A couple of months ago, my friend Roger and I, plus our families, went to visit the Amon Carter Museum in Forth Worth, Texas. Out goal was to see the collection of works by the two great artists of the American West, Frederic Remington and Charles Marion Russell. We spent a couple of hours at the museum.
A typical Eastern European, I was interested in the paintings and sculptures simply for their inner harmony and beauty. The points I was looking at were the geometric perspective, the expression of movement, the use of colors and shades to depict perspective and light. I noticed the influence of Impressionism (which I happen to like a lot), and the earlier more academic, but less skillful forms. The places, the people, and the historical accuracy of the paintings were of no consequence to me. I knew a lot about the history of the American frontier but the specific context of the paintings wasn’t of a great interest to me. I would have been just as excited if the works depicted not cowboys and horses but Tusken Raiders and Banthas. After all, art is worth when its art for the sake of art, and its inner harmony and beauty is what matters the most.
Roger, on the other hand, is American, educated in an American school, then joined the US Army (the Green Berets) for 20+ years, then after retiring from the Army, went to an American seminary. He is a descendant of cowboys who moved west as the frontier moved, until they reached California. Every painting and sculpture had a little plate on the wall next to it, with particulars facts about its setting and history. Roger didn’t head out for the exit until he read every single one of those plates, learning much about the particulars for every one of the art works displayed in the museum. Whether he was impressed by the academic value of the paintings I can’t tell; I am certainly sure he didn’t care for the influence of impressionism. But the fact that it was art about cowboys and frontiersmen, about his particular heritage, was very important to him. Art is valuable for its contribution to our understanding of history, as far as Roger is concerned, as part of something that practically inspires us how to live today.
Roger and I exemplify the differences I gave in the previous article, “Math Education: The One and the Many,” between a “Greek” and a “Roman” model for education and understanding of reality. I am “Greek.” Roger is “Roman.”
Whose view is better for appreciating art?
James Nickel’s book, Mathematics: Is God Silent?, is the only non-fiction book that I have read three times from cover to cover. There are some non-fiction books which I have read twice; there are more that I have read once; many are those I haven’t even finished, judging them to be worthless of my time. Nickel’s book has a special place in my library. On one hand, it is about mathematics, which has always been my favorite subject. On the other hand, it talks about mathematics from the perspective of the Trinity, the very foundation of the Biblical worldview. The two make an exciting combination, as far as I am concerned.
Nickel’s main point in his book is this: Mathematics is not neutral. Our view of mathematics depends on our general worldview; and therefore our understanding and development of mathematics depend on our worldview. Different cultures do not have the same view of mathematics, neither do they have the same mathematics or math education. The partial successes of civilizations in history in the development of mathematics were due to the partial consistency of their religious worldviews with the Biblical worldview. But when those pagan worldviews grew epistemologically self-conscious and reached the point of final antithesis with the Christian worldview, mathematics reached a dead end. The rationalistic Greeks and the pragmatic Romans are among the many examples. Other examples abound in the Muslim world, India, China, and other civilizations. It is only when the Christendom consistently developed and applied the Trinitarian model to the fields of knowledge, science, and education, the world saw its first revolution in scientific advance and educational development. Consequently, with the loss of the Biblical worldview in the West, mathematical thought and education gradually lost the momentum they had inherited from the Christian centuries. The old models of learning and education – rationalist and pragmatic, “Greek” and “Roman” – crept back in, and mathematics once again reached a dead end, and children come out of school ignorant about mathematics.
The ultimate question of any worldview and any philosophy is this: Is the world essentially “one,” or is it essentially “many”? Is unity or plurality ultimate? Is there an underlying reality that transcends all individuality and diversity, or is diversity reigning supreme, with no existing or recognizable patterns or principles that govern reality? The answer to this question will give us the answer to the foundation of our philosophy of math education.
The Biblical worldview has the following answer: There is equal ultimacy of the one and the many in the Godhead, and therefore there is equal ultimacy of unity and plurality in the universe. God is both One and Many, and therefore the world He has created is both unity and plurality. Neither the “Greek” nor the “Roman” model is valid in the Biblical worldview. The Trinitarian model must transcend those two models.
James Nickel gives an excellent summary of the components of a worldview; a worldview must have its own metaphysics, epistemology, and ethics. These three – taught consistently with the Biblical worldview – must be present in the whole system of Christian education, applied to the specific fields of study. Without one or two of them, a curriculum would be lacking. We will now see how they apply to the field of study of mathematics and math education.
The greatest metaphysical question about mathematics is this: Is mathematics a separate reality, with its own logic, beauty and inner harmony, or is it just a property of the physical reality itself, without independent existence? To put it in religious terms, as some mathematicians like to do, is mathematics “the mind of God,” or is it just a product of His works?
The answer to this question will determine the position of the child in the process of math education: Whether he is a passive mind which passively gets filled with the impersonal, rigorous law-order of mathematics, or an active explorer, a wanderer who has a world of things to discover but knows not where to start and what he is looking for.
If mathematics is a separate reality, it must have its own system of law that keeps it together, and that system of law is not dependent of the human mind and its operations, neither is it dependent on the vagaries of the material reality. Our curriculum then should not allow for much freedom of exploration and discovery by the child because such freedom will necessarily be subject to the failures and infirmities of a mind locked in a material body. Such a mind will have problems accessing the lofty heaven of mathematical knowledge without thorough discipline. Math education should then be conducted with the discipline and rigorousness of a Prussian military academy to make sure that the student stays on track, and can logically develop every concept of math he is learning. But such discipline is well rewarded given the fact that once the mind of the child is trained to think along the lines of abstract mathematical logic, it will be able to grasp the principles of mathematics and spot discrepancies and errors, thus protecting his mathematical knowledge from losing its systematic foundation.
But there is a problem. Mathematics as a pure reality without relation to space and time is unintelligible and therefore is not knowledge; or, rather, it is knowledge without significance. If a curriculum is focused on teaching math as an abstract reality only, the child won’t know what to do with it; and he will certainly lack any motivation to experiment with it. In fact, experimentation with math will be illegitimate under such system; experimentation by default presupposes we at least allow for the possibility of results different from the expected. But unexpected results can not be accepted in such a system; the assumption will always be that the student has made a mistake, that he has left the orderly world of logic and has either messed up the experiment or has misinterpreted the results. Education based on this premise will necessarily degenerate into educational legalism which kills any initiative and experimentation.
The opposite metaphysical assumption – that mathematics is simply a property of the physical reality around us – will help us unshackle the math curriculum from the restraints of an inexorable, inflexible law, and will require us to send the child on a journey of discovery. Any hands-on experience then will make the child learn about mathematics. If matter “contains” mathematics in itself one of its properties – just as color, taste, firmness, etc. – then the child learns best when he “experiences” matter. Math education then can become fun, with games, exploration activities, hands-on projects, everyday informal activities. Cooking, shopping, games, driving, can become an integral part of the education and help the child learn without the legalistic of logical formalism.
There is a problem here too: a world without a fixed law is impossible to explore. While we can “experience” matter and play with it to discover math in it, we have no assurance that any system of mathematical law will come out of it. If mathematics is only a property of reality that we discover, then whatever existing higher general principle for mathematics that exists must be non-mathematical, and therefore not provable by the laws of logic. How do we know if a principle that the child seems to have discovered is not simply the wrong interpretation of a limited set of experiences? Is any “principle” we discover a true principle? How do we know? If we find a set of mathematical laws in physics, is it possible that we find another set of mathematical laws in business, and then another in chemistry? Can all these laws be so independent of each other as are the practical fields they come from?
The problem here is between legalism and license in the curriculum. A legalistic math education may prevent the discovery of things like irrational and complex numbers, or develop the laws of probability and statistics, or non-Euclidean geometries. On the other hand, a curriculum that emphasizes too much freedom may send the child well beyond the limits of the reasonable, experimenting with nonsensical concepts like π = 4 or with self-contradictory laws of logic.
The Trinitarian model will acknowledge that math has both independent existence in the mind of God (although it is not the mind of God itself), and therefore it is also interwoven in the creation as one of its properties (although creation is not strictly mathematical in nature). Thus it will be build around the idea of the ultimacy of laws that govern mathematics together with the understanding that reality contains these laws in its own structure. Math education will then follow the pattern of the Law of God in the ethical sphere: a fence that delineates what is rational, but with large room for experimentation within the limits of that rational knowledge.
The epistemological question in math education is this: How do children learn math? Do they learn it from particulars to principles, or the other way around? Do they first learn dollar signs and cents, and eventually go to the more abstract areas of math, or should they be taught axioms, definitions, and theorems first before they can embark on applying that knowledge to the practical world of particular cases?
The development of knowledge from the principles to the particulars gives the children a grid which is necessary. Without such a grid facts make no sense. Facts mean nothing to us without interpretation, and mathematical facts without the general frame of mathematical theory will eventually be forgotten. So theoretical knowledge seem to be necessary as a skeleton for the rest of the body of knowledge to be built upon.
The problem there, of course, is that theory without system of practical laws that demonstrate it and support it is called a myth. Ancient Greek education was based on myths. These myths were about gods and heroes of the past who had different circumstances than the circumstances of the hearers of the myths; and therefore there was no system of predictable application of those myths to the lives of the present generation. When the same model is applied to math, the child’s mind will tend to be enclosed in a mode of abstract thinking; but the skills of application will lag behind.
Of greater interest to us today is the opposite model, where the child is expected to first learn the particulars, and then develop principles from them. This seems to be the established wisdom today. Even the Classical model of education, applied today, departs from the standard ancient Greek system in the first stage of learning, that of Grammar. While “grammar” for the Greeks was the learning of language for communication – the language as a comprehensive coherent system – in the modern interpretation of the Classical model, “grammar” for any field of knowledge is learning as many facts as possible as a stepping stone for the next stage, Logic. In the Logic stage, the child is expected to start making the connections between these raw facts and build a system of general principles. From facts to principles, is the direction the child’s mind is supposed to be going. Supposedly, a younger mind can not comprehend abstract principles until it has enough particulars to connect. The modern Classical model is not alone; all education in the US today is based on the idea that children can not comprehend, and therefore can not learn abstract principles. Give them enough facts to play with, and eventually they will build a system of principles from them.
But such reasoning is fallacious. Facts do not speak for themselves, and a mind that is not trained to think in principles will never develop principles, no matter how many facts it learns. Besides, there is only so many facts a child can learn; but from a limited set of facts an unlimited number of principles can be derived. Does the fact of lightning and thunder itself speak about electricity, or does it speak about personal demons that play with occult forces? When we say that 2 apples plus 2 apples make 4 apples, does this give us a principle about apples or about the number of apples, or both?
Also, it is a false statement that a child can’t learn general principles at an early age. In fact, a child learns both particulars and principles from the very beginning. At a very early age a child knows that a a chair is chair, even if he sees many different chairs of different colors, makes, and shapes. Apparently, if a three-year old child can recognize a garden chair made of wood and an office chair as under the same category of “chair,” then the child already has a concept of the abstract category “chair” that is not dependent on the specific particulars. Children who may have only seen a cat on a photo can recognize a “cat” even in the stylized drawings in a children’s book; there is no need for them to be told that that specific drawing represents a cat. They somehow know it. Children do not wait until later age to develop skills for abstract thinking; they have them from the very beginning.
A Trinitarian model for math education will recognize that the child learns principles and particulars together, not separated in time. The practical consequence of such recognition will be that from an early age both theory and application will be taught to the children. As we will see in the next article, that means that the child’s mind can be developed from an early age both to evaluate quantitatively and do logical proofs; both to understand symbolic characters and build visible material and geometric representations, both to do abstract algebra and applied arithmetic.
The ethical question is: What do we do with the math we learn? Do we learn it to enjoy its majesty, harmony and beauty? Or do we learn it to use it a practical tool in our job? Is it to be good at chess and other games? Or is it to pass exams, get in college and get a good job in the future?
The issue is an issue of motivation. Motivation is what is not included in most curricula today; modern education assumes that motivation and the right use of knowledge must come from outside sources, and the curriculum is supposed to provide only the bare knowledge of facts and theory. This problem is even stronger in math education; since modern scientists believe that mathematics of all disciplines is morally and religiously neutral, they have no use for moral issues and ethical motivation when teaching math. Math is something a student just “has” to learn because “without it, you can’t have education.” This failure to provide ethical motivation for learning math leads to what is the worst problem of our schools today: “Why do I need to learn this?” This is the question that as a math instructor at college I have heard the most; of what I know from friends teaching in high school, they have heard it even more often than I. Students approach the subject of math as something that sadistic educators have imposed on them with the sole purpose of tormenting them and consuming their time, which could have been spent learning better things, or simply in leisurely inactivity. No curriculum I have seen ever includes the ethical questions of “What math is for and why you need it.”
The ethical requirements of the Bible can be summarized in two words: obey and enjoy. We are commanded to obey God and apply His Law to our everyday life. But this obedience is not the joyless toil of a slave in a Roman stone quarry. It comes with the joy of watching God’s mighty works in His creation, and taking pleasure at participating in God’s plan. God has both work and pleasure for us. In fact, the more we draw closer to God, the more work is joy, and the more our burden is made lighter.
A Trinitarian curriculum therefore will have to, first, teach mathematics from an ethical perspective; i.e., what we can and should do with mathematics. Second, it will present mathematics as both a source of aesthetic pleasure, and also a practical tool for conquering the world.
Mathematics must be presented as a painting, or a literary work, and the student must be taught to appreciate it just as much as he is taught to appreciate any work of art. Math has its own beauty and harmony, it is a world of its own, and in it there are numberless areas that can make men enjoy that world in itself. There are beautiful intricacies that can capture the eye of a connoisseur; complex connections that resemble the beautiful details of Renaissance buildings. There are curious interdependences between structures in different fields; there are unexpected discoveries of strange regularities; there are beautiful graphs of functions that seem rather dull when written in symbols. There is order in apparent chaos, and there is wild variety of outcomes in areas that on the surface seem rigid and inflexible. There are games like chess, and checkers, and gambling that present mathematical principles in a fun and entertaining way; number games that reveal exciting qualities of numbers that the untrained eye can not spot. There are codes and cyphers and puzzles, and even mathematical poems. There are interesting mathematical trivia in the Bible, and in history, and in the universe. There are stories from the history of mathematics which rival any fiction stories; and there are characters who can be role models just as much as heroes from the military history or history of the church. Mathematics is art, it was made for the purpose that man can enjoy the beauty and the riches of God’s creation in the intricate order He has built into it. A curriculum must make sure it has this ethical purpose integrated in it.
But mathematics also must be taught as a tool for taking dominion over the world. God requires obedience, and obedience means taking dominion under the Law of God, and mathematics is one of those areas where man encounters the Law of God in the very law that governs mathematical elements and operations. Just like the general principles of God’s Law (the Ten Commandments) must be accompanied by specific case applications (the case law in Exodus 21-40) to form one Law, so theoretical math must be taught together with its applications in home economics, business, physics of movement, construction, dynamics, electricity, etc. Business math can’t be separated from theoretical math; business statistics can’t be separated from theoretical statistics and probability. Math textbooks must include sections on practical applications of theoretical principles into mathematical models for physics, engineering, navigation, etc. (I never learned the practical applications of much of calculus and probability until learned navigation and naval weapons in the naval academy; and there are areas of math whose application I never learned at school, and only my father’s skill of teaching made me understand how to apply them.) Current statistics must be analyzed as part of the curriculum on the basis of general principles so that the student can learn to apply those principles to future events. Taking dominion over nature is a reward in itself; when the child is taught to take dominion based on the theoretical principles he learned from the textbook, this will create an additional motivation for him to excel in both knowledge and application.
Thus, a Trinitarian math curriculum based on the equal ultimacy of the One and the Many must teach math as both a separate reality and a property embedded in the material world, thus making the child both a recipient of general principles from above and yet an active explorer in the world below; it must start from an early age with both principles and particulars; and it must teach math both for aesthetic enjoyment and entertainment, and as a tool for practical action.
In the next article, we will see how this will be applied at an early age. We will identify the areas of unity and plurality within the very discipline of mathematics. And will see the technical principles to be applied in math education to make a child understand math, and enjoy it.