We are getting now to the question about particulars: What should we teach? Even when we know how to teach and what the goals of our curriculum are, this knowledge can only become flesh when we have the specific material for our curriculum. Start with the general components, and then work down to the details of each course.
The components of mathematics, as they are taught today, are clear: arithmetic, algebra, geometry, logic, number theory, calculus, statistics, etc. Some students go through those and even excel in them. Some even develop proficiency to use much of what they learned in their jobs. But very few know the purpose and the proper place of each one of those components of mathematics in the discipline of mathematics itself, and in the broader world of math applications. I have had students who repeatedly asked me, “What do we need algebra for?” I have also had a co-worker at college, a math professor, who admitted to me he didn’t understand mathematical statistics well enough to teach it; he just never understood what the practical value and the meaning of all the different concepts was. Not knowing the place of a discipline in the overall grid of our learning and knowledge certainly doesn’t help. Therefore we must analyze mathematics itself, see what it is composed of, and see the place of the different components in our curriculum
Vern Poythress, in his path-breaking essay in The Foundations of Christian Scholarship, gave us the profound insight that even the simplest mathematical statement contains in itself the truth of the one and the many, and therefore doesn’t make sense in any other worldview except in a self-consciously Trinitarian worldview:
2 + 2 = 4
From a strictly monist perspective, the numbers 2 and 4 make no sense, as Poythress demonstrated. The very concept of a number presupposes some existing plurality in the universe, separate identities for persons and things. A Zen Buddhist, for example, can not accept this as representing reality but only as a symbolic image of an illusion of identity that is not there in the first place. “What do you mean by 2?”, he will ask, if he is consistent with his view. (Of course, they seldom are. They need to borrow from the Christian worldview if they are to remain sane.) Therefore the study of the properties of numbers will be a study of the properties of plurality in the universe. We call this study of numbers arithmetic.
From a strictly pluralistic perspective, the signs + and = don’t make sense. The very concept of operations between individual entities presupposes some kind of unity between the different entities. If one is a consistent polytheist or pluralist, or even a consistent agnostic, one chair plus one chair can not make two chairs because we can not define an abstract category called “chair” and then unite different chairs under it. One chair plus one chair will always equal one chair plus one chair, and no unity between the different chairs can be postulated. Therefore the study of the properties of mathematical operations will be a study of the properties of unity in the universe. We call this study of operations algebra.
Arithmetic is properly the study of numbers. In our classification based on the one and the many, arithmetic therefore is the study of particulars.
American math education, as based on a pluralist/particularist worldview and educational model (see my first article of this series, “Math Education: The One and the Many”) has always emphasized arithmetic to the exception of all other branches of mathematics. Arithmetic is taught almost exclusively in the first 7 or 8 grades, with much less attention to other fields like geometry, logic, or algebra. Many of the homeschool curricula also focus on arithmetic and basic calculations in the first several years. Some educational models openly state that simple rules for addition, subtraction, multiplication, and division is all that a child needs well into his teens. Early “business math” for many is preferable to algebra or formal logic; and “business math” is simply adding dollars and cents, or learning to work with measures and weights. Families have told me that using basic arithmetic in cooking and shopping for the girls and repair work for the boys is all they had as math for many years.
In the formal course of education, even where algebra is taught, there is still much preoccupation with arithmetic in what I consider an obsession with application problems. Algebra in many textbooks is taught and practiced as simply substituting numerical values for variables in equations; which of itself is reducing algebra to simple arithmetic. Geometry is also reduced to arithmetic – calculating areas, perimeters, and lengths of sides by simple application of formulas.
And of course, that glorious crown of the American education – the multiple option tests – mainly encourage learning of calculations and very little else.
But reducing arithmetic to calculations, simple or advanced, is an erroneous approach. Arithmetic is not simply numerical calculations, it is the study of the mathematical properties of the plurality in the universe. And if so, then our modern arithmetic is lacking severely.
From a philosophical and ethical point of view, the plurality as an ultimate reality in our world requires exploration. Proverbs 25:2 says, “It is the glory of God to conceal a matter, But the glory of kings is to search out a matter.” Of course, the verse doesn’t speak about the general principles and the ethical laws He has given man; God doesn’t expect man to discover what is good and what is bad on his own. But in terms of the particulars of God’s universe, God as if conceals things from man, and man searches them, and grows in knowledge and practical wisdom through that exploration.
Arithmetic, then, being the study of the properties of plurality, must be viewed and taught as a field of exploration. The world of numbers and combinations of numbers is huge and full of all kinds of diversity and new things to explore and play with. There are certain properties of numbers that are common to all numbers. But there are also properties that are specific to specific numbers or groups of numbers. Numbers have an amazing ability to present us with all kinds of new discoveries – sometimes it is regularities, sometimes it is the chaotic nature of some numbers that deserve exploration. Some of those properties present themselves in geometric figures (think of the golden section). Others can be found by using simple mathematical operations that lead to amazing harmonies (the Fibonnaci sequence). And so on.
Of course, this starts with simple calculations at an early age. A child must learn to do 1 + 1, 2 + 5, 10 – 3, etc. Then he must learn the multiplication table. Then division. Even before this process has started, he must learn that numbers have symbols which we use when writing. This in itself is a serious task and children who start late sometimes have problems with the position system that is so foundational for our modern arithmetic. In the process, the child can learn to see regularities that later will help him manipulate numbers much easier. For example, all the multiples of 9 in the simple multiplication table follow certain pattern that is particular only to 9 and no other number: 09 18 27 36 45 54 63 72 81 90. Added, the digits of each product give us 9. The multiplication table can also be used to present another concept: even and odd numbers. Both groups have particular qualities that can also be explored. Then the divisibility rules can be established, and practiced abundantly. And so on.
Of course, this all must happen in the context of practical applications. While arithmetic must be taught as a study of numbers for themselves, as a world of its own, a Trinitarian curriculum must apply that knowledge to the real world and prepare a number of practical applications for what the student is learning.
So far there is nothing new. Our present system is doing the same in the first three grades. The problem is, the next 4 or 5 grades it doesn’t do much more than that.
Numbers can be used and explored even further. There are constructs of numbers that can be built easily by the students and their properties explored. For example, the Pascal Triangle will be used later in algebra and number analysis but students must be introduced to it very early. In my experience, we were taught to use the Pascal Triangle to construct complex substitution ciphers. Encryption ciphers are always fun to work with, especially for children, and integrating a number tool like the Pascal Triangle in games of encryption makes children learn more about the properties of numbers.
Drills and exercises are important when it comes to learning particulars. The problem is, when arithmetic is only viewed as a discipline of numerical calculations, drills and exercises tend to be focused only on calculating a result, multiple times with different numbers. While children learn a pattern, this seldom teaches them the properties of the numbers themselves. Exercise by itself doesn’t teach, unless it is geared towards more exploration into the nature of numbers.
This has a solution. The world of math books is full with interesting examples of math problems that require not only repetitive drill but also exploration on the part of the student. One example that is my favorite to give to students (and very few actually solve it!) is this old problem:
“How old are your three daughters, Mr. Jones?”, the tax-collector asked. “Here,” Mr. Jones replied, “if you multiply their ages, you’ll get 72, and if you add them, you’ll get the street number of my house.” The tax collector looked at the plate with the street number and said, “That’s not enough information, Mr. Jones.” “Well,” Mr. Jones replied, “I can’t tell you anything else except that my youngest one has a dog with a wooden leg.” “Thank you, Mr. Jones,” the tax-collector said, “I now know how old your daughters are.” What are the ages of Mr. Jones’s daughters?
A problem like this requires understanding of the properties of the number 72, and requires multiple repetitive arithmetic operations in a context of discovery and fun rather than dull drill. It also requires resourcefulness, especially when the useful information needs to be separated from the chaff – what the mathematical significance of the statement “my youngest one has a dog with a wooden leg” can be. And there are literally hundreds or even thousands of similar problems that children can be given to try their teeth on, and in the process learn the properties of numbers.
Then arithmetic goes into some even more exciting fields. One is the study of numbers in higher-order operations – powers and radicals. This will bring the child to the knowledge of rational and irrational numbers (of course, assuming that he was taught fractions and decimals before that). Another one is the study of integers, which is properly called number theory. While the name “number theory” may sound scary, in reality it is pretty simple in the beginning; in fact, some of the concepts are taught in 4th and 5th grade, like the greatest common divisor and the least common multiple. And in fact, there is a special part of number theory which is called “recreation number theory”; the name shows that it can be used for fun and still help the child learn. We won’t have enough space to explore all the possibilities here.
The most important principle of all is that arithmetic must be taught as a field of experimentation and discovery for the child. It shouldn’t be reduced to simple calculations. While calculations are important, they must be all geared toward discovering new properties of the different numbers. The beauty of arithmetic is not in simply adding and multiplying numbers – in fact, it is this reduction of arithmetic that has made it look ugly to students. The beauty of arithmetic is in training the child to explore the properties of numbers, and of groups of numbers, and find pleasure in the discoveries of new patterns, regularities, and qualities. When a child learns to appreciate the beauty of numbers, then he will be able to easily manipulate numbers for his practical solutions.
All exploration and discovery must be made in the framework of fixed laws and principles, otherwise discoveries mean nothing and will soon be lost. There must be unity in mathematics, a grid of fixed mathematical laws that give meaning to the particulars.
Algebra is the study of that unity. It is the study of the properties of the operations that unite quantities with quantities and values with values. It studies the place of addition and subtraction, multiplication and division, powers and radicals, powers and logarithms. It studies the relationships between those operations. It is not concerned with the individuality of numbers; in fact, that’s why we use letters in algebra so profusely, because we do not care what specific numbers participate in the operations. We want to find the properties for every operation for any set of numbers that it may be applied to.
The American educational theory has it that algebra must wait till later. Students usually start learning algebra in earnest very late in their school years. By that time their brain has been conditioned into thinking only simple arithmetic. When the first abstract rules of algebra are introduced, the student is confused. Students who have been taught only particulars can’t easily understand abstract values like variables and identities which are part of the study of unity in mathematics. The question I always encountered, even in college, was, “What is x?” When a student is told that it is a variable, that its value is not fixed, his brain blocks. “How do you calculate anything if you don’t know the value?” That’s the point. We don’t calculate in algebra. We analyze expressions, and we reduce them or rework them. We solve equations. And so on. Calculation is only a small part in algebra. Algebra is about learning principles.
Of course, most people do algebra everyday, they just don’t realize it. A good example is the price war between a Shell and a Chevron gas stations about 2 miles from my home, on state road 529 and Eldridge in NW Houston, two years ago. We had the lowest gas prices in the nation on that junction because of the rivalry between the two Indian owners. Eventually, the Shell owner put up a sign that covered the price for Regular, and it said, “1¢ lower than Chevron across the street.” The owner forced everyone of us to do algebra. He didn’t give us a specific number; he gave us a formula. Now we had to apply that formula to any number we got from the other side, in order to calculate his price. Simple algebra, but still algebra.
Do we have to wait until late to start teaching the child algebra? That wouldn’t correspond to a Trinitarian curriculum. Just like we start with simple arithmetic from the very beginning, we should also start with simple algebra. Remember, the child is not dumb, he can think abstract from the very beginning. He knows the general category of “chair” even if a hundred chairs are presented to him, wildly different in shape, material, and color. He knows to recognize a cat even from a stylized drawing in a coloring book. Therefore, if the child is started early with the principles of algebra, he will be able to learn the principles much earlier than the modern education theory has it.
To start teaching those principles, we need to use that marvelous algebraic tool, the equation. The simple arithmetic problems given to children can be used for that:
8 + 5 = □
Then it will be redone in the following way:
8 + □ = 13
Then – and this is the most important part of algebra – it will be shown that this equation is identical to another equation which is simple arithmetic:
13 – 8 = □
Now the child will be able to make the connection between the operations of “+” and “–” in the same equation which means he will learn something about the properties of the very operations themselves. He will discover that “+” can be transformed into “–” for any combination of numbers, and this will give him knowledge about the unity, the uniform laws that rule math and do not change with the change of the particulars. Similar problems will be given to see the connection between multiplication and division. Then the empty box will be replaced with x and y, and more complicated equations will be given for learning the relations between several operations in the same equation. Then, once the concept of variables symbolized by letters has been introduced, other concepts like simplification of algebraic expressions (x + 3x + 2x = 6x) can be introduced, or the first steps into rational expressions can be made (expressions in the form of fractions).
By fifth or sixth grade, the child can have a very solid foundation in algebra if the education in algebra is not separated from the education in arithmetic; and if arithmetic is used as a tool for algebra, and vice versa. We won’t have the room to explore all the possibilities here.
Again, the overall idea when teaching algebra will be developing the knowledge of the unity, the fixed laws that govern the world of mathematics. The properties of mathematical operations and their relation to each other is the goal of education in algebra are the properties of unity in the world of mathematics, and they tell us something about the properties of unity in the world around us.
We will leave geometry, logic, statistic, calculus, for another article in the future.