Americans in general like technology but seldom have a grasp of the science behind it. And the math that is behind science and technology is regarded as even more mysterious, like an inner sanctum into which only initiates may gain entry. It is far more than just a few who shudder with distaste at the thought of unpleasant memories of trying to learn a subject that remained inscrutable and therefore uninteresting. We are enamored of space shuttles, the Internet, fast computers, jumbo jets, microwave ovens, washing machines, answering machines, and any other kind of machine that will make material life easier. Unlike some of the ancients who revered mathematics for its beauty and mental exercise, Americans tend as a culture to have a very pragmatic attitude toward math. Many might appreciate its value for what material results it can bring but regard it as almost a necessary evil towards the goal of producing beneficial visible results.
These see the rich and nourishing technological fruit on this tree of knowledge as precious, but they see no deeper than the surface branches and twigs on which these fruits grow. To them, the region behind this exterior of the tree, where the trunk and limbs grow without any fruit on them, is pointless and purposeless. "What's the use of math?" is the common query. "I'll never use it." When a nation's leaders are composed primarily of lawyers, business people, administrators, military men and stars of the entertainment industry more than statesmen, philosophers, the wise, the spiritual, and the men and women of science and mathematics, then it should be no surprise that there is little grasp of the simple reality that one cannot dispense with the trunk and limbs and still continue to enjoy the fruit.
However much we might delight in the fruit of technological development, it's essential to realize that all of this derives originally from a world of abstraction that has no immediate and apparent material fruit. To fail to grasp this simple principle is to kill the goose that lays the golden eggs. Even mathematicians locked away in their cubicles may forget the extremely important role their own field plays in providing the strong sturdy limbs of abstraction that is required to do any kind of clear thinking at all. Our modern-day world is not characterized by a great deal of clear rational thinking, otherwise, if we were to come to good conclusions and apply them sensibly, our newspapers would certainly not be filled with pages recounting one self-defeating insanity after another.
The weight lifter might exercise daily for no other purpose than to have a well-muscled body that to him and his fellows seems beautiful. But there's another reason for such exercise: to develop strength, endurance and health to do productive work. The pure mathematician might seek only the beauty in his work and have little thought for how his abstractions might benefit the world, but such exercise of the mind does in fact prove useful.
Others are loath to do any such exercise, whether physical or mental, and question its practical value. But who doubts that either kind of exercise is ultimately very productive and conducive to a sound mind and body. Aside from its immediate practical value in bringing us answers that save and make money, that help us in industry, that enable us to chart the course of space probes and create amazing computer graphics, mathematics and its logical underpinnings are unquestionably the best exercise of the mind known to us. Whoever lacks the skills to reason well may well not have bothered to do well with this exercise of the mind.
What is it that would cause us to focus only on this external fruit of material development and play down the antecedent realms of abstraction that lie deeper and possibly hidden behind the more apparent material foliage and fruit? It would be good to find a word less condemning than "superficiality", but how else can this tendency be described in a word? Perhaps facing up to the ugly side of this word can stir us into action to remedy what seems to be an extremely grave crisis in Western education--the American system in particular. Inasmuch as a society's very foundations are passed on to succeeding generations, this has very threatening consequences for all that we call "modern civilization" now.
When a people, a culture, a civilization, loses touch with its roots, its intangible foundations, with the original ideals and vision that powerfully motivated it to climb upwards in development, when it comes to see only the final material products of that civilization, then it gradually lets go of its hold on the original sources of that material development.
There is a great deal of evidence that a civilization that passes over a number of different cultures evolves through several distinct stages of growth. Arnold Toynbee, the great historian who devoted his life to the study of civilizations carefully studied 22 of the more than 30 civilizations of which we have record and noted that, without exception, every great process of civilization originated with a people who are suddenly moved by a set of high-minded ideals, with a vision for a glorious future, something he simply called "religious fervor". Next comes a stage of intellectual development with little immediate application of this knowledge. Then arrives a stage of material development where this abstract knowledge is applied. Lastly, there is a long period of slow social decline accompanied by continued material development until a sudden collapse brings the entire structure down.
The first stages deal with foundations, with the trunk and limbs at the center of the tree of human social development. The last stages focus on the outermost branches and twigs with their material fruit where, by ignoring or destroying the very source of the development they so revere, reach a limiting point and suddenly find themselves with a dead and fruitless tree.
Rome took an intellectual civilization from Greece and developed with it materially with a very pragmatic attitude. It had only a limited grasp of the foundations underlying its material success. Greece acquired its rise into intellectual civilization through an influence from the Hebrews just to the east of it. Early philosophers such as Socrates taught very unGreek and very Hebrew ideas of one God, a life after death, and so on. But for those who protest against this last notion since it is not a popular one, let's take note that this is not an essential part of the argument. For those who are not perturbed by this evidence, we will point out that Hebrew civilization provided the first religious stage of this process that finally culminated in Rome.
Let's now make a parallel with our own modern society. This current civilization is clearly in a stage of material development with little real reverence paid to its intellectual underpinnings and almost none to its original spiritual foundations. This immense process started in Europe during that great rebirth called the Renaissance. The evidence that this was due to the influence of Islam through Islamic Spain and Sicily as well as the Crusades is hard to avoid. Though Europe rejected the tree of Islamic religion and its seed Muhammad, it avidly partook of the fruit. The torch was passed on to a previously benighted Europe by an Islamic society that lost touch with its roots and fell into decline as Europe rose. After this European Renaissance, the Industrial Revolution occurred with its many new inventions that greatly boosted the progress of the exploration and colonization of the world--by which this modern material civilization was disseminated to most of the world.
It is just as distressing for some moderns to find overwhelming evidence that modern civilization is not "Christian civilization" at all--which had its death in the Dark Ages--but in fact originated with the religion of Muhammad, just as it is for others to face the evidence that Greek and Roman civilization had their origins in a Hebrew religious civilization that began through the influence of Moses. If these thoughts are unacceptable, then please reject them--they are not essential to our argument here and have been added only to complete two pictures for those who find these notions acceptable.
The point of this is that we are now in a stage of civilization not unlike what the Romans had, that we display the same pragmatic materialist worldview, the same unwarranted self-confidence in the midst of many indications of having lost control, and, as many point out, the same disdain for the moral and ethical standards that seem to be characteristic of each civilization on the rise.
Are we fated to repeat history by being unable to see and learn lessons repeated time and again? Whatever the case, our modern tendency towards superficiality, to seeing only what is materially tangible and ignoring the mental and spiritual basis of this, could likely bring us the same destiny the Romans saw--unless corrective action is taken.
A BROAD SOLUTION
A society immersed in the outermost regions of the tree of social development will tend to look towards material, mechanical solutions to its problems. Are our students not learning math well? Then devise new methods of teaching with various plastic, wood and metal devices to help convey the ideas. Of course, such mechanical techniques are important and valuable, but what about students who say they hate math and teachers who neither grasp nor love what they teach? What about classrooms filled more with the rebellious spirit of MTV than with a thirst to learn? What about teachers who are more like day-care personnel who can barely teach a handful who are interested in learned while the rebellious majority find joyous entertainment in disrupting a classroom? What about poor students who should fail who receive high grades from timid teachers who are as terrified of principals who are terrified of students with so many rights as they are of the students themselves?
When students are given effective control of a classroom with loose and lenient support of school heads, then all the wonderful techniques for teaching math are almost nullified.
And why do students enter school with such relaxed, casual and rebellious attitudes? It's because the process is started at home with the same culture of permissiveness. At this point or earlier some will protest that we are going astray of the topic. This is precisely the point here. The topic cannot be dealt with microscopically as though the first stages of parental education have nothing to do with the education that occurs after this. The tree is one living interconnected whole, not a collection of separate branches that have been stuck together. The illusion of separateness derives from seeing only a confusion of branches, twigs and leaves on the outside without seeing how these proceed from larger branches that derive from major limbs that all grow from one central trunk. Superficiality leads to many illusions.
In short, what happens during the child's earliest education in the family will largely determine how well teachers in schools will be able to do their work. That is, this is not a problem for a teacher, or a school, or a Board of Education, or a government to solve. It's a problem that the entire society must solve as a cooperative team. And the first step towards doing this is to recognize the deceptive illusions bred by seeing only the surface of issues, of seeing only a myriad of small areas to be dealt with by specialists, one for each area. Piecemeal superficiality won't work. The solution requires a concerted effort by an entire people. That's how civilizations rise.
If a culture has sufficient depth to grasp the extreme importance of education, such as we see in the Far East, then teachers will be considered highly, have a high social status and be paid well. If education is not highly regarded, then schools will crumble from neglect; teachers, aside from those few devoted souls, will be teachers because they couldn't get any other work, or because it's just a stepping-stone to something better; and many will have just a smattering of a subject gained through a few education courses and have no real love for the subject. They will pass this same lack of love and shallowness of knowledge on to their students. This is a program for the slow and gradual march towards the suicide of a civilization.
Why do Chinese and Japanese excel in American schools and universities? Confucius had a lot to say about education that still echoes down to the present day. In Japan a teacher has a special title "sensei" added after his name and is shown the greatest respect. No East Asian student would dream of showing the disrespect and disobedience to teachers that is so common in the USA. And these Asian students must work hard for high grades. The free "A"s and "B"s to students who have learned very little is very rare if not nonexistent in the Far East. We've had to swallow the bitter pill of seeing these Asians surpass the USA in matters of business and industry through their greater sense of cooperation and service. Now we need to take a sober look at lifting a rapidly-declining Western-American educational system up out of the muddy waters into which it has sunk. Teaching must become a noble and highly respected profession to which the very most qualified must strive to attain.
As one ancient stated, teaching is not a matter of pouring knowledge from one mind into another as one pours water from one glass into another. It is more like one candle igniting another. Each candle burns with its own fuel. The true teacher awakens a love for truth and beauty in the heart--not the mind--of a student after which the student moves forward with powerful interest under the gentle guidance of the teacher. (Isn't it interesting how the mention of these two most important goals of learning--truth and beauty--now evokes snickers and ridicule, almost as if by instinct, from those who shrink from all that is not superficial.) These kinds of teachers will inspire love of math, while so many at present diffuse a distaste for it through their own ignorance and clear lack of delight in a very delightful subject
SPECIFIC SOLUTIONS -- THE MECHANICS
When the comprehensive plan to create an excellent environment for learning has been approached by an entire society, when teachers are in control of classes and supported by their superiors without fear of being sued by students or parents, when there is an atmosphere of learning among a class rather than arrogance and rebellion, then the mechanics of teaching are important too and must be carefully considered. Here are the thoughts of one teacher of mathematics whose experience is largely in Africa, Asia and the Pacific--and sometimes in an American system.
THE COURSE OF LEARNING MATH
Mathematical proof occurs deductively. Learning occurs mostly inductively--particularly in the earlier stages. The failure to grasp this has led to mass failure to learn mathematics. The "New Math" of a few decades ago failed largely because of this. If Set Theory comes first in teaching, then why was it developed only recently? We need a better model of the course of development of mathematics so as to best teach it.
The development of mathematics through the centuries might be compared to a seed of a tree that starts its growth near the surface of the earth. From this point it grows in two opposite directions: up and down. Mathematics can be thought of as the art of concise symbolized abstraction. We start from the ground level of concrete experience and then develop upwards into practical developments and extensions and downwards towards logic, towards ever more fundamental bases on which all above it is built.
Deductively, we move from bottom to top. But we learn from the ground level of concrete experience towards abstraction in two different directions. This is how we learned as a species. Why shouldn't this be how we learn as individuals? Shouldn't the course of historical development give us the pattern by which individual human minds are to rapidly retrace this evolution? Our bodies as embryos seem quickly to retrace the long evolution of our species. Perhaps the development of the thinking mind might best follow this course as well.
When we strip the cultural flesh off of the living, growing body of knowledge and present only a dangling lifeless skeleton, we should not be surprised when young minds feel uninterested. When we plunge right into abstract concepts without starting from concrete examples that give rise to these abstractions, should we be disheartened that the students have learned little more than memorizing how to manipulate symbols without understanding what they are doing? Why is it that the mention of "word problems" elicits groans and moans from so many classes if not because of this? To bring math to life we need to follow the course of history letting students make the discoveries ancient Egyptians or Mesopotamians did. There is no reason history must be separated from math and science. And there is no reason that learning math has to occur only inside a classroom.
What better way to arouse a fascinated interest in young students by taking them out into a field and giving them challenges to solve, perhaps in groups, such as devising a system of distance and area measurement, laying out a rectangle and a square and measuring areas with string and pegs while telling them that they are reliving the experience of the ancients. Then, when these students have the concrete experience firmly in mind and are excited with the challenges and discovery, then the return to the classroom where these ideas are given an abstract symbolic form will dawn more easily.
As the teaching of mathematics unfolds ever more time can be spent in the classroom with symbolic representations and manipulations. But perhaps the earlier stages are best spent with concrete experience. And movement towards the logical roots of mathematics should occur apace with its practical development in the other direction--just as we learned it collectively over the ages.
Though others have already made a strong case for students working on problems in teams, it won't hurt to underline this important idea. How will students learn to work on teams in the workplace if they don't get some experience before this? The extreme emphasis on grading a student as an individual is often an obstacle to this. But if we re-evaluate our reasons for grading and think of quizzes and tests as learning devices more than as means of evaluation and then reserve final grades as based only on a few examinations, we might get a fairer appraisal and surmount these obstacles.
There are more points to be made but these are sufficient for a brief essay. As a final note, let it be re-emphasized that important as these mechanics mentioned at the end are, it is the social foundations that are reflected in the attitudes of an entire culture that determine how the course of education develops. When these essential tentpoles are slowly eaten away by the termites of negligence the whole tent of human civilization reaches a limiting point and suddenly comes down. That has happened to each of the thirty or so civilizations of the past, so we are not justified in an casual dismissal of the notion without any serious consideration of the matter.
We haven't discussed the means by which entire societies are imbued with a new set of positive constructive attitudes that move this whole group up to higher levels of advancement in civilization. That is another topic outside the scope of this essay.
David R. Garcia
Manzini, Swaziland (Africa)
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