cross.pngWhen 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.

26 thoughts on “WHAT IS MATHEMATICAL PROOF? DOES 2 + 2 = 4?

  1. “Will, I am not sure what you mean by education. I’ve studied world religions and history.”

    Science helps, too.

    The problem, Tom, is that you don’t know anything to be true. That’s the point. And four apples is four apples regardless of their state of decay. I wouldn’t buy four rotten apples to consume, but that’s a completely different issue.

  2. kgotthardt – One thing I hope I have learned from reading the Bible is a bit of humility. When you suggest that we might know what God intended, I think of the Book of Job and laugh.

    What did God intend? I would imagine that what is is what God intended, but I do not know why He did things the way He did them.

  3. “Well, my guess that it is pointless to try to argue or pester anyone into heaven..”

    Yes, but also, you can’t MAKE someone believe in a specific heaven or hell. My mother has tried for years. It doesn’t work : )

    Would you try to make a Hindu believe in your version of heaven or hell, for example? Sure, you could talk about ideas from your belief system (and I personally might welcome it), but the idea of what I call “persuasive conversion” has never appealed to me. In other words, “Don’t badger me!” Does that make sense? Not everyone is Christian, after all. And I don’t think God INTENDED everyone to be Christian.

  4. Will – Is art done for its own sake? Do people do math just for the sake of doing math? Well, I suppose a few do, but most of us do math with a purpose in mind. If you do not consider the nature of the apples you count worthy of your notice, what is the point of counting them? Would you pay the same price for four rotten apples that you would pay for four sweet and juicy apples?

    Religion such as you describe has little to do with an orderly universe. Perhaps that has to do with your understanding of “faith.”

    Faith is word that means different things in different contexts. In one context, faith is a virtue. Consider what it means to have faith.

    Imagine a difficult moment. You have trained for a difficult task, and you are about to be tested. Now, to overcome your fears and concentrate on the task at hand, you must have faith in yourself. You know you can do what is necessary, but you must still overcome your doubts. You must have faith in yourself.

    We are each fragile. The universe is beyond our knowing. What good does it do to have faith in a mere man or woman, that face that slowly ages before us each morning in the mirror? When know our failure is altogether certain, what cause is there to have faith? So we strive for something greater than ourselves.

    Christians say God known to them. So they describe faith in God as trusting in what we know to be true. Such faith allows Christian to concentrate on the task at hand, to live the life that God would have us live, in the service of others.

    kgotthardt – Although Jesus said we must each make our own choice to accept salvation, he also gave us the Great Commission. So there is a balance. We cannot be indifferent. So what is the correct balance? Well, my guess that it is pointless to try to argue or pester anyone into heaven, and I suppose that is what you meant by not getting too hung up.

  5. BTW, Will, I will assume you must have been an infant at ONE time or another. Since you are apparently concerned with origins, I am surprised you dismiss your own so easily.

  6. Tom, a recent study (albeit somewhat limited) published in the Washington Post showed atheists largely believe in a higher power. They might not call it “God” but there is something there. Agnostics generally don’t know what they believe and are comfortable with that. And in my mind, anyone who has taken the time to think about the existence of a God probably is somewhat concerned about the meaning of life and what it should mean. I try not to get hung up on anyone’s journey towards truth because it’s highly personalized. So long as they don’t impede on anyone else’s journey and try to do right in life, I have no problem with them.

    Will, I am not sure what you mean by education. I’ve studied world religions and history. I’ve done some time in Catholic School and “taste tested” a number of Christian churches as well. I don’t think education necessarily leads one away from the idea of God. In fact, education can strengthen belief because education challenges people to think for themselves and sort out, “Is this really what I believe?” Most teens and people in their 20’s go through this process with or without organized religion and education, so we might as well encourage thought and inquiry. As you can see, there is plenty of it going on in this blog.

    Religious history and thought are fascinating aspects of our culture and I wouldn’t want to do without. That said, limiting ourselves to learning one perspective is just that–limiting ourselves.

  7. Just talking about apples. Not apples in the abstract, and neither size nor taste have anything to do with it. Take 2 apples out of your refrigerator. Put them on the counter. Now take 2 more out of the refrigerator and put them on the counter, too. And now you’ll have 4 apples on your counter. Or chair apples, if you prefer.

    And sure, maybe infants have to learn numbers and math, but, well, I’m not an infant. So.

    And actually, I didn’t introduce the invention of God; I merely extrapolated what you were saying. You said: “The belief that the universe is orderly and follows rules actually began as religious belief.” I was just pointing out that you were putting the cart before the horse, because, in fact, religious belief began as the desire for orders and rules, and not vice-versa (as the above statement implies). The Egyptians wanted rules for their universe, for their sunrise, so they invented Ra and etc. The Norse wanted rules for the afterlife and the end of the world, so they invented Valhalla and Fenrir. The Native Americans wanted an explanation for how the world began, so they came up with the turtle with the world on its back; the Hebrews, when faced with the same task, came up with a paternal father figure who said “Let there be Light.” And when certain people believed they required salvation they couldn’t earn on their own, they ascribed the mythological stories of Baldur and Dionysus to a humble carpenter from a Jewish city under Roman rule.

    Faith is a story we tell ourselves to explain the world.

    What’s unexplained or puzzling? When I hear people say anything is inexplicable, I tend to wonder if they know enough to explain it.

    Really. School helps. Education is awesome.

  8. “God is the term we use to explain the inexplicable.”

    kgotthardt – I think I know where you are coming from, and perhaps that statement is partly true. However, for atheists, I fear that statement is the beginning and the end of God. For an atheist, God is just a dodge for being unable to explain the inexplicable.

    While much is unexplained and ever so puzzling, God has told the important things. He has told us He exists. He has told us why He created us. He has told us how He redeemed us. And He told us what we must do.

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