From The Point, on a recent meeting of the 25,000-member American Education Research Association:
[Gabriel] Reich was trying to explain … why it was presumptuous for professional mathematicians (and many parents) to be up in arms about the currently fashionable constructivist idea that instead of explaining to youngsters, say, how to do long division, teachers should let them count, subtract, make an educated guess, or otherwise figure out their own ways to solve division problems. College math professors may complain that young people taught the constructivist way arrive in their classrooms unable to perform the basic operations necessary to move on to calculus, but so what? “Why should we privilege professional mathematicians?” Reich asked…. “Most of the people here at this meeting don’t think of themselves as good at math, and they don’t think math is creative.”
Let them “count, subtract, or otherwise figure out their own ways to solve division problems” to make it more creative, he says. Does that sound off the wall to you? Not necessarily so to everyone. I refer you to one of the stranger moments in the history of this blog as evidence, a strange moment that was echoed in this researchers’ meeting (caution: sexual themes in the linked page).
There are no wrong answers in constructivist theory, so Reich, pursuing his mathematical theme, had a tough sell the next day when he presented a paper to his fellow educators arguing that the principles of constructivism should be modified a bit in teaching arithmetic. “I know some constructivists might take issue with what I’m saying,” was his delicate way of telling his audience that when a student says two and two equals five, there might be a problem, if only with the child’s non-constructivist parents who might have “right-answer” concerns. Reich was suggesting that the youngster’s incorrect (or “incorrect”) answer be “vetted by the class” to see if it “works.” That way, he explained, “the students are learning to act as members of a mathematical community–they are becoming mathematicians.”
Now, why does this matter to a blog on Christian thinking? First and most obviously, it ought to be of concern to anyone, not just Christians. Students left to figure out long division on their own will either flail about uselessly until they despair, or they will “make an educated guess,” for which I will offer my own educated guess: they’re almost always going to guess wrong. (More on that in a moment.)
Second, those at this meeting who aren’t good at math (most of the people there at the meeting) “don’t think math is creative,” and one of them, Gabriel Reich, asked, “Why should we privilege professional mathematicians?” Well, why should we privilege professional musicians? I studied trombone in college and played professionally for several years. I still play occasionally, and the rules haven’t changed. I have to play in the same key as everybody else. If I’m going to play a low A-flat, I have play it in third position, and if I miss the right spot on the slide by as little as half an inch, it’s going to sound wrong. If I’m going to be really creative and ad lib a jazz solo, I’m going to be given a set of chord changes to follow. I need to know what the notation needs, and I need to stick with those chords, or it’s going to sound wrong.
Creativity depends on discipline, in other words. It only works within a structure. I wasn’t much good as a visual artist in school, but I recall my art teachers telling me the same thing: you need to learn the basics of lines, colors, and form before you can succeed in abstract art.
Ravi Zacharias has tells in one of his podcasts that another writer has summarized:
A friend of mine told me that when Christian apologist and author Ravi Zacharias visited Columbus to speak at Ohio State University, his hosts took him to visit the Wexner Center for the Arts. The Wexner Center is a citadel of postmodern architecture. It has stairways leading nowhere, columns that come down but never touch the floor, beams and galleries going everywhere, and a crazy-looking exposed girder system over most of the outside. Like most of postmodernism, it defies every canon of common sense and every law of rationality.
Zacharias looked at the building and cocked his head. With a grin he asked, “I wonder if they used the same techniques when they laid the foundation?”
(“Postmodern” as used there is roughly synonymous with Reich’s “constructivist.”) The building’s zany creativity depended on applying discipline at the foundation. Now, do you suppose the architect might have employed some mathematics in engineering the walls, the floors, the stairways and galleries, the roof? Or just educated guesses? The answer is obvious: his or her creativity was enabled by the discipline and structure of real mathematics.
So we must ask, does Gabriel Reich really think math can become more creative by letting students figure it all out on their own, or make their own guesses? Let’s ask him how long it would take these students to design a Wexner Center that way. Or to write software for a creative new computer game. Or even to compose a song—for there is a lot of mathematics in music.
One of the characteristic stances of postmodernism (or constructivism) is to set aside the importance of right and wrong. Truth is relative, especially in matters of religion, values, and ethics, but why stop there? As noted in my already-linked strange moment, 2+2=5 is an “illegitimate” answer, but it’s “not necessary” to use the terms “right and wrong.” This leads me to my third response: there is truth in religion, values, and ethics. Jesus Christ is the Truth (John 14:6), and God’s word is truth (John 17:17). There is life, first of all, that comes from recognizing that truth, and then a creativity-begetting discipline, a genuine freedom built on truth and expressing itself in love (Galatians 5:13-14).
Math, music, art, and architecture—and religion, too—none of these are simply matters of “educated guesses.” They all depend on yielding our opinions to an existing reality. There is a structure of truth under each of them: a structure that unleashes life and creativity.
Remember: it was those who didn’t consider themselves good at math who didn’t think it was creative. I’m quite sure that for them it isn’t. But why should we stifle a generation’s creative life by denying them the discipline of the truth?
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Possibly related posts (automatically generated):
Postmodernism: perhaps the one thing that religious apologists and religious skeptics tend to agree on.
I realize that this is off topic, but as a former professional mathematician, let me write briefly in defense of “discovering” mathematics. I think that most people would agree that in many spheres of life, we lead students to discover lessons, rather than just telling them what they need to know. It is one thing to hear someone say, “look before you leap,” it is much more vivid when you get hurt because you didn’t look first. I think my little boy learns more by building towers that don’t stand than by my telling him the right way to build a tower.
In mathematics, I have seen elementary students discover ways to add that are for them, faster and clearer than I could explain. I’ll never forget the day I “discovered” the fundamental theorem of calculus. Discovery, experimentation, discussion are all much more fun, more creative, and can be more instructive than merely learning what to do and then doing it perfectly in order to be judged correctly. Certainly the method described in the article can be used badly, but it can also be used successfully.
That said, I wholeheartedly agree with your comment that the structure of truth under mathematics unleashes creativity, rather than stifling it.