×óÓÒ»¥²«¡ª¡ªÀÏÍçͯ

来源: BlogBus 原始链接: http://www.blogbus.com:80/blogbus/blog/index.php?blogid=1388&pg=11&cat= 存档链接: https://web.archive.org/web/20040510125404id_/http://www.blogbus.com:80/blogbus/blog/index.php?blogid=1388&pg=11&cat=

×óÓÒ»¥²«¡ª¡ªÀÏÍçͯ ¹ØÓÚÈËÉú¡¢Éç»á¡¢ÊÀ½çµÄ˼¿¼¡£ÓÈÆäÊÇ×éºÏÊýѧ¡¢Êýѧ½ÌÓý¡¢Ë¼¿¼ÎÊÌâµÄ·½·¨µÄÌÖÂÛ¡£ Ê×Ò³ Why The Professor Can (16) ·½·¨ (28) ÃîÎÄʰȤ (33) ÍøÕ¾ÈÕÖ¾ (5) Ó°ÊÓÒôÀÖ (8) Êýѧ (3) ¾­µäÐÀÉÍ (11) С˵Á¬ÔØ (34) ÆÀÂÛ (8) 2004 Äê 5 Ô 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 ×îºó¸üРɳɮÈռǡª¡ªÐ㶺ǰÄê9ÔÂ30ÈÕ ´Ë¼äµÄÉÙÄꡪ¡ªÇǷ壨£É£©£ºÒ»´ÎÀºÇòÈü ɳɮÈռǡª¡ªÐ㶺ǰÄê9ÔÂ28ÈÕ ´Ë¼äµÄÉÙÄꡪ¡ªÇǷ壨£É£©£º¹ØϵÌú ɳɮÈռǡª¡ªÐ㶺ǰÄê9ÔÂ27ÈÕ ´Ë¼äµÄÉÙÄꡪ¡ª¹ù¾¸£¨£É£©£ºÍ¼Êé¹Ý ɳɮÈռǡª¡ªÐ㶺ǰÄê9ÔÂ14ÈÕ ´Ë¼äµÄÉÙÄꡪ¡ª¹ù¾¸£¨£É£©£ºÅóÓÑ ÇÔϲ¼´Íµ»¶ ·çÓêÔٷýðÎÄÃ÷ ×îÐÂÆÀÂÛ Àä¾þÉ¢ÊÆ : ×óÓÒ»¥²«¾ÍÊÇ×Ô¼º. ½ðÓ¹ : ¶àÉÙ·çÇ飬 . ä¸ä¸Á÷Ë® : ÕâƪÎÄÕ²»³¤£¬µ«. darkevan : ¸ã¸öÁ´½Ó²»¾ÍÐÐÁË. ivy : that is a good o. mojaves : ÓÃÁËÎÒ×îϲ»¶µÄº£. isgaryzhu : ÄãÊǼÍÏþá°¾ç×éµÄ. ´æµµ ÄÚÈË E-Mail to ÀÏÍçͯ ÎÒµÄÁ´½Ó ·ÖÒ³: µÚÒ»Ò³ [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] ³ÉÓïºÍѧϰ¡ª¡ª×íÎÌÖ®Òâ²»ÔÚ¾Æ - 2003-05-27 07:01 22£®×íÎÌÖ®Òâ²»ÔÚ¾Æ ¡¡¡¡³ÉÓï¡°×íÎÌÖ®Òâ²»Ôھơ±Ô´ÓÚËγ¯Å·ÑôÐ޵ġ¶×íÎÌͤ¼Ç¡·£º¡°Ì«ÊØÓë¿ÍÀ´ÒûÓÚ´Ë£¬ÒûÉÙéü×í£¬¶øÄêÓÖ×î¸ß£¬¹Ê×ÔºÅÔ»×íÎÌÒ²¡£×íÎÌÖ®Òâ²»Ôھƣ¬ÔÚºõɽˮ֮¼äÒ²¡£¡±Òâ˼ÊÇÕæÒâ²»ÔںȾƣ¬¶øÔÚÓÚÐÀÉÍɽ¾°¡£±ÈÓ÷±¾Òâ²»Ôڴ˶øÊÇÁíÓÐËùͼ¡£ ¡¡¡¡µ±Ç°£¬È«Éç»á¶¼ÔÚ¹ØÐÄÎÒÃÇÏÖÐеĽÌÓý×´¿öºÍ½ÌÓý¸Ä¸ïµÄÎÊÌ⡣ѧУµÄ½ÌʦºÍѧÉú×ÔÈ»»Ø±Ü²»ÁË˼¿¼ÕâЩÎÊÌâ¡£×î½ü£¬ÓÐÒ»·Ýµ÷²éÏÔʾ£¬´ó¶àÊýÖÐѧÉú¸Ðµ½ÔÚѧУÀïѧµÄ֪ʶ¡°Ëùѧ·ÇËùÓᱡ£Ò²ÐíÕâ·Ýµ÷²é²»ÄÜÍêÈ«·´Ó³ÖÐѧÉúµÄÕûÌåÏë·¨£¬µ«ÊÇÖÁÉÙÄÜ·´Ó³Ò»Ð©È˵Ŀ´·¨¡£²»´í£¬Ñ§ÉúÔÚѧУµÄÒ»¸öÖ÷ÒªÈÎÎñÊÇÕÆÎÕ½ñºóѧϰ¡¢Éú»îºÍ¹¤×÷ÖÐÐèÒªµÄ֪ʶ¡£Ê²Ã´Êǽñ±ØÐèµÄ֪ʶ²»ÊÇÒ»¸öÁ½¸öר¼Ò»ò²¿·ÖѧÉúËùÄܿ϶¨µÄ¡£ÓÈÆäÊǽñºóÉç»áÌṩÎÒÃǵÄÑ¡Ôñ»ú»á½«·Ç³£¹ã·º£¬¶øÇÒÒ»¸öÈËÒ»ÉúËùÄÜÑ¡ÔñµÄ»ú»áÒ²²»ÊÇÒ»Á½¸ö£¬ÓÐЩ֪ʶ¿ÉÄÜÊÊÓÃÓÚһЩ¹¤×÷£¬¶øÁíÍâһЩ֪ʶ¿ÉÄÜ»áÊÊÓÃÓÚÆäËü¹¤×÷¡£³ýÁË´«ÊÚËùνÓÐÓõÄ֪ʶ֮Í⣬ѧУ´«Êںܶà֪ʶµÄÄ¿µÄÊÇ ¡°×íÎÌÖ®Òâ²»Ôھơ±£¬¶øÔÚÓÚÅàÑøѧÉúµÄÐÞÑøºÍÄÜÁ¦¡£ÎÒÃDZØÐëÇåÐѵØÈÏʶµ½£¬Ñ§ÉúÔÚѧУµÄÁíÒ»¸öÖ÷ÒªÈÎÎñÊÇÌá¸ß×Ô¼ºµÄÐÞÑøºÍÄÜÁ¦¡£Ò»¸öÈ˸÷·½ÃæµÄÐÞÑøºÍÄÜÁ¦ÔÚÇóѧʱÆÚ¾ÍÓ¦¸ÃÒ»µãµãÅàÑø£¬Ò»µãµãÌá¸ß¡£Åà¸ùÔÚ¡¶Ì¸¶ÁÊé¡·ÖоÍ˵¹ý£º¡°¶ÁʷʹÈËÃ÷ÖÇ£¬¶ÁʫʹÈËÁéÐ㣬ÊýѧʹÈËÑÏÃÜ£¬ÎïÀíѧʹÈËÉî¿Ì£¬Â×ÀíѧʹÈËׯÖØ£¬Âß¼­Ñ§¡¢ÐÞ´ÇѧʹÈËÉƱ磻·²ÓÐËùѧ£¬½Ô³ÉÐÔ¸ñ¡£¡±½²µÄ¾ÍÊÇÕâ¸öµÀÀí¡£ ¡¡¡¡ÔÚÖÚ¶àµÄÄÜÁ¦ÖУ¬Ñ§Éú×ÔÎÒѧϰÐÂ֪ʶµÄÄÜÁ¦Ó¦¸ÃÊǺÜÖØÒªµÄÒ»¸ö¡£´ÓÇ°ÃæÌáµ½µÄµ÷²é¿ÉÒÔ¿´³ö£¬´ó¶àÊýÖÐѧÉúÖªµÀÔÚѧУѧµÄ֪ʶÊDz»¹»ÒÔºóʹÓõġ£ÕâÒ»µã²î²»¶àÊÇËùÓÐÀ뿪ѧУµÄÈ˵Ĺ²Ê¶¡£Ò²¾ÍÊÇ˵һ¸öÈË×ß³öУÃźó×Ü»áÓöµ½Ñ§Ï°ÐÂ֪ʶµÄÎÊÌ⣬²»ÊÇÓÐÄÇôһ¾ä»°Â𣺡°Êéµ½ÓÃʱ·½ºÞÉÙ¡±¡£Æäʵ£¬¶Ô´ó²¿·ÖÈËÀ´Ëµ£¬µ±½ñµÄʱ´úµ¹²»ÊǺÞÊéÉÙµÄÎÊÌ⣬¶øÊÇÓÐÊé²»ÖªµÀÈçºÎÈ¥ÕÒ¡¢ÈçºÎ¶Á¶®µÄÎÊÌâ¡£»»¾ä»°Ëµ£¬¾ÍÊÇ×ÔÎÒѧϰж«Î÷µÄÄÜÁ¦²»¹»¡£ ¡¡¡¡ÅàÑøѧÉú×ÔÎÒѧϰÄÜÁ¦µÄ·½·¨»áÓв»ÉÙ£¬µ«ÊÇÄÄÒ»ÖÖ·½·¨¶¼Àë²»¿ªÖªÊ¶£¬¶¼»áÈÃѧÉúѧµ½Ò»Ð©ÖªÊ¶¡£ÕâÑùµÄ»°£¬¼´±ãÊÇÕâЩ֪ʶÍêÈ«ÎÞÓÃÓֺηÁ? ¡¡¡¡ÏÖÔÚÎÒÃÇÔÙÀ´Ì¸Ì¸Êýѧ֪ʶ¡£ÔÚÎÒ¹úµÄ³õ¸ßÖУ¬ÊýѧÄÚÈÝËƺõÉîÁËÒ»µã£¬¹ãÁËÒ»µã¡£Èç¹û¿¼Âǵ½ÎÒ¹úÓкܶàѧÉúÉϲ»ÁË´óѧµÄÏÖʵ£¬ÕâÑù×öÒ²ÊDZØÒªµÄ¡£Ò²ÐíÒ»¸ö¸ßÖбÏÒµÉúÒÔºóÓÀÔ¶²»»áÓõ½ËûËùѧ¹ýµÄһЩÊýѧ֪ʶ£¬µ«ÊÇѧÁËÕâô¶àµÄÊýѧ×ÜÓ¦¸ÃÄÜÅàÑøËûµÄÊýѧÐÞÑøºÍ˼άÄÜÁ¦°É£¬Æ©ÈçÂß¼­Ë¼Î¬ÄÜÁ¦¡£ ¡¡¡¡¼ÇµÃÔÚµçÊÓµÄÒ»´ÎÌåÓý½ÚÄ¿ÖÐÓÐÒ»¸öѧ¾­¼ÃµÄÅ®´óѧÉúÙ©Çǵ¤µÄÊ¡£ËýÈÏΪNBA½ñºó²»»áÔÙ³öÏÖÇǵ¤Ê½µÄÈËÎÀíÓÉÈçÏ£ºÇǵ¤µÄ³É¹¦²»½öÔÚÓÚÇǵ¤±¾Éí£¬¶øÇÒ»¹ÔÚÓÚÇǵ¤Ê±´úµÄ¾­¼Ã×´¿ö¡£Ò²Ðí£¬NBA»¹»áÓÐÏóÇǵ¤µÄÌì²Å³öÏÖ£¬µ«ÊÇËû²»¿ÉÄܳÉΪÇǵ¤Ê½µÄÈËÎÒòΪÇǵ¤Ê±´úµÄ¾­¼Ã×´¿ö²»»áÔÙÓÐÁË£¬ÎªÊ²Ã´Çǵ¤Ê±´úµÄ¾­¼Ã×´¿ö²»ÄÜÔÙÏÖ£¬ËýûÓÐ̸¡£´Ëʱ£¬µçÊÓÀïµÄÆäËûÈËÎï´óΪÅå·þ¡£µçÊÓ±¨½éÉܸýÚÄ¿µÄÎÄÕÂÒ²¸Ð¿®ËýµÄÉƱ硣Ëý´Ó¾­¼Ã×´¿ö½Ç¶ÈµÄÙ©·¨ÊÇÓÐÐÂÒâµÄ£¬ÖµµÃÔÞÉÍ¡£µ«ÊÇÓеãÂß¼­Ë¼Î¬ÄÜÁ¦µÄÈ˶¼²»»á±»Ëý˵·þ£¬ÒòΪûÓгä·ÖÀíÒÔ±íÃ÷Çǵ¤Ê±´úµÄ¾­¼Ã×´¿ö²»ÄÜÔÙÏÖ£¬³ý·ÇÄãÏàÐÅÕâÒ»µã¡£ ¡¡¡¡±ØÐë³ÐÈÏ£¬Ä¿Ç°Ñ§Ð£Àï´«ÊÚµÄ֪ʶÓÐЩÊDz»ºÏÊʵģ¬ÓÐЩÀÏʦÍêÈ«²»ÖªµÀÈçºÎͨ¹ý´«ÊÚ֪ʶÅàÑøѧÉúµÄÄÜÁ¦¡£Èç¹ûÄܱæ±ð³öʲôÊÇ֪ʶÀ¬»øʲô²»ÊÇ£¬Ñ§ÉúÃDz»¾Í¾ßÓÐÁËÒ»¶¨µÄÄÜÁ¦Âð?ÀÏʦ²»ÖªÈçºÎÅàÑøѧÉúµÄÄÜÁ¦£¬Ñ§Éú×Ô¼º¾Í²»ÄÜÊÔÊÔÂð?ÈçºÎÔÚ²»ÀûµÄ»·¾³Ï¾¡¿ÉÄܵØÄ¥Á·×Ô¼º²»Ò²ÊÇÒ»ÖÖÄÜÁ¦Âð? C&O ·¢±íÓÚ 07:01 | ÔĶÁÈ«ÎÄ | ÆÀÂÛ(0) | ÒýÓÃ(trackback0) Why The Professor Can't Teach¡ª¡ªµÚÆßÕ¡¡´¿´âµÄÊýѧ¼Ò - 2003-05-27 06:53 CHAPTER 7 The Undefiled Mathematician. The most vitally characteristic fact about mathematics is, in my opinion, the quite peculiar relationship to the natural sciences, or, more generally, to any science which interprets experience on a higher than a purely descriptive level. ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡John von Neumann Until about one hundred years ago all mathematicians would have accepted von Neumann?s conception of mathematics. Certainly the three men whom mathematicians nominate as the greatest of all time - Archimedes, Newton, and Gauss - did more scientific than mathematical work and in fact justified their mathematical investigations by mentioning or describing the applications that warranted the mathematical research. Even those whom many cite as pure mathematicians - Carl Jacobi, Karl Weierstrass, and Bernhard Riemann - not only applied mathematics, but pursued mathematics proper to clarify, rigorize, and extend the theory and technique they already knew to be applicable. This is not to deny that some of these men engaged in subjects, such as the theory of numbers, whose attraction is primarily aesthetic value or intellectual challenge; but one need only count the years they devoted to purely aesthetic subjects as opposed to those bearing on science to determine which research they regarded as more important. The situation is quite different today. Though the greatest mathematicians of recent times - Hermann Weyl, David Hilbert, Felix Klein, and Henri Poincar¨¦ - would have endorsed von Neumann's characterization of mathematics, now only about one out of ten mathematicians devotes himself to problems of the physical and social sciences, and many of these are employed in governmental and industrial laboratories. Among professors, about 5 percent do applied work. The rest are totally ignorant of science and do not undertake any problems bearing on it. The days when mathematicians saw the hand of God in the motion of the planets and the stars are gone. Especially in view of the facts that science and technology have expanded at least as much as mathematics and that social and biological problems are now being tackled mathematically, this reversal calls for an explanation. In part we have already given it. Professors are expected to publish. The older applied fields - mechanics, elasticity, hydrodynamics, and electromagnetic theory- have been explored for one, two, or three centuries and the outstanding problems are no longer simple. The newer applied fields - quantum mechanics, magnetohydrodynamics, solid state physics, meteorology, physical chemistry, and molecular physics - presuppose an extensive background in physics. As for the social and biological sciences, these are more complicated, and so far successes have defied the best brains. What, then, should professors, especially young people who have yet to earn rank and tenure, publish? The obvious answer is to pick some specialty in pure mathematics and to invent problems that can be solved. Since the editors of the journals come from the same milieu as the professors, this artificial research is as readily publishable today as are the most profound papers of mathematical science. What has this alteration in the nature of research to do with pedagogy? The answer is that most mathematics professors no longer teach either the uses of mathematics in science nor how to apply mathematics to scientific problems. Perhaps the best example of the detachment of mathematics from science is furnished by the teaching of calculus. This subject is the crux of applied mathematics, and next to the liberal arts course it is the one that serves the most students. Prospective engineers, physical and social scientists, actuaries, technicians, and medical and dental students take it to learn how to apply the subject. How then do the professors teach calculus? During the first four decades of this century the calculus course was a series of mathematical techniques taught mechanically and imitated by the students. The students worked but didn't have to think. Of course, this type of course was neither very helpful nor enlightening to the students, but in view of the level of mathematical knowledge in the United States it was about as good as could be expected. As more students flocked to college, and as a college education became a prerequisite for a good job or a professional career, the meaninglessness of the calculus course became more apparent. Though the professors had become more knowledgeable in mathematics proper, they were not prepared to teach a calculus course suited to the interests and needs of the students. The subsequent squirming and twisting reveal the modern professor's evasion of his obligations. Until about 1945 mathematics students took analytic geometry before calculus. Analytic geometry deals with a new and vital idea, the coordination of curve and equation, an idea that is used extensively in calculus. To "improve" the calculus course mathematics professors decided to start students with calculus and to teach the requisite analytic geometry as it was needed. Analytic geometry consequently got short shrift. This consolidation also meant asking the student to learn two major techniques simultaneously. Moreover, since the study of analytic geometry obliges students to utilize algebra and trigonometry, when they took analytics before calculus they were better prepared for the necessary uses of these tools in calculus. Subordinating analytic geometry to a topic in the calculus course deprived students of a sorely needed background. Most, therefore, did poorly. The professors "saw" the remedy for this trouble. They decided to incorporate more algebra and trigonometry, along with the analytic geometry, in the calculus course. This move proved to be still more disastrous, and the professors backtracked. Now the algebra, trigonometry, and analytic geometry are packed into a one- or two-semester course that is called precalculus - a nice semantic device to avoid admission of the original error. Some professors took another tack. In addition to including algebra, trigonometry, and analytic geometry, they adopted a rigorous approach to calculus. That is, they included the theory as well as the technique. This move, they evidently believed, would make the calculus course understandable. But the theory of the calculus is highly sophisticated. The best mathematicians, from Newton and Leibniz, who worked in the late seventeenth century, to Cauchy, who worked in the early nineteenth century, struggled to understand the logical foundation of the calculus, and Cauchy was the first to make the proper start. A sound foundation was not achieved until Karl Weierstrass, fifty years after Cauchy's breakthrough, cleared up the mess. Certainly, then, the theory is not easy for beginners to grasp. One may be sure that the very same teachers who believe that students beginning calculus can absorb a theoretical foundation would have been swamped, in their own student days, by such a presentation. Nevertheless, having finally grasped the theory after some years of study, they forgot their own experiences and acquired a missionary zeal to spread the light. The proper pedagogical approach to any new subject should always be intuitive. The strictly logical foundation is an artificial reconstruction of what the mind grasps through pictures, physical evidence, induction from special cases, and sheer trial and error. The theory of the calculus is about as helpful in understanding that subject as the.theory of chemical combustion is in understanding how to drive an automobile. This approach through theory, which had its heyday in the mid-1960s, had to be abandoned. The prewar teaching of pure technique was not successful; the inclusion of algebra, trigonometry, and analytics was no more so; and the inclusion of rigor was soon found to be a disaster. What measure could the mathematicians adopt? Since the precalculus course, when instituted, took care of the elementary material needed for calculus, the obvious move was to include material beyond calculus. Bits of linear algebra, vector analysis, differential equations, and other topics (some of which have no relevance to calculus) were therefore included in the calculus course, which is now a hodgepodge of topics and a m¨¦lange of unrelated techniques. (See also Chapter 10 on texts.) In all these maneuvers the professors have avoided the one measure that would make the calculus course meaningful and serve the purpose for which it is intended - namely, to make it the introduction to applied mathematics. Mathematics proper and calculus especially are mazes of symbols and manipulations of symbols. As such, these have no meaning or purpose. They are the shadows of substance and have as much meaning as the notes of a musical score to one who cannot hear the composition they describe. The symbols have no life, but properly interpreted they can tell us about the vital forces that affect almost every aspect of our lives. Only the applications supply meaning and motivation. Since most calculus students will be engineers or physical scientists and intend to use calculus, what better insight could they be given than examples of where calculus achieves results? Calculus offers excellent opportunities not oniy to apply mathematics but also to show how physical arguments suggest deep mathematical results. For example, we know that a ball thrown into the air rises and then falls. At the highest point in its path the velocity must be zero, else it would continue to rise. Because the velocity is the rate of change of distance with respect to time, what this physical happening suggests is that the rate of change of one variable with respect to another must be zero at the maximum value of the first variable. This is a basic theorem of calculus. There is, however, an obstacle to the introduction of physical problems that might supply motivation, meaning, and application: Most mathematics professors know no science and will not extend themselves to learn it. Those few who know enough to present the simplest physical applications fear the questions that may ensue. Many professors realize that calculus proper is dull and meaningless but, not prepared to offer real applications, they put on a pretense of doing so. They assign the following typical problem: Find the velocity of an object that moves with an acceleration of a To find velocity knowing the acceleration is indeed an application of calculus, but what object in this universe moves with the acceleration stated? Perhaps a drunken driver. Some professors are more realistic. It happens that objects moving near the surface of the earth, if one neglects the resistance of air, are subject to a downward acceleration of thirty-two feet per second each second. Hence, it does make some sense to pose problems of motion involving this acceleration. However, such problems are simple and do not exhibit the power of the calculus. Motion in a vacuum should be followed by more realistic problems involving motion in an atmosphere. The parachutist who could not rely upon the resistance of air would not, after one drop, have to rely upon anything. Perhaps the professors who confine their applications to motion in a vacuum are preparing students for life on the moon, which has no atmosphere, while convin¨¨ing them that life on earth is intolerable. Calculus texts offer other "real" applications. A man six feet tall is walking away from a street light at the rate of five feet per second; the problem asks how fast the man's shadow is lengthening when he is ten feet away from the light. The problem deals only with the shadow of reality. Professors also introduce problems in which physical terms such as "center of gravity" and "moment of inertia" are used. But the physical meanings of these concepts and their uses are not taught. The consequence is that the gravity of these problems produces moments and even hours of inertia in the students. Of course, curiosity might induce some people to solve any problem. But curiosity not only kills cats; it kills interest in mathematics courses that pose pointless problems. To teach calculus without real applications is to ask people to sit down at a table set for a dinner where no food is served, or to teach grammar but never to mention literature. Why are professors content to teach artificial, dull and pointless applications? Such problems have been in calculus texts for fifty or more years. The professors learned how to solve them when they were students. Why bother to dig up new and more significant ones and incur much more work in learning to present them if there is no pressure to improve the course? Surely it is boring to repeat the same deadly material year after year; but then all of teaching is a chore to be disposed of as quickly as possible. The deficiencies in the calculus course are exemplary of a glaring deficiency in the entire mathematics program, graduate and undergraduate. Though the major reason that students take mathematics is to use it, only a few undergraduate and graduate schools offer applied mathematics. There are courses and texts that are titled "Applied Mathematics," but these are pitiful. They offer mathematics that can be applied, for example differential equations, but at best they mention where the topics are applied. They omit the problem of analyzing physical phenomena to determine which factors or features can be neglected and which must be incorporated in the mathematical formulation; and they fail to teach at all the process of translating physical facts into mathematical language. Since no physical problems are treated, the payoff - what one learns through the mathematics about physical phenomena - is missing. Moreover, most of the texts contain tidbits from various mathematical areas used in applications but no one topic is pursued in depth. If these texts make any impression on the students it is only to bewilder them. They are shifted from topic to topic so quickly that nothing sticks. One is reminded of the whirlwind tours of Europe which cover ten countries in ten days. The tourists return home uncertain as to whether the Eiffel tower is in Paris or Prague. Mathematicians are, of course, aware of the existence and importance of applied mathematics, and they are sensitive to the charge that they are neglecting it. They justify their purely mathematical offerings on the ground that they are teaching students how to build models for the solution of real problems. As new physical problems are tackled, presumably all the applied mathematician or scientist will have to do is run through his files and select the model that fits his problem. But this type of model-building is a waste of time, and advocacy of purely arbitrary mathematical creations reveals ignorance of what applied mathematics involves. God may have designed the world mathematically but evidently He did not intend to make that design readily accessible. Mathematics is not emblazoned on the face of nature. Several crucial and difficult steps are necessary to mathematize genuine physical problems. Any real situation contains dozens of elements whose relevance must be considered. If one is studying the motion of a ball, the color can surely be neglected, but the shape and size may not be negligible. On the other hand, if one is studying the reflection of light from some surface or the transmission of light through some translucent material, the color of the surface or the material may be critical. In the study of the motion of a planet around the sun, both the planet and the sun may be regarded as point-masses, that is, the mass of each can be regarded as concentrated at one point. The reason is simply that their sizes are small compared to the distance between them. On the other hand, if one is studying the motion of the moon around the earth, the size and shape of the earth must be taken into account. Should the attraction of both bodies by the sun also be taken into account? That depends upon the problem to be solved. To predict the tides of the oceans on the earth, the sun's attraction does matter. However, to study the precession of the earth's axis, that is, the change in the direction of the imaginary line through the North and South Poles, the sun?s attraction can be ignored. The more exact the answer required, the more care must be exercised to be sure that the relevant factors are taken into account. Simplification of a problem by discarding the irrelevant factors is a crucial step, and it presupposes an insight that may, of course, be deepened by experience. After simplifying a problem one must apply physical principles. (In studying the motion of the earth around the sun, the law of gravitation is a fundamental principle.) Such principles are usually supplied by physicists, but the translation of the physical principles and other relevant information about the particular problem into the language and concepts of mathematics must be done by mathematicians. The concepts may not be available and may have to be created. In fact, precisely the need to treat problems of motion motivated the creation of the calculus. New concepts are added constantly as new problems are tackled. Once the mathematical formulation is achieved, the next stage is the solution of a mathematical problem. There are times when the applied mathematician is lucky. The mathematical problem may have been solved in the course of some earlier study. And one may be sure, if this does prove to be the case, that the solution was originally sought in behalf of a real problem. More often, unfortunately, the mathematical problem is a new one that calls for original work. Other problems and processes, such as approximation adequate for the use to which the solution is to be put, enter into applied mathematics, but the major point is that mathematical models cannot be constructed a priori and then called upon when needed. One cannot prefabricate useful models. The mathematics involved in real problems is far too complex and special to be conjured up by the free play of the imagination. A physical problem comes to the hands of mathematicians as a rock encrusted with sediment and mud. It is up to the mathematician to remove the dross, chip away the encrustations, polish the rock, and bring forth ultimately a blazing gem of physical truth. There is an art of applying mathematics and an art of teaching that art. One cannot expect students to solve new applied problems. But they will be required to do it in their professional work; therefore, they should be taught all that is involved in the entire process. To delude students into believing that the study of solely mathematical structures and processes suffices is to falsify the account. The professors? contention that the study of mathematical models prepares students for applying mathematics is, at best, wishful thinking to rationalize their own ignorance and, at worst, conscious deceit. When challenged that the values of mathemafics proper do not mean much to potential users, many professors retort that students will learn the applications in other courses. But to ask students to take seriously theorems and techniques whose worth will be apparent one, two, or several years later is a grievous pedagogical error. Such an assurance does not stir up incentive and interest and does not supply meaning to subject matter. As Alfred North Whitehead has advised, whatever value attaches to a subject must be evoked here and now. Still another argument offered by professors against the teaching of applications is that it imposes a heavy burden on the students; they must learn the mathematics and the relevant physics, say, and they must also learn to relate the two. But this argument is specious. Carefully chosen applications do not require much extramathematical background, and the little that is required can readily be included in the mathematics course. Moreover, the teaching of mathematics is expedited by tying it in with applications. These provide motivation, which mathematics proper does not. Equally important, the only meaning the concepts had for the mathematicians who created them and the only meaning students will find in courses such as calculus derive from physical or, more generally, real situations. Professors also use the argument that they cannot cover the syllabus if they include real applications. But even if this argument has force, and it does have some, in what sense is the ground covered? The professors cover the topics in the syllabus but the students are buried so deeply under an landslide of ideas and techniques that they no longer see light. The ground is covered over the students. In view of the fact that the application of mathematics to science and engineering is its most vital and widespread use, the absence of applied mathematical courses is as deplorable as the absence of honesty in our political leaders. But since most mathematicians are no longer capable of offering applications, they shun them as infections in a sound body. The intransigence of mathematics departments in meeting their obligations to students majoring in areas such as science and engineering is notorious. They teach as though mathematics is all we know and all we need to know. Unfortunately, it is in the most prestigious universities that the professors are allowed to take the position that they are authorities unto themselves. They are autonomous and teach what they will. Syllabi for courses and a planned sequence of courses, essential in a cumulative subject such as mathematics, are detested and ignored by the high and mighty. Professors often palm off the teaching of courses addressed to science and engineering students on the younger members of the faculty or on graduate students, who have no choice of the courses they must teach. What subject matter do professors teach? Except in a course such as calculus, where the content is prescribed, they favor their own specialties. These in turn are determined by their research. It is not surprising to learn that courses in mathematical logic, abstract algebra, topology, the theory of numbers, functional analysis, and axiomatics dominate the undergraduate and graduate curricula. The technical nature of these subjects need not be examined. What matters is that these subjects, very fashionable today, constitute a one-sided account of mathematics. All are pure; that is, devoid of applications. Professors prefer virginity to bedding with science. Professors are specialists and they tend to view the world of mathematics through the medium of their own specialty. Certainly there should be courses for prospective pure mathematicians. But all mathematicians should be informed about the chief value of mathematics - namely, its interplay with science and the amazing fruitfulness of that interplay. Here man demonstrates the magical power of his mind, and how mathematics bridges the gap between his mind and the real world. Or, as Max M. Schiffer, professor of mathematics at Stanford University, has pointed out: The miracle of mathematics is that paper work can be related to the world we live in. With pen or pencil we can hitch a pair of scales to a star and weigh the moon. Such possibilities give applied mathematics its vital fascination. Can any subject give the would-be mathematician - initially at least - a stronger and more natural interest? And what about the non-mathematician? Deny him introduction to this subject, and his appreciation of our cultural heritage must inevitably be inadequate. For mathematics in the broadest sense is instrumental not only to our understanding, but also to our changing the world we live in. Mathematics majors may be free later to pursue any branch of pure mathematics, but not to know the chief role of their subject is to be ignorant, no matter how many research papers they may write later. A goodly number of the courses should deal with applications to science, and some physics should be part of the education of every mathematician. Further, since the prospective mathematicians are most likely to become university teachers, they should be prepared to teach prospective scientists and engineers. That the typical mathematics professor was not required to learn any science can be charged to the graduate school professors, who (with few exceptions) are a collection of specialists in various areas of pure mathematics. Thdse professors, who face the task of educating scientists, engineers, and future teachers of such students and refuse to do so, are irresponsible. Were they in an industrial or commercial organization, they would be fired. That such dismissals do not take place in the universities is due only to the fact that mathematicians as a group are nearly all in the same position, and not even the chairman, who is one of the group, would wish to take action. Many professors claim that they take pride in their work and do fulfill their obligations as teachers. But their actions belie their words. What is wrong, then, is not that professors do not know what they are teaching but that they do not know how to relate their subject to the rest of knowledge and to life. Education is evaded within academic walls as well as without, and it is more often the professors rather than the students who do so. Professors may not be consciously dishonest or aware of their pedagogical ineptness. However, the lack of clear standards of teaching, the professors? ignorance of pedagogy, and the obligation, which many take too literally, to cover ground prescribed in syllabi produce the same effect as incompetence and dishonesty. The practices of the past are not re-examined, and the challenges that true education might pose are ignored. The student who goes to college to prepare for a career is certainly justified in doing so, and the colleges have been implicitly accepting such a goal while explicitly talking about liberal education. Hence, courses for business-oriented students, statisticians, actuaries, and scientists must be given the attention other courses get. True, vocational or professional interests clash to some extent with purely academic interests, but the former should not be submerged or ignored. Nevertheless, myopic professors impose their own interests on the students, with the result that their courses are largely useless to most students and to society. Students are accused of resistance to intellectuality, but their resistance is to arrogant and indifferent professors who, in the name of academic freedom, serve themselves. Mathematicians have abandoned science in an age whose major achievements are scientific and most of whose principal problems will be solved largely by resort to science. They live in a self-imposed exile from the real world. Blinded by a century of ever purer mathematics, the professors have lost the will and the skill to read the book of nature. Like the mathematicians Gulliver met in his voyage to Laputa, they live on an island suspended in the air and leave to others the problems of earthly society. The extent to which they have abandoned science is perhaps best indicated by the words of Marshall Stone, who served over the last few decades as a professor at Yale, Harvard, and Chicago: Nevertheless the fact is that mathematics can equally well be treated as a game which has to be played with meaningless pieces according to purely formal and essentially arbitrary rules, but which become intrinsically interesting because there is such a great fascination in discovering and exploiting the complex patterns of play permitted by the rules. Mathematicians increasingly tend to approach their subject in a spirit which reflects this point of view concerning it. . . . I wish to emphasize especially that it has become necessary to teach mathematics in a new spirit consonant with the spirit which inspires and infuses the work of the modern mathematician, whether he be concerned with mathematics in and for itself, or with mathematics as an instrument for understanding the world in which we live. . . . In fact, the construction of mathematical models for various fragments of the real world, which is the most essential business of the applied mathematician, is nothing but an exercise in axiomatics. . . . When an acceptable modern curriculum has been shaped in terms of its mathematical content, one must still be concerned with the spirit which animates the subject and the manner in which it is taught. It is here that it is highly appropriate to demand that, even in the earliest stages, an effort should be made to bring out both the unity and abstractness of mathematics. Stone's words have not gone unchallenged. Professor Richard Courant, formerly head of the pre-Hitlerian world center for mathematics at the University of Gottingen and more recently the founder and head of what is now called the Courant Institute of Mathematical Sciences of New York University, has denounced this abrogation of the essence of mathematics: A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from the definitions and postulates that must be consistent but otherwise created by the free will of the mathematician. If this description were accurate, mathematics would not attract any intelligent person. It would be a game with definitions, rules and syllogisms without motive or goal. The notion that the intellect can create meaningful postulational systems at its whim is a deceptive half-truth. Only under the discipline of responsibility to the organic whole, only guided by intrinsic necessity, can the free mind achieve results of scientific value. The various sins of pedagogy were attacked again by Courant: Perhaps the most serious threat of one-sidedness is to education. Inspired teaching by broadly informed, educated teachers is more than ever an overwhelming need for our society. True, curricula are important; but the cry for reform must not be allowed to cover the erosion of substance, the propaganda for uninspiring abstraction, the isolation of mathematics, the abandonment of the ideals of the Socratic method for the methods of catechetic dogmatism.. . . At any rate it would be without doubt a radical and vitally needed remedy for many ills in our schools and colleges if a close interconnection between mathematics, mechanics, physics, and other sciences would be recognized as a mandatory principle which must be vigorously embraced by the coming generation of teachers. To help such a reform is a solemn obligation of every scientist. The abandonment of applied mathematics by most mathematicians is a blow not merely to pedagogy. It is a threat to the very existence of mathematics itself. Problems which stem from the real world are the lifeblood of mathematics. It must remain a vital strand in the broad stream of science or it will become a brook that disappears in the sand. The entire history of mathematics shows that physical science has supplied the inspiration, vitality, and fruitfulness of this subject. Nor should one overlook the value of applied mathematics to technology and thereby to humanity. That men and women now work thirty-five or forty hours a week instead of eighty, that their homes are better built and more comfortable, that they enjoy quality phonograph records and television, that they can receive medical treatments which cure diseases or at the very least prolong their lives, these and a multitude of other benefits are due in large measure to mathematics. To speak of an applied mathematics program as though it were one of many programs or as though there were two kinds of mathematics, pure and applied (or monastic and secular, as some would put it), is a concession to the current practice. But it is a misrepresentation of mathematics and mathematics education. There is just one subject: mathematics. The chief function of that subject, and its chief claim to support by society and to an important role in education, reside in what it does to help man understand the worlds about him¡ªphysical, social, biological, and psychological. So far the successes have been mainly in the physical sciences, but judged by the time scale of civilizations, mathematics is young. Many mathematicians today and of recent years would dispute this evaluation. Mathematics, they say, is what mathematicians do, and since most mathematicians have no interest in anything but the subject itself the relationship to other fields is irrelevant. No doubt many are sincere. Since they do not know the magnificent and powerful uses of mathematics they can be indifferent to them. But they should heed more of von Neumann's words: As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from "reality," it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l'art pour l'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that. . . at a great distance from its empirical source, or after much "abstract" inbreeding a mathematical subject is in danger of degeneration. . . . In any event when this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the rein jection of more or less directly empirical ideas. lam convinced that this was a necessary condition to conserve the freshness and vitality of the subject and that this will remain equally true in the future. Does the abandonment of science by most mathematicians mean that science will be deprived of mathematics? Not entirely. The Newtons, Laplaces, and Hamiltons of the future will create the mathematics they need, jusf as they did in the past. Formerly such men, though they were honored as mathematicians and served as mathematics professors, were basically physicists. In today's world they are cast out by the mathematicians but they find their place in science departments. Nevertheless, the services of many able people are now lost to science and to the necessary pedagogy. The failure of mathematics departments to cater to science, engineering, and social science students has had the expected effect. The physical science, social science, and engineering departments of many universities offer their own mathematics courses. In some institutions, statistics and probability courses are given in half a dozen departments. In almost all colleges and universities computer science has broken away from mathematics and is a separate department. Clearly, the users of mathematics have decided that mathematics is too important to be left to the mathematicians. Should this practice expand, it will be the end of mathematics education as such. Curiously, though some mathematics professors resent this competition for students and the takeover of what they regard as their province, others exhibit a remarkable "broadmindedness." Their reaction is, Well, now we don't have to teach the useful mathematics; we can teach what we please. Though the teaching of mathematics courses by other departments and schools such as engineering may seem to resolve the problem of giving students the kinds of mathematics courses that would be more useful to them, it is not the realistic solution. A relatively minor objection is that it leads to endless duplication of courses, which the universities can ill afford. More consequential is the fact that mathematics, physics, economics, and the various branches of engineering are now vast domains and a professor is hard put to it to master even a portion of any one of these domains. A physics professor, for example, may know selected portions of mathematics, but he cannot teach mathematics effectively. He does well if he can teach successfully his area of physics. If, on the other hand, these several users of mathematics were to hire mathematicians to teach the mathematics they want taught, they may not require the services of a full-time professor and may settle for a graduate student or adjunct. However, even if a number of full-time professors are employed but are segregated in nonmathematical departments, they will be isolated and will fail to benefit from contact and communication with colleagues who can stimulate and educate each other. The departmental organization of disciplines has drawbacks, but it is the best we have. Mathematics is far too important to be taught in dribs and drabs. The neglect of applied courses and the poor teaching by mathematicians has had the consequence that more than half of the students who enter college as science majors either drop out or turn to other fields. About 50 percent of the engineering students do not graduate. Other factors enter, of course, but poor mathematics education is certainly a major cause. The courses are lethal weapons that snuff out the intellectual interests and sometimes the academic lives of the students. One who peruses the current mathematical journals will find articles advocating the institution of applied mathematics programs in the undergraduate and graduate schools, and may therefore infer that mathematicians are now genuinely concerned with this obligation. But closer investigation will reveal that this sudden interest is caused not by any recognition of student needs or guilt about the poor courses that have been offered but is, rather, selfishly motivated. Academic positions for Ph.D.'s in mathematics are now far fewer than the number of Ph.D.'s being turned out by the graduate schools. However, industry and the federal government do employ applied mathematicians. Hence, graduate students and Ph.D.'s are being urged to prepare for the available jobs. Were they to accept this recommendation, most would have great difficulty in acquiring the proper training. If one were to judge mathematicians by their pedagogy, one could take either of two positions, depending upon whether he was charitably or critically disposed toward them. Under the former, they would be characterized as so introspective as to be unworldly, deeply in love with their subject, intensely concerned with the progress of mathematics, and so far superior to ordinary mentalities that they cannot appreciate normal problems. Under the second view, their preference for their own values and interests marks them as selfish, overly absorbed in themselves, culturally narrow, and indifferent to society?s interests and to the needs of students. In either case the mistakes made by professors bury their pupils just as surely as the mistakes of doctors bury their patients. Nevertheless, the professors who teach pure, abstract mathematics and maintain that they are successful do provide some lift to all of us. We can be quite sure once again that there is a heaven and that there are angels. Certainly the students must be angels to be able to absorb what the professors say they teach them, and the professors evidently derive their sustenance from heaven because they have their heads in the clouds. C&O ·¢±íÓÚ 06:53 | ÔĶÁÈ«ÎÄ | ÆÀÂÛ(0) | ÒýÓÃ(trackback0) Ò²ÎÞ·çÓêÒ²ÎÞÇç - 2003-05-26 04:07 ¶¨·ç²¨¡¡ËÕéø ÈýÔÂÆßÈÕɳºþµÀÖÐÓöÓê¡£Óê¾ßÏÈÈ¥£¬Í¬ÐнÔÀDZ·£¬Óà¶À²»¾õ¡£ÒѶøËìÇ磬¹Ê×÷´Ë´Ê¡£ ĪÌý´©ÁÖ´òÒ¶Éù£¬ºÎ·ÁÒ÷Ð¥ÇÒÐìÐС£ ÖñÕÈâЬÇáʤÂí£¬Ë­Å£¿Ò»ËòÑÌÓêÈÎƽÉú¡£ ÁÏÇÍ´º·ç´µ¾ÆÐÑ£¬Î¢À䣬ɽͷбÕÕÈ´ÏàÓ­¡£ »ØÊ×ÏòÀ´Ïôɪ´¦£¬¹éÈ¥£¬Ò²ÎÞ·çÓêÒ²ÎÞÇç¡£ ×¢£º¹éÈ¥£¬Ò²ÎÞ·çÓêÒ²ÎÞÇ磬ÕâÊÇÎÒÏÖÔÚ´òËã»Ø¹úʱµÄÐÄ̬¡£ C&O ·¢±íÓÚ 04:07 | ÔĶÁÈ«ÎÄ | ÆÀÂÛ(0) | ÒýÓÃ(trackback0) ´ËÐÄ°²´¦ÊÇÎáÏç - 2003-05-26 04:01 ¶¨·ç²¨¡¡ËÕéø ¡¡¡¡ ÄϺ£¹éÔùÍõ¶¨¹úÊÌÈËÔ¢Äï ³£ÏÛÈ˼ä×ÁÓñÀÉ£¬ÌìÓ¦ÆòÓëµãËÖÄï¡£ ¾¡µÀÇå¸è´«ð©³Ý£¬·çÆð£¬Ñ©·ÉÑ׺£±äÇåÁ¹¡£ ÍòÀï¹éÀ´ÑÕÓúÉÙ£¬Î¢Ð¦£¬Ð¦Ê±ÓÌ´øÁë÷Ïã¡£ ÊÔÎÊÁëÄÏÓ¦²»ºÃ£¬È´µÀ£º´ËÐÄ°²´¦ÊÇÎáÏç¡£ ×¢£º´ËÐÄ°²´¦ÊÇÎáÏ磬ÕâÊÇÎÒ³ö¹úʱµÄÐÄ̬¡£ C&O ·¢±íÓÚ 04:01 | ÔĶÁÈ«ÎÄ | ÆÀÂÛ(1) | ÒýÓÃ(trackback0) Ç©Ãûµµ2 - 2003-05-26 03:49 £±£®±ä̬ÊÇÎҵij£Ì¬£¬³£Ì¬ÊÇÎҵıä̬¡£ £²£®¶ÔÓÚÎÒÀ´Ëµ£¬»î×ű¾Éí¾ÍÒѾ­ÊǶÔÀÏÌìÒ¯µÄÒ»ÖÖÍ×ЭÁË¡£ £³£®Ó׶ùÔ°µÄʱºòÎÒ²»Ì¸Áµ°®£¬ÒòΪ²»ÖªµÀʲôÊÇÔô¡£Ð¡Ñ§µÄʱºòÎÒ²»Ì¸Áµ°®£¬ÒòΪ֪µÀûÓÐÔôÐÄҲûÓÐÔôµ¨¡£³õÖеÄʱºòÎÒ²»Ì¸Áµ°®£¬ÒòΪÓÐÔôµ¨Ã»ÔôÐÄ¡£¸ßÖеÄʱºòÎÒ²»Ì¸Áµ°®£¬ÒòΪÓÐÔôÐÄûÔôµ¨¡£´óѧµÄʱºòÎÒ²»Ì¸Áµ°®£¬ÒòΪÓÐÁËÔôÐÄ£¬Ò²ÓÐÁËÔôµ¨£¬ÔôȴûÁË¡£ £´£®·¢Á˲ÅÊÇÓ²µÀÀí£¡ £µ£®ÄîÁËÊ®¼¸ÄêÊ飬»¹ÊÇÓ׶ùÔ°±È½ÏºÃ»ì¡£ £¶£®Á½¸öÅ®ÓÑ£¬¶ÔÓÚÄÐÈ˾ÍÊǺìõ¹åÓë°×õ¹å¡£¡£¡£ Èý¸öÅ®ÓÑ£¬¶ÔÓÚÄÐÈ˾ÍÊÇÐÇÐÇÔÂÁÁºÍÌ«Ñô¡£¡£¡£Ò»ÈºÅ®ÓÑ£¬¶ÔÓÚÄÐÈ˾ÍÖ»ÊÇÒ»¸öÌøÔéÊг¡¡£¡£¡£ £·£®ÄÄÒ»ÌìÎÒ²»ÔÚÁË£¬Öйú¾ÍºÃÏóʧȥÁËÁé»ê£¬ ÕâÖÖ¿´·¨²»ºÃ¡£ £¸£®ÉíÌ彡¿µ£¬ÔÓʳ¶¯ÎÊÊÓ¦ÐÔÇ¿£¬ÐÔϲ°²¾²£¬ÖҳϿɿ¿£¬»¶Ó­ÁìÑø¡£ £¹£®¸ÃMM°ÔÖ÷Ò»Ã棬Ϊ»öËÄ·½¡£´ýÎÒÁ·¾Í±¾Ê£¬ÊÂÒµÓгɣ¬µÃÒÔ±¨Ð¢¸¸Ä¸£¬¶¨È»ÉáÃüÉúÇÜÖ®£¬ÑºÖÁÃñÕþ¾Ö·ü·¨£¡£¡ £±£°£®6Ëê֮ǰרעÓÚ³ÔÄÌ£»6Ëê¶ÏÄ̽øÈëСѧѧϰ£¬Ñ§»áÎüÑÌÐï¾Æ¿¼ÊÔ×÷±×£»12Ëê¶Á³õÖУ¬²»¾ÃÖ®ºóÒòÔڽ̹¤ËÞÉáæÎËÞ±»ÐÌʾÐÁô£»15Ëê¶Á¸ßÖУ¬³ÁÄçÓÚÔçÁµ¶ø¶ÁÁËÒ»¸ö¸ßËÄ£»19Ëê½øÈëÇ廪£¬´óһѧ»á¹àË®£¬´ó¶þѧ»áqq£¬´óÈýµ±ÉÏ°æÖ÷£¬´óËÄÒò·´¶¯ÑÔÂÛ±»É±µµ£»23ËêÔÚ±±ÃÅÍ⿪ÁËÒ»¼ä»ð¹øµê£»29ËêÌêÁËÒ»¸ö¹âÍ·£¬×¨ÐÄ¿à¶Á¾­¼Ãѧ£» 39Ëê»ñµÃŵ±´¶û¾­¼Ãѧ½±£»¶øºó¿ªÊ¼40ÓàÄêµÄÈ«ÇòÑݽ²ÉúÑÄ£¬ÆÚ¼äÔø³é¿ÕÔÚÒâ´óÀûάÂÞÄÉ¿´¹ýÒ»³¡Òâ¼×µÂ±È´óÕ½£»80ËêÔÚƽ¹ÈÂòÏÂÈý¼ä´óÍß·¿£¬°üÁËÒ»¸öÇ廪µÄ¶þÄÌ¡£¡£¡£ C&O ·¢±íÓÚ 03:49 | ÔĶÁÈ«ÎÄ | ÆÀÂÛ(0) | ÒýÓÃ(trackback0) Ç©Ãûµµ1 - 2003-05-26 03:30 £±£®ÇëÔ­ÁÂÎÒ»¹»î×Å¡£ £²£®Èç¹û£¬ÎÒÊÇ·¨¹Ù£¬ÎÒ½«ÅоöÄ㣬ÖÕÉí¼à½û£¬¼à½ûÔÚ£¬ÎÒµÄÐÄÀï¡£ £³£®Âí°¡£¬ËÄÌõÍÈ°¡£»º£°¡£¬È«ÊÇË®°¡¡£ £´£®ÓÚÎÒ¶øÑÔ£¬PPMM¾ÍÈçͬʵº¯Êý¿Õ¼äÖд¦´¦Á¬ÐøÈ´´¦´¦²»¿Éµ¼µÄº¯Êý¡£ÀíÂÛÉÏ˵ÔÚÈÎһСÇøÓòÄÚÒ»×¥Ò»°Ñ£¬¿ÉÊǵ±ÎÒÏëÈ¥×¥µÄʱºòÈ´´ÓÀ´¶¼×¥²»µ½¡£ £µ£®ÎÒÃÇÕâ¸öʱ´ú£¬ÓÐÌ«¶àÈËϲ»¶Ã°×ÅɱͷµÄΣÏÕ˵ÖÚËùÖÜÖªµÄµÀÀí£¬¶øÇÒ£¬ÈËÃÇϲ»¶°ÑÕâÑùµÄÈ˽Ð×öÓÂÊ¿£¬°ÑÕâÑùµÄ»°½Ð×ö˼Ïë¡£ £¶£®´Ó¿ªÊ¼¿Þ׿ɶʣ¬±ä³ÉÁËЦ×ÅÏÛĽ¡£Ê±¼äÊÇÔõôÑùÅÀ¹ýÁËÎÒƤ·ô£¬Ö»ÓÐÎÒ×Ô¼º×îÇå³þ ¡£ £·£®ÎÒÃÇÄÇ¿ªÊ¼¶¼ÖÖÓñÃ׵ģ¬ºóÀ´¸ã¸Ä¸ï¶¼ÖÖÁ˲¤ÂÜ£¬Ô¶Ô¶ÍûÈ¥£¬Ò»´óƬһ´óƬµÄ¡£ Áìµ¼ÃÇÀ´ÊӲ죬˵£ººÜºÃºÜºÃ£¬ÕâÀﶼ³ÉÁ˲¨Â޵ĺ£ÁË¡£¡£¡£ £¸£®ÔÚ½ÌÊÒ˯¾õ£¬ÔÚͼÊé¹Ý³Ô¶«Î÷£¬ÔÚʳÌÃ×ÔÏ°£¬ÔÚÇÞÊÒ¶ÁÊé¡£ £¹£®¶ÍÁ¶¼¡È⣬·ÀÖ¹°¤×á¡£ £±£°£®¾ÝÎÒËùÖª£¬×Ô´Ó´´ÊÀÖ®³õ£¬ÖªÊ¶·Ö×Ӿͱ»ÈË¿´²»Æð¡£Ö±µ½ËûÃÇÔì³öÁËÔ­×Óµ¯£¬Ê¹È«ÊÀ½ç»Ì»Ì²»¿ÉÖÕÈÕ£¬ÕâÖÖÇéÐβÅÓÐËù¸Ä±ä¡£ ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡ ¡ª¡ªÍõС²¨¡¤Çàͭʱ´ú C&O ·¢±íÓÚ 03:30 | ÔĶÁÈ«ÎÄ | ÆÀÂÛ(0) | ÒýÓÃ(trackback0) ÔöÉèÐÂÀ¸Ä¿ - 2003-05-26 03:21 ¡¡¡¡Í»È»ÐÄѪÀ´³±£¬ÏëÐÂÔö¼¸¸öÐÂÀ¸Ä¿£¬½«ÎÒƽÈÕÔÚÍøÉÏÑ°µ½µÄºÃÎÄÊÕÈëÆäÖУ¬ÌùÔÚBLOGÉϹ²Ïí¡£ ¡¡¡¡¡¶¾­µäÐÀÉÍ¡·Ö÷ÒªÊÇÊ«´Ê¡¢¶ÔÁª¡¢¹ÅÎÄÃûƪµÈ¡£ ¡¡¡¡¡¶ÃîÎÄÊ°È¡¡·ÔòÊÇÍøÂçÉÏÁ÷Ðеġ£ÎÒÓÈÆäÏëÊÕ¼¯ÕªÂ¼Ò»Ð©ÓÐÒâ˼µÄÇ©Ãûµµ¡£ C&O ·¢±íÓÚ 03:21 | ÔĶÁÈ«ÎÄ | ÆÀÂÛ(0) | ÒýÓÃ(trackback0) ³ÉÓïºÍѧϰ¡ª¡ª°ë²¿ÂÛÓïÖÎÌìÏ - 2003-05-26 02:34 µÚÁùƪ й¶Ìì»ú ¡¡¡¡ÀÏʦµÄ±¸¿Î±Ê¼Ç¡¢½Ì°¸Í¨³£ÊÇÀÏʦ×Ô¼ºÓõģ¬½Ìʦ½Ìѧ²Î¿¼ÊéµÄ¶ÁÕßÓ¦¸ÃÊǽÌʦ¡£²»Öª´Óʲôʱºò¿ªÊ¼£¬ÄãÃÇѧÉúÒ²¿ªÊ¼¶Ô½ÌʦµÄ½Ìѧ²Î¿¼Êé¸ÐÐËȤÁË¡£ ¡¡¡¡Èç¹ûÏëѧΪʦ֮µÀµ¹Ò²²»»µ£¬µ«ÊÇÊÂÇé²¢·ÇÈç´Ë¡£ ¡¡¡¡¼ÈÈ»ÄãÃÇѧÉúÒ²¿´½ÌʦµÄ½Ìѧ²Î¿¼Ê飬²»·ÁÈÃÄãÃÇÖªµÀÒ»µãÒ»¸ö´óѧÊýѧÀÏʦµÄ½Ì°¸¡£ 21£®°ë²¿ÂÛÓïÖÎÌìÏ ¡¡¡¡³ÉÓï¡°°ë²¿ÂÛÓïÖÎÌìÏ¡±µÄÒâ˼ÊÇ£¬ÕÆÎհ벿¡¶ÂÛÓï¡·£¬È˵ÄÄÜÁ¦¾Í»á¸ßÇ¿£¬¾ÍÄÜÖÎÀí¹ú¼Ò¡£¾Éʱ£¬¸Ã³ÉÓïÓÃÀ´Ç¿µ÷ѧϰÈå¼Ò¾­µäµÄÖØÒª¡£ ¡¡¡¡ÎÒ¶ÁÊéʱÓÐÕâôһ¾ä»°:ѧºÃÊýÀí»¯£¬×ß±éÌì϶¼²»Å¡£Ç¿µ÷Êýѧ¡¢ÎïÀíºÍ»¯Ñ§ÖªÊ¶µÄÖØÒªÐÔ¡£ ¡¡¡¡¶þÊ®ÄêÇ°¿ªÊ¼µÄ¶ÔÍ⿪·Å£¬Ê¹µÃÍâÎÄ(ÓÈÆäÊÇÓ¢ÎÄ)±äµÃÔ½À´Ô½ÈÈ¡£½üÄêÀ´Ëæןßм¼Êõ²úÒµ£¬ÓÈÆäÊǼÆËã»ú¼¼ÊõµÄ·¢Õ¹ÒÔ¼°¼ÆËã»úµÄÆÕ¼°£¬Óë¼ÆËã»úÏà¹ØµÄ¶«Î÷±äµÃÖËÊÖ¿ÉÈÈ¡£ÎÒ¹úһλ¿Æѧ¼Ò¾ÍÔø˵¹ý£¬ÏÖÔڵĿÆѧ¼¼Êõ¹¤×÷ÕßÓ¦¸ÃÕÆÎÕÈýÖÖÓïÑÔ£¬Ò»ÖÖ¶«·½ÓïÑÔ¡¢Ò»ÖÖÎ÷·½ÓïÑԺͼÆËã»úÓïÑÔ¡£Ç¿µ÷ÖÐÎÄ£¬Ó¢ÎĺͼÆËã»ú֪ʶµÄÖØÒªÐÔ¡£ ¡¡¡¡µ±½ñ£¬ÈËÀàÖǻ۵Ľᾧ´óÌå¿ÉÒÔ¹é½áΪ¿Æѧ¡¢¼¼ÊõºÍ¹¤³ÌÈý¸ö·½Ãæ¡£¼¼ÊõÒÔ¿ÆѧΪ»ù´¡£¬¹¤³ÌÒÔ¿Æѧ¼¼ÊõΪ»ù´¡¡£ ¡¡¡¡¿Æѧ¿ÉÒÔ·Ö³ÉÈËÎÄ¿Æѧ¡¢Éç»á¿ÆѧºÍ×ÔÈ»¿ÆѧÈý´óÀà¡£ÆäÖУ¬ÈËÎÄ¿Æѧ°üÀ¨ÎÄѧ¡¢ÀúÊ·¡¢ÕÜѧµÈѧ¿Æ£¬Éç»á¿Æѧ°üÀ¨ÕþÖΡ¢¾­¼Ã¡¢·¨ÂɵÈѧ¿Æ£¬×ÔÈ»¿Æѧ°üÀ¨Êýѧ¡¢ÎïÀí¡¢»¯Ñ§¡¢ÌìÎÄ¡¢µØÀí¡¢ÉúÎïµÈѧ¿Æ¡£ ¡¡¡¡´ºÇïÕ½¹úʱÆÚÊÇÎÒ¹úÈËÎÄ¿ÆѧºÍÉç»á¿Æѧ·¢Õ¹µÄÒ»¸ö¶¦Ê¢½×¶Î£¬ÐγÉÁËÖî×Ó°Ù¼ÒµÄ˼Ïë¡£ÆäÖÐÒÔ¿××ÓºÍÃÏ×ÓΪ´ú±íµÄÈå¼Ò˼Ïë¶ÔºóÀ´µÄÖйúÉç»áÓ°Ïì×î´ó¡£ËùÒÔÓС°°ë²¿ÂÛÓïÖÎÌìÏ¡±Ò»Ëµ¡£ ¡¡¡¡½üÈý°ÙÄêÊÇ×ÔÈ»¿Æѧ·¢Õ¹µÄÖØÒª½×¶Î£¬µ±½ñÎ÷·½·¢´ï¹ú¼ÒµÄ×ÔÈ»¿ÆѧÑо¿¼°Æä¼¼ÊõˮƽÒÑ´ïµ½ºÜ¸ßµÄµØ²½¡£ÏÖÔÚÎÒÃǹú¼ÒÒ²ÔÚÖÂÁ¦ÓÚ¿Æѧ¼¼ÊõµÄ·¢Õ¹¡£ ¡¡¡¡ÁíÍ⣬»¹ÓÐÒ»ÖÖ·¢Õ¹Ç÷ÊÆ£¬¾ÍÊÇÈËÎÄ¡¢Éç»áºÍ×ÔÈ»¿ÆѧµÄѧ¿ÆÏ໥Éø͸¡¢Ï໥½áºÏ¡£ÏÖÔÚÒÑÓеÄһЩËùν±ßÑØѧ¿Æ¾ÍÊÇÕâÖÖÉø͸ºÍ½áºÏµÄ½á¹û¡£ ¡¡¡¡ËùÒÔ˵´Ó´ó´¦½²£¬Ñ§ÉúµÄ֪ʶ½á¹¹Ó¦¸ÃÎÄÀí¼æ±¸¡£ÐγÉÕâÑùµÄ֪ʶ½á¹¹ÓÐûÓС°°ë²¿ÂÛÓÄØ?»»¾ä»°Ëµ£¬Ê²Ã´Ñ§¿Æ֪ʶÔÚÎÄÀí¼æ±¸µÄ֪ʶ½á¹¹ÖÐ×î»ù±¾¡¢×îÖØÒª?ÎÒÈÏΪ£¬ÓУ¬¾ÍÊÇÓïÎĺÍÊýѧ¡£Ò»¸ö×î¼òµ¥µÄÖ¤¾Ý¾ÍÊÇ´ÓСѧһÄ꼶µ½¸ßÈý£¬ÓïÎĺÍÊýѧһֱÊÇ×îÖØÒªµÄÁ½Ãſγ̡£ÏÖÔÚÓв»ÉÙ´óѧ£¬Àí¹¤¿ÆѧÉú»¹ÒªÑ§´óѧÓïÎÄ£¬ÎĿƵÄѧÉú¿ªÊ¼Ñ§¸ßµÈÊýѧ¡£ ¡¡¡¡ÈËÎĺÍÉç»á¿ÆѧÊÇ´ÓÈ˵ĽǶȳö·¢ÈÏÖªÊÀ½çµÄ£¬×ÔÈ»¿ÆѧÊÇ´Ó×ÔÈ»½çµÄ½Ç¶È³ö·¢ÈÏÖªÊÀ½çµÄ¡£ËüÃǵĽ»²æ¡¢½áºÏ¾ÍÊÇËùνµÄ¡°ÌìÈ˺ÏÒ»¡±¡£ÈËÎĺÍÉç»á¿Æѧ×î»ù´¡µÄѧ¿ÆÊÇÓïÎÄ£¬×ÔÈ»¿Æѧ×î»ù´¡µÄѧ¿ÆÊÇÊýѧ¡£ ¡¡¡¡ÎÒÃÇÏÈ̸̸ÓïÎÄ¡£Ê×ÏÈÖÐÎÄÊǹ¤¾ß£¬ÊÇ˼άºÍ½»Á÷µÄ¹¤¾ß¡£È˵Ä˼ά½¨Á¢ÔÚÓïÑԵĻù´¡ÉÏ£¬Ë¼Î¬ÊÇÓÃÓïÑÔ½øÐеġ£Èç¹ûÈÃÒ»¸ö³ÉÈËÓÃËû²»»á»ò½ö»áÒ»µãµÄÓ¢ÎĽøÐÐ˼ά£¬Ò²¾ÍÊǪ«Thinking in Englishª«£¬ÄÇËûÁ¬Ò»¸öÈýËê¶ùͯ¶¼²»Èç¡£ËùÒÔ£¬ÖÐÎĵÄÌý¡¢Ëµ¡¢¶Á¡¢Ð´ËÄÖÖÄÜÁ¦ÊÇѧÉú±ØÐëÕÆÎյġ£Æä´Î£¬Ò»¸öÃñ×åµÄÎÄÃ÷ÌåÏÖÔÚËýµÄÓïÑÔÖ®ÖС£Öйú¼¸Ç§ÄêÎÄÃ÷³ÁµíÔÚÖÐÎĵÄÓïÑÔÖУ¬Ëý×ÔȻҲ°üÀ¨ÎÒ¹úÈËÎĺÍÉç»á¿Æѧ˼Ïë¡£ËùÒÔ£¬Ñ§Ï°ÓïÎIJ»µ«¿ÉÒÔÅàÑøѧÉúµÄÎÄѧÐÞÑøºÍÄÜÁ¦£¬»¹¿ÉÒÔÅàÑøÈËÎĺÍÉç»á¿ÆѧµÄÐÞÑøºÍÄÜÁ¦¡£ ¡¡¡¡ÏÖÔÚÀ´Ì¸Êýѧ¡£ÊýѧÊ×ÏÈÒ²Êǽ»Á÷¹¤¾ß¡£×÷Ϊ½»Á÷¹¤¾ßÊýѧ²»½ö½öÄܽÌÈËÔÚÈÕ³£Éú»îÖлáËã¼ÛÇ®£¬¶øÇÒ»¹ÄܽÌÈËÁì»áÐí¶àÊýѧ¸ÅÄÈçÂß¼­¡¢¸ÅÂÊ¡¢Í¼ÏóµÈµÈ¡£µ±½ñÉç»áµÄ½»Á÷ÖÐÐèÒªÊýѧµÄ¶ÁдÄÜÁ¦¡£»¹ÓиüÖØÒªµÄÒ»µã£¬ÊýѧµÄ˼ά·½·¨ÊÇÅàÑøѧÉú·¢ÏÖÎÊÌâ¡¢Ìá³öÎÊÌâºÍ½â¾öÎÊÌâÄÜÁ¦µÄ×î¸ù±¾µÄ·½·¨¡£ÏóÖÐÎÄÒ»Ñù£¬ÊýѧÊÇ¿Æѧ¼¼ÊõµÄÓïÑÔ¡£ËùÒÔѧϰÊýѧ²»µ«¿ÉÒÔÅàÑøѧÉúÊýѧÐÞÑøºÍÄÜÁ¦£¬»¹¿ÉÒÔÅàÑø×ÔÈ»¿ÆѧµÄÐÞÑøºÍÄÜÁ¦¡£ ¡¡¡¡¼ÇµÃÓÐÈË˵¹ýÕâÑùµÄ»°£¬Ê«È˺ÍÕÜѧ¼ÒÖ®¼äÓÐһƬÂ̵أ¬Ê«ÈË×ß¹ýÈ¥¾ÍÊÇÏÈÖªÏȾõ£¬¶øÕÜѧ¼Ò×ß¹ýÈ¥¾Í±ä³ÉÁËÊ¥ÈË¡£»¹ÓÐÈË˵¹ý£¬Êýѧ¼Ò¾ÍÊÇÕÜѧ¼Ò¡£ÕâЩ˵·¨¶¼Ç¿µ÷ÁËÒ»¸öÈ˵Ä֪ʶºÍÄÜÁ¦µÄ×ÛºÏÐÔ¡£´ïµ½ÕâһĿ±êµÄÇ°ÌáÊǾßÓÐÁ¼ºÃµÄÎÄѧºÍÊýѧµÄÐÞÑøºÍÄÜÁ¦¡£ÏÖÔÚ´¥Ä¿½ÔÊǵÄÊÇ£¬Ñ§ÎĵÄÂÔÊä×ÔÈ»£¬Ñ§Àí¹¤µÄÉÔÑ··çɧ£¬¶¼²»äìÈ÷£¬¶¼²»¡°¿á¡±¡£ÎÒÏë½èÓÃëÔ󶫵Ĵʾ䣺¡°¾ãÍùÒÓ£¬Êý·çÁ÷ÈËÎ»¹¿´½ñ³¯¡£¡±±í´ï¶ÔÄãÃǵÄÏ£Íû¡£ ¡¡¡¡²»ÉÙÈËÈÏΪ¼ÆËã»ú֪ʶºÜÖØÒª£¬µ«ÊǼÆËã»ú֪ʶûÓÐÖØÒªµ½ÓïÎĺÍÊýѧµÄ»ù´¡µØλ¡£¶ÔÓÚ´ó¶àÊýÈËÀ´½²£¬¼ÆËã»úÖ»ÊÇÒ»¸ö¹¤¾ß¶øÒÑ£¬Ö»Òª»áÓþͿÉÒÔÁË¡£ÁíÒ»¸ö·½Ã棬¼ÆËã»úÒªÏëµÃµ½¸ü¹ã·ºµØʹÓ㬱ØÐëÔ½À´Ô½Éµ¡£²»ÄÜɵµ½Ïóɵ¹ÏÏà»úÒ»Ñù£¬ÖÁÉÙÒ²µÃÏóÆû³µÒ»Ñù¡£´ïµ½Õâ¸öÄ¿±ê»¹Óкܳ¤µÄ·Ҫ×ߣ¬ÕâÒ²ÊǼÆËã»ú²úÒµ¾ßÓо޴ó·¢Õ¹Ç°Í¾µÄÔ­ÒòÖ®Ò»¡£ ¡¡¡¡¹ØÓÚÓ¢ÎÄ£¬ÎÒ²»µÃ²»ÎÞ¿ÉÄκεسÐÈÏ£¬ÎÒÃÇÏÖÔÚ»¹ºÜÐèÒªÕâ¸ö½»Á÷¹¤¾ß¡£×îÖØÒªµÄÔ­ÒòÊÇÎÒÃÇÒª°ÑÂäºóµÄ×æ¹ú·¢Õ¹³ÉÇ¿´óµÄ¹ú¼Ò¡£Ê¹µÃ×Ô¼º¹ýÉϸüºÃµÄÈÕ×Ó¡£ÔÚÍâ¹úÈËÖ÷¶¯Ñ§Ï°ÖÐÎÄÄǸöʱºò֮ǰ£¬ÎÒÃÇÖ»ÄÜÎÔн³¢µ¨¡£ C&O ·¢±íÓÚ 02:34 | ÔĶÁÈ«ÎÄ | ÆÀÂÛ(0) | ÒýÓÃ(trackback0) ³ÉÓïºÍѧϰ¡ª¡ªµÇɽС³ - 2003-05-26 02:29 20£®µÇɽС³ ¡¡¡¡Â³£ºÖܳ¯¹úÃû£¬ÔÚ½ñɽ¶«¾³ÄÚ¡£³ÉÓï¡°µÇɽС³¡±Ô´ÓÚ¡¶ÃÏ×Ó¡¤¾¡ÐÄÉÏ¡·£¬¡°¿××ӵǶ«É½¶øС³£¬µÇ̩ɽ¶øСÌìÏ¡£¡±Î½µÇ¸ßÍûÔ¶¶øÒÔ×ãÏÂÖ®´¦ÎªÐ¡¡£ºóÓÃÓÚ±ÈÓ÷ѧÎʼ´¸ßÄÜÈÚ»á¹áͨ£¬ÑÛ¹âÔ¶´ó¡£ ¡¡¡¡³ÉÓï¡°µÇɽС³¡±½«µÇɽÓë×öѧÎÊÁªÏµÔÚÒ»Æ𡣡°ÊéɽÓзÇÚΪ¾¶¡±½«¶ÁÊé±È³ÉµÇɽ¡£×öѧÎÊÉÙ²»Á˶ÁÊ飬ËùÒÔɽÓë¶ÁÊéÓв»½âÖ®Ôµ¡£ ¡¡¡¡ÓÐÒâµÇɽÕß»òÔç»òÍí»áÑ¡¶¨Ò»×ùɽ¿ªÊ¼ÅÀ¡£Í¨³££¬ÓÐÒâÓÚijɽµÄʱ¼äÊÇÔÚ¸ßÖпì±ÏÒµµÄʱºò¡£ÔÚ¿ªÊ¼µÇɽǰµÄÒ»Çж¼ÊÇ×ö×¼±¸£¬ÆäÖаüÀ¨½«×ÔÉíÖÃÓÚ¾¡¿ÉÄܸߵĺ£°ÎµØµã¡£ ¡¡¡¡ÏÖÔÚ£¬Ò»Ð©ÓÐʶ֮ʿ¾õµÃÉÏ´óѧ¾Í¿ªÊ¼ÅÀÑ¡ºÃµÄɽÔçÁ˵㣬Ӧ¸ÃÔÙ»¨·ÑһЩʱ¼ä×¼±¸£¬ÔÙ¿¼ÂÇ¿¼ÂÇÅÀʲôɽ¡£µ«ÊÇ£¬ÁíÍâһЩÓÐʶ֮ʿÔÚ¸ßÖеÄij¸öʱ¼ä¾Í°ÑѧÉú·Ö³ÉÁËÁ½²¦£¬Ò»²¦ÅÀÎÄɽ£¬ÁíÒ»²¦ÅÀÀíɽ¡£ÓÐЩѧÉú¾Í¸üÔ翪ʼÅÀijɽ¡£ ¡¡¡¡»¹ÓÐÒ»¸ö±È½ÏÆÕ±éµÄÏÖÏ󣬾ÍÊÇ°Ñ´óѧҲµ±×÷Ò»×ùɽÀ´ÅÀ¡£Êâ²»Öª£¬´óѧ²»ÊÇɽ£¬³äÆäÁ¿²»¹ýÊǵÇɽµÄÒ»¸ö´ó±¾Óª¶øÒÑ¡£ ¡¡¡¡µÇɽºÎµÈ¼èÄÑ¡£¡°»áµ±Áè¾ø¶¥¡±¸üÐèÄ¥ÄÑ¡£Ã»ÓмáÇ¿µÄÒâÖ¾¡¢·á¸»µÄÖǻۺͽ¡¿µµÄÌåÆÇ£¬Ò²Ðíδµ½¡°»ØÐÄʯ¡±¾ÍÒѵôÍ·ÁË£¬Ò²Ðí²îÒ»²½¾Íµ½ÁËɽ¶¥È´ÎÞÂÛÈçºÎÒ²ÔÙ²»ÄÜÂõ³öÄÇÒ»²½ÁË¡£ ¡¡¡¡ÅÀɽµÄ¾­Ñé̸֮ÊÇ£º²»ÅÂÂýÖ»ÅÂÕ¾¡£µÇɽµÄ·²»ÊÇ·É¿ìµÄ¼¸Ê®²½¡¢¼¸°Ù²½¾Í¿ÉÒÔ×ßÍêµÄ¡£¾­³£ÓÐÆø´­ÐêÐêµÄÈËÕ¾ÔÚÄÇÀÍû×ÅÀ´Õß·¢´ô£º¸Õ¸ÕÒѾ­³¬¹ýÁËËûÃÇ£¬Ôõô¾ÍÒ»»á¶ù¹¤·òÓÖ±»ËûÃǸÏÉÏÁË¡£ ¡¡¡¡ÈËÃdz£³£ÓÐÒ»Öָоõ£¬±Ëɽ×ܱȴËɽ¸ß¡£»¨Á˲»ÉÙÁ¦ÆøµÇÉÏÁËÒ»×ùɽµÄɽ¶¥£¬¾ÙÄ¿ËÄÍû£¬¸Ð¾õ»¹ÓÐɽ±ÈÕâɽ¸ß£¬³É¹¦µÄϲÔöÙʱ¼õÉÙ¡£Æäʵ£¬Ëüɽ²¢²»Ò»¶¨±ÈÕâɽ¸ß£¬µ«ÊÇÄÜÓÐÕâÖָоõ¾Í˵Ã÷´Ëɽ¾ÍÊǸßÒ²¿Ï¶¨²»»á±ÈËüɽ¸ßµ½ÄÄÀï¡£ÔÚ´ËÇé¿öÏ£¬ÓÐÈ˾ͻá¼ûÒì˼Ǩ¡£Ò»¸öÈËÒ»Éú²»±ØÖ»ÅÀÒ»×ùɽ£¬Ö»ÒªÐÄδÀÏ£¬¶àÅÀ¼¸×ùɽҲδ³¢²»¿É¡£ ¡¡¡¡¡°µÇɽÈÝÒ×£¬ÏÂɽÄÑ¡±¡£¹âÒõËƼý£¬ÈË×ÜÒª³öɽ¡£ÔÚɽÖÐÁ·¾ÍµÄÒ»Éí¹¦·òÊÇ¡°»¨È­ÐåÍÈ¡±£¬ÔÚÉç»áÉÏÉÙ²»ÁËÅö±ÚµÄÈÕ×Ó¡£¾ÍÊÇÓÐÒ»ÉíÓ²¹¦·òÒ²µÃ¼ÙÒÔʱÈÕ¡¢¼ÙÒÔÄ¥Á·¡¢¼ÙÒÔ»ú»á²ÅÄÜÔÚ½­ºþÉÏÑïÃûÁ¢×ã¡£ ¡¡¡¡Ë×»°Ëµ£º¡°¸ôÐÐÈç¸ôɽ¡±¡£´Ë»°¶ÔÓÚÖ»ÔÚɽµ×»ò°ëɽתÓƵÄÈËÀ´½²µÄÈ·Èç´Ë¡£ÒòΪ¡°²»Ê¶Â®É½ÕæÃæÄ¿£¬Ö»ÔµÉíÔÚ´ËɽÖС£¡±¶ø¶ÔÓÚÓС°»áµ±Áè¾ø¶¥£¬Ò»ÀÀÖÚɽС¡±¾³½çµÄÈËÀ´½²£¬ÆßÊ®¶þÐÐΪһÐС£ ¡¡¡¡µ±È»£¬ÄÜ´ïµ½¿××Ó¡°µÇɽС³¡±¾³½çµÄÈ˱Ͼ¹ÊÇÉÙÊý¡£¶ÔÓÚ¾ø´ó¶àÊýÈËÀ´½²£¬¡°É½²»Ôڸߣ¬ÓÐÏÉÔòÁ顱ҲÊǺܺõÄ×·ÇóÂï¡£ C&O ·¢±íÓÚ 02:29 | ÔĶÁÈ«ÎÄ | ÆÀÂÛ(0) | ÒýÓÃ(trackback0) Why The Professor Can't Teach¡ª¡ªµÚÁùÕ¡¡ÏÁ°¯µÄÊýѧ¼Ò - 2003-05-26 01:42 CHAPTER 6: The Illiberal Mathematician Education is not in reality what some people proclaim it to be in their statements. ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡Plato The liberal arts colleges - whether integral parts of universities or independent entities - now cater to a variety of student interests. Adequate instruction calls for courses that, while conforming to the values and objectives of a liberal arts education, serve these interests. Mastering the requisite materials and incorporating them in suitable pedagogical format constitute the major tasks of educators. But the professors are under pressure to do research, and the graduate students, to obtain a Ph.D. Both are narrowly educated, though in different respects and for different reasons. Thus, both groups of teachers, whether engaged in or training to do research, know some mathematics but are ignorant in science and pedagogy, and both are concerned almost entirely with content and values that only mathematicians prize. Yet both must tackle the courses that are supposed to serve many different student interests. Just what is it that professors and graduate students, taking their cue from professors, teach in some of the most fundamental courses? The largest single group taking mathematics in the colleges and universities consists of liberal arts students who register for a mathematics course only to meet the requirements for a degree. The typical registrant is indifferent to mathematics or actively dislikes it. Nevertheless, the teaching of these nonspecialists is probably the most important task of the professional mathematician. It is important from the broad sociological point of view and even from the standpoint of mathematics, because many of these students will become leaders in our society and will decide how much support to give to the subject. Also, since the segregation of students into prospective users and nonusers of mathematics is based on interest rather than ability, the nonuser group contains some of the most worthwhile students. To cater to the nonusers most colleges, whether independent or part of a university, offer what is called a liberal arts mathematics course. Despite the importance of this course, most mathematicians despise teaching it. It is beneath their dignity to bother with nonmathematicians and to waste their precious time and knowledge on mathematical "nonentities." However, many professors are obliged to teach the course in order to fulfill the required number of teaching hours. What, then, do they teach? Because most mathematicians are narcissists, they offer reflections of themselves. They prefer pure mathematics, and even in that area only topics that strike their fancy. Let us examine some of these topics. Their favorite topic by far is the theory of sets. A set is no more, of course, than a collection of objects. But students are asked to learn specific operations with sets such as union, intersection, and complementation, and are also asked to learn properties of these operations. Set theory also includes infinite sets, such as the set of all even integers. All this material has no, or at best trivial, bearing on elementary mathematics or on real phenomena. In addition, the concept of an infinite set baffled and was rejected by the best mathematicians until the 1870s and is still unacceptable to many today. Since the study of sets, and particularly infinite sets, does not return even infinitesimal riches to the student, he does not see why he should attempt to make his reach exceed his grasp. Another favorite topic is the theory of numbers, wherein one studies unusual properties of the integers. Some numbers, such as 6, are "perfect" because each is the sum of its divisors (other than the number itself), as 6 is the sum of 1, 2, and 3. "Unfortunately" there are very few known perfect numbers; so the course soon proceeds to study prime numbers. A number is prime if it is divisible only by itself and 1. Thus 7 is a prime, whereas 6 is not. Any number of theorems treat properties of prime and nonprime (composite) numbers. To some mathematicians prime numbers are delightful, intriguing members of the number system. To the students they are hostile strangers. When they learn that there is an infinity of prime numbers they become convinced that the world is full of enemies. The theory of numbers also includes what is called the theory of congruences, and this topic seems to be a must in liberal arts courses. The theory of congruences concerns an arithmetic that is suggested by how our clocks record time. Six hours after nine o'clock the clock reads three o'clock; that is, 9 + 6 = 3. In other words, twelve and any multiples of twelve are discarded or counted as zero. Certainly this is a peculiar arithmetic. The theory of congruences studies many such varieties. But even ordinary arithmetic is dull, however practical the knowledge. What, then, can students find interesting in clock arithmetic? Moreover, since many are still insecure about the operations of ordinary arithmetic, the new arithmetic shatters what little confidence they had. A favorite topic in liberal arts courses is axiomatics (see Chapter 3). Because the axiomatic approach to teaching is now the popular one in many courses, this topic warrants special attention here. Every branch of mathematics is founded on axioms, the prototype being Euclidean geometry. There is much to be gained from a study of the axiomatic basis of any branch. Indeed, the most momentous development of the nineteenth century - non-Euclidean geometry - resulted from a change in the axioms of Euclidean geometry. But what features of the study of axiomatics are presented in a liberal arts course? Some professors introduce a system of axioms and merely discuss what properties the system should possess. One of these, for example, is independence; that is, it should not be possible to prove any one of the axioms on the basis of the others, for in that case the provable axiom is more properly a theorem. The study of properties of an axiom system is of great interest to specialists in the foundations of mathematics, but to tyros the axioms are the least significant part of any mathematical development. They are the seeds from which the fruit eventually emerges. Hence, a treatment of the properties of the axioms themselves has little value. Another type of "play" with axioms is to show students that the set required to develop a branch of mathematics is not unique. One can change the axioms and still deduce the same body of theorems. Or one can in some branches reduce the number of axioms, though this may involve the need for more complicated proofs of the theorems. The latter activity is relatively simpleminded and usually profitless. One is not surprised to find, then, that students are not exhilarated when they are shown than ten axioms can be replaced by nine or nine and one-half. One truly liberal arts value to be derived from the study of axiomatics is to help people become conscious of their actual assumptions when they make a decision or adhere to a belief in any sphere. But this "carry-over" of mathematics education to other areas of our culture is never mentioned. The teaching of axiomatics per se reinforces the contention that mathematics does not teach critical thinking. For if it did, professors would certainly ask themselves why they teach axiomatics in a liberal arts course. Some liberal arts courses and certainly the more advanced courses do indeed derive the theorems that are implied by the axioms, just as the high school course in Eucidean geometry does. This deductive approach to a branch of mathematics is surely the elegant one. Unfortunately, it is almost always a distortion of the more natural thinking that led to the theorems and proofs. The necessity to establish a theorem on the basis of the axioms and previously established theorems obliges the mathematician to recast his original argument to force the theorem into the most suitable place in the logical sequence. This recast proof may be far from the original thoughts that convinced the creator his theorem was correct. Moreover, after at least one successful proof is obtained, its creator or his successors, now able to see how the essential difficulty was overcome, can usually devise a more ingenious or more direct proof. Most theorems have been reproven several times, each successive proof a remodeling of the previous one and often including generalizations or stronger results. Hence, the final theorem and proof are far from the original thoughts. Indeed, they are often shorn entirely of their intuitively grasped form. Some of the logic supplied to shore up the original intuition is entirely artificial and so trumped up and stilted as to preclude understanding. Over one hundred years ago Augustus De Morgan, one of the founders of modern logic, warned, "The student must not believe that theorems have been invented or perfected by the methods in which it is afterwards most convenient to deduce them. The march of the discoverer is generally anything but on the line on which it is afterward convenient to cut the road." Because the deductive approach is not the understandable one, and especially because it is a distortion of the natural, intuitive approach, its value to the student is inversely proportional to its elegance. As far as the students can see, the axioms are handed down ex cathedra and then the logical crank is turned to grind out theorem after theorem. The students do not know where the axioms came from, why the particular ones were chosen, and where the seemingly interminable sequence of theorems is going. Even though a student may get to the point of verifying each step in a proof, he usually does not understand the rationale, the basic thought or method behind the proof. Why is this particular chain of steps used rather than some other perhaps more easily understood sequence? The student has a case. Facing such a reworked, more sophisticated, and possibly more complicated result, he cannot grasp it at all. Henri Poincar¨¦, the leading mathematician of the late nineteenth and early twentieth centuries, makes this point: "Can one ever understand a theory if one builds it up right from the start in the definitive form that rigorous logic imposes, without some indications of the attempts which led to it? No. One does not really understand it; one cannot even retain it or one retains it only by learning it by heart." Following a proof step by step has been compared to the way a novice at chess observes two masters play. The novice recognizes that each player makes a move that conforms to the rules, but he will not understand why a player makes a particular move rather than any one of a dozen others available to him. Nor will he perceive the overall strategy that suggests a series of moves. Similarly, watching a mathematical demonstration being made step by step, each of which is justified by some axiom or previously established theorem, does not in itself convey the plan of the demonstration, nor the wisdom of that entire method of proof as opposed to some other method. Because he may be called upon to reproduce the proof, the student is reduced to memorizing the prescribed series of steps. Many professors, having delivered a series of theorems and deductive proofs, walk out of their classrooms very much satisfied with themselves. But the students are not satisfied. They were not involved in the real thinking and derived no stimulation from a presentation they did not understand. The learning of such proofs calls for much activity but little cerebration or celebration. The logical order of mathematical presentations is about as helpful pedagogically as the alphabetic order of words in a literary work. Colleges and universities state in their catalogues that their first objective is to encourage students to think for themselves; yet professors presentations promote not the emancipation but the enslavement of minds. Mathematicians have a naive idea of pedagogy. They believe that if they state a series of concepts, theorems, and proofs correctly and clearly, and with plenty of symbols, they must necessarily be understood. This is like an American speaking English loudly to a Russian who does not know English, in the belief that his increased volume will ensure understanding. The deductive presentation of mathematics is psychologically damaging because it leads students to believe that mathematics is created by geniuses who start with axioms and reason directly and flawlessly to the theorems. Given this impression of elevated, far-ranging minds, the student feels humbled and even depressed about his own capacities, especially when the obliging professor presents the material as though he too is genius in action. The logical or deductive approach does not convey understanding. As Galileo put it, "Logic, it appears to me, teaches us to test the conclusiveness of an argument already discovered and completed, but I do not believe that it teaches us to discover correct arguments and demonstrations." The logical formulation does dress up an intuitive understanding, but it conceals the flesh and blood. It is like the clothes that make the woman, or make the man want to make the woman, but are not the woman. Logic may be a standard and an obligation of mathematics, but it is not the essence. Nevertheless, the deductive approach to mathematics is almost universally adopted by professors. The major reason for its popularity is that is is easier to teach. The entire body of material is laid out in a complete, ready-made sequence and all the teacher has to do is to repeat it. On the other hand, to know the intuitive meaning of a concept or proof, to penetrate to the basic idea of a proof, and to know why one proof is preferable to another call for depth of understanding. Even if the professor does acquire this understanding, it is far more difficult to impart it to students. Many teachers complain that students, particularly engineers, wish to be told only how to perform the processes they are asked to learn and then want to hand back the processes. But the teachers who offer only the logical presentation because it avoids the real techniques of teaching - leading students to participate in a constructive process, explaining the reasons for proceeding one way rather than another, and finding convincing arguments - are more reprehensible. Since the logical approach to mathematics does not convey understanding and even distorts the original thinking, should it be presented at all? The answer is affirmative, and the reasons have already been at least implied. Proof is a check on our intuition. It also refines and sharpens the intuition, much as argumentation with an adversary on, say, a political issue often reveals defects in our thinking. Whether or not some liberal arts courses do much with axiomatics per se, all do offer as a prime example of "superb" mathematics the logical development of the real number system. Real numbers include the positive and negative whole numbers, fractions, and the irrational numbers such as and the like. Some history here is relevant. These types of numbers were introduced into mathematics on an intuitive and pragmatic basis. Thus, mathematicians learned to add 1/2 and 1/4 to obtain 3/4 because one-half of a pie and one-quarter of a pie amount to three-quarters of a pie. Mathematicians worked successfully with these various numbers for over five thousand years without much, if any, concern about precise definitions or the logical development of their properties. For purely professional reasons mathematicians decided in the late nineteenth century that a logical structure based on a clear, axiomatic foundation should be provided. Of course, the logical structure had to sanction what had already been established on empirical grounds. It proved to be highly artificial, contrived, and complicated. From a logical standpoint, irrational numbers in particular are intellectual monsters, and most who study this apparent aberration wryly appreciate the mathematical term "irrational." Pascal's maxim, "Reason is the slow and tortuous process by which those who do not understand the truth arrive at it," is most appropriately applied to the logical development of the real number system. Many teachers might retort that the college student has already learned the intuitive facts about the number system and is ready for the appreciation of the deductive version, which exemplifies mathematics. If the student really understands the number system intuitively the logical development will not only not enhance his understanding, it will destroy it. As an example of mathematical structure no poorer choice could be made, because the construction is highly contrived. The development not only stultifies the mind but obscures the real ideas. Yet this topic has become the chief one in college mathematics courses. One may well conjecture that some teachers enjoy presenting the intuitively familiar facts about the number system in the recondite axiomatic approach because they understand the simple, underlying mathematics and yet can appear to be presenting profound material. To defend teaching the logical development of the real number system many professors extol it as an example of how mathematics builds models for the solution of real problems. This is not the place to discuss applied mathematics (see the next chapter) but it is very clear that the professors who make such a statement haven't the least idea of how mathematics is applied. The example is absurd on many accounts. Let us note two. The real number system had been in use since about 3000 B.C., roughly over five thousand years before the logical "model" for it was constructed. Fortunately, no one waited for the availability of this model to apply real numbers. Nor would anyone use the model today, because the artificial, logically complex construction is as far removed from reality as heaven from earth. No one would ever think of using it even to predict anything about real numbers, let alone for a physical application. The reason for constructing the logical foundation of the real number system had nothing to do with real problems. A few professors may be aware of these facts but perhaps introduce the word "model" in this context because it has another, more pleasing association. But the logical structure in question lacks flesh and blood. No liberal arts course is considered complete without symbolic logic. This topic, which presents the ordinary principles of reasoning in symbolic form, supposedly teaches reasoning; actually, it is farcical for this purpose. To know what symbolism to use and how to manipulate it, one must already know the common meanings of "and," "or," "not," and "implies." But the students do not have these clearly in mind, and symbolic logic only conceals them under meaningless symbols. How ridiculous to teach symbolic logic to students who still confuse "All A is B" with "All B is A." This topic is of significance only to specialists in the foundations of mathematics. Boolean algebra, which is closely related to symbolic logic, is frequently included in liberal arts courses because it can be applied to the design of switching circuits, and presumably the liberal arts students in question are going to be electronics engineers. To the students the mention of this application may well suggest switching courses. The liberal arts courses purport to teach the power of mathematics, and they do this by teaching abstract structures such as groups, rings, and fields. A group, for example, is any collection of objects, such as the positive and negative integers, and an operation that performed on any two members produces a member of the collection. The operation, like the addition process applied to the positive and negative integers, must possess certain additional properties that are the abstract analogues of the familiar properties: 3+ (4+5) = (3+4) +5; there is a zero; and to each integer, 2 say, there is another, -2, whose sum is zero. (See also Chapter 3.) Abstraction is indeed a valuable feature of mathematics. It reveals properties common to many concrete structures just as knowledge of the structure of mammals teaches us much about hundreds of varieties of mammals. Moreover, one who knows the abstraction can often see at once that it applies to a totally new phenomenon. Abstractions do lay bare the logical structure of several kindred concrete systems, but they are an impoverishment of the concrete as surely as the bone structure of the human body fails to present the whole man. But, some mathematicians rejoin that after presenting the abstract structure they give concrete examples. However, the concrete cases must be thoroughly understood before one introduces the unifying abstraction. To introduce as examples concrete material not yet familiar to the student is of no help in making the abstract concept clearer. In every case learning proceeds from the concrete to the abstract and not vice versa. To see the forest by means of the trees is pedagogically the only sound approach. Many teachers favor an abstract theme, such as group theory, because they believe it to be an efficient way of imparting much knowledge in one swoop. They are under the impression that if a student is taught group theory he will automatically learn the properties of the rational, real, and complex numbers, matrices, congruences, transformations, and other topics. But a student who learns only the abstract group theory could not on this basis add fractions. Abstractions do relate and unify many seemingly unrelated developments. However, for young people who possess little background, nothing is unified and illuminated by the abstraction. The abstract structures in question are so remote from their mathematical experiences that the values such structures grant to mathematics are no more evident than the power of philosophy to run a spaceship. To the student the abstractions are shadows that can be perceived only dimly and induce a feeling of mystification and even apprehension. Just as the human body struggles for breath in a rarefied atmosphere, so the mind strains to grasp abstractions. They may pervade the teaching but they evade the student. What is the major problem facing this nation today? Is it inflation? Unemployment? The absorption of minorities? Women's rights? Retaining the respect of other nations? If one were to judge by the contents of the liberal arts courses, it is the Koenigsberg bridge problem. As we have related (Chapter 1), some two hundred years ago the citizens of the village of Koenigsberg in East Prussia amused themselves by trying to cross seven nearby bridges in succession without recrossirig any one. The problem attracted Leonhard Euler, certainly the greatest eighteenth-century mathematician, and he soon showed that the attempt was impossible. But mathematicians will not let the dead rest in peace, and they revive the problem as though it were the most momentous one facing our civilization. No worse a collection of dull, remote, useless, or sophisticated topics could have been chosen for a liberal arts course. Many of these topics come from the foundations of mathematics, where only specialized and professional needs justified their creation. With a few exceptions they are late nineteenth- and twentieth-century products that came long after most of the greatest mathematics was created. The best mathematicians of the past - Archimedes, Descartes, Newton, Leibniz, Euler, and Gauss - used almost none of them, for the simple reason that they didn't exist. And even the great mathematicians of the present use most of them only in specialized foundational studies. A liberal arts course that includes such topics must devote a great deal of time to convincing students that they should learn what the entire mathematical world did not miss for thousands of years, and what very few mathematicians need even today. The topics have about as much value as learning to dig for clams has for people who live in a desert. (See also Chapter 10.) A common alternative to the melange of topics such as set theory, axiomatics, symbolic logic, and a rigorous treatment of the number system is a presentation of technical mathematics that starts about where the high school courses leave off and covers more advanced techniques. This type of course continues an old tradition. The colleges used to require that all students take more algebra and trigonometry, a requirement reminiscent of the treatment doctors of the Middle Ages prescribed for all illnesses. They "cured" every illness by bleeding the patient. Just as the cure relieved the suffering of those who bled to death, the modern technique course drains students of any vestige of respect for mathematics. Another alternative, very popular today, is known as finite mathematics. Just what is finite about it, except perhaps the students' attention to it, is not clear. It does not include any calculus, but it does use real numbers, complex numbers, and algebraic processes and theorems that involve infinity in several ways. The content, like that of the typical liberal arts course, is a conglomeration of topics having little relationship to each other and little significance for the students to whom it is addressed. This hodgepodge of topics, set theory, symbolic logic, probability, matrices, linear programming, and game theory (we shall not here undertake to explore the nature of these topics) is one of the fads that constantly sweep through mathematics education. Some such course, if the topics are properly chosen, might be useful to social science students. Presumably then it would contain applications to the social sciences. On examination one finds a mathematical system that describes the marriage rules of a primitive Polynesian society. To select the proper topics the organizers of the course would have to take time to find out what is really useful to the social scientists, but mathematics professors do not do this. Finite mathematics is not just a fad; it is a fraud. In any case it is not a liberal arts course. The liberal arts courses described above give a low return on the investment of the hard work called for. Students learn brick-laying instead of architecture and color-mixing instead of painting. If these courses exhibit the liberal arts values of mathematics, then certainly student disregard and contempt for mathematics are justified. Since all this material and its purported values fail to win over students, some professors concentrate on those who, for whatever reason, accept and pursue the material that is taught. With unconscious immodesty the professors label these students keen or bright. The "less intelligent" ones, quantitatively about 98 percent of the total, do want to know why they should learn seemingly useless material. But these professors do not recognize that they have an obligation to teach all the students. No matter what the choice of topics, a fundamental objection to the usual liberal arts course in mathematics is that all the topics are devoted to mathematics proper. And mathematics proper is remote, unworldly, and even otherworldly. This thought was expressed by one of the greatest mathematicians of recent times, Hermann Weyl: One may say that mathematics talks about things which are of no concern at all to man. Mathematics has the inhuman quality of starlight, brilliant and sharp, but cold. But it seems an irony of creation that man’s mind knows how to handle things the better the farther removed they are from the center of his existence. Thus we are cleverest where knowledge matters least: in mathematics, especially in number theory. Moritz Pasch, a leading mathematician of the late nineteenth century, even contended that mathematical thought runs counter to human nature. The unnaturalness of mathematics is attested to by history. Dozens of civilizations that have existed, some celebrated for their literature, religion, art, and music, did create practical rules of arithmetic and a mixture of correct and incorrect rules for areas and volumes of common figures; but only one¡ªthe ancient Greeks¡ªenvisioned and created mathematics as the science that establishes its conclusions by deductive reasoning. Even the Greeks regarded mathematics as only a means to an end: the understanding of the physical world. Only one other civilization, the modern European civilization, has surpassed the Greeks in depth and volume of new results, and western Europe learned mathematics at the feet of the Greeks. The subject matter of mathematics proper cannot be very attractive to most people. It deals with abstractions and this is one of the severest limitations. A discourse on the nature of man can hardly be as rich, satisfying, and life-fulfilling as living with actual people, even though one may learn a great deal about people from the discourse. Beyond the fact that the subject matter is abstract, it is in itself hardly relevant to life. All of mathematics centers on number, geometrical figures, and generalizations thereof. But number and geometric description are insignificant properties of real objects. The rectangle may indeed be the shape of a piece of land or the frame of a painting, but who would accept the rectangle for the land or the painting? Moreover, the likelihood of interesting students in the material we have described is poor, especially in view of their prior education. The student who is about to begin a college course in mathematics has met the subject to some extent in his earlier education. Unfortunately, the average student leaves these earlier experiences with limited ideas as to what mathematics is and what it has accomplished in our civilization. As far as he knows the subject matter is a series of techniques for solving problems; certainly the techniques are isolated from any easily conceivable use. The failure to see values in mathematics has generally caused the student to do poorly in it, deprecate it as worthless, and shrink from further involvement. Many professors would argue that the content of the liberal arts course is almost irrelevant. The course teaches students precise reasoning; this usually means deductive reasoning. A course in mathematics proper may teach sharper reasoning, but the students have already had three or more years of mathematics in high school, and it would seem that whatever mental training mathematics can supply would have been supplied already. Actually, the vaunted value of deductive reasoning is grossly exaggerated. In daily life, business, and most professions deductive reasoning is practically useless. There are no solidly grounded axioms from which one can deduce what career to pursue, whom to marry, or even whether to go to the movies. On the other hand, the distinctions that must be made in analyzing character, personality, values, and good and bad behavior are far more subtle and call for a more highly perceptive and critical faculty than anything mathematics will ever develop. Deductive reasoning is not the paradigm for the life of reason. In fact, whatever faculties equip a man to understand and judge wisely about human problems are not more widely found among mathematicians nor does the study of mathematics contribute to the acquisition of such faculties. Newton was certainly not the critical thinker when he wrote about the prophecies of Daniel. In the contention that mathematics serves to train the mind, even the students smell an aroma of professional humbug. Moreover, deductive reasoning can be learned more readily in many other contexts. The student who is asked to recognize that because the base angles of an isosceles triangle are equal it does not follow that equality of the base angles makes the triangle isosceles, must first learn the meaning of the terms involved. He can learn the same point from the example that good cars are expensive does not imply that all expensive cars are good. Actually, most mathematics courses do not teach reasoning of any kind. Students are so baffled by the material that they are obliged to memorize in order to pass examinations. Perhaps the best evidence for this assertion is supplied by the professors themselves. When asked why they do not allow students to use books in examinations, they usually reply, "What, then, can I ask in the examination?" When the defense of mathematics as training in reasoning is deflated, the professors fall back on the aesthetic satisfactions mathematics offers. Though portions of mathematics are beautiful and could be presented in a liberal arts course, there are two limitations. The first is that mathematics lacks the emotional appeal of painting, sculpture, and music. The second is that students who are repelled by the mathematics they are compelled to learn in elementary and high school will not feel moved to pursue the subject to reach the few beautiful themes. The attempt to sell the beauty of mathematics to liberal arts students is doomed to fail. Many professors proclaim that the goal of a liberal arts course should be to teach what mathematics is and what mathematicians do. No more effective means of driving students away from mathematics has ever been devised. What is mathematics? It is a collection of abstractions remote from life. What do mathematicians do? They strive for personal success, and to arrive at it some even cheat in all sorts of ways, including neglecting the interests of the very students they say they want to attract. But mathematicians create. Do these courses then teach the fumbling, the guessing, the blundering, the mental struggles, the testing of hypotheses, the frustrations, the false proofs, the insights, and other acts of the creative process? No. They teach precise definition, theorem, and proof as though God inspired the elect to proceed directly to the finished product. Intellectual challenge and thrill of accomplishment are other values cited for mathematics. Mathematicians respond to intellectual challenge much as businessmen do to the excitement of making money. They enjoy the fascination of the quest, the sense of adventure, the thrill of discovery, the satisfaction of mastering difficulties, the pride and glory of achievement -or, if one wishes, the exaltation of the ego and the intoxication of success. Such values are present in mathematics more than in any other subject because it offers sharp, clear problems. But to obtain these values one must be interested in the subject and have already acquired some facility in it. An occasional, exceptional student persevering on his own or fortunate enough to have had one or two fine teachers may come to enjoy these values of mathematics. But such students are exceptions and are almost certainly not to be found among the students taking the liberal arts course. Moreover, people’s intellects make their own claims about what they find challenging. Many find more significant challenge in law and economics. What should a college course in mathematics for liberal arts students offer? The answer is contained in the question. The liberal arts values of mathematics are to be found primarily in what mathematics contributes to other branches of our culture. Mathematics is the key to our understanding of the physical world; it has given man the conviction that he can continue to fathom the secrets of nature; and it has given him power over nature. We now understand, for example, the motions of the planets and of electrons in atoms, the structure of matter, and the behavior of electricity, light, radio waves, and sound. And we can use this knowledge in man's behalf. Some uses of this knowledge are familiar to all of us: the telephone, the phonograph, radio and television are achievements of mathematics. Mathematics, especially through statistics and probability, is becoming increasingly valuable in the social sciences and in biological and medical research. The search for truth in philosophy or the social sciences cannot be discussed without involving the role that mathematics has played in that quest. Painting and music have been influenced by mathematics. Much of our literature is permeated with themes treating the implications of mathematical achievements in science and technology. Indeed, it is impossible to understand some writers and poets unless one is familiar with mathematical influences to which they are reacting. Religious doctrines and beliefs have been dramatically altered in the light of what mathematics has revealed about our universe. In fact, the entire intellectual atmosphere, the Zeitgeist, has been determined by mathematical achievements. These are the liberal arts values of mathematics and should constitute the essence of a liberal arts course. Though this is not the place even to sketch the contents of such a course, a few elaborations may help to make clear what it can offer. The average person thinks of science rather than mathematics as providing the explanation of natural phenomena. Yet mathematics is the essence of science. Let us consider an example. The force of gravity is involved in all phenomena of motion. The action of gravity presumably explains why the planets and their moons keep to their periodic paths and why space ships can be sent to the moon. In all motions on earth - as we walk along a level road or up or down a hill, ride in an automobile or airplane, rise from a sitting position or sit down; in the whirring of machinery; and even in the flow of blood in our bodies - the action of gravity is involved. Presumably an understanding of the force of gravity would clarify all these motions. One might argue that the mathematical law that describes quantitatively the action of gravity is useful, but that gravity is a physical phenomenon. However, to emphasize the physical force of gravity and to regard the mathematical law as an aid in analyzing and predicting the physical action is to miss the main point. How does the sun's gravitational attraction keep the planets in their appointed paths? Is there a steel cable stretching from the sun to the earth that keeps the earth from flying off into space and confines it to its elliptical path? We have no idea of how gravity acts physically. In fact, there is no force of gravity. As Russell Baker once remarked, "You cant buy it any place and store it away for a gravityless day." It is a fiction introduced to supply some intuitive understanding of the various motions we perceive and undertake. How, then, has science been able to treat gravity, to make such precise predictions of eclipses of the sun and moon, and even to send men to the moon? The answer is that the mathematical law of gravitation is all that we know about this force. By means of the mathematical law and deductions from it, we can describe and predict the behavior of thousands of objects. In fact, one of Newton' s great accomplishments was to show that this very law applies to both terrestrial and celestial motions. Mathematics, then, is not only a key to our understanding of motion, it is the only knowledge we have. The same can be said of light, radio waves, television waves, X rays, and in fact all of the waves of what is called the electromagnetic spectrum. The student who enjoys radio reception of music, from Beethoven to the Beatles, should bless mathematics. Physical science has reached the curious state in which the firm essence of its best theories is entirely mathematical, whereas the physical content is vague, incomplete, and in some cases, self-contradictory. This science has become a collection of mathematical theories adorned or cluttered with a few physical observables. To use Alexander Pope's words, the mighty maze is not without a plan, and the plan is mathematical. In fact, it is not hard to maintain that our knowledge of the entire physical world must reduce to mathematics. As Sir James Jeans has put it, "All the pictures which science draws of Nature, and which alone seem capable of according with observational facts, are mathematical pictures." More than that, the pictures are made by man. There is no known physical, objective universe. We, not God, are the lawgivers of the universe. These examples of the liberal arts values of mathematics have admittedly been drawn from the physical rather than the social sciences. The theory and predictions that mathematics supplies in the former field do describe what takes place. In the latter we have models of what could happen but doesn't; however, the social sciences are young. Liberal arts values are so numerous and so monumental that only another sample or two can be presented here.*

• For a fuller exposition see the author’s Mathematics in Western Culture. In the sixteenth and seventeenth centuries, thanks primarily to the work of Copernicus and Kepler, astronomical theory was converted from geocentric to heliocentric, primarily because the mathematics of the latter was simpler. Under the older view the earth was the center of the universe, and since man was obviously the most important creature on earth, man's life, goals, and activities were the most important concerns. If there is a God and prior to the seventeenth century no one professed to doubt it - He certainly would be concerned about humans and had evidently designed the world to favor and further man's interests. But the heliocentric theory shattered all such beliefs. The earth became just one of many planets, all revolving about the sun, and man just one of many insignificant creatures on earth. How, then, at least on the basis of astronomical theory, could one believe in a God concerned with a mite, a speck of dust in a vast universe? As Matthew Arnold put it: The Sea of Faith Was once, too, at the full, and round earth's shore Lay like the folds of a bright girdle furled, But now I only hear Its melancholy, long withdrawing roar... For the world, which seems To lie before us like a land of dreams, So various, so beautiful, so new, Hath really neither joy, nor love, nor light, Nor certitude, nor peace, nor help from pain... An even more momentous theme, which certainly belongs in any liberal arts course, is the pursuit of truth. From prehistoric times onward man has sought truths, whether through religion, philosophy, science, or mathematics. Beginning with Greek times, the one universally accepted body of truths was mathematics. The significance of this fact extended far beyond mathematics. The acquisition of some truths gave man the evidence that he could acquire them; and it gave him the courage and confidence to seek them in political science, economics, ethics, and the arts. But the creation of non-Euclidean geometry shattered centuries of confidence in man's intellectual potential. Mathematics was revealed to be not a body of truths but a man-made, approximate account of natural phenomena, subject to change and having only pragmatic sanction. Though mathematics is a product of cultures, it in turn fashions cultures, notably our own. Just as the meaning of good literature lies beyond the collection of words on paper, so the true significance of mathematics consists in what it accomplishes for our society, civilization, and culture. In particular, mathematics is man's strongest bridge between himself and the external world. It is the garment in which we clothe the unknown so that we may recognize some of its aspects, and it is the means by which, to use Descartes’ words, we have become the possessors and masters of nature. Mathematics proper may be a monument to human inventiveness and ingenuity, but it is not in itself an insight into reality. Only insofar as it aids in understanding reality is it important. And this is what we must teach. Thus the prime goal of a true liberal arts course should not be mastery of purely mathematical concepts or techniques, but an appreciation of the role of mathematics in influencing and even determining Western culture. Appreciation, as well as skill, has long been recognized as an objective in literature, art, and music. It is equally justifiable as an objective in mathematics. The cultural aspects of mathematics, rather than the narrow viewpoint of the specialist, should be stressed in the liberal arts course to achieve an intimate communion with the main currents of thought in other fields. Is it surprising that mathematics and the other major branches of our culture are inextricably involved with each other? Knowledge is a whole and mathematics is part of that whole. However, the whole is not the sum of its parts. The present procedure in the liberal arts course is to teach mathematics as a subject unto itself and somehow expect the student who takes only one college course in the subject to see its significance for the general realm of knowledge. This is like giving him an incomplete set of pieces of a jig-saw puzzle and expecting him to put the puzzle together. Liberal arts mathematics must be taught in the context of human knowledge and culture. Professors must learn that mathematics proper is not the most important subject for the nonprofessional. Even some of the best professional mathematicians did not grant the subject supreme importance. Newton regarded religion as more vital and said that he could justify much of the drudgery in his scientific work only on the ground that it served to reveal God’s handiwork. But of course Newton was just a lowly physicist. Gauss ranked ethics and religion above mathematics, but Gauss, too, was as much a physicist and astronomer as a mathematician. Weyl's words - it is an irony of creation that man is most successful where knowledge matters least, in mathematics - bear repetition. Why do mathematics professors teach pointless material to liberal arts students and ignore the truly cultural values of mathematics? The sad fact is that most professors are themselves ignorant of these values. Some may be powerful engines of mathematical creativity but limited to their tracks. They work in their own mental grooves and naively assume that what they value is eminently suitable. If the thought of including, say, applications to science should occur, they would banish it because they know they might be embarrassed by the students' questions. The elitist, narcissistic mathematician who presents his own values, curiosities, and trick problems is totally unfit to be a teacher in any course. The charge that most mathematics professors are culturally narrow may seem incredible. But one must remember that mathematics is in large part a technical subject in which one can be highly proficient as a cabinetmaker is proficient among carpenters; and one would not necessarily be surprised to learn that a cabinet-maker is neither cultured nor a pedagogue. Competent researchers need not and generally do not know the broader values of mathematics. Certainly most do not care about the art of pedagogy. These professors present to all students just those values that they as professional mathematicians see in their subject - and they do not question their own values. They wish to have students appreciate the abstractions, the rigorous reasoning, the logical structure, the crystalline purity of mathematical concepts, and the presumed beauty of proofs and result. Non-Euclidean geometry, the most dramatic and shocking event in recent intellectual history, is to them just another topic. The damage done by such professors extends beyond what they inflict on students. Because they are so involved in their own research, they ignore and even disparage anything outside their own specialty. This attitude discourages young professors, many of whom, less indoctrinated, do recognize the need for a truly liberal arts course but refrain from any action for fear that the older men will regard them as trafficking in trivialities. They are made to feel that any talk about music or philosophy is a default on their real obligation to teach mathematics, and in fact to train future mathematicians. Hence, departure from the norm is discouraged. It is tragic that professors teaching in a liberal arts college, which is purportedly devoted to educating the whole person and to instilling interests and attitudes, are not themselves interested in learning material closely related to their own subject. C.P. Snow, in his famous lecture, The Two Cultures and the Scientific Revolution, deplored the gulf separating the scientists and humanists. The former, self-impoverished, disdain the humanistic culture and even take pride in their ignorance of it. The latter respond by wishful thinking to the effect that science is not part of culture but rather mechanization of the real world. They wish neither to understand nor to sympathize with the nature and goals of the scientific enterprise, and they pity the scientist who does not recognize a major work of literature. Snow's more severe criticism is directed toward the humanists who pretend that their cultural interests are the whole of culture and that the exploration of nature is of no consequence. Actually, it would have been more just to rail at the scientists, particularly the mathematicians. One cannot expect them to make exposition of their subject their mission in life; but where they have the chance, even the obligation, to offer a liberal arts course to about two million freshmen each year and thereby make the meaning and significance of their subject clear to themselves and the students, they default. The students in a liberal arts course, many of whom are the most intelligent of our youth, are our best bets for producing broadly educated men and women who would be unhampered by petty parochialism and fully alive to the interrelationships of not just two cultures but of all knowledge. Some might even become members of a now rare breed of politicians who have some idea of what scientists are doing. Surely the mathematics course should aim at such goals. Mathematicians are digging their own graves. Student protests for relevant education, though not always wisely formulated, are fully justified in the case of mathematics. A course in college mathematics is fast disappearing as a general requirement, and mathematicians will lose not only jobs but also any significant role in the liberal arts college. The mere fact that mathematics has been taught for centuries certainly is no assurance that it will be retained. Both Latin and Greek are the languages of the cultures that have contributed most to the fashioning of Western culture, and Latin was the international language of educated people until about one hundred years ago. But both languages have practically disappeared from modern education. Indeed, even the study of classical Greek and Roman culture has practically disappeared.The danger that mathematics, too, may be dropped from the liberal arts curriculum has already occurred to many mathematicians, and some are growing concerned about the public image of their subject. But apparently they are not sufficiently aroused to utilize the best medium already at their disposal to improve that image. Socrates was condemned to death for corrupting the morals of the youth of Athens. What punishment should be meted out to professors who degrade mathematics, cause the students to hate the subject or intensify the already existing hate, and in many cases poison the students' minds against all learning? And should the universities be exculpated for entrusting the education of liberal arts students to specialists or graduate students who are the living refutation of liberally educated people? C&O ·¢±íÓÚ 01:42 | ÔĶÁÈ«ÎÄ | ÆÀÂÛ(0) | ÒýÓÃ(trackback0) 16.79981303215