Restructuring Math Education: The One and the Many

Math Education: The One and The Many

By Bojidar Marinov

 I am not surprised by the SAT scores comparison between the three groups: homeschoolers, religious-based private schools, and public schools. Homeschoolers beat the other two groups in every category, and consistently score above the 84th percentile in every category. Only 29 per cent of the homeschooling fathers and 13 per cent of the mothers have Master’s or Doctor’s degree. In the institutional schools these days, Master’s degree is the 100% requirement for most schools for the main subjects. Homeschooling families have more children in average (3.5 per family), and therefore they have more economic and financial problems to solve in addition to education. Given these differences, homeschoolers should have lower scores. But they consistently beat the institutional schools, defying the “expertise” of the “experts” and the “wisdom” of the compromisers in the Christian community.

I read my Bible, and I am not surprised. Education is more than delivery of information; it takes the right environment too for a child to learn self-control, discipline, self-motivation, responsibility, and the ability to make the right decisions in different situations. That environment is called family, as far as the Bible is concerned. The SAT, being a serious test not only for the knowledge the students have accumulated but also for their discipline and ability to think, shows the expected results for those of us who read our Bibles.

I am especially pleased that homeschoolers score so high above the other groups in the subject that requires the most effort in terms of discipline and logic: Mathematics. While reading and writing can be used and practiced in a variety of different ways, mathematics as a theoretical discipline requires much more. It requires the student to self-consciously apply himself to spend countless hours learning the theory and solving problems. In the schools, the student’s time is regimented; but homeschoolers need to organize their own time to be able to excel in math. The fact that homeschoolers score so high in math is just another proof that only in the home the skills of self-discipline can be taught.

But homeschoolers shouldn’t rest on their laurels. Being better than the institutional schools in the US – public and private – is not a great achievement, yet. The math education in the US in general is seriously lagging behind other nations. The comparison with Chinese and Indian public school students shows large discrepancies. In this situation, homeschool students in the US, while vastly superior to the private and the public schools in terms of SAT scores, still fail to reach the levels of mathematical knowledge and skills necessary to capture the academic field of mathematics and establish a clear superiority there. Passing SAT is not enough; a principled knowledge of mathematics in depth is what is necessary. And this is what the current math education in the US fails to supply.

Having taught and tutored my children, other homeschooled students, as well as public and private school graduates at a local college, I have been asked many times by homeschooling mothers, “What math curriculum do you find best?” Of the existing ones, I have my preferences; the textbooks I like are the Chicago University School of Mathematics. They are the closest to the way I was taught Math back in Eastern Europe. But I think even they are inadequate to creating a true, working, Christian Math curriculum that will both help children understand and appreciate mathematics in all its beauty, and also build in them the skills to pass any exams and solve any practical problems they may encounter at any level of their academic or professional development. As it is of now, this hasn’t been easy, at least not for a sufficient majority of American students.

I am beginning to work on this project, with a group of like-minded friends. I hope God will give us the inspiration and the self-discipline to be able to turn out a useful product in time for our readers to use.

As a Reformed Christian, I know where to start: If there is a practical problem, it must have a theological root. If ideas have consequences, then the problem with the math education in the US must be rooted in the wrong worldview in the foundation of the very understanding of mathematics and of education. And since the Christian worldview starts with the Triune God, the ultimate One and the Many, then the philosophy of our math curriculum must start from the One and the Many. We have some beginnings in Vern Poythress’ essay in Foundations of Christian Scholarship[1] and in James Nickel’s excellent study, Mathematics: Is God Silent? But a comprehensive philosophy for math education is yet to be developed; as well as a curriculum that follows from it.

In this and the following several articles I will try to lay out my ideas of how the Biblical foundation of the Trinity must be made the basis of our teaching mathematics. My conviction is that only when we base our curriculum on such foundation, having a worldview rooted in our knowledge of God, we can successfully teach math.

The Greek and the Roman Models of Education

There are two basic models for education that are used today around the world. I call them “Greek” and “Roman.” You can call them “platonic” and “aristotelian”; or “rational” and “empirical”; or “realist” and “nominalist”; the names don’t matter. What matters is that they are based on the philosophical emphasis on unity (Plato) and the philosophical emphasis on plurality (Aristotle). Of course, as is with everything, nowhere can either of them be found in a pure form; the difference is usually in the degree one or the other is emphasized.

The Greek Model

The Greek model is the model of science as an ivory tower, far exalted above the mundane world of everyday practice. Under this model a subject is viewed as a world of its own, with its own inner harmony, beauty, and self-existence. Mathematics is then a set of “eternally existing ideas” which have no predictable relation to the practical world of senses, work, or applied science and technology. Man understands those eternal ideas through setting his mind on a higher plane where only pure and perfect forms exist, with only pure and perfect relations between them. Such approach places heavy emphasis on the theoretical learning of the subject. Practical examples are seldom emphasized; “practice” is usually contemplative rather than applied to the real world. Application problems or projects are usually constructed to be ideal and demonstrative of a theoretical principle, not of real situations in practice.

Such approach produces good theoreticians but poor engineers. While not always openly expressed, it looks down on the “dirty” world of practical applications.

A perfect example for such approach to mathematics is the school of Pythagoras and his followers. In the worldview of the Pythagoreans, the world could not contain anything else but pure forms, numbers, and ratios. So deep and unshaken was their faith in the rationality of the world in terms of numbers that they refused to accept any evidence – material or logical – that there are entities and ratios in the world that are not rational. When Hipassus, one of the Pythagoreans, eventually proved that the ratio of the diagonal to the side of the square (or of the hypotenuse of an isosceles right triangle to the leg) was an irrational number, i.e. not expressed by a simple fraction of two whole numbers, his comrades reacted in horror. According to one legend, they threw him overboard at sea; according to another, they built him a tomb while he was still alive and pretended he was dead.

Very much the same approach was used by the schools in Eastern Europe back in the 1980s; I learned under such approach. The theoretical side of math was well developed and taught quite thoroughly in the schools. While there were specialized high schools for learning math and sciences, the requirements and the expectations and the requirements were high for everyone. Calculus, geometry, and logic were an integral part of the high-school curriculum; students were expected to master the algebra of functions and geometrical proofs before they graduated from high school. Of course, this attention to theory tended to de-emphasize the practical applications to technology and engineering. This de-emphasis continued through higher education. At the end of the day, while the Soviet Union had a number of bright nuclear scientists of world class, it lacked nuclear engineers to man the Chernobyl nuclear plant. Of what I know of the math education in China, India, and Japan, it is not much different. Theory is emphasized, application is neglected. With all the bright mathematicians and scientists coming out of these regions of the world, their contribution to innovation and engineering is still negligible, compared to the United States and Europe.

A current example in the US today of a “Greek” approach to education is the Classical approach of the Trivium. I understand that most of the Christian homeschoolers that use the Classical approach are working to redeem the approach; I am not a stranger to the Classical approach, having translated Harvey and Laurie Bluedorn’s Teaching the Trivium into Bulgarian, and also having taught at a Classical homeschool co-op. But the attempts to make it less “Greek” notwithstanding, the very structure of the Classical approach presupposes an “ivory tower” approach to all knowledge. According to the very philosophy of it, the main body of learning in the Classical approach – the Trivium – ends with Rhetoric, i.e. the ability of the student to verbally convey an idea, rather than apply it to technology or practice. (Some modern Christian Classical educators include practical training in the Rhetoric phase.) Only then a Classical student goes to the next level, the Quadrivium, where some more practical knowledge is accumulated, and it is still under the same trivium-structure, and that only as a preparation for future theoretical studies into philosophy and theology. Applied science and mathematics, geometry, engineering, etc., have to be taught as a “side dish.” In fact, I have heard at least one Classical educator declare that mathematics and engineering were taught to the slaves in the ancient world, while the “liberal sciences” were for the free citizens, and therefore he recommended that humanities are emphasized and that math and sciences can be postponed till very late in the child’s teenage years. It is not a mere coincidence that Dorothy Sayers, the modern prophet of the Classical approach, had played with unitarian/modalist views concerning the nature of God in her theological writings.

The Roman Model

This model is based on the die-hard pragmatism of the Romans: If I can’t use it here and now to build a road, a bridge or an aqueduct, I don’t need it. The value of all knowledge is established by its immediate practical results. Theoretical speculations and generalizations are to be avoided since they make the brain leave the world of the concrete and material. Application is everything. Laws of logic, of mathematics, of nature have no meaning since they are by nature mental constructs; man discovers knowledge by the process of trial and error. And even when he discovers knowledge, he has no assurance that it is valid because in the world of pragmatism there is no discernible pattern that can help a man predict the exact outcome of of the same action in different circumstance. Such prediction will take mentally constructing ideal patterns; but since patterns do not have material existence, they are worth nothing.

Of course, much of our modern science needs theoretical reasoning before it can develop further. In mathematics, areas like calculus, number theory, and even computer science have first appeared as purely theoretical constructs before they could be applied to practice. While every mathematical discipline has real applications in the real world, not every such application is directly obvious to the human senses; sometimes theoretical speculations are necessary before the understanding of the practical application can be acquired. No wonder that the Romans, for all their love for practical achievements, never invented anything of their own; until the end of the Roman Empire in the 5th century, all the technologies it used were taken from other nations – Phoenicians, Etruscans, Greeks, Egyptians. Even the famous Roman Law, the greatest Roman achievement, was only a motley collection of isolated customs, and mythological precedents; not until the Christian Emperor Theodosius did the Classic world have a unified law code; and not until the Renaissance a unified theory of the Roman Law was constructed and taught as a system.

The best example of the difference between the Roman and Greek model is seen in the death of Archimedes in the siege of Syracuse by the Romans. Even though he was a Greek theoretician, Archimedes had the rare ability to apply his theoretical knowledge to practice – especially to build siege machines that wrought havoc among the Roman armies besieging the city. The Romans never cared to understand the theoretical knowledge that made Archimedes such a successful inventor but they could pragmatically appreciate when a device worked. The Roman general Marcellus had issued an order that Archimedes shouldn’t be harmed and brought alive to him to be used for his technological skills; but the Roman soldier who entered his house obviously saw nothing to tell him he was at Archimedes’s house. Archimedes was busy drawing circles in the sand, thinking about some theoretical problem. “Don’t disturb my circles,” were his last words, before the soldier killed him to rob him of his mathematical instruments which he thought to be some tools of practical value. Later Marcellus had the soldier executed for disobedience.

Modern American education – and especially math education – is heavily “Roman.” It is entirely focused on the pragmatic value of knowledge. Math is separated into “theoretical” and “practical” math, and emphasis is laid on the latter over the former. Even in colleges, there is this peculiar American discipline that is not known anywhere else in the world: “Business math.” The obsession with “application problems” is also very specifically American. The underlying motive is that if the children have enough application problems to solve, this will necessarily build in them the ability to apply math in practice, which is all that matters. The exams are focused on specific applications; and the multiple option tests are built in such a way as to completely ignore the theoretical part of math education. (Our exams in Eastern Europe back in the 1980s were predominantly theoretical, even in the problems section.) The books that give preparation for the exams are also based on that Roman model – very little teaching of patterns and unified principles, and a focus on the problems themselves.

The favorite math curriculum to most homeschoolers, Saxon Math, is the Roman model on steroids. The very organization of the material is structured not to create an ordered system of theoretical math knowledge so that the student understands mathematics as a system of its own. The strategy of Saxon Math is rather [to] train the instincts of the student to deal with a problem when he encounters it at the exam. The goal is strongly pragmatic: Pass the exams. I have talked to students who have passed exams after having used Saxon Math all the way; while they were successful in passing the exams, they still had problems understanding basic relations in mathematics, or even mathematics as a whole. And Saxon Math is not the only one. The public and private schools approach is just as Roman, and that makes the majority of students quite unable to comprehend algebra at higher levels; and, as we will see later, algebra is the study of unity and patterns in mathematics, and is by nature a theoretical study before it can have practical applications.

Toward a Trinitarian Model of Math Education

The two models I outlined above correspond to the two extremes all non-Christian philosophies fall into: either monism (a belief in the ultimate oneness of reality) or polytheism (a belief in the ultimate manyness of reality). While I am not going to go into details about those two extremes, an important book to read in this regard is Rushdoony’s The One and the Many.


  1. Gary North, ed., (Vallecito, California: Ross House Books, [1976] 2001), ch. IX, “A Biblical View of Mathematics.” []

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