When I investigated the Math Investigations program being implemented by the Prince William County Public Schools (PWCS), I discovered Contructivism as the underlying basis for this program. Constructivists believe we learn by building upon the knowledge that results from previous experience. I found this idea intriguing. As I studied the ideas of the Constructivists, however, I notice a gaping hole in how they put their ideas into practice. What role does religion have as a foundation for learning?
Religion and Logic
Consider the issue of proof. In this article (here), two of the academics who initiated Investigations in Number, Data, and Space® (PWCS’ current textbook series for elementary school children), Michael T. Battista and Douglas H. Clements, discuss how to teach children the concept of mathematical proof. In this article, they approach the issue at a rather high level. For example, they assume children will accept the notion that something can be logically proved.
The belief that the universe is orderly and follows rules actually began as religious belief. Our ancestors originally believed in a universe ruled by numerous gods and spirits (Roman Mythology, for example). With such beliefs they found any odd happenstance easily explained. “The gods did it.” Nature had no rules, just fickle gods and spirits. The belief that there is only one God and that this God loves us provided a new perspective. It made sense to apply Aristotelian logic. In fact, some people even went so far as to assert that the existance of orderly universe proves the existence of God (see here).
The Validity of Mathematical Proof
It might also be interesting to hear Battista and Clements discuss the validity of mathematical proof. Mathematics is very much like a language. Consider the first task God set for Adam (Genesis 2:19-20). Adam named the creatures God had created, and with this task God set man above all the other animals.
When we think rationally, we think in words. Each noun provides us a symbol denoting a model of something we find in the real world. Each verb provides a symbol that models a process we can find in the real world. Adjectives and adverbs narrow the meaning of the words they “modify.” If I say “tall building,” then the adjective “tall” helps the listener better understand something about a particular building. Similarly, if I say a cup quickly overflowed, my listeners know more about how fast the water was running.
Like any other language, mathematics provides us symbols and rules that help us to model the real world in greater detail and with greater accuracy. Also like any other language, mathematical symbols are only abstractions. Only in the abstract does 2 + 2 = 4.
When we take a measurement in the real world, that measurement is an approximation. We cannot exactly measure two pounds, two inches or two quarts. We can only measure abstractions with infinite accuracy. Thus, because money represents an abstract concept, two dollars is exactly two dollars. So while two dollars plus two dollars is four dollars, two miles plus two miles is only approximately four miles. Because the term “miles” models a real distance, our measurement must contain some degree of error. Because we use mathematics to model the real world, what we find in the real world, not with mathematics, sets the standard of proof.
Abstraction Versus Reality
What do children learn about proof? With television, computers and even books, we immerse our children in abstractions. In the abstract world of fiction, we teach children of an imaginary universe of perfect heroes, of talking animals, and of spotless worlds. In the real world, our children learn that is impossible for them to draw a perfectly straight line. In the real world, children learn that their teachers are flawed, animals bite, and the water they drink contains poison. As part of a flawed real world, do our children learn that their own imperfections contribute to the world’s imperfections? Are we providing our children proof that they are unworthy?
Properly taught, mathematics helps children learn to deal with the frustrations posed by abstractions. When we teach children how to relate mathematics to the real world, we help children to learn abstractions are not real. Abstractions are merely tools. We help children to understand that an abstraction serves no useful purpose unless it helps us to understand God’s creations. The only proof that matters is how well an abstraction models the universe God made for us.