruction should be planned to illustrate simply a succession of ideas of obvious importance. In another essay of 1912, Mathematics and Liberal Education (published in his Essays in Science and philosophy), Whitehead goes further: Elementary mathematics. . . must be purged of every element which can only be justified by reference to a more prolonged course of study.... The elements of mathematics should be treated as the study of a set of fundamental ideas, the importance of which the student can immediately appreciate; every proposition and method which cannot pass this test, however important for a more advanced study, should be ruthlessly cut out.... simplify the details and emphasize the important principles and applications. In 1912 Whitehead was addressing himself to essentially the same curriculum that we teach today. The criticisms and positive recommendations still apply and, in fact, with all the more force because the high schools now teach a more diverse group. To present mathematics as a liberal arts subject requires a radical shift in point of view. The traditional approach presents those topics of algebra, geometry, intermediate algebra, and trigonometry that are necessary to further students' progress in mathematics per se. The new approach would present what is interesting, enlightening, and culturally significant, with the inclusion of only those concepts and techniques that will serve to further the liberal arts objective. No technique for the sake of technique should be presented in the first two or three years. In other words, the material should be objective-oriented rather than subject-oriented. But what do we do for the future professional user of mathematics? Admittedly a small but appreciable percentage of the students will become mathematicians, physicists, chemists, engineers, social scientists, technicians, statisticians, actuaries, and other specialists whose work requires mathematics. Of course these students, too, should know the cultural significance of mathematics. Moreover, students who are already inclined toward a specific career will certainly take an interest in mathematics if they see how the subject is involved. But if by pursuing the cultural objective we do curtail somewhat the technical preparation of those students who will need mathematics later, what can we do for them? Those who are convinced by the end of the eleventh year of schooling that they do wish to pursue mathematics for some professional use should be offered an optional technical course in the twelfth year. Because these students know that the career they intend to follow will require mathematics, they will be better motivated and, most likely, quite capable of rapid progress. They should be able in one year to acquire a considerable technical background that might well include far more than they may acquire in the present traditional curriculum. In fact, these students, mixed in at present with indifferent or poorly prepared high school students, don?t go far in the ninth and tenth grades, and in the usual eleventh-grade class they are bored by repetition intended for the poorer students. Beyond content there are many pedagogical considerations - such as the treatment of proof; getting students to enter into the discovery of results; the use of laboratory materials, among which for present purposes we may include the computer; and testing. However, these pedagogical problems lie beyond our present concern. Our discussion of content may be sufficient to indicate that all is not well in high school education. More relevant, in view of the notoriously poor results of mathematics education, is the question of why the basic content of the high school mathematics curriculum has remained fixed. The answer is that for many generations the mathematics departments of the universities and of most four-year colleges have taught the same subject matter to all prospective high school mathematics teachers. In fact, on the whole, the prospective teachers are taught the same subject matter that is taught to all mathematics students. The professors project their own values and interests; in the present case, in which courses are directed to prospective high school teachers, student needs do not count. To study the problems of high school education and to fashion courses that would enable prospective teachers to meet those problems would call for the full-time efforts of at least one professor in each college or university department. But the archenemy of all undergraduate education - the pressure to do research - precludes such attention. Though the professors of the schools or departments of education do teach prospective teachers how to teach, they are unable to counter the domination of content by the academic departments. Hence, their impact on curriculum is nil. Consequently, the high school teacher is limited in knowledge and restricted to goals and values set by mathematics professors and administrators. However much he may sense the poverty of the material he is teaching, he has neither the expertise nor the power to change the curriculum. Also, unsurprisingly, curriculum reform is not often welcomed by older teachers. Indoctrinated only in traditional, isolated, unmotivated mathematics, the teacher confronted with the challenge to teach more meaningful and more purposive material shrinks back in fear. When asked, for example, to teach applications involving acceleration, he throws up his hands and argues that the concept is too difficult for the students or that it presupposes a knowledge of physics. However the students and the teachers ride in automobiles. To get a car moving and to stop it one must accelerate or decelerate. A moving automobile rarely travels at a constant speed but accelerates and decelerates constantly; hence, the notion is certainly intuitively familiar. The students, who are still young and open to ideas, would not find the concept difficult - nor would the teachers, had they not been conditioned to concentrate solely on mathematics. In fact, it is even vital to teach acceleration. It is too rapid deceleration that causes many passengers in automobiles to go through the windshield. To require teachers to teach applications does not impose any real hardship. Actually, very little knowledge of science is required to teach applications of high school mathematics; every teacher can acquire it. Prospective mathematics teachers should certainly study some science during their undergraduate days. A rich, vital, and attractive high school curriculum can be fashioned. Reform should be led by experienced, knowledgeable, broadly educated professors. Specialists might serve as consultants but certainly should not lead. It would be equally important to have cultured nonmathematicians participate. Their judgment as to what the future citizen would find valuable should take precedence over that of the specialists. High school mathematics that consists of algebraic techniques, proofs of geometrical theorems, and mazes of abstract concepts and symbols will continue to reduce the students to a state of bafflement and loathing. At present, many conclude that they do not have mathematical minds when, in fact, they have not had informative, inspiring, and stimulating mathematics education. Though sincere teachers have been imparting what they themselves have learned - skills and proofs - unconsciously and sometimes consciously they work hocus-pocus on their students, presenting and repeating opaque formulas, sometimes to the admiration but almost always to the bewilderment of their charges. The current high school curriculum is a threat to the life of mathematics. If the schools do not offer a more rewarding and meaningful curriculum, then rather few students will take mathematics. Requirement for admission to college, which is the main factor keeping academic mathematics alive in the secondary schools, may not sustain high school mathematics in the future. Almost certainly, it will cease to be a requirement. If high school mathematics is not made more attractive and significant to the student, mathematics as an integral part of general education will die quietly and its soul will rise to heaven through an atmosphere of irrelevance. C&O 发表于 10:48 | 阅读全文 | 评论(0) | 引用trackback(0) | 编辑 另类婚姻观 - 2003-05-27 07:16 另类婚姻观 A．延长爱情的唯一方法，是推迟结婚时间。 B．喂，小伙子，结婚是仓促不得的，除非你的女朋友已经怀孕了。 C．关于结婚这件事，我对一般的朋友会表示祝贺，对最好的朋友则表示同情。 D．不论从哪种角度来看，爱情只是一种脑力劳动，而婚姻则是一种体力劳动。 E．爱情仿佛打桥牌，全靠算计；婚姻仿佛打麻将，全靠运气。 F．人生的滋味大致可概括为：酸甜苦辣。仔细品味，爱情的滋味又酸又甜，婚的滋味又苦又辣。 G．女人再婚是为了赌气，男人再婚是为了碰运气。 H．结婚的历史意义在于：连结你们的那条"红线"（千里姻缘一线牵）从此会变一条绳子。 I．当女人走投无路时，她会和一个男人结婚；当男人走投无路时，一个女人会她离婚。 J．女人具有水的属性，男人具有火的属性，婚姻就是试图使水火相容的一种尝。 K．爱情使人睡不着，婚姻使人打瞌睡。 L．爱情的烦恼多，婚姻的麻烦多。 M．二人世界太大，一人世界太小，据说一个半世界最好。 N．没有婚姻的爱情使你终生遗憾，没有爱情的婚姻使你天天遗憾。 O．婚姻可是一部《神曲》。不过，这部神曲与但丁那部刚好相反。它的上册"天堂"，中册是"炼狱"，下册是"地狱"。 P．爱是浪漫主义，结婚是现实主义，独身是超现实主义。 Q．娶漂亮女人做老婆是一种艳福，娶不漂亮的女人做老婆是一种幸福。 R．现代人谈话恋爱的时间缩短了，认识两三个月就可以做爱；婚姻的时间也相缩短了，结婚三四年就该闹离婚了。 S．医学专家发现：爱情会刺激人体分泌一种使人兴奋的化学物质氨基丙苯，婚会刺激人体分泌一种使人镇压静的化学物质内啡肽。假如我们在热恋之中能打几针"内啡肽注射液"，那就会少干一些傻事；假如我们在结婚之后能服几粒"氨基丙苯胶囊"，那就会多留一些激情。遗憾的是，这两种药至今尚未问世。 T．可能是"爱情鸟"飞错了地方，正当谈婚论嫁的大男大女"不谈爱情"、"懒得结婚"，而少男少女们的"早恋"，有夫之妇、有妇之夫的"婚外恋"却搞得热火朝天。 U．"单亲家庭"的增加证明：城里的人已经逃出来了。"单身贵族"的增多证明：城外的人并不想冲进去。由此可见，婚姻即将沦为一座"空城"。 V．据说，婚姻是一双鞋，舒不舒服只有脚知道。对这双鞋我所知道的是―― 1．一只是40码的，另一只是37码的。 2．这双鞋只能容纳两只脚，容不得第三者"插足"。 3．大凡不合脚的鞋，穿起来不那么舒服，脱起来也没那么容易。 W．男人结婚前易患"相思病"，婚后易患"妻管严"。人类社会的进程曾经经历"母系社会"、"父系社会"，如今又进入第三个阶段"妻系社会"。 X．不幸的婚姻天天吵架，幸福的婚姻隔一天吵一架。 Y．随着离婚率的持续上升，现在朋友见了面，最流行的问候语已由"吃了吗"改为"离了吗"，有些前卫分子甚至会问："离第几次了？"而最酷的回答据说是："我也想不起第几次了。" Z．"婚"字的一半是女人，一半是昏。如果让测字先生来解释，"婚"字的含义很可能就是女人昏了头。 C&O 发表于 07:16 | 阅读全文 | 评论(0) | 引用trackback(0) | 编辑 踏遍青山人未老 - 2003-05-27 07:09 清平乐　毛泽东 会昌　一九三四年夏 东方欲晓，莫道君行早。 踏遍青山人未老，风景这边独好。 会昌城外高峰，颠连直接东溟。 战士指看南粤，更加郁郁葱葱。 C&O 发表于 07:09 | 阅读全文 | 评论(0) | 引用trackback(0) | 编辑 成语和学习――学而优则仕 - 2003-05-27 07:06 23．学而优则仕 成语“学而优则仕”源 于《论语・子张》：“子夏曰：‘仕而优则学，学而优则仕。语中的“优”字是指有余力，这个成语的意思是学习了还有余力就去做官。后指学习成绩优秀然后提拔为官。 我认为 “学而优则仕”的 “学”字不是学习的意思，而是做学问的意思，虽然做学问少不了学习。看一下与该成语在一起的话就可以说明我的想法。“学而优则仕”本来的意思是做学问时还有余力的话，那么去做官吧。“仕而优则学”的意思恰好把学问和官的位子互换。为此我们可以得出，做学问和做官是“平起平坐”的。事实上，老百姓对于好官和有学问的人都是很尊重的。 然而，后来的 “学而优则仕”就变味了，学习不知为什么变成了戴乌纱帽的敲门砖。一旦当了官，书中的“颜如玉”和“黄金屋”也就显形了。进入了知识经济时代，掌握知识的作用不仅仅可以为官，而且许多知识可以 “学而优则富”了。于是知识也被一些人按着功利思想分了类和级别，例如，强势知识、弱势知识等等。家长们为了望子成龙，或者为了孩子将来不至于囊中无钱，死逼孩子往肚子里填知识。看看大学现在的热门专业，计算机、经济、企业管理，那一门不是冲着现在可以挣钱较多的行业。 我是一个做学问的教书匠，自然希望有才华、有能力的学生将来也做学问。但是我也明白不可能有的人都做学问，那样话大家就不会有饭吃了。做学问、做官、挣钱的人都是有出息的，都可以成为成功人士。人各有志，我们的学生们选择哪一行都是对的。只是要考虑适合和不适合的问题。 有人认为，许多人之所以找不到工作，并非没有知识，而是学而无用。哪怕你有硕士、博士文凭，只要你掌握的知识在现实社会中无用，你就可能找不到工作。我不知道什么知识会在现实社会中无用，也不知道有多少上过大学的人后悔他在大学里学了那么多无用东西，只想谈谈所谓有用的知识。 首先，我们说说高中里学的知识。上高中的学生少有不想上大学的，现在高中的老师教给学生的东西不管是不是知识，对上大学应该是有用的。但是，每年都有很多高中生考不上大学。 再说热门专业知识，就说现在很多人想学的计算机知识吧。如果掌握计算机专业知识的人多到一定程度，那么在他们当中一定会有人找不到必须用计算机专业知识干活的职业。注意，这里谈的是专业知识而不是属于一般常识的东西。 能否找到工作的关键不在于所学知识的所谓有用，至少“学而优则仕”是一个原因。 对于一个人来讲，他所掌握的知识是双刃剑。一方面，所谓艺不压身，一个人学过、研究或用过的东西总是有用的，哪怕是备而不用。另一方面，如果不能灵活运用知识且被所学知识禁锢住，那么最大的障碍便是这些已学的知识了。 C&O 发表于 07:06 | 阅读全文 | 评论(0) | 引用trackback(0) | 编辑 分页: 第一页 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]